Frontiers in Mathematics
Advisory Editorial Board Luigi Ambrosio (Scuola Normale Superiore, Pisa) Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît Perthame (Ecole Normale Supérieure, Paris) Gennady Samorodnitsky (Cornell University, Rhodes Hall) Igor Shparlinski (Macquarie University, New South Wales) Wolfgang Sprössig (TU Bergakademie Freiberg)
Birkhäuser Verlag Basel • Boston • Berlin
Author’s address: Rolf Sören Krausshar Department of Mathematical Analysis Ghent University Galglaan 2, Building S-22 9000 Ghent Belgium e-mail:
[email protected] 2000 Mathematical Subject Classification 11F03, 11F55, 30G35, 11G15, 32A25, 53C27, 42B35 Library of Congress Cataloging-in-Publication Data Krausshar, Rolf Sören, 1972Generalized analytic automorphic forms in hypercomplex spaces / Rolf Sören Krausshar. p. cm. – (Frontiers in mathematics) Includes bibliographical references. ISBN 0-8176-7059-9 (acid-free paper) – ISBN 3-7643-7059-9 (acid-free paper) 1. Automorphic forms. 2. Hardy spaces. 3. Functions of complex variables. I. Title. II. Series Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de.
ISBN 3-7643-7059-9 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2004 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Cover design: Birgit Blohmann, Zürich, Switzerland Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 3-7643-7059-9 987654321
www.birkhauser.ch
To Cristina, Francisca and my parents in love
Contents Introduction 1
Function theory in hypercomplex spaces 1.1 Hypercomplex numbers and Clifford Algebras . . . 1.2 Vahlen groups and arithmetic subgroups . . . . . . 1.3 Differentiability, conformality and analyticity in hypercomplex spaces . . . . . . . . . . . . . . . 1.4 Basic theorems of Clifford analysis . . . . . . . . . 1.5 Orders of isolated a-points, an argument principle and Rouch´e’s theorem . . . . . . . . . . . . . . . . 1.6 The generalized negative power functions . . . . .
ix
. . . . . . . . . . . . . . . . . .
1 1 5
. . . . . . . . . . . . . . . . . .
9 17
. . . . . . . . . . . . . . . . . .
28 35
2
Clifford-analytic Eisenstein series associated to translation groups 49 2.1 Multiperiodic Mittag-Leffler series . . . . . . . . . . . . . . . . . . 49 2.2 Some results on the zeroes of the generalized cotangent and tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.3 Liouville type theorems for generalized elliptic functions . . . . . . 60 2.4 Series expansions, divisor sums and Dirichlet series . . . . . . . . . 65 2.5 The integer multiplication of the Clifford-analytic Eisenstein series 72 2.6 Characterization theorems . . . . . . . . . . . . . . . . . . . . . . . 79 2.7 Lattices with hypercomplex multiplication . . . . . . . . . . . . . . 83 2.8 Bergman kernels of rectangular domains . . . . . . . . . . . . . . . 97 2.9 Szeg¨o kernels of strip domains . . . . . . . . . . . . . . . . . . . . . 108 2.10 Boundary value problems on conformally flat cylinders and tori . . 110 2.11 Order theory and argument principles on cylinders and tori . . . . 113
3
Clifford-analytic Modular Forms 3.1 Rotation and translation invariant Eisenstein series . . . . . . . . . 3.2 Clifford-analytic modular forms in one hypercomplex variable . . . 3.3 Clifford-analytic modular forms in two and several hypercomplex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117 118 124 131
viii
Contents 3.4 3.5
Some remarks on Clifford-analytic modular forms in real and complex Minkowski spaces . . . . . . . . . . . . . . . . . . . . . . . 144 Some Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Bibliography
151
List of Symbols
163
Index
165
Introduction Automorphic functions and automorphic forms are functions that show an invariance behavior or quasi-invariance behavior, respectively, under the action of a discrete group. For a number of reasons one is in particular interested in automorphic forms that are endowed with certain analytic properties. Examples of holomorphic automorphic forms and functions of one complex variable appeared first systematically in the paper [41] by G. Eisenstein (1847) and in lectures from K. Weierstraß in 1863. In [41] G. Eisenstein introduced the function series 1 1 1 n = 1, z+ z+m − m m∈Z\{0} (1) (1) n (z; Z) := 1 n ≥ 2, (z+m)n m∈Z
and (2) n (z; Ω)
:=
1 z2
+
ω∈Ω\{0} ω∈Ω
1 (z+ω)2
−
1 ω2
n = 2, n ≥ 3,
1 (z+ω)n
(2)
where Ω denotes a two-dimensional lattice in C. These series are automorphic functions for a translation group generated by one or two generators. They are sometimes called meromorphic Eisenstein series for translation groups. They provide building blocks for the complex holomorphic trigonometric and elliptic functions in one complex variable. The Laurent expansion of the series 2 (z; Z + Zτ ) leads in turn to the function series (cτ + d)−n Im(τ ) > 0 n ≥ 4, n ≡ 0 mod 2 (3) Gn (z) = (c,d)∈Z×Z\{(0,0)}
which represent holomorphic functions on the upper half-plane H + (C) := {z ∈ C | Im(z) > 0} that satisfy for all z ∈ H + (C) the transformation law a b f (z) = (f |n M )(z), for all M = ∈ SL(2, Z), c d where (f |n M )(z) := (cz + d)−k f
(4)
az + b
. cz + d The Eisenstein series Gn (z) provide thus examples of holomorphic automorphic forms for the full modular group SL(2, Z). A more systematic method to construct examples of holomorphic functions satisfying (4) is to start with a holomorphic function f˜ : H + (C) → C that is
x
Introduction
funcbounded on H + (C) and that belongs already to the class of automorphic 1 1 tions for the translation group T , the group generated by the matrix 0 1 which induces the translation z → z + 1. Summing the expressions (f˜|n M )(z) over a complete system of representatives of right cosets of SL(2, Z) modulo the translation invariance group T , i.e., (f˜|n M )(z), (5) M :T \SL(2,Z)
gives a convergent series for sufficiently large n whose limit function is then a modular form with respect to the full modular group SL(2, Z). These types of function series are often called Poincar´e series in a wider sense. The simplest non-trivial examples can be obtained by putting f˜ ≡ 1, thus yielding — up to a normalization factor — the classical Eisenstein series Gn (z). In turn the classical Eisenstein series Gn serve as building blocks for all modular forms of even positive entire weight n ≥ 4 for the full modular group SL(2, Z). However, the construction method proposed in (5) has a significant advantage: it can easily be adapted to the framework of other discrete groups, for instance, important congruence subgroups. The systematic development of the general theory of holomorphic automorphic forms of one complex variable is mainly due to H. Poincar´e, F. Klein and R. Fricke. See for instance [127, 81]. The theory of holomorphic automorphic forms and functions turned out to be a fundamental tool for solving many fundamental problems in number theory, including the determination of explicit relations between sums of divisors and representation numbers of quadratic forms, the construction of class fields, and even Fermat’s last theorem. Further areas of applications within mainstream mathematics include the study of Bergman and Hardy spaces of polyhedron type domains in C, the representation of Riemann surfaces and related partial differential equations and problems from index theory and algebraic geometry. Physicists use automorphic forms for instance in Yang–Mills theory and quantum gravity. To get a general idea about applications to physics, see for instance [67, 19] among many other contributions. Looking for higher dimensional analogues of the classical holomorphic automorphic forms of one complex variables is therefore strongly motivated under a lot of different viewpoints. There are several possibilities to extend classical function theory to higher dimensions. On the one hand there is the theory of several complex variables (see e.g., [76]) which is endowed with the ordinary regularity concept of complex holomorphy in each complex variable. Several complex variables theory has already become a well established branch inside mainstream mathematics since the 1930s and 1940s.
Introduction
xi
A further and different approach considers Clifford algebra-valued functions of one hypercomplex variable (see e.g. [13]) that are in the kernel of higher dimensional Cauchy–Riemann type operators. This function theory is today most often called Clifford analysis. Especially in the previous two decades Clifford analysis has emerged more and more as a valuable counterpart to the theory of several complex variables. It offers a number of generalizations of classical theorems from complex analysis, including a Cauchy integral formula, a residue theory, and even an argument principle. Already in the first half of the twentieth century higher dimensional (but complex-valued) generalizations of the classical automorphic forms from the plane have been developed within the framework of several complex variables theory. This line of investigation started with O. Blumenthal [9] in 1904 and continued with the school of C. L. Siegel and H. Maaß in the 1930s and early 1940s [153, 115]. One of the first authors to consider automorphic forms in a hypercomplex variable was R. Fueter. In 1927 R. Fueter constructed in his paper [54] automorphic forms and functions related to Picard’s group in a three-dimensional hypercomplex variable. However, the functions treated in [54] and also in his follow-up paper [55], are not in kernels of higher dimensional Cauchy–Riemann type operators. In 1949 H. Maaß proceeded to introduce in [117] another type of automorphic forms in a hypercomplex variable. The class of automorphic forms introduced by him consists of complex-valued non-analytic eigenfunctions to the higher dimensional hyperbolic Laplace–Beltrami operator. They are often called Maaß wave forms. They provide a natural generalization of their well-known two-dimensional non-analytic analogues in the complex upper half-plane which were introduced only a short time earlier by H. Maaß in [116]. All these works had a remarkable influence: Afterwards many authors extended these theories under numerous different aspects. Just to give a short overview over some of the most important mainstreams and recent developments, we recall that for instance, on the one hand, variants of holomorphic Siegel modular forms that are related to quaternionic symplectic groups or, more generally to orthogonal groups, have been studied, as for example by A. Krieg [99], by E. Freitag and C. Hermann [50] or very recently by e.g., F. B¨ uhler [17]. On the other hand for example E. K¨ ahler [80], J. Elstrodt, F. Grunewald, J. Mennicke [42, 45] as well as A. Krieg [100, 101] and V. Gritsenko [65] and others considered in several works complex-valued generalizations of the non-analytic automorphic forms with respect to discrete subgroups of the real Vahlen group in higher dimensional real half-spaces. Very recently O. Richter et al. for example considered also automorphic forms on m-fold Cartesian products of quaternionic half-spaces (see e.g., [131, 132]). Most of these generalizations defined in higher dimensional spaces are scalar valued. Vector-valued variants became of a growing interest, too (see e.g., [39, 15, 131]).
xii
Introduction
None of the generalizations of automorphic forms in a hypercomplex variable listed so far fit within the regularity concepts of Clifford analysis. They are not nullsolutions to higher dimensional Cauchy–Riemann type equations. In the period of 1998–2002, we were concerned to develop also a general theory of Cliffordvalued automorphic forms within the special framework of Clifford analysis and its regularity concepts. As far as we know, before 1998 research in this direction was primarily focussed on the construction of higher dimensional versions of the doubly periodic holomorphic Weierstraß functions, as for instance in [40, 61, 63, 67, 133, 68]. Additionally, in [128] a class of k-fold periodic hypercomplex-meromorphic functions on sector domains in IRk+1 has been considered — however, under rather different viewpoints and with different intentions. In [165] G. Z¨ oll made an attempt to construct monogenic functions that show an invariance behavior under discrete subgroups of Vahlen’s group that have the property of containing only finitely a b many matrices with c = 0. This attempt however is very restrictive. c d Restricting to matrix groups with this property, rules out all hypercomplex generalizations of the classical modular group immediately. However, G. Z¨ oll’s aim was primarily to give one example that is different from the generalized Weierstraß elliptic functions. Unfortunately, the only concrete example that he gave then explictly turned out to coincide with the zero function. Nevertheless, this attempt provided a motivation for doing a deeper and more systematic research in this direction. In the previous five years, we started then to fill this gap step by step and to develop systematically a theory of monogenic and polymonogenic hypercomplexvalued automorphic forms in the setting of Clifford analysis in real Euclidean spaces and in arbitrary real and complex Minkowski types spaces. In [84, 85, 87, 92] for example we studied intensively function theoretical and number theoretical properties of monogenic and s-monogenic vector-valued Eisenstein series in IRk+1 with s ≤ k that are associated to arbitrary discrete translation groups. On the one hand we hereby extended the theory of generalized elliptic functions under several viewpoints. On the other hand we introduced in IRk+1 , to any arbitrary p-dimensional lattice in IRk+1 with 1 ≤ p ≤ k, p-fold periodic monogenic and polymonogenic generalizations of several classical trigonometric functions, including the tangent, the cotangent, the secant and the cosecant function. The results provide moreover a counterpart and a complement to the recent papers [105, 106] in which analogues of the Weierstraß ζ and ℘-functions were constructed in the context of the Fueter–Sce equation D∆m f = 0 in the particular space IR⊕ IR2m+1 . We showed that the s-monogenic generalized trigonometric and Weierstraß functions have many characteristic properties in common with the classical ones. In particular, they satisfy similar characteristic duplication formulas. More generally, we developed general integer multiplication formulas for s-monogenic Eisenstein series related to arbitrary translation groups in IRk+1 . We proved that a
Introduction
xiii
non-constant polymonogenic function that satisfies one of these specific integer multiplication formulas of the s-monogenic Eisenstein series must have singularities. Furthermore, it coincides, at least up to a constant, with one of those polymonogenic Eisenstein series whenever the singularities are all distributed in the form of a lattice and have all the same order and principal parts. Moreover, we studied the hypercomplex multiplication of the s-monogenic translative Eisenstein series in arbitrary finite space dimensions and derived explicit formulas for the trace of their hypercomplex division points. The transfer of the concept of complex multiplication to the setting of Clifford-analytic multiperiodic functions focuses on paravector-valued lattices whose paravector components stem from multi-quadratic number fields and are characterized in terms of special integral conditions. This in turn links special functions from Clifford analysis with particular algebraic number fields. In our joint paper with T. Hempfling [74] some results on isolated zeroes were developed for this function class. In particular we gave a very first insight in the fundamental value distribution of isolated a-points and showed for some special cases that the number of isolated poles and isolated a-points, counting multiplicity of the generalized monogenic elliptic functions, stands in a close relationship with each other. One thus obtains at least under particular conditions a certain analogue of the third Liouville theorem. The main tool for this analysis is the generalized argument principle for isolated a-points of monogenic paravector-valued functions that was developed before in [74]. In joint works with D. Constales [25, 24] and J. Ryan [96] applications of the Clifford-analytic translative Eisenstein series to function spaces, spin geometry and partial differential equations arising in Clifford and harmonic analysis were developed. To go a bit more into detail, in [25] it has been shown that the p-fold periodic monogenic Eisenstein series (1 ≤ p ≤ k + 1) of the pole order k + 1 are the building blocks for the reproducing kernel functions of the space of monogenic functions that are square-integrable over a rectangular domain in IRk+1 that is bounded in k + 1 − p directions. In [24] an explicit formula of the reproducing kernel function of the space of monogenic functions in a strip domain IRk+1 which have L2 -boundary values is derived in terms of the monogenic cosecant function. In [96] s-monogenic Eisenstein series of the pole order k are used to construct Cauchy and Green kernels on conformally flat cylinders and tori which arise from factoring out IRk by the translation group. They in turn admit the study of a number of classical boundary value problems on these manifolds, including the Dirichlet problem. Further to this, they lead to the development of useful techniques for the study of Hardy spaces that arise in this context, including explicit Plemelj projection formulas and Kerzman–Stein formulas. In [93] we developed explicit argument principles and a Rouch´e type theorem for monogenic functions with isolated a-points that are defined on the surface of such a cylinder or torus. In [84, 86] we used the monogenic translative Eisenstein series to introduce kfold periodic generalizations of the Eisenstein series (3) to Clifford analysis. Their
xiv
Introduction
Fourier expansions involve explicit representation numbers of sums of divisors and a vector-valued variant of the Riemann zeta function which is built with monogenic functions and defined on arbitrary finite-dimensional lattices in IRk+1 . In [23] a detailed study on the structure of this variant of the Riemann zeta function was provided and explicit formulas in terms of Dirichlet type series with polynomial coefficients were established. Returning to the monogenic generalizations of the Eisenstein series (3) that we defined in [84, 86], we observed that these series have several number theoretical properties in common with their classical counterparts from complex analysis, as for instance the structure of the Fourier expansion. However, in contrast to the classical complex case, only their set of singularities shows an invariance behavior under the inversion. On the half-space these generalizations are not quasi-invariant under inversions. In [88, 89, 90, 91, 93] we finally managed to construct step by step also s-monogenic automorphic forms that show a quasi-invariance under larger arithmetic subgroups of the Vahlen group, in particular under hypercomplex generalizations of the modular group and their principal congruence subgroups. To this end, we could once more use special s-monogenic translative Eisenstein series, in particular the generalizations of the series (3) introduced before in [84, 86], as generating functions in Poincar´e type series which provide us with non-trivial Clifford and vector-valued s-monogenic automorphic forms for larger arithmetic subgroups. In [90] we also introduced Clifford-valued s-monogenic Hilbert modular forms for products of hypercomplex arithmetic subgroups acting on Cartesian products of higher dimensional hyperbolic spaces. All the constructions can be adapted to the context of arbitrary real and complex Minkowski type spaces. These results open systematically the door to extend the applications of the s-monogenic translative Eisenstein series to function spaces, boundary value problems and spin geometry to a more general context. Partial results in this sense could be obtained in [27] and [97]. This book gives a comprehensive summary on the research results in this field developed over the last five years, either published only in form of articles or not yet published at all. As indicated in the particular text passages, some parts of the book contain results from joint articles with D. Constales, R. Delanghe, T. Hempfling, H. Malonek, and J. Ryan. The text is composed into three chapters. The first chapter deals with results of general function theoretic interest. We introduce the most important notions and explain the basic context. Then we treat fundamental questions related to concepts of differentiability, conformality and analyticity in the context of Clifford analysis. After this we recall basic theorems from Clifford analysis. Then we turn to fundamental questions from order theory and
Introduction
xv
to the development of a higher dimensional analogue of an argument principle. Next we give some useful representation formulas on elementary s-monogenic rational functions of negative homogeneity degree providing the building blocks of s-monogenic automorphic Eisenstein and Poincar´e series. Chapter 2 deals with s-monogenic Eisenstein series associated to translation groups and their applications where we treat in the topics mentioned here in passing. In Chapter 3 we treat s-monogenic automorphic forms for more general polyhedron type arithmetic subgroups of Vahlen groups and to their principal congruence subgroups. This book is based on my Habilitationsschrift [94] which I have prepared in the period October 2000–May 2003 at the Department of Mathematical Analysis of Ghent University (Belgium) and defended at the Technical University of Aachen (Germany) in an extremely friendly atmosphere at both places. To both the members of the Department of Mathematical Analysis (Ghent University) as of the Faculty of Mathematics, Natural Sciences and Computer Sciences (RWTH Aachen) and the further exterior referees of this work I am therefore deeply indebted. In particular I wish to express a special gratitude to Denis Constales for the numerous valuable discussions on a number of parts of this book, to Alan Offer for all his help on specific linguistical and particular typsetting questions as well as to the staff of Birkh¨auser Verlag, in particular to Thomas Hempfling, for expert advise in editing and publishing. Last but not least I wish to thank very much all my family and friends, especially my wife Cristina and my daughter Francisca, for all their love, patience, understanding and support in any concern during all the time. Rolf S¨ oren Kraußhar, Ghent, December 2003
Chapter 1
Function theory in hypercomplex spaces 1.1
Hypercomplex numbers and Clifford Algebras
Hypercomplex numbers are, roughly speaking, generalizations of the complex numbers in the sense of having several imaginary units. While a complex number represents a two-dimensional vector from IR2 , a hypercomplex number represents a vector in spaces of higher dimensions. The treatment of vectors from IR2 in terms of complex numbers has the additional advantage that a multiplication operation can be defined on IR2 . The systematic development of hypercomplex number systems can be traced back until the first half of the 19th century. A rather detailed historical survey about the first steps can be found for instance in the first part of [4]. In the first half of the 19th century, R. W. Hamilton looked for a threedimensional analogue of the complex numbers for a rather long time. However, a three dimensional analogue of the complex number field with the structure of a division algebra does not exist. To meet these ends one has to work at least in four dimensions. In this sense, R. W. Hamilton introduced in 1843 the quaternions IH which are numbers of the form q = q0 + iq1 + jq2 + kq3 where the imaginary units satisfy i2 = j 2 = k 2 = −1 and ij = k = −ji, jk = i = −kj, ki = j = −ik. In contrast to the complex number field the quaternions are not commutative and form a skew field. J. T. Graves and A. Cayley invented independently in 1843 and 1844, respectively, the eight-dimensional octonions O which are also often called the Cayley numbers, today. The octonions are numbers that consist of a real part and seven imaginary parts. They have the form z = x0 i0 + x1 i1 + · · · + x7 i7 , where i1 , i2 are the quaternionic units i, j, respectively, i3 = w is a further independent unit,
2
Chapter 1. Function theory in hypercomplex spaces
i4 = k, i5 = iw, i6 = jw and i7 = kw. The octonions are closed under multiplication. However, they are both non-commutative and non-associative. All the imaginary units il with l ∈ {1, . . . , 7} satisfy i2l = −1, i0 il = il i0 = il and il im = −im il for all m ∈ {1, . . . , 7} with m = l. The multiplication in O is explained completely by the following multiplication table of the imaginary units: · i1 i2 i3 i4 i5 i6 i7
i1 −1 −i4 −i5 i2 i3 i7 −i6
i2 i4 −1 −i6 −i1 −i7 i3 i5
i3 i5 i6 −1 i7 −i1 −i2 −i4
i4 −i2 i1 −i7 −1 i6 −i5 i3
i5 −i3 i7 i1 −i6 −1 i4 −i2
i6 −i7 −i3 i2 i5 −i4 −1 i1
i7 i6 −i5 i4 −i3 i2 −i1 −1
The multiplication of the element il with im is the element in the l-th row and the m-th column (not counting the first row above the horizontal line and the first column left of the vertical line). For the basic properties of octonions, see for example [4]. Due to the famous theorem of Frobenius, respectively Hurwitz [51], the algebras IR, C and IH are (up to isomorphy) the only real associative division algebras, while IR, C, IH, O are the only normed real algebras. In the following period the development of hypercomplex numbers continued mainly in two directions. One approach aimed in the direction of generalizing the quaternions by hypercomplex numbers that form associative algebras which culminated in the invention of the Clifford numbers by W. K. Clifford in 1879. A second approach was based on the so-called Cayley–Dickson construction which leads to 2k -dimensional both non-commutative and non-associative algebras extending the octonions. In this work we will mainly focus on the treatment of higher dimensional spaces by using Clifford algebras. For our needs we recall briefly the most important notions about Clifford algebras over general real and complex vector spaces. For more details we refer the reader for example to [13] and [37]. Let IK stand for the field of real numbers IR or complex numbers C. Let k be a positive integer, and let p, q be non-negative integers with p + q = k. In what follows we consider a k-dimensional vector space over IK that is endowed with a non-degenerate quadratic form Q of signature (p, q). The attached bilinear form B is then given by B(x, y) :=
1
Q(x + y) − Q(x) − Q(y) 2
x, y ∈ IKk .
Let e1 , . . . , ek be the standard basis in the associated quadratic space IKp,q so that Q(e1 ) = · · · = Q(ep ) = −1, Q(ep+1 ) = · · · = Q(ek ) = 1 and B(ei , ej ) = 0 for i = j.
1.1. Hypercomplex numbers and Clifford Algebras
3
The attached Clifford algebra Clp,q (IK) is then the free algebra that is generated by IKk modulo the relation x2 = −Q(x)e0
x ∈ IKk
(1.1)
where e0 stands for the neutral element in the Clifford algebra. The relation (1.1) involves the following multiplication rules for the basis elements of the underlying vector space: e2i = 1 for i = 1, . . . , p, e2i = −1 for i = p + 1, . . . , k and ei ej + ej ei = 2δij e2i . A IK-basis for Clp,q (IK) is given by the set {eA : A ⊆ {1, . . . , k}} with eA = el1 el2 · · · elr , where 1 ≤ l1 < . . . < lr≤ k, e∅ = e0 = 1. The scalar and the vector part of an arbitrary element a = A⊆{1,...,k} aA eA where aA ∈ IK will be denoted by Sc(a) and by V ec(a). A Clifford number from Clp,q (IK) that has only a scalar and a vector part is called a paravector. The set of paravectors in Clp,q (IK) will be denoted by Ak+1 (IKp,q ). Every paravector from Ak+1 (IKp,q ) will be represented in the form z = x0 + x where the vector part will be denoted by a bold face letter. In the particular case IK = IR and p = 0, we simply write Ak+1 for the associated space of paravectors. Each element a ∈ Clp,q (C) can be written in the form a = a0 + ia1 with real Clifford numbers a0 , a1 from Clp,q (IR) where i denotes the complex imaginary unit. We write further Re(a) = a0 and Im(a) = a1 . One can subdivide the Clifford algebra Clp,q (IK) into an even and an odd part. The even part Cl+ p,q (IK) consists of Clifford numbers from Clp,q (IK) that can be represented in the form aA eA a= A,|A|≡0 mod 2
where aA are elements from the underlying field IK and where |A| stands for the + cardinality of A. The odd part Cl+ p,q (IK) is defined analogously. Clp,q (IK) is a k−1 . subalgebra of Clp,q (IK) and its dimension over IK is 2 The complex number field is realized by C ∼ = Cl01 (IR) and is (up to isomorphy) the only non-trivial commutative Clifford algebra over IR. The subalgebra Cl+ 03 (IR) is isomorphic to the Hamiltonian skew field IH in the sense of an algebra isomorphism. One identifies the bivector units of Cl+ 03 (IR) with the quaternionic units i := e12 , j := e13 and k := e23 . The complexification Cl+ 03 (C) realizes then similarly the complex quaternions which were probably first used by W. K. Clifford in the period of 1873–1876. The “Clifford” conjugation in Clp,q (IK) is defined by a = A aA eA where eA = elr el−1 · · · el1 ,
ej = −ej for j = 1, . . . , k and e0 = e0 = 1.
4
Chapter 1. Function theory in hypercomplex spaces
|A|(|A|−1)/2 aA eA . Furthermore, the main The reversion is defined by a∗ := A (−1) ∗ |A| involution is defined by a := A (−1) aA eA and one has a = a = a∗ . These (anti) automorphisms act on the basis elements eA ; they leave the complex unit i invariant. The “complex” conjugation, mapping an element a = a0 + ia1 from Clp,q (C) onto a0 − ia1 , shall be denoted by a in what follows. The Euclidean (Hermitian) scalar product in IKp,q is defined by z, w =
p
k
xj wj −
j=1
xj wj .
j=p+1
It extends to Clp,q (IK) by a, b = Sc(ab ) and induces a pseudo semi-norm
a = |Sc(aa )| a ∈ Cl0k (IK) which is commonly called the Clifford norm. However, if a, b are arbitrary elements from Cl0k (IK), then we have only ab ≤ 2k/2 a
b in general. Let us turn again to the subspace of paravectors. In Clp,q (IK) a paravector z ∈ Ak+1 (IKp,q ) is invertible if and only if the expression N (z) := zz = x20 −
p i=1
x2i +
k
x2i
i=p+1
does not vanish. In this case z −1 = z/N (z). The set S0 := {z ∈ Ak+1 (IKp,q ), N (z) = 0} has in general the geometrical structure of a cone. The case IK = IR with p = 0 is a special case. In this case N (z) = x20 + · · · + x2k = z 2 for a paravector in Ak+1 , so that the null cone S0 in Ak+1 reduces to a single isolated point. Hence every non-zero paravector in Ak+1 is invertible. Let us further use the notation Sz˜ := {z ∈ Ak+1 (IKp,q ), N (z − z˜) = 0} for the singularity cone with center z˜. Here again Sz˜ reduces to the single point z˜ in the particular context of working in real Euclidean spaces. Every paravector z ∈ Ak+1 (IKp,q ) satisfies a quadratic equation of the form z 2 − S(z)z + N (z) = 0 where S(z) = 2x0 stands for the trace of the paravector.
1.2. Vahlen groups and arithmetic subgroups
5
In what follows we will mainly concentrate on working in Clifford algebras over real Euclidean spaces with p = 0, since they play a special role for the reasons that we pointed out. Generalizations to the more general setting of Clifford algebras over Minkowski type spaces are pointed out in some particular passages. In this text the symbol Z stands for the set of integers, IN stands for the set of positive integers and IN0 for that of non-negative integers. We will use the and standard multi-index notation. For multi-indices n = (n0 , n1 , . . . , nk ) ∈ INk+1 0 and paravectors z ∈ Ak+1 (IKp,q ) we write as usual j := (j0 , j1 , . . . , jk ) ∈ INk+1 0 z n := xn0 0 · · · xnk k , n! := n0 ! · · · nk !, |n| := n0 + · · · + nk , nk n n0 ··· , j ≤ n :⇔ j0 ≤ n0 , . . . , jk ≤ nk . := j j0 jk By τ (j) we further denote the multi-index n = (n0 , n1 , . . . , nk ) for which ni = δij , δij standing for the Kronecker symbol. We also write (a)p for the Pochhammer symbol a(a + 1) · · · (a + p − 1). Following for instance [120], the permutational product of arbitrary Clifford numbers a1 , . . . , an is defined by a1 × a2 × · · · × an :=
1 n!
ai1 · ai2 · · · · · ain .
perm(i1 ,...,in )
Let us further use the abbreviation a × · · · × a1 × · · · × an × · · · × an = [a1 ]k1 × [a2 ]k2 × · · · × [an ]kn = [a]k .
1
k1 times
kn times
Notice that the permutational product of Clifford numbers is commutative but not associative. In order to distinguish powers in terms of the permutational product from powers in the usual sense, we set round brackets whenever we mean ordinary powers. In this sense we write for example [a1 ]2 ×a2 = a1 ×a1 ×a2 , but (a1 )2 ×a2 = (a1 · a1 ) × a2 .
1.2
Vahlen groups and arithmetic subgroups
In this section we briefly recall the basic definitions of the higher dimensional analogues of the classical groups GL(2, C), GL(2, R), SL(2, R) and its arithmetic subgroups in the setting of using Clifford algebra-valued matrices that act on domains of IRk or Ak+1 , respectively. Approaches in this direction can be traced back to the beginning of the twentieth century. In 1902, K. Th. Vahlen discovered in [161] that one can describe M¨ obius transformations in higher dimensional
6
Chapter 1. Function theory in hypercomplex spaces
Euclidean spaces in terms of special (2 × 2) Clifford matrices. Unfortunately his ideas were forgotten for a rather long time. H. Maaß rediscovered K. Th. Vahlen’s approach in 1949 (cf. [117]). L. V. Ahlfors provided in the first half of the 1980s several more extensive contributions to the description of M¨ obius transformations by Clifford numbers and their geometrical mapping properties. His papers (see for instance [1]) contributed to a remarkable renaissance of K. Th. Vahlen’s ideas to a broader community. Many authors started to use more extensively Vahlen matrices in hyperbolic geometry, number theory and Clifford analysis. Just to point out some of the numerous contributions that followed shortly, see for instance [43, 44, 45, 165, 138] among many other important related works. Following the above mentioned references, we introduce Definition 1.1 (Clifford group). The Clifford group (C, IRk ) resp. (C, Ak+1 ) is defined as the set of elements a ∈ Cl0k (IR) which can be written in terms of a finite product of non-zero vectors from IRk , or paravectors from Ak+1 , respectively. Remark. In the case of working in arbitrary Minkowski type spaces, one considers instead of products of non-zero vectors or non-zero paravectors, vectors or paravectors that satisfy Q(x) = 0 or N (z) = 0, respectively. See for instance [141] and [29]. Next we recall the definition of Vahlen groups. Following [29] we introduce a b Definition 1.2 (General Vahlen group). A matrix M = with coefficients c d k a, b, c, d out of (C, IR )∪{0} or from (C, Ak+1 )∪{0}, belongs to the general Vahlen group GV (IRk ) or GV (Ak+1 ), respectively, if additionally ad∗ − bc∗ ∈ IR\{0}, −1
a
−1
b, c
d ∈ IR
k
(1.2)
resp. ∈ Ak+1 , if c = 0 or a = 0, respectively. (1.3)
The general Vahlen group generalizes the group GL(2, C) to higher dimensions and permits a similar representation for M¨ obius transformations in IRk or Ak+1 as in the complex case. We recall Theorem 1.3. A function T : IRk ∪{∞} → IRk ∪{∞} is a M¨ obius transformation if and only if it can be written in the form T (x) = (ax+b)(cx+d)−1 with coefficients a, b, c, d ∈ (C, IRk ) that satisfy (1.2) and (1.3). obius transformation if A function T : Ak+1 ∪ {∞} → Ak+1 ∪ {∞} is a M¨ and only if it can be written in the form T (z) = (az + b)(cz + d)−1 with coefficients a, b, c, d ∈ (C, Ak+1 ) that satisfy (1.2) and (1.3). In view of z ∗ = z for all paravectors from Ak+1 (in particular, also for all vectors from IRk ) one can write T (z) = (az +b)(cz +d)−1 = (zc∗ +d∗ )−1 (za∗ +b∗ ). In the setting of arbitrary real and arbitrary real and Minkowski type spaces, (1.3) is replaced by ac∗ , cd∗ , db∗ , bc∗ ∈ IKp,q , permitting an analogous representation of
1.2. Vahlen groups and arithmetic subgroups
7
M¨obius transformations in arbitrary real and complex Minkowski type spaces, as mentioned e.g., in [141, 29]. Next we need: Definition 1.4 (Special Vahlen group). The special Vahlen group of SV (IRk ) or SV (Ak+1 ) is the subgroup of matrices belonging to GV (IRk ) or GV (Ak+1 ), respectively, whose coefficients satisfy moreover ad∗ − bc∗ = 1. The special Vahlen group can be regarded as a generalization of SL(2, C). As shown for instance in [43] the group SV (IRk ) or SV (Ak+1 ) is generated by the inversion matrix 0 −1 J := 1 0 and by translation matrices of the form 1 Ta := 0
a 1
where a ∈ IRk or a ∈ Ak+1 , respectively. The subgroup SV (IRk−1 ) acts transitively on the upper half-space of IRk , H + (IRk ) = {x ∈ IRk | xk > 0} by SV (IRk−1 ) × H + (IRk ) → H + (IRk ), x → M x := (ax + b)(cx + d)−1 . Consequently, all its subgroups leave the upper half-space invariant. The subgroup SV (Ak ) acts transitively on the upper half-space of Ak+1 , H + (Ak+1 ) = {z ∈ Ak+1 | xk > 0} by SV (Ak ) × H + (Ak+1 ) → H + (Ak+1 ), z → M z := (az + b)(cz + d)−1 . In view of the particular structure of the paravector space, it is sometimes advantageous to work instead on the right half-space of Ak+1 : H r (Ak+1 ) = {z ∈ Ak+1 | x0 > 0}. The appropriate analogue of SV (Ak ) for the right half-space is the modified special Vahlen group M SV (Ak+1 ) (see also [99, 100, 101] for the quaternionic case) which is generated by the modified inversion matrix 0 1 Q := 1 0
8
Chapter 1. Function theory in hypercomplex spaces
and by translation matrices of the form Ta where a ∈ IRk . Notice that M SV (Ak+1 ) is not a subgroup of SV (Ak+1 ), but a special subgroup of GV (Ak+1 ), since all matrices of M SV (Ak+1 ) have the property that their coefficients a, b, c, d satisfy either ad∗ − bc∗ = 1 or ad∗ − bc∗ = −1. Arithmetic subgroups of Vahlen groups that act discontinuously on the upper half-space were for instance considered in [44, 45]. We first recall the definition of the rational Vahlen group in V = IRk or V = Ak+1 , respectively. In what follows the set of rational numbers will be denoted as usual by Q. Definition 1.5. (see e.g., [44, 45]) The rational Vahlen group SV (V, Q) is the set a b of matrices from M at(2, (C, V )) that satisfy c d (i) ad∗ − bc∗ = 1, (ii) ab∗ = ba∗ , cd∗ = dc∗ , (iii) aa, bb, cc, dd ∈ Q, (iv) ac, bd ∈ V , (v) axb + bx a, cxd + dx c ∈ Q (∀x ∈ V ), (vi) axd + bx c ∈ V
(∀x ∈ V ).
Next we need Definition 1.6. (cf. e.g., [44, 45]) A Z-order in a rational Clifford algebra is a subring R such that the additive group of R is finitely generated and contains a Q-basis of the Clifford algebra. The simplest examples of Z-orders in Cl0k (IRk ) are the standard Z-orders Op := ZeA p ≤ k. A⊆P (1,...,p)
The following definition provides us with a number of arithmetic subgroups of the Vahlen group which act on the respective upper half-spaces. Definition 1.7. (cf. e.g., [44, 45]) Let I be a Z-order in Cl0k (IR) which is stable under the reversion and the main involution of Cl0k (IR). Then SV (V, I) := SV (V, Q) ∩ M at(2, I). For an n ∈ IN the principal congruence subgroup of SV (V, I) of level n is defined by a b SV (V, I; n) := a − 1, b, c, d − 1 ∈ nI . c d The following groups provide some special examples:
1.3. Differentiability, conformality and analyticity in hypercomplex spaces
9
Definition 1.8 (Special hypercomplex modular groups and their principal congruence subgroups). For p < k, resp. p < k + 1, let Γp (IRk ) := J, Te1 , . . . , Tep , resp. Γp (Ak+1 ) := J, Te1 , . . . , Tep . When itis clear which space IRk or Ak+1 is considered we simply write Γp . Let Op := A⊆P (1,...,p) ZeA be the standard order in Cl0p (IR). Then the associated principal congruence subgroups of Γp of level n are defined by a b Γp [n] := ∈ Γp a − 1, b, c, d − 1 ∈ nOp . c d We have Γp [1] = Γp . All the groups Γp [n] are of course discrete and act all discontinuously on the upper half-space H + (IRk ) or H + (Ak+1 ), respectively. The element J has the order 4, i.e., J 4 = I, where I denotes the identity matrix. The group Tp . In elements T1 , . . . , Tp have infinite order and generate the translation 0 u∗ turn these matrices generate together with the matrices of the form 0 u−1 where u ∈ {±eA : A ⊆ {1, · · · , p}}, the subgroup
a b ∈ Γ Γ∞ := | c = 0 . (1.4) p p c d 1 b The elements from Tp with b ∈ nOp , n ∈ IN will be denoted by Tp [n] 0 1 in what follows. We further denote the translation group, associated to a general lattice Ωp = Zω1 + · · · + Zωp that is contained in the subspace IRk or Ak+1 , respectively, by T (Ωp ). Notice that the larger group generated by the matrices J, Tω1 , . . . , Tωp leaves the upper half-space H + (IRk ), resp. H + (Ak+1 ), invariant whenever Ωp ⊂ IRk−1 or Ωp ⊂ Ak respectively. However, this group does not act discontinuously on the respective half-space for all choices of generators of Ωp . Finally let us also introduce into a general p-dimensional lattice Ωp ⊂ Ak+1 the transversion group V (Ωp ) (cf. [66]) as 1 0 1 0 . ,··· , V (Ωp ) := ωp 1 ω1 1 Notice that the transversion group V (Ωp ) is conjugated with the translation group T (Ωp ). The transversion group associated with the orthonormal lattice will consequently be denoted by Vp .
1.3
Differentiability, conformality and analyticity in hypercomplex spaces
This section describes the transfer and the adaptation of the concept of classical complex holomorphy to functions of a hypercomplex variable. To this end let us
10
Chapter 1. Function theory in hypercomplex spaces
recall that the concept of complex holomorphy in the plane can be introduced in several equivalent ways. Among them there is the Cauchy approach which introduces holomorphy of a complex function f at a point z0 by the criterion of the existence of the limit of a complex differential quotient in an open neighborhood around z0 . Geometrically, non-constant holomorphic functions are precisely those functions that are locally orientation and angle preserving maps, i.e., locally conformal mappings in the sense of Gauss that preserve orientation. A third way is to introduce holomorphy by the criterion of the existence of a unique Taylor series representation of f around z0 . This approach is called the Weierstraß approach. Furthermore, there is the Riemann approach which provides an access to introduce holomorphic functions as real analytic functions that are in the kernel of the ∂ ∂ + i ∂y . Cauchy–Riemann operator D = ∂x G. Scheffers [147] was one of the first to consider quaternionic differential quotients of quaternion-valued functions of the form
−1 (1.5) lim f (z + ∆z) − f (z) ∆z ∆z→0
or
lim
∆z→0
∆z
−1
f (z + ∆z) − f (z)
(1.6)
at the end of the 19th century. As a consequence of the lack of commutativity this approach does not offer a rich function theory in quaternions. The set of quaternionic functions that satisfy (1.5) is restricted to linear affine functions of the form f (z) = az + b a, b ∈ IH while (1.6) implies that f (z) = za + b
a, b ∈ IH.
A complete proof for these statements was first provided by N. M. Krylov in 1947 (cf. [102]) and his student A. S. Melijhzon in 1948 (see [121]). The only functions for which both limits exist are functions of the form f (z) = αz + b where α ∈ IR and b ∈ IH. See also J.J. Buff’s paper [16] from 1970. A very elegant proof which uses the embedding of IH in C2 is given in [160]. The second approach, introducing hypercomplex-analyticity by conformality in the sense of Gauss, does not lead to a rich function theory, either. For convenience recall (cf. e.g. [7, 8]) that a real differentiable function f : G → IH, where G ⊆ IH is a domain, is called conformal in the sense of Gauss if there is a strictly positive real-valued continuous function λ : G → IR>0 , z → λ(z), such that
df 2 = λ(z) dz 2
1.3. Differentiability, conformality and analyticity in hypercomplex spaces
11
where 3 ∂f dxi df = ∂x i i=0
and
dz = dx0 + idx1 + jdx2 + kdx3 .
However, J. Liouville’s famous theorem tells us Theorem 1.9 (Liouville’s theorem). Let G ⊂ IRn be a domain. A C 1 function obius transformation f : G → IRn is a conformal map if and only if f is a M¨ whenever n ≥ 3. J. Liouville proved this theorem in 1850 (cf. [111]) under the assumption that f is a C 3 homeomorphism. More than one hundred years later, P. Hartman managed to prove this assertion in [69] for C 1 homeomorphisms. These proofs use essentially methods from differential geometry. Very recently a new and rather compact proof of this theorem, which uses primarily methods from Clifford analysis, has been given in [30]. In [83, 95] and in the recent overview paper [36] we have shown that one can extend the definition of the differential quotients (1.5) and (1.6) so that all M¨obius transformations and constant functions are included. This modification is based on the use of variable structural sets. A quaternionic structural set is a set of four quaternions that are mutually orthonormal. Structural sets are often used for example in works of M. Shapiro, N. Vasilevski and V.V. Kravchenko (cf. e.g., [98, 151]). What follows provides in particular an extension of [5]. For our needs it is crucial to use continuously variable structural sets, which may be interpreted geometrically as continuous moving frames. To be more precise: Definition 1.10. A variable continuous structural set in an open set U ⊆ IH is a set of four C 0 -functions Ψ0 (z), . . . , Ψ3 (z) that satisfy Ψi (z), Ψj (z) = δij at each z ∈ U. This tool in hand gives rise to the following notion of quaternionic differentiability in a more general sense which was introduced in [83, 95]. See also [36]. Definition 1.11. Let U ⊂ IH be an open set and let z˜ ∈ U with z˜ = x˜0 + ix˜1 + j x˜2 + k x˜3 . Then f : U → IH is called left quaternionic differentiable at z˜ if there exist three C 0 (U )-functions Ψ1 , Ψ2 , Ψ3 : U → spanIR {i, j, k} satisfying at each z˜ z ), Ψk (˜ z ) = δjk and such that the relation Ψj (˜ lim (∆z [Ψ] )−1 (∆f )
∆z [Ψ] →0
exists. Here, ∆z [Ψ] = ∆x0 +
3
(1.7)
∆xi Ψi (˜ z ) with ∆xk = xk − x˜k .
i=1
f is called left quaternionic differentiable in U if f is left quaternionic differentiable at every point z ∈ U .
12
Chapter 1. Function theory in hypercomplex spaces
In an analogous way, right quaternionic differentiability is defined for f : U → IH, namely by requiring that for each z ∈ U , there exist three C 0 (U )-functions Φ1 , Φ2 , Φ3 : U → spanIR {i, j, k} such that Φj (z), Φk (z) = δjk and lim (∆f )(∆z [Φ] )−1
(1.8)
∆z [Φ] →0
exists. The limit lim (∆z [Ψ] )−1 (∆f )
∆z [Ψ] →0
may be considered as a linearization of the function f at the point z˜ with respect to the orthonormal basis [1, Ψ1 (˜ z ), Ψ2 (˜ z ), Ψ3 (˜ z )] and it is equal to the expression ∂f (˜ z ). It may be regarded as the left quaternionic derivative of f at the point z˜. ∂x0 One obtains the following nice analogy to the complex case: Theorem 1.12. Let G ⊂ IH be a domain. Then the set of left quaternionic differentiable functions in G coincides with the set of right quaternionic differentiable functions in G. A quaternionic differentiable function in G is either a conformal map in the sense of Gauss or a constant function in G. From Theorem 1.12 follows that one can associate with a quaternionic differentiable function two structural sets [Ψ] and [Φ] such that for k = 1, 2, 3, ∂f ∂f = Ψk (z) , ∂xk ∂x0
(1.9)
∂f ∂f = Φk (z). ∂xk ∂x0
(1.10)
and
In general, [Ψ] = [Φ], as a consequence of the non-commutativity. Proof of Theorem 1.12. If f is a constant function, then f is obviously both right and left quaternionic differentiable. Let us thus suppose that f : G → IH is a conformal map in G. According to the definition, there is a strictly positive continuous function λ : G → IR>0 such that
df 2 = λ(z)
3
dx2k .
k=0
In view of df 2 = df df one hence obtains the relation 3 3
3 3 ∂fr 2 2 ∂fr ∂fr dxi + 2 dxj dxi = λ(z) dx2k . ∂x ∂x ∂x i j i i=0 r=0 j
(1.11)
1.3. Differentiability, conformality and analyticity in hypercomplex spaces
13
By a comparison of coefficients, one can thus characterize the conformality of f in terms of the following system of differential equations: 3
∂fi 2 i=0
∂xk
3 ∂fi ∂fi ∂x j ∂xk i=0
= λ(z),
=
0,
k = 0, 1, 2, 3
j < k = 0, 1, 2, 3.
(1.12)
(1.13)
In other words, a C 1 (G)-function f is conformal in G if and only if it satisfies at each z ∈ G the two relations ∂f 2 ∂f 2 ∂f 2 ∂f 2 (1.14) = = = = λ(z) > 0 ∂x0 ∂x1 ∂x2 ∂x3 and
∂f ∂f =0 i<j Sc ∂xi ∂xj
i, j = 0, 1, 2, 3.
(1.15)
Next one defines for k = 1, 2, 3 at each point z ∈ G the functions Ψk (z) :=
∂f ∂f −1 . ∂xk ∂x0
(1.16)
∂f = 0 The functions Ψk (z) are actually well defined elements of C 0 (G), since ∂x 0 in G. This follows from the strict positivity of the function λ. (1.14) implies that
Ψk (z) = 1 at each z ∈ G. As a consequence of (1.15), one further obtains
Sc{Ψk (z)}
∂f ∂f −1 = Sc ∂xk ∂x0 1 ∂f ∂f = 0. = Sc ∂f 2 ∂xk ∂x0
∂x
0
The functions Ψk (z) take thus all their values in spanIR {i, j, k}. To show that the functions Ψk (z) form an orthogonal system at every single point z ∈ G one considers 2 Ψi (z), Ψj (z)
= = + =
Ψi (z)Ψj (z) + Ψj (z)Ψi (z) ∂f ∂f −1 ∂f −1 ∂f ∂xi ∂x0 ∂x0 ∂xj
∂f ∂f −1 ∂f −1 ∂f ∂xj ∂x0 ∂x0 ∂xi ∂f −2 ∂f ∂f 2 = 0, Sc ∂x0 ∂xi ∂xj
14
Chapter 1. Function theory in hypercomplex spaces
where we again applied (1.15). The system [Ψ(z)] := (Ψ1 (z), Ψ2 (z), Ψ3 (z)) thus actually represents at each single point of G an orthonormal frame of quaternions from spanIR {i, j, k}. By construction, the frame Ψk (z) satisfies indeed the system (1.9). As a consequence, the limit (1.7) exists. Hence f is left quaternionic differentiable. Let us now conversely suppose that f is a non-constant left quaternionic differentiable function in the whole domain G. As a consequence, one can thus associate with each point of G an orthonormal system of quaternions [Ψ] = [Ψ(z)] with values in spanIR {i, j, k} such that for k = 1, 2, 3 the system (1.9) is satisfied. To show the conformality of f in the whole domain G, we verify that f satisfies the system of differential equations (1.14) and (1.15) in G. Since Ψk (z) = 1 at each z ∈ G property (1.14) follows immediately. To show (1.15), consider the following two expressions in which we apply (1.9): ∂f ∂f = , 2 ∂x0 ∂xk = =
∂f ∂f ∂f ∂f + ∂x0 ∂xk ∂xk ∂x0 ∂f ∂f ∂f ∂f Ψk (z) + Ψk (z) ∂x0 ∂x0 ∂x0 ∂x0 ∂f 2 2 Sc{Ψk (z)} = 0. ∂x0
Here we used the property that the functions Ψk (z) take only values in spanIR {i, j, k} in the whole domain G. Further, ∂f ∂f , 2 ∂xj ∂xk
=
∂f ∂f ∂f ∂f + (j = k, j, k = 0) ∂xj ∂xk ∂xk ∂xj
∂f ∂f ∂f ∂f Ψj (z) Ψk (z) + Ψk (z) Ψj (z) ∂x0 ∂x0 ∂x0 ∂x0 ∂f 2 = 2 Ψj (z), Ψk (z) = 0, ∂x0
=
since (Ψi (z))i=1,2,3 is an orthonormal system at each single point z ∈ G. In view of (1.14) and (1.15) we can conclude that that f is conformal at each z ∈ G. Similarly, one can show that a conformal map in G is characterized by the system (1.10). To this end one simply defines in the proof the frame [Φ] to be Φk (z) :=
∂f −1 ∂f for every k = 1, 2, 3. ∂x0 ∂xk
(1.17)
In complete analogy one thus establishes that a right quaternionic differentiable function is either a conformal map in G or a constant function in G. The set of left quaternionic differentiable function thus coincides with the set of right quaternionic differentiable functions.
1.3. Differentiability, conformality and analyticity in hypercomplex spaces
15
Remarks. 1. If f is a linear fractional function with c = 0, then with each point a different structural set is associated. Only if f has the form f (z) = (az + b)d−1 (i.e., if c = 0), then the associated functions Ψi (z) are constants which however, turn out to be different from the canonical structural set [1, i, j, k] in general. For details we refer to [95]. 2. If one restricts oneself to the constant standard structural set [Ψ1 , Ψ2 , Ψ3 ] = [i, j, k], then exactly the functions of the form f (z) = az + b or f (z) = za + b will be obtained. The use of structural sets thus provides an extension of the strict definition of differentiability given in (1.5) or (1.6). 3. According to [121] a function f (z) = az + b where a ∈ IH\IR is only left differentiable with derivative equal to a, but not right differentiable in the sense of (1.6). However, Theorem 1.12 tells us that every left quaternionic differentiable function in the more general sense of Definition 1.11 is also right quaternionic differentiable with the same derivative. One might think at the very first that this result yields a contradiction. However, it does not in the context working with structural sets. Indeed each function f (z) = az + b where z = x0 + x1 i + x2 j + x3 k can be written in the form f (z) = z [Ψ] a + b where z [Ψ] = x0 + x1 Ψ1 + x2 Ψ2 + x3 Ψ3 is another structural set. Therefore, f is also right quaternionic differentiable with the same derivative, however, with respect to another orthonormal basis. Only when a is real, one obtains [Ψ1 , Ψ2 , Ψ3 ] = [i, j, k] whence we are in the case treated by J. J. Buff in [16]. Although this generalized notion of quaternionic differentiability gives a nice characterization of conformality, it does not offer a rich function theory, either, due to the fact that conformal mappings are restricted to the set of M¨ obius transformations up from three dimensions. At this point we wish to point out that it is also possible to characterize M¨obius transformations in the space by the property of being annihilated by certain generalized Schwartzian derivatives. See [140] for details. The third possibility of generalizing complex holomorphy to higher dimensions by the Weierstraß approach considering power series representations involving ordinary monomials of the form a0 za1 za2 · · · zan fails, too (cf. e.g., [31]). This ansatz leads to a too large function class, namely to all real-analytic functions. A.C. Dixon [40] (1904), G.C. Moisil and N. Theodorescu [122] (1931), R. Fueter ([56]–[63]) (1932–50), his students P. Boßhard [10] (1940) and W. Nef [125] (1944), V. Iftimie [79] (1965) and R. Delanghe [32] (1970) are some of the most important creators of a generalized function theory of functions in hypercomplex variables in the sense of the Riemann approach. First studies were restricted to quaternion-valued functions that are defined in open subsets of spanIR {i, j, k} and that are nullsolutions to the Dirac operator DIR3 :=
∂ ∂ ∂ i+ j+ k ∂x1 ∂x2 ∂x3
16
Chapter 1. Function theory in hypercomplex spaces
(e.g., [40, 122]), or, quaternion-valued functions defined in open subsets of IH that are annihilated by the quaternionic Cauchy–Riemann operator DIH :=
∂ ∂ ∂ ∂ + i+ j+ k, ∂x0 ∂x1 ∂x2 ∂x3
as did the Fueter school in the first ten years. Only later could researchers start to also consider Cl0k (IR)-valued functions that are defined in open subsets of the Euclidean spaces IRk or Ak+1 that are annihilated by the Dirac operator in IRk , i.e., DIRk :=
k ∂ ei ∂x i i=1
or the Cauchy–Riemann operator in Ak+1 , DAk+1 =
∂ + DIRk , ∂x0
respectively. As far as we know, this line of investigation started with P. Boßhard in 1940 (cf. [10]). W. Nef introduced the analogue for arbitrary real Minkowski type spaces in 1944 (cf. [125]). See also [63] p. 264. Also complexifications of the Cauchy–Riemann system were already considered by the Fueter school, however, without going into detail. See [63] p. 273. Unfortunately, after R. Fueter’s death these results were neglected for a long time and remained unknown for a broader community. With R. Delanghe’s papers around 1970 [32, 33, 34] the study of Clifford-valued functions in the kernels of Euclidean Dirac operators and Euclidean Cauchy–Riemann operators, respectively, had a remarkable renaissance and made this theory more popular. This function theory is today often called Clifford analysis. In the 1980s extensive works on complexified Dirac and Cauchy– Riemann systems have been provided, as for example by V. Souˇcek in 1982 in the special setting of complex quaternions, and more generally and extensively by J. Ryan in several papers, see for example [134]. Let us first again restrict ourselves exclusively to the context of Euclidean spaces and let us write down the explicit definition of the related regularity concept of the function class under consideration. Definition 1.13. Let U ⊂ IRk (resp. U ⊂ Ak+1 ). Then a real differentiable function f : U → Cl0k (IR) is called left monogenic, if Df = 0 (resp. Df = 0). A real differentiable function f : U → Cl0k (IR) is called right monogenic, if f D = 0 (resp. f D = 0). The set of left monogenic functions in IRk in terms of the Dirac operator is isomorphic to the set of right monogenic functions Ak+1 in terms of the Cauchy– Riemann operator, as mentioned for instance in [66]. For some problems it is advantageous to work in the vector formalism, i.e., with the Dirac operator, for other problems the paravector formalism attached to the Cauchy–Riemann operator has advantages. In the first two chapters and Chapter 3.1 of this book we
1.4. Basic theorems of Clifford analysis
17
prefer to work in the paravector formalism and in Chapter 3.2–3.4 in the vector formalism. The set of left monogenic functions is in general different from the set of right monogenic functions. However, both sets can be treated in a similar way. In the special case where the range of values is contained in Ak+1 , the set of left monogenic functions coincides with that of right monogenic functions. In contrast to complex function theory, the set of left (right) monogenic functions forms only a right (left) Clifford module. In general, the product or the composition of two monogenic functions does not give a monogenic function again. Since Clifford analysis has been established as a generalization of complex analysis in the sense of the Riemann approach, it is natural to ask whether this extension of function theory is also related to any kind of notion of differentiability in the sense of linear approximability. A positive answer and related concept has been developed by H. Malonek in [118, 119]. This is based on consideration of the isomorphism IRk+1 ∼ = Hk = { z : zj = xj − x0 ej ; x0 , xj ∈ IR, j = 1, . . . , k} which defines a second hypercomplex structure on IRk+1 , different from that endowed by Ak+1 . In this framework H. Malonek introduced in [118, 119] the following notion of hypercomplex differentiability: Definition 1.14. Let U ⊂ Hk be an open set and let z ∈ U . Suppose that f : U → Cl0k (IR) is continuous. Then f is called left hypercomplex differentiable at the k point z if there is a left Cl0k (IR)-linear map Ll = i=1 zi Ai such that |f ( z + ∆ z) − f ( z) − Ll (∆ z)| =0 ∆ z →0
∆ z lim
(1.18)
where the elements A1 ,. . ., Ak are uniquely defined Clifford numbers from Cl0k(IR). Similarly, right hypercomplex differentiability is defined by regarding analok gously a right linear map of the form Lr = i=1 A˜i zi . Indeed, this notion provides a characterization of the class of monogenic functions in terms of differentiability in the sense of linear approximation, as stated in the following theorem proved in [118, 119]: Theorem 1.15. A continuously differentiable Cl0k (IR)-valued function in U ⊂ Ak+1 ∼ = Hk is left (right) hypercomplex differentiable if and only if f is left (right) monogenic in U .
1.4
Basic theorems of Clifford analysis
The Riemann approach offers a function theory of Clifford-valued functions defined in higher dimensional spaces — for example in higher dimensional Euclidean spaces
18
Chapter 1. Function theory in hypercomplex spaces
or higher dimensional conformal manifolds — which includes generalizations of many classical theorems from complex analysis. For details, see for instance [63], [13] and [37]. For our purposes, we recall: Theorem 1.16 (Cauchy’s theorem). Let U ⊂ Ak+1 be a non-empty open set and let S ⊂ U be a compact, orientable, (k + 1)-dimensional differentiable manifold with boundary. Assume that C is a (k + 1)-chain on S. If f : U → Cl0k (IR) is left (right) monogenic, then dσ(z)f (z) = 0, resp. f (z) dσ(z) = 0, ∂C
where dσ(z) =
k j=0
∂C
j = dx0 ∧ · · · ∧ dxj−1 ∧ dxj+1 ∧ · · · ∧ dxk j with dx (−1)j ej dx
stands for the k-dimensional oriented Lebesgue measure. Theorem 1.17 (Cauchy’s integral formula). Let U ⊂ Ak+1 be a non-empty open set and let again S ⊂ U be a compact differentiable k + 1 dimensional oriented manifold with boundary. Assume further that z ∈ U \∂S and that f : U → Cl0k (IR) is left monogenic. Then ζ −z 1 f (z), z ∈ int S, dσ(ζ)f (ζ) = (1.19) 0, z ∈ U \S, Ak+1
ζ − z k+1 ∂S
where Ak+1 stands for the surface measure of the k + 1-dimensional unit ball. More generally, we have Theorem 1.18 (Generalized Cauchy’s integral formula). Let U ⊂ Ak+1 be a nonempty open set. Let Γ be a k-dimensional nullhomologous cycle in U . Assume further that z ∈ U \Γ and that f : U → Cl0k (IR) is left monogenic. Then 1
Ak+1 Γ
ζ −z dσ(ζ)f (ζ) = wΓ (z)f (z),
ζ − z k+1
(1.20)
where wΓ (z) stands for the winding number of Γ with respect to z. The (generalized) Cauchy integral formula provides the basis for the development of many important theorems in Clifford analysis. From Cauchy’s integral formula it is easy to derive a mean value theorem which further allows us readily to deduce the maximum modulus theorem. See [13] pp. 54–56 for details. In contrast to classical complex function theory of the plane, the powers of the hypercomplex variable z → z n are not in the kernel of the Cauchy–Riemann operator (except for n = 0). The positive powers are substituted by the following polynomials, first mentioned in [58] for the quaternionic case, and therefore
1.4. Basic theorems of Clifford analysis
19
sometimes called Fueter polynomials: Vn (z) :=
1 |n|!
Zπ(n1 ) Zπ(n2 ) · · · Zπ(nk ) .
π∈perm(n)
Here perm(n) stands for the set of all permutations of the sequence (n1 , . . . , nk ), Zi := xi − x0 ei
for i = 1, . . . , k
and V0 (z) := 1. Following H. Malonek [118, 120], these polynomials can be written very elegantly in the form 1 Vn (z) = Z n n! where Z n stands for [Z1 ]n1 × · · · × [Zk ]nk . If one had commutativity, this product would simplify to a usual product of powers of the variables Zi . The permutational product compensates the non-commutativity in the higher dimensional case. In terms of the functions Vn one obtains a generalization of the Taylor series expansion theorem for monogenic functions. Next and in all that follows we use the notation B(˜ z , R) := {z ∈ Ak+1 | z − z˜ < R} for the open ball in Ak+1 with radius R > 0 centered around z˜. The symbol B(˜ z , R) stands for the closed ball around z˜ with radius R. Theorem 1.19 (Taylor expansion). Let f : B(˜ z , R) → Cl0k (IR) be left (right) monogenic. Then in each open ball B(0, r) with 0 < r < R, f (z) =
∞ |n|=0
Vn (z − z˜)an
or
f (z) =
∞
an Vn (z − z˜)
|n|=0
where the coefficients are uniquely given by a0 = f (˜ z ) and an = Here n stands for multi-indices of the form n = (0, n1 , . . . , nk ).
∂ |n| ∂xn f (z)
z=˜ z
.
The negative powers are replaced by the functions q0 (z) :=
z ∂ |n| ∂ n1 +···+nk and q (z) := q (z) = q0 (z), n ∈ INk+1 , |n| ≥ 1. n 0 0
z k+1 ∂xn ∂xn1 1 · · · ∂xnk k
The function q0 (z) coincides in the planar case k = 1 with the function z −1 . It is the fundamental solution to the D-operator and q0 (z − ζ) is the Cauchy kernel. The functions qn (z) with |n| ≥ 2 coincide in the case k = 1 up to a normalization constant with z −n for n ≥ 2. They turn out to provide the counterpart of the functions Vn in the Laurent expansion of monogenic functions with singularities. For convenience we recall:
20
Chapter 1. Function theory in hypercomplex spaces
Definition 1.20. A point z˜ ∈ Ak+1 is called a left (right) regular point of a left (right) monogenic function f , if there exists an ε > 0 such that f is left (right) monogenic in B(˜ z , ε). Otherwise, z˜ is called a singular point of f . z˜ is called an isolated left (right) singularity of f if it is a left (right) singular point of f for which one can find an ε > 0 such that f is left (right) monogenic in B(˜ z , ε)\{˜ z }. In analogy to classical function theory of one complex variable one can classify the singularities into three essentially different types, namely into removable singularities, unessential and essential singularities. However, in contrast to the classical case, a monogenic function in Ak+1 can have manifolds of singularities of dimension 0 (isolated singularities), and additionally of dimension 1, . . . , k−1. As a consequence of the Cauchy–Riemann equations, k-dimensional manifolds of singularities cannot appear. In order to recall the definition of essential and unessential singular sets of monogenic functions, which was first introduced for the quaternionic case by R. Fueter [61] and W. Nef [123, 124], we first need the following notions. Suppose that U ⊂ Ak+1 is an open set and that S ⊂ U is a closed subset. Let us consider around each s ∈ S a ball with radius ρ > 0 and let us denote by Hρ the hypersurface of the union of all balls centered at each s ∈ S with radius ρ. In the case where S is just a single isolated point, Hρ is simply the sphere centered at s with radius ρ. If S is a rectifiable line, or, more generally, a p-dimensional manifold with boundary, then Hρ is the surface of a tube domain. Let us denote by J(ρ) the limit inferior of the volumes of all closed orientable hypersurfaces HR with continuous normal field that contains S in the interior where we suppose that inf{ s − z˜ ; s ∈ S, z˜ ∈ HR } ≥ ρ. In terms of these notions one can now give the following definition: Definition 1.21. Let U ⊂ Ak+1 be an open set and let S ⊂ U be a closed subset. Suppose that f is left (right) monogenic in U \S and that f has singularities in each s ∈ S. Let Hρ ⊂ U be a hypersurface as defined above. The singular point s ∈ S is then called an unessential singularity of f if there is an r ∈ IN and an M > 0 such that z ) < M z˜ ∈ Hρ . (1.21)
ρr J(ρ)f (˜ In this case the minimum of the parameters r is called the order of the singular point s. If no finite r with this property can be found, then s is called an essential singularity. Functions that have at most unessential singularities and that are left (right) monogenic elsewhere are called left (right) meromorphic (in the sense of Fueter). In the case where S is an isolated point singularity, a rectifiable line or a manifold with boundary, one can substitute the condition (1.21) by z ) < M.
ρr f (˜
1.4. Basic theorems of Clifford analysis
21
Remarks. In classical complex function theory of functions in one complex variable any meromorphic function satisfies limz→˜z f (z) = ∞ at a pole z˜, independently on which path one approximates the singularity. In general, this is not true in higher dimensions (cf. [60, 129, 13, 136]). As a simple counterexample we note the function q1,1 (z) which has limit value zero when approaching the singularity 0 along the x0 -axis. In connection with this limit behavior J. Ryan has proved in [136] in the case k = 3, that provided f : A3 \{0} → Cl02 is a left monogenic function having a negative degree of homogeneity, then the set of lines radiating from the origin on which f vanishes, can at most have a finite cardinality. Let z˜ be an isolated singularity of f and let γ be an arbitrary path with γ(0) = z˜ and where f (γ(t)) is left (right) monogenic for all t > 0. For the quaternionic case, the Fueter school has provided examples where the set of limit values at a non-essential point singularity z˜ of a monogenic function f , i.e., {L ∈ IH ∪ {∞} |L = lim f (γ(t))} t→0
is a 2- or 3-dimensional manifold in IH ∪ {∞}. See for example [63] pp. 151–158. As a consequence of the Cauchy–Riemann system, 1-dimensional manifolds of limit values can never appear. This was proved in 1948 by Hs. Reich in [129]. Hs. Reich showed furthermore in his thesis [129] that this result remains even valid for the case where z˜ is a non-essential singularity lying on a two-dimensional manifold of non-essential singularities. For monogenic functions with isolated singularities one has the following kind of Laurent expansion theorem: Theorem 1.22 (Laurent expansion). Let f be left monogenic in the annular domain B(˜ z , R)\B(˜ z , r) ⊂ U with 0 < r < R. Then f has a Laurent series expansion of the form ∞ ∞ Vn (z − z˜)an + qn (z − z˜)bn , (1.22) f (z) = |n|=0
|n|=0
where both series converge normally in B(˜ z , R), resp. in Ak+1 \B(˜ z , r) and where n are multi-indices of the form (0, n1 , . . . , nk ). The coefficients an , bn are uniquely determined Clifford numbers and have the integral representation 1 1 qn (z) dσ(z)f (z), bn = Vn (z) dσ(z)f (z). an = Ak+1 Ak+1 ∂B(˜ z ,ρ)
∂B(˜ z ,ρ)
for an arbitrarily chosen real ρ within r < ρ < R. The coefficient b0 =: res(f ; z˜) is called the residue of f at z˜. Similarly to the complex case one can establish Theorem 1.23 (Residue Theorem). Suppose that U ⊂ Ak+1 is an open set. Let S ⊂ U be an oriented (k + 1)-dimensional compact differentiable manifold with
22
Chapter 1. Function theory in hypercomplex spaces
boundary. Assume that f : U → Cl0k (IR) is left monogenic in U with exception of a finite number of isolated points, say in y 1 , . . . , y r , such that y 1 , . . . , y r ∈ ∂S and that y 1 , . . . , y s ∈ int S where 1 ≤ s ≤ t. Then
1
dσ(z)f (z) =
Ak+1
s
res(f ; y i ).
i=1
∂S
An important tool for our needs is Theorem 1.24 (Mittag-Leffler theorem). Suppose that (ai )i is a sequence of distinct points in Ak+1 having no accumulation point in Ak+1 , so that we can associate to each point ai a non-negative integer pi and a function Qi (z) =
pi
(i) qn (z − ai )bn ,
(1.23)
|n|=0 (i)
where the elements bn are arbitrary Clifford numbers from Cl0k (IR). Then there is a function f : Ak+1 \ i {ai } → Cl0k (IR) which is left meromorphic in the whole space Ak+1 having poles of order (pi + k) at the points ai with the corresponding singular parts Qi , respectively. This function is uniquely defined up to a function that is left monogenic in the whole space. Remarks. W. Nef established in [123, 124] for the quaternionic case a more general version of the Laurent series expansion theorem and Mittag–Leffler theorem for functions f that have also sets of non-isolated singularities. These theorems in turn can also be generalized directly to arbitrary dimensions, as mentioned for instance in Section 1.6 of [84]. In these cases the expressions of the form qn (z − z˜)an in (1.22) and (1.23) are substituted more generally by qn (z − c)d(an (c)),
(1.24)
S
involving the Lebesgue–Stieltjes integral over the singular set denoted by S. Also the residue theorem can be generalized to the more general context dealing with manifolds of singularities. See [61, 63] for the quaternionic case and [38] for the case in arbitrarily finite-dimensional real Euclidean spaces. In this case the residue of a function f at the singularity set S is said to be the first generalized Laurent coefficient in the sense of (1.24), i.e. d(a0 (c)). Res(f ; S) = S
These kinds of residues are also known as Leray–Norguet residues (cf. [38]).
1.4. Basic theorems of Clifford analysis
23
Remark. All these theorems have complete analogues in the vector formalism in terms of the Dirac operator. In the vector formalism in IRk , the function q0 (z) is replaced by x q0 (x) := −
x k and the functions qm (z) are substituted by partial derivatives of q0 (x). In the vector formalism in IRk , one can restrict consideration to the partial differentiations in k − 1 directions, say for instance in the x1 , . . . , xk−1 directions. Similarly the analogues of the Fueter polynomials Vn (z) in the vector formalism are given by Vm1 ,...,mk−1 (x) :=
1 (xσ(i) + xk ek eσ(i) ) . . . (xσ(i) + xk ek eσ(i) ) m!
(1.25)
where m := m1 + · · · + mk−1 and σ(i) ∈ {1, . . . , k − 1} and the summation runs over all distinguishable permutations of the expressions (xσ(i) +xk ek eσ(i) ) without repetitions. One important property of the Dirac and Cauchy–Riemann operator in IRk and Ak+1 is that they factorize the Euclidean Laplace operator, viz DIRk = −∆k and DAk+1 DAk+1 = ∆k+1 , respectively. The monogenic functions are thus annihilated by the Euclidean Laplace operator, and every real component of a monogenic function is harmonic. More generally, let us consider Definition 1.25 (s-monogenic functions). Let U ⊂ IRk be an open set. Let s be a positive integer. Suppose that f : U → Cl0k (IR) is a C s -function. Then f is called left or right s-monogenic in U if DIsRk f = 0 or f DIsRk = 0, respectively. Sometimes we say for simplicity Clifford-analytic or polymonogenic for smonogenic. If s is even, then f ∈ Ker ∆s/2 . The analogue of the set of s-monogenic functions in the paravector formalism is the set of C s -functions in U ⊂ Ak+1 that satisfy D[s] f = 0, resp. f D[s] = 0, where we mean D[s] = ∆s/2 if s is an even positive integer and D[s] = D∆(s−1)/2 if s is an odd positive integer. We will call functions in Ker D[s] also s-monogenic and sometimes also simply Clifford-analytic. The analogues of the function q0 (x) in the framework of Ker Ds (vector formalism in IRk ) are given by x s odd with s ≤ k − 1, k+1−s (s)
x (1.26) q0 (x) := 1 s even with s ≤ k − 1. k−s
x The s-monogenic analogues of qm (x) for |m| ≥ 1 are given in terms of (s) (x) := qm
∂ |m| (s) q (x). ∂xm 0
24
Chapter 1. Function theory in hypercomplex spaces
Whenever s > 1, we cannot restrict ourselves to only consider indices m with mk = 0. The analogues of the function q0 (z) in the framework of Ker D[s] (paravector formalism in Ak+1 ) are given by
(s) q0 (z)
(s)
q0
z
z k+2−s := 1
z k−s (s)
s odd with s ≤ k, s even with s ≤ k.
and its partial derivatives qm (z) :=
∂ |m| (s) ∂z m q0 (z)
=
∂ n0 +···+nk (s) (z) n q n ∂x0 0 ···∂xk k 0
(1.27)
provide
canonical generalizations of the usual negative power functions in Ker D[s] from several different viewpoints. Again, whenever s > 1, one cannot restrict oneself exclusively to indices m with m0 = 0. Special attention shall be paid to the case where s = 2m + 1 and k = 2m + 2 in the context of the paravector spaces Ak+1 . The associated function class, first investigated by R. Fueter in 1931 [55] and M. Sce [146], is today often called the class of the Fueter–Sce solutions or the class of holomorphic Cliffordian 2m+2 (2m+1) (x0 + j=1 xj ej ) functions, see [104]. In this framework the functions q0 and its partial derivatives with respect to x0 coincide (up to a constant) with the ordinary negative powers of the hypercomplex variable z, see for example [104]. All 2m+2 positive and negative powers of the paravector variable z = x0 + j=1 xj ej are annihilated by this operator. It should be pointed out that the set of Fueter–Sce solutions contain the set of the hypermonogenic functions studied by H. Leutwiler et al. (see e.g., [107, 72, 47]), i.e., functions that are in the kernel of the hyperbolic Cauchy–Riemann operator Dhyp := xk D + (d − 1) as a special subset. The power functions of the paravector variable are also annihilated by the hyperbolic Cauchy-Riemann operator. The crucial point is that one can reconstruct locally every s-monogenic function from a finite number of 1-monogenic functions. One obtains a very simple representation in the vector formalism: Theorem 1.26 (Almansi type decomposition). (cf. e.g., [144]) Let B(0, R) ⊂ IRk be the open ball with radius R > 0 having its center at the origin. Let f : B(0, R) → Cl0k (IRk ) be a left s-monogenic function in B(0, R). Then there are s uniquely defined functions f1 , . . . , fs that are 1-monogenic in B(0, R) such that f (x) =
s j=1
xj−1 fj (x).
1.4. Basic theorems of Clifford analysis
25
In the paravector formalism one gets a similar but slightly more complicated representation for a function in Ker D[s] of the form f (z) = f1 (z) + zf2 (z) + z 2 f3 (z) + z z 2 f4 (z) + · · · where the functions f1 , f2 , . . . are uniquely defined and satisfy Df2j−1 = 0, resp. Df2j = 0, where j is a positive integer with j < (s + 1)/2 if s is odd and j < s/2 is s is even. The Almansi decomposition leads to Green formulas for s-monogenic functions. To this end notice there are real constants C1 and C2 so that Dq0j+1 (z) = C1 q0j (z) and
(j)
∆q0j+2 (z) = C2 q0 (z) both for the paravector and the vector formalism. In order to get simple Green for(2) (3) (2) (3) mulas let us introduce normalizations of the functions q0 , q0 , . . . say q˜0 , q˜0 , . . . so that (2)
(3)
(4)
(5)
q0 (z) = Dq˜0 (z) = ∆˜ q0 (z) = D∆˜ q0 (z) = ∆2 q˜0 (z) = . . . and similarly for the vector formalism so that (j+1)
q0 (x) = Dj q0
(x)
for j < k. These functions serve as Green kernels for s-monogenic functions. In the vector formalism one obtains again very simple Green type formulas. Theorem 1.27 (Green type formulas for s-monogenic functions in IRk ). (cf. e.g. [144]) Let 0 < r < R and suppose that f : B(0, R) → Cl0k (IRk ) is left s-monogenic. Then s−1 1 (j+1) ˜0 q (y − x)dσ(y)f (y). f (x) = Ak j=0 ∂B(0,r)
In the paravector formalism working in Ak+1 one gets a similar formula of the slightly more complicated form (1) (2) Ak+1 f (z) = q0 (ζ − z)dσ(ζ)f (ζ) + q˜0 (ζ − z)dσ(ζ)Df (ζ) ∂B(0,r)
∂B(0,r) (3)
q˜0 (ζ − z)dσ(ζ)∆f (ζ) +
+ ∂B(0,r)
(4)
q˜0 (ζ − z)dσ(ζ)D∆f (ζ) + · · ·
∂B(0,r)
In particular one obtains a Green formula for the class of holomorphic Cliffordian functions which include the class of hypermonogenic functions (compare with [104, 47]).
26
Chapter 1. Function theory in hypercomplex spaces
All s-monogenic functions are totally invariant under the translation group. If f is s-monogenic in the variable z or x respectively, and if T is a matrix from T (Ωp ), then f (T z ) or f (T x ) remains s-monogenic. Moreover, the composition a b of a left (right) s-monogenic function with a general matrix M = from c d k GV (IR ), resp. GV (Ak+1 ), remains left or right s-monogenic if one multiplies from (s) (s) the left with the factor q0 (cz + d) or from the right with the factor q0 (zc∗ + d∗ ), respectively. The following theorem gives a more precise formulation and provides a slight adaptation of Theorem 9 from [141] to the context of groups acting on the upper half-space H + (Ak+1 ). a b Theorem 1.28. Let M = be a matrix from SV (IRk−1 ), resp. from c d SV (Ak ). Suppose f : H + (IRk ) → Cl0k (IRk ) is a solution to DIsRk f (x) = 0 or s analogously that g : H + (Ak+1 ) → Cl0k (IRk ) is a solution to DA g(z) = 0. Then k+1 (s) (1.28) DIlRk q0 (cx + d)f (M x ) = 0 for all l ≥ s for all x ∈ H + (IRk ) and all M ∈ SV (IRk−1 ). Similarly, [l] (s) DAk+1 q0 (cz + d)g(M z ) = 0 for all
l ≥ s.
(1.29)
A similar result is obtained for the right s-monogenic case by instead multiplying (s) (s) the factor q0 (xc∗ + d∗ ) or q0 (zc∗ + d∗ ), respectively, from the right. Remark. Notice that z = z ∗ holds for all paravectors from Ak+1 . All vectors from IRk satisfy x = −x. However, the only paravectors that satisfy z = −z are vectors from IRk . This leads to an advantage in the vector formalism concerning the conformal invariance formula Theorem 1.28. If (c, d) is the second row from a Vahlen matrix SV (IRk ), then (cx + d)∗ = ±(cx + d) for each vector x ∈ IRk . Therefore we can substitute in the vector formalism (1.28) equivalently by (s) DIlRk q0 (cx + d)∗ f (M x ) = 0
for all l ≥ s.
(1.30)
This causes a slight symmetry break which we will need later in Chapter 3.2–3.4 for the construction of non-vanishing s-monogenic Hilbert modular forms. In the following sections in Chapter 1, in Chapter 2 and Chapter 3.1 we will exclusively work in the paravector formalism. Everything can be developed analogously in the vector formalism in which one often gets simpler formulas. The (s) (s) main thing is to switch from the functions qm (z) to qm (x), etc.. In Chapter 3.2– 3.4 we prefer then to work in the vector formalism in order to make use of the
1.4. Basic theorems of Clifford analysis
27
mentioned symmetry break by interchanging the conjugation with the reversion in the conformal invariance formulae. Of course, everything that we develop in Chapter 3.2–3.4 can be developed similarly in the paravector formalism. However, in this case one needs to take instead of the reversion another anti-automorphism, [s] or to consider also anti s-monogenic functions, i.e., functions in Ker D . Remarks. A number of the central theorems presented in this section can be adapted quite nicely in a similar form to the more general setting of real and complex Minkowski type spaces. For details, see for instance [135, 134, 141, 29]. (s) In the real Minkowski type spaces Ak+1 (IRp,q ) the functions q0 (z) are replaced for many of those applications by the expressions 1 s ≡ 0 mod 2, (k+1−s)/2 (s)
N (z) Q0 (z) = z s ≡ 1 mod 2, (k+2−s)/2
N (z) (s)
(s)
(1)
and the functions qm by the partial derivatives of Q0 . The functions Qm are annihilated from the left and from the right by the associated Cauchy–Riemann operator in Ak+1 (IRpq ), i.e., DAk+1 (IRpq ) :=
p k ∂ ∂ ∂ − ei + ej . ∂x0 i=1 ∂xi ∂xj j=p+1
(2)
The functions Qm are special nullsolutions to the wave operator ∆Ak+1
(IRp,q )
p k ∂2 ∂2 ∂2 = − + . 2 2 ∂x0 i=1 ∂xi ∂x2j j=p+1 (s)
In the complexified case Z ∈ Ak+1 (IRp,q ) with k ≡ 1 mod 2 the analogues of qm are given by 1 s ≡ 0 mod 2, (k+1−s)/2 [N (Z)] (s) Q0 (Z) = Z s ≡ 1 mod 2 (k+2−s)/2 [N (Z)] (1)
and their partial derivatives. The functions Qm are annihilated from the left and (2) from the right by the complexified Cauchy–Riemann operator and Qm by the [s] complexified Laplace operator. Holomorphic functions in the kernel of DA (Cp,q ) k+1 are consequently called complex s-monogenic. In [135, 134, 139] complete analogies of the integral theorems given above (s) were established in terms of the functions Q0 (Z) in the setting of complexified Clifford analysis for the cases where k ≡ 1 mod 2. Notice that if both p and q differ
28
Chapter 1. Function theory in hypercomplex spaces (s)
from zero, then the functions Q0 do not only have a point singularity at zero. They become singular in the whole zero-quadric. In this context one therefore has to restrict oneself to the integration over special regions which are sometimes called real domain manifolds. Following the above cited papers, a real domain manifold M ⊂ Ak+1 (Cp,q ) is a smooth real (k + 1)-dimensional manifold, if for each Z ∈ M the tangent space T Mz is spanned by (k + 1) paravectors {cj,Z ej }kj=0 where each ˜ cj,Z ∈ C\{0} and where SZ˜ ∩ M = {Z}. In view of Theorem 10 from [141], one obtains analogies of Theorem 1.28 in special circular half-cones in any type of real or complex Minkowski space IKp,q (s) (s) in terms of the functions Q0 (z) and Q0 (Z), respectively. This will be explained (s) in a more detailed way later in Chapter 3.4. The use of the functions Qm (z) (s) and Qm (Z) admits furthermore a generalization of Mittag-Leffler’s theorem to the framework of arbitrary real and complex Minkowski-type spaces. The func(s) (s) tions Qm (z) and Qm (Z) provide thus important building blocks for s-monogenic functions in Ak+1 (IKp,q ) with prescribed singularity cones. However, in contrast to the Euclidean case, additional restrictions concerning the locations of the centers of the singularity cones have to be made in order to preserve convergence. In the complexified case for instance, the centers of the singularity cones are supposed to lie once more on IRk+1 -like domain manifolds. Since the singularity cones Sz˜ reduce always to a single point in the context of real Euclidean spaces, all the special additional restrictions disappear in Ak+1 . This reflects once more the special role of real Euclidean spaces to which we will return now again.
1.5
Orders of isolated a-points, an argument principle and Rouch´e’s theorem
Many classical theorems of classical complex analysis, in particular of value distribution theory, are established on order relations of a-points of holomorphic functions for whose quantitative description the argument principle plays the central role (cf. e.g., [78]). Classical complex analysis offers several approaches to define the order of an a-point. A lot of textbooks introduce the order of an a-point (a = ∞) of a holomorphic function f at a point c as the smallest non-negative integer for which one obtains a decomposition of the form f (z) − a = (z − c)k g(z)
(1.31)
with a holomorphic function g that has no zeroes within a sufficiently small chosen neighborhood around z = c. Alternatively, one can also introduce the order in the
1.5. Orders of isolated a-points, argument principle, Rouch´e’s theorem
29
following ways: ord(f − a; c)
:=
ord(f − a; c)
:=
ord(f − a; c)
:=
dk (f (z) − a) = 0 , min k ∈ IN0 ; k dz z=c 1 1 dw, 2πi f (γ) w − a 1 f (z) dz. 2πi γ f (z) − a
(1.32) (1.33) (1.34)
In (1.33) and (1.34) γ stands for a simple closed curve around c which has no other a-points in its interior and no a-points and poles on its path. (1.33) shows that the order of the a-point of f at c is the winding number of f ◦ γ counting how often the imagined curve f ◦ γ wraps around a. An important question is to analyze whether generalizations can also be developed in the Clifford analysis setting. The development of generalizations in this sense is far from being a straightforward extension from two to higher dimensions. On the one hand a monogenic function in Ak+1 needn’t have only isolated apoints. A non-constant 1-monogenic function may possess p-dimensional manifolds of a-points where 0 ≤ p ≤ k−1. Due to the Cauchy–Riemann system the case p = k appears only for constant functions. If s ≥ 2, then one can also find s-monogenic non-constant functions that have a k-dimensional manifold of a points. This is an important difference to the classical complex case. On the other hand it does not seem very sensible to introduce the notion of the order of an isolated a-point of a monogenic function simply by generalizing (1.31), since the multiplication of a polynomial with a general monogenic function does in general not give a monogenic function again. At this point it is natural to ask whether one can introduce the notion of the order of an isolated a-point of a monogenic function by generalizing (1.32). However, G. Z¨oll has shown in [165] that such an approach would in general not provide a notion of the order in the sense of the topological mapping degree, so described for instance in [2]. In this sense, G. Z¨oll gave the following example within the framework of quaternionic analysis (see for details [165], p. 131): f (z) = (x0 + ix1 )2 + (x2 − ix3 )3 j = −2V2,0,0 (z) + 6V0,3,0 (z)j + 6V0,0,3 (z)k − 6V0,2,1 (z)k − 6V0,1,2 (z)j. The function f (z) has an isolated zero point at the origin. It is exclusively composed by homogeneous terms of degree 2 and degree 3. However, following G. Z¨ oll, the topological mapping degree around zero is −6. In [59, 63] R. Fueter suggested defining the order of an isolated quaternionic a-point of a quaternionic monogenic function in the sense of generalizing the approach (1.33). This order definition describes then indeed the topological mapping degree of the function. In 2000, T. Hempfling extended this definition to the setting of Clifford analysis in arbitrary finite dimensional Euclidean spaces (cf. [73]).
30
Chapter 1. Function theory in hypercomplex spaces
This approach enables indeed a theoretical treatment of orders of isolated a-points of monogenic functions that take values in Ak+1 . However, the appearing integral in [73] is in general rather difficult to compute explictly, since one has to perform the integration over the image of the sphere under the given arbitrary function. A further important question that arose in that context was to ask whether it is possible to set up an explicit argument principle for monogenic functions in Ak+1 from this order definition. In the paper [74], jointly written with T. Hempfling, an explicit argument principle for isolated a-points of monogenic functions that take values in the paravector space Ak+1 has been set up for arbitrary dimensions. This argument principle in turn involves furthermore integrals which are easier to evaluate. It provides also a generalization of the particular reformulated order formula for quaternionic functions given in [59] p. 88 (Formula (5)) and in [63] on p. 199. The argument principle provided the basis for the treatment of a number of rather central questions concentrated around isolated a-points of monogenic functions. In particular, a generalized version of Rouch´e’s theorem could be developed in the context of isolated a-points of monogenic functions. Furthermore, it gave some first insight in the value distribution of monogenic generalizations of elliptic functions. For some particular cases we could show with this argument principle that there is an explicit balance relation between the a-points and the poles of the monogenic generalized elliptic functions. In this section we summarize some of the general results from [74]. The specific applications to generalized elliptic functions will be discussed in Chapter 2. In order to start we first recall the basic definition of isolated a-points of monogenic functions and the order definition (cf. [59, 63, 73]): Definition 1.29 (Isolated a-points). Let U ⊂ Ak+1 be an open set and f : U → Ak+1 be a function. Let a ∈ Ak+1 . Then f is said to have an isolated a-point at c ∈ U , if f (c) = a and if there is an ε > 0 such that f (z) = a for all z ∈ B(c, ε)\{c}. Remark. A point c ∈ U that satisfies limz→c f (z) = ∞, independently from the path, is called an ∞-point of f . As a consequence of the implicit function theorem one obtains Proposition 1.30 (Sufficient criterion for isolated a-points). (cf. [73]) Let f : U → Ak+1 be a local diffeomorphism. Let c ∈ U and f (c) = a. If the Jacobian determinant (det Jf )(c) = 0, then c is an isolated a-point of f . It shall be noticed that this criterion is again only a sufficient criterion. One can show more generally, that if rank(det Jf (c)) = k + 1 − p, then c lies on a p-dimensional manifold of a-points. See [73] for more details. This is a sufficient criterion, too. Within the framework of isolated a-points, T. Hempfling introduced in [73] the following definition of the order of an isolated zero point of a monogenic function that takes values in Ak+1 :
1.5. Orders of isolated a-points, argument principle, Rouch´e’s theorem
31
Definition 1.31. Let U ⊂ Ak+1 be a non-empty open set. Assume that f : U → Ak+1 is monogenic and that c ∈ U is an isolated zero of f . Let ε > 0 such that B(c, ε) ⊆ U and f |B(c,ε)\{c} = 0. Then the integer 1 q0 (y)dσ(y) (1.35) ord(f ; c) := Ak+1 f (∂B(c,ε))
is called the order of the zero point of f at c. The expression ord(f ; c) is actually an integer. The proof is an application of the generalized Cauchy’s integral formula which tells us that 1 q0 (z − y)dσ(z)g(z) = w∂G (y)g(y), Ak+1 ∂G for any function g that is left monogenic in a simply connected domain G ⊂ Ak+1 with sufficiently smooth boundary conditions. In particular, one obtains by putting g ≡ 1 and G = B(c, ε) that 1 q0 (z − y)dσ(z) = w∂B(c,ε) (y). (1.36) Ak+1 ∂B(c,ε) Next one substitutes in (1.36) y by f (c) and ∂B(c, ε) by f (∂B(c, ε)). Following for instance [2] p. 470, the image f (∂B(c, ε)) is actually again a k-dimensional cycle under the given conditions. This leads consequently to 1 q0 (z − f (c))dσ(z) = wf (∂B(c,ε)) (0). (1.37)
Ak+1 f (∂B(c,ε)) =0
The expression wf (∂B(c,ε)) (0) counts how often the image of the sphere around the zero point wraps around zero and is hence an integer. It should be noticed for all that follows, that one may substitute more generally the sphere in formula (1.35) by a nullhomologous k-dimensional cycle parameterizing a k-dimensional surface of a (k + 1)-dimensional simply connected domain inside U that has c in its interior, no further zeroes in its interior and no zero on the proper cycle. This definition generalizes the classical definition of the order in the sense of (1.33). For the special quaternionic case it has already been formulated by R. Fueter in [59, 63]. For topological mapping reasons, it is important to claim that the range of values of f is contained in the paravector space Ak+1 when working in arbitrarily finite dimensional spaces. The quaternionic case turns out to play once more a very special role. In the case where f (c) = 0 one chooses ε sufficiently small so that f |B(c,ε) = 0. Then, (1.35) defines also the order of f at c. In this case ord(f ; c) = 0, which follows as a consequence from Cauchy’s integral theorem. Suppose that the function
32
Chapter 1. Function theory in hypercomplex spaces
g(z) = f (z) − a has an isolated zero at z = c. Then c is an isolated a-point of f . Substituting in (1.35) the function f by g = f − a gives then consequently the order of the isolated a-point of f at c. A contrast to the classical case consists of the fact that it is possible to have ord(f ; c) = 0 even if f (c) = 0 (see [63]). This phenomenon appears already in the quaternionic case. As mentioned previously one needs to perform in (1.35) the integration over the image of the sphere which is very difficult in most cases. To get simpler integrals which are easier to determine, one can apply the following transformation rule for differential forms proved by G. Z¨ oll ([165], §3.2): Lemma 1.32 (Transformation formula). Let G ⊂ Ak+1 be a domain and f : G → Ak+1 be continuously differentiable. Then dσ(f (z)) = [(Jf )∗ (z)] ∗ [dσ(z)]
(1.38)
where (Jf )∗ stands for the adjoint matrix of Jf . The multiplication ∗ denotes here the matrix-vector multiplication. In order to avoid confusion with the usual Clifford multiplication, we shall use the symbol ∗ when matrix-vector multiplication is meant and use brackets additionally. Proof. Following [165], let us write
∂f ∗
∂f (z) := (−1)i+j det (z) (Jf )(z) = Adj ∂xj ∂xj i,j i,j
∗ ∂f where ∂x (z) denotes the matrix that is deduced from the Jacobi matrix Jf j by deleting the i-th row and the j-th column. One obtains: dσ(f (z))
=
=
k
k ∂(f0 , . . . , fi−1 , fi+1 , . . . , fk ) dxj (z) (−1) ei ∂(x 0 , . . . , xi−1 , xi+1 , . . . , xk ) i=0 j=0 i
k k i=0 j=0
=
Adj
=
∂f0 ∂x0 (z)
..
. Adj
∂f ∗ j (z) (z) (−1)j dx ∂xj
∂f0 · · · Adj ∂x (z) 0 (−1)0 dx k .. .. . .
. k ∂fk (−1) dxk · · · Adj ∂xk (z)
(−1)i+j ei det
∂fk ∂x0 (z)
[(Jf )∗ (z)] ∗ [dσ(z)].
If one puts (1.38) for y = f (z) into (1.35), then one obtains the following formula for the order of an isolated a-point (a ∈ Ak+1 ) which is easier to evaluate:
1.5. Orders of isolated a-points, argument principle, Rouch´e’s theorem
33
Theorem 1.33. (cf. [74]) Let G ⊂ Ak+1 be a domain. Suppose that f : G → Ak+1 is monogenic in G and that c ∈ G is an isolated zero point of f . Let ε > 0 so that B(c, ε) ⊆ G and f |B(c,ε)\{c} = 0. Then ord(f ; c) =
1
q0 (f (z)) (Jf )∗ (z) ∗ dσ(z) .
Ak+1
(1.39)
∂B(c,ε)
Formula (1.39) provides a generalization of the second quaternionic order formula given in [59] p. 88 (Formula (5)) and in [63] p. 199. An interesting effect that appears in formula (1.39) is that both matrix-vector multiplication and Clifford multiplication operations have to be performed. This effect did not appear that clear and explicit in R. Fueter’s quaternionic order formula, since he used quaternionic determinants instead. Let us also analyze how this formula generalizes the classical formula from complex analysis, i.e., ord(f ; c) =
1 2πi
∂B(c,ε)
f (z) dz. f (z)
We observe that in the higher dimensional case the derivative f (z) is generalized 1 is replaced by q0 (f (z)), by the adjoint matrix of the Jacobian. The expression f (z) which is rather natural since q0 (z) provides the monogenic generalization of the complex analytic function z1 . Next we may finally deduce the following argument principle for isolated a-points of monogenic functions. Theorem 1.34. Let G ⊂ Ak+1 be a domain and let f : G → Ak+1 be a monogenic function. Suppose that Γ is a nullhomologous k-dimensional cycle parameterizing a k-dimensional surface of a (k+1)-dimensional simply connected bounded domain E ⊂ G. If f has only isolated a-points in the interior of E and furthermore no a-points on ∂E, then c∈E
ord(f − a; c) =
1
Ak+1
q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)].
(1.40)
Γ
Proof. Since E is a bounded domain, f can at most have a finite number of isolated a-points in E; we denote them by c1 , . . . , cm . Next take sufficiently small numbers ε1 , . . . , εm > 0 such that all the sets B˜1 := B(c1 , ε1 )\{c1 }, . . . , B˜m := B(cm , εm )\{cm } do not contain any a-points of f . Since f has no a-points and no
34
Chapter 1. Function theory in hypercomplex spaces
singularities in E\
m
B˜i , one obtains consequently 1 q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]
i=1
Ak+1
Γ
=
1
i=1
Ak+1
=
m
q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]
∂B(ci ,εi )
ord(f − a; c).
c∈E
Remark. Recall that the formula (1.38) simplifies to dσ(f (z)) = det Jf (z)[((Jf )−1 )tr (z)] ∗ [dσ(z)]
(1.41)
whenever f is a local diffeomorphism, where tr means to take the transpose of the matrix. See [165] for instance. As a consequence of this, we can then rewrite the argument principle (1.40) in the form 1 ord(f − a; c) = q0 (f (z) − a) det Jf (z)[((Jf )−1 )tr (z)] ∗ [dσ(z)] (1.42) Ak+1 c∈E
Γ
whenever Γ is a cycle that has at most a countable number of points at which det Jf (z) = 0. The argument principle allows us to set up a generalized version of Rouch´e’s theorem for monogenic functions with isolated zeroes. In [74] we proved: Theorem 1.35 (Generalized Rouch´e’s Theorem for isolated zeroes). Let G ⊂ Ak+1 be a domain and let Γ be a nullhomologous k-dimensional cycle parameterizing a k-dimensional surface of a (k + 1)-dimensional simply connected domain E ⊂ G. Let us assume that f, g : G → Ak+1 are monogenic functions that have both only a finite number of zeroes in E and no zeroes on the boundary ∂E. If
f (z) − g(z) < f (z) then
c∈E
ord(f ; c) =
for all z ∈ ∂E,
(1.43)
ord(g; c).
(1.44)
c∈E
Proof. Let λ ∈ [0, 1]. Next define the function hλ := f + λ(g − f ) and consider ord(hλ ; c) N (λ) := c∈E
=
1
Ak+1 Γ
q0 (f (z) + λg(z) − λf (z))[(J(f + λg − λf ))∗ ] ∗ [dσ(z)].
1.6. The generalized negative power functions
35
The integrand depends continuously on λ. N (λ) is thus a continuous function. However, N (λ) ∈ Z, so that N (λ) does not depend on λ. Therefore, N (λ) is a constant, and consequently, N (1) = N (0). The proof is hereby finished. With special monogenic automorphic forms which will be described in the following chapters, one can extend these techniques to the more general framework of conformally flat spin manifolds. In Chapter 2.11 we will present argument principles and Rouch´e type theorems explictly on conformally flat cylinders and tori.
1.6
The generalized negative power functions (s)
The generalized negative power functions qn (z) provide elementary building blocks for the construction of higher dimensional Clifford-valued monogenic, Euclidean harmonic and polymonogenic Eisenstein and Poincar´e series and related variants of Riemann zeta type functions. Adequate representation formulas for these functions are needed for a quantitative understanding and description of the associated function classes that will be developed in the following two chapters. In [13] and [37] one can find some representation formulas of the monogenic functions qn (z) in terms of an infinite series of Lagrange or Gegenbauer polynomials, respectively. However, they do not give a description of the functions qn in terms of a finite explicit sum of explicitly determined functions. In the recent papers [86] and [23] new and essentially different formulas were developed which meet most of our ends for the monogenic case. In [86] we developed an explicit recurrence formula in terms of finite permutational products of 2k hypercomplex variables. This formula leads nearly immediately to an impor(1) tant fully explicit upper bound estimate on the monogenic qn functions. This estimate in turn will provide the central tool for the convergence analysis of the related Eisenstein and Poincar´e series. In [23] also a non-recurrent finite closed representation formula for the func(1) tions qn is developed in particular. By means of this formula, we will later be able to give a fully explicit description of the variants of Riemann zeta functions which appear in the framework of monogenic and polymonogenic automorphic forms and functions. In [86, 23] these formulas have been established in the context of 1-monogenic functions in real Euclidean spaces. In [89] we have seen that they extend naturally to the framework of complexified Clifford analysis, although there asymptotic and growth behavior is different from the real Euclidean case. This is due to the fact that the functions qn have in the complexified case a singularity cone in the space, and not only one isolated singularity at the origin. This behavior is then reflected in the corresponding estimate on the qn -functions, which is more complicated in the complexified case than in the case of real Euclidean space. One obtains a similar result for Minkowski type
36
Chapter 1. Function theory in hypercomplex spaces
spaces with arbitrary signature (p, q) where both p and q are different from zero. This case can be treated in complete analogy to the complexified case discussed in [89]. In this section we will stick to the setting in real Euclidean space. In addition to [86] and [23] we will provide here a detailed description of the s-monogenic negative power functions for s < k + 1, too. Notice that in the case s > 1, the (s) functions qn with n = (n0 , . . . , nk ) where n0 > 0, are in general not linear (s) combinations of qn functions with n0 = 0. Therefore, in addition to [86, 23], (s) we also develop representation formulas for qn with n0 > 0. We start with the treatment of the cases where s is odd, which includes in particular the monogenic case s = 1. We begin by showing Lemma 1.36. Assume that s is an odd positive integer with s < k + 1. Let n ∈ IN. Then n−1
∂ n (s) q (z) = ∂xn1 0
j=0
n−1 (s) j! q(n−(j+1))τ (1) (z) j ' ( k − s −1 j+1 k+2−s )(z e1 ) )(e1 z −1 )j+1 . · ( + (−1)j+1 ( 2 2
Proof. We prove this lemma by induction. A direct computation gives (s)
qτ (1) (z)
e1 ze1 z e1 = − z k+2−s + k+2−s − k+2−s 2 2 z k+2−s z k+4−s (s) −1 = q0 (z) k−s e1 ) − k+2−s (e1 z −1 ) . 2 (z 2
The assertion holds for n = 1. Now let n ≥ 1. For n ∈ IN we obtain by induction ∂n {z −1 } ∂xn 1
= n!(z −1 e1 )n z −1
and
∂n {z −1 } ∂xn 1
= (−1)n n!z −1 (e1 z −1 )n .
(1.45)
Hence, (s)
q(n+1)τ (1) (z) =
∂ ∂x1
n−1 j=0
n−1 j
(s)
j! q(n−(j+1))τ (1) (z)
−1 e1 )j+1 + (−1)j+1 ( k+2−s )(e1 z −1 )j+1 · ( k−s 2 )(z 2 n−1 n−1 (s) −1 j! q(n−j)τ (1) (z) ( k−s = e1 )j+1 + (−1)j+1 ( k+2−s )(e1 z −1 )j+1 j 2 )(z 2 j=0 (s) −1 e1 )j+2 +q(n−(j+1))τ (1) (z) ( k−s 2 )(j + 1)(z +(−1)j+2 ( k+2−s )(j + 1)(e1 z −1 )j+2 2
1.6. The generalized negative power functions =
n−1 j=0 n
n−1 j
+
j=1
=
j=0
(s) −1 j! q(n−j)τ (1) (z) ( k−s e1 )j+1 + (−1)j+1 ( k+2−s )(e1 z −1 )j+1 2 )(z 2
n−1 j−1
n n j
37
(s) −1 j! q(n−j)τ (1) (z) ( k−s e1 )j+1 + (−1)j+1 ( k+2−s )(e1 z −1 )j+1 2 )(z 2
(s) −1 j! q(n−j)τ (1) (z) · ( k−s e1 )j+1 + (−1)j+1 ( k+2−s )(e1 z −1 )j+1 . 2 )(z 2
The lemma is hereby proven. n
(s)
∂ For the functions ∂x with i = 2, . . . , k we obtain an analogous recurrence n q0 i formula. We only have to replace the element e1 by ei . In view of
∂ n −1 ∂ n −1 {z } = n!(z −1 (−e0 ))n z −1 and {z } = (−1)n n!z −1 (e0 z −1 )n , (1.46) n ∂x0 ∂xn1 n
(s)
∂ we get a slightly different formula for ∂x n q0 (z). This is an effect of the symmetry 0 break in the paravector formalism in which the x0 -direction is distinguished. If we perform the same calculations as in the previous lemma, then we arrive at
Lemma 1.37. Assume that s is an odd positive integer with s < k + 1. Let n ∈ IN. Then n−1 n − 1 ∂ n (s) (s) q (z) = j! q(n−(j+1))τ (0) (z) j ∂xn0 0 j=0 ' ( k − s −1 k+2−s )(z (−e0 ))j+1 + (−1)j+1 ( )(e0 z −1 )j+1 . · ( 2 2
The next step is to develop such a recurrence formula, where n is a general \{0}. multi-index of INk+1 0 First we treat all multi-indices (n0 , n1 , . . . , nk ) where n0 = 0. If we deal with mixed derivatives, then permutational products appear. We will illustrate this by the following examples. We assume that i, j ∈ {1, . . . , k} are pairwise distinct. Then we obtain ' ( ∂ ∂ (s) k − s −1 k+2−s (s) −1 )(z ei ) − ( )(ei z ) q (z) = qτ (j) (z) ( ∂xj ∂xi 0 2 2 ' ( k − s −1 k+2−s (s) −1 −1 −1 + q0 (z) ( )(z ej )(z ei ) + ( )(ei z )(ej z ) . 2 2
38
Chapter 1. Function theory in hypercomplex spaces
Furthermore, (s)
∂ 3 q0 (z) ∂xj ∂x2i
' ( k − s −1 k+2−s (s) )(z ei ) − ( )(ei z −1 ) = qτ (j)+τ (i) (z) ( 2 2 ' ( k − s k +2−s (s) + qτ (i) (z) ( )(z −1 ej )(z −1 ei ) + ( )(ei z −1 )(ej z −1 ) 2 2 ' ( k − s k + 2 − s (s) + qτ (j) (z) ( )(z −1 ei )2 + ( )(ei z −1 )2 2 2 k−s (s) + 2 q0 (z) ( )(z −1 ej ) × (z −1 ei ) · (z −1 ei ) 2 k+2−s )(ei z −1 ) · (ei z −1 ) × (ej z −1 ) . −( 2
For the iteration of this procedure we deduce by a simple induction proof the following formulas: ∂ {[z −1 ei ]ni × [z −1 ej ]nj · (z −1 ej )}, ∂xj = (ni + nj + 1)[z −1 ei ]ni × [z −1 ej ]nj +1 · (z −1 ej ) ∂ {(ei z −1 ) · [ei z −1 ]ni × [z −1 ej ]nj } ∂xj = −(ni + nj + 1)(ei z −1 ) · [ei z −1 ]ni × [ej z −1 ]nj +1 .
(1.47)
With these formulas in hand we can finally show the following theorem using the nk n1 n2 notation := ··· : 0≤j≤n
j1 =0 j2 =0
jk =0
Theorem 1.38. Suppose that s is an odd integer with s < k + 1. Let n := (n0 , n1 , n2 , . . . , nk ) ∈ INk+1 with n0 = 0. Then 0 (s)
qn+τ (1) (z) =
n (s) |j|! qn−j (z) j 0≤j≤n k−s )[z −1 ek ]jk × · · · × [z −1 e1 ]j1 · (z −1 e1 ) · ( 2 k+2−s )(e1 z −1 ) · [e1 z −1 ]j1 × · · · × [ek z −1 ]jk . +(−1)|j|+1 ( 2
Proof. For multi-indices of the form n := (0, n1 , 0, . . . , 0) with n1 ∈ IN0 , the statement has been proved in Lemma 1.36. Let us now assume that n is a multi-index of the form n := (0, n1 , n2 , 0, . . . , 0) where n2 ∈ IN0 \{0}. By a direct computation one verifies immediately that the assertion is true for n2 = 1. In the sequel we assume that n2 ≥ 1 and compute
1.6. The generalized negative power functions
39
(s) ∂ ∂x2 q0,n1 +1,n2 ,0,...,0 (z)
=
n j
0≤j≤n
(s)
|j|!q0,n1 −j1 ,n2 +1−j2 ,0,...,0 (z)
−1 e2 ]j2 × [z −1 e1 ]j1 · (z −1 e1 ) · ( k−s 2 )[z
+
+ (−1)|j|+1 ( k+2−s )(e1 z −1 ) · [e1 z −1 ]j1 × [e2 z −1 ]j2 2 n (s) j |j|!q0,n1 −j1 ,n2 −j2 ,0,...,0 (z)
0≤j≤n
−1 · (|j| + 1) ( k−s e2 ]j2 +1 × [z −1 e1 ]j1 · (z −1 e1 ) 2 )[z
=
+(−1)|j|+2 ( k+2−s )(e1 z −1 ) · [e1 z −1 ]j1 × [e2 z −1 ]j2 +1 2 n (s) j |j|!q0,n1 −j1 ,(n2 +1)−j2 ,0,...,0 (z)
0≤j≤n
−1 · ( k−s e2 ]j2 × [z −1 e1 ]j1 · (z −1 e1 ) 2 )[z +(−1)|j|+1 ( k+2−s )(e1 z −1 ) · [e1 z −1 ]j1 × [e2 z −1 ]j2 2
j2 ←j2 +1
+
τ (2)≤j≤n+τ (2)
n j−τ (2)
(s)
|j|!q0,n1 −j1 ,n2 +1−j2 ,0...,0 (z)
−1 · ( k−s e2 ]j2 × [z −1 e1 ]j1 · (z −1 e1 ) 2 )[z
=
+(−1)|j|+1 ( k+2−s )(e1 z −1 ) · [e1 z −1 ]j1 × [e2 z −1 ]j2 2 n+τ (2) (s) |j|!q0,n1 −j1 ,(n2 +1)−j2 ,0,...,0 (z) j
0≤j≤n+τ (2)
−1 · ( k−s e2 ]j2 × [z −1 e1 ]j1 · (z −1 e1 ) 2 )[z
−1 −1 j1 −1 j2 . +(−1)|j|+1 ( k+2−s )(e z ) · [e z ] × [e z ] 1 1 2 2
The assertion turns out to be true for (n2 + 1). Hence, the formula is proved for all indices n of the form (0, n1 , n2 , 0, . . . , 0) with (n1 , n2 ) ∈ IN20 . Next we assume that n has the form (0, n1 , n2 , n3 , 0, . . . , 0), with (n1 , n2 , n3 ) ∈ IN30 . For multi-indices n := (0, n1 , n2 , n3 , 0, . . . , 0) with n3 = 0 the validity of the formula has been proved by the previous induction step. With induction over n3 one verifies, analogously to the previous induction procedure, that the formula holds for all multi-indices of the form (0, n1 , n2 , n3 , 0, · · · , 0). Subsequently, one proceeds to consider multi-indices n of the form (0, n1 , n2 , n3 , n4 , 0, . . . , 0), establishes the formula by induction over n4 analogously as we have shown in the second step, and proceeds with further induction steps until one finally obtains the validity of the formula for all multi-indices (0, n1 , n2 , . . . , nk ) ∈ INk0 .
40
Chapter 1. Function theory in hypercomplex spaces
\{0} is an index with n1 = 0, one In the case where (0, n1 , n2 , . . . , nk ) ∈ INk+1 0 chooses an α ∈ {2, . . . , k} with mα = 0. In the associated formula for the function (s) qn the index α plays then the role of the index 1 in Theorem 1.38, so that we get more generally: Corollary 1.39. Let s be an odd integer with s < k + 1. Let α ∈ {1, . . . , k} and n ∈ INk+1 with n0 = 0. Then 0 n (s) (s) qn+τ (α) (z) = |j|! qn−j (z) j 0≤j≤n k−s )[z −1 ek ]jk × · · · × [z −1 e1 ]j1 · (z −1 eα ) · ( 2 k+2−s )(eα z −1 ) · [e1 z −1 ]j1 × · · · × [ek z −1 ]jk . +(−1)|j|+1 ( 2 Now we see how to extend this formula to the general case involving also indices with n0 = 0. If n = (n0 , n1 , . . . , nk ) is a general index from INk+1 \{0} 0 with n1 = · · · = nk = 0, then we are in the situation of Lemma 1.37. Let us thus suppose that there is at least one α ∈ {1, . . . , k} for which nα = 0. We can hence write ∂ n0 (s) q (z) qn(s) (z) = ∂xn0 0 m+τ (α) where m is a multi-index with m0 = 0. For qm+τ (α) (z) Corollary 1.39 provides us with a recurrence formula. Thus we need only to perform differentiations with respect to x0 on the formula in Corollary 1.39. Applying (1.46) to Corollary 1.39, then an analogous induction argument leads to the final result: Theorem 1.40. Let s be an odd positive integer with s < k + 1. Let α ∈ {1, . . . , k} and n = (n0 , . . . , nk ) ∈ INk+1 . Then 0 (s)
qn+τ (α) (z)
n (s) = |j|! qn−j (z) j 0≤j≤n k−s )[z −1 ek ]jk × · · · × [z −1 e1 ]j1 × [z −1 (−e0 )]j0 · (z −1 eα ) · ( 2 k+2−s )(eα z −1 ) · [e0 z −1 ]j0 × [e1 z −1 ]j1 × · · · × [ek z −1 ]jk . +(−1)|j|+1 ( 2
All the configurations of multi-indices n are now treated for the cases where s is odd. Notice that the terms with k−s 2 in Theorem 1.40 vanish if k = s and one obtains a much simpler formula. In the case where s is odd and where additionally (k) s = k, we are in the holomorphic Cliffordian case whence the functions qn (z) simplify to the partial derivatives of z −1 . In view of (1.47) we thus obtain in this
1.6. The generalized negative power functions
41
special case the holomorphic Cliffordian negative basis polynomials from [104] which read in our notation: [z −1 e0 ]n0 × [z −1 e1 ]n1 × · · · × [z −1 ek ]nk z −1 . Let us now turn to the cases where s is even. We shall see that we can treat these (s) cases rather similarly. Notice that for even s the functions qn are all scalar-valued. To derive similar formulas for even s with s < k +1, let us again first consider is a multi-index of the form nτ (i) where i ∈ {1, . . . , k}. the case where n ∈ INk+1 0 We compute 1 ∂ (s) qτ (i) (z) = 2 ∂xi x0 + · · · + x2k (k−s+1)/2 1 (k − s + 1)xi = − 2 2
z (x0 + · · · + x2k )(k−s+1)/2
k − s + 1 (s) q0 (z) z −1 ei − ei z −1 , = (1.48) 2 in view of xi = − 12 (zei + ei z) for i ∈ {1, . . . , k}. If we next apply on (1.48) the formulas (1.45), then we arrive by an absolutely analogous induction argument as in Lemma 1.36 at the following representation formula: Lemma 1.41. Let s ∈ 2IN with s < k + 1. Let i ∈ {1, . . . , k}. Then for all n ∈ IN, n−1 n − 1 (s) qnτ (i) (z) = j!q(n−(j+1))τ (i) (z) j j=0
k − s + 1 (z −1 ei )j+1 + (−1)j+1 (ei z −1 )j+1 . · 2 Turning to the case i = 0, then we observe first that (s)
qτ (0) (z) = −
(k − s + 1)x0
z 2
1
(k−s+1)/2 x20 + · · · + x2k
z z k − s + 1 (s) q0 (z) = (−1) · + 2 2
z
z 2 k − s + 1 (s) −1 q0 (z (−e0 )) + (−1)(e0 z −1 ) , (1.49) = 2 in view of x0 = 12 (z + z). Applying next on (1.49) the formulas (1.46), then an analogous induction argument as in Lemma 1.37 leads finally to Lemma 1.42. Let s ∈ 2IN with s < k + 1. Then for all n ∈ IN, n−1 n − 1 (s) qτ (0) (z) = j!q(n−(j+1))τ (0) (z) j j=0
k − s + 1 (z −1 (−e0 ))j+1 + (−1)j+1 (e0 z −1 )j+1 . · 2
42
Chapter 1. Function theory in hypercomplex spaces
In complete analogy to the proof of Theorem 1.40 one can deduce with an absolutely analogous induction argument, by applying the differentiation formulas (1.47) on the expressions in Lemma 1.41 and in Lemma 1.42, the following general (s) recurrence formula for the functions qn for even s with s < k + 1: Theorem 1.43. Let s be an even positive integer with s < k + 1. Let α ∈ {1, . . . , k} . Then and n = (n0 , . . . , nk ) ∈ INk+1 0 n (s) (s) qn+τ (α) (z) = |j|!qn−j (z) j 0≤j≤n
k − s + 1 [z −1 ek ]jk × · · · × [z −1 e1 ]j1 × [z −1 (−e0 )]j0 · (z −1 eα ) · 2 +(−1)|j|+1 (eα z −1 )[e0 z −1 ]j0 × [e1 z −1 ]j1 × · · · × [ek z −1 ]jk .
We proceed to show that we can now easily derive from Theorem 1.40 and (s) Theorem 1.43 the following estimates on the functions qn . the following Proposition 1.44. Let s ∈ {1, . . . , k}. For all multi-indices n ∈ INk+1 0 estimate holds for all z ∈ Ak+1 \{0}: (k + 1 − s)(k + 2 − s) · · · (k + |n| − s) ∂ |n| (s) . (1.50) n q0 (z) ≤ ∂z
z k+|n|+1−s Proof. We first restrict ourselves to the cases where n = (0, n1 , 0, . . . , 0). A simple calculation shows that the estimate holds for n1 = 0 and n1 = 1, both for even and odd s. In the sequel we assume n1 ≥ 1. We obtain both for even and odd s: n +1 n1 ∂ 1 (s) ((k−s)+(n1 −j))! n1 +1 q0 (z) ≤ z k+1−s k+n1 +2−s n1 ! (k−s)!(n1 −j)! ∂x1
j=0
=
k+1−s z k+n1 +2−s
n1 !
n1 (k−s)+n1 −j j=0
n1 −j
k+1−s+n1
=
k+1−s z k+n1 +2−s
=
(k+n1 +1−s)! k+1−s z k+n1 +2−s (k+1−s)!
=
(k + 1 − s)(k + 2 − s) · · · (k + n1 + 1 − s) z k+n11 +2−s .
n1 !
n1
Thus, the assertion holds for indices of the form n := (0, n1 , 0, . . . , 0). With the estimates (e1 z −1 ) · [e0 z −1 ]j0 × [e1 z −1 ]j1 × · · · × [ek z −1 ]jk ≤ e1 z −1 |j|+1 , −1 j [z ek ] k × · · · × [z −1 e1 ]j1 × [z −1 (−e0 )]j0 · (z −1 e1 ) ≤ z −1 e1 |j|+1 and the formula
n0 nk |n| ··· = j j0 jk k+1
j∈IN0 |j|=j
1.6. The generalized negative power functions
43
in combination with the statements of Theorem 1.40 and Theorem 1.43, respectively, we get furthermore, by applying a simple induction argument, that (s)
qn(s) (z) ≤ q0,|n|,0,...,0 (z)
∀n ∈ INk+1 0
and the assertion is shown.
Remarks. This formula provides actually a more precise estimate than that given in [13]. The estimate in (1.50) is furthermore stronger than the estimate proved in [63] by R. Fueter for the quaternionic case in the framework s = 1, i.e.,
qn(1) (z) ≤ (|n| + 2)! z −(|n|+3) .
(1.51)
R. Fueter’s method for his proof is based on the formula qn(1) (ζ) = ζ −1
∂ |n| [∆z {(zζ −1 )n+2 }], ∂xn
z ∈ IH, ζ ∈ IH\{0},
(1.52)
where ∆z denotes the Laplace operator with respect to the variable z. Switching from the complex holomorphic case, to the higher dimensional 1-monogenic cases, then the ordinary positive power functions are replaced by permutational products of the hypercomplex variables Zi = xi − ei x0 . See [58], and in particular [118, 120]. Theorem 1.40 provides us with a similar representation formula of the negative power functions that is built with permutational products of the special hypercomplex variables ζi := z −1 ei and ηi := ei z −1 . Theorem 1.40 and Theo(s) rem 1.43 are explicit recurrence formulae for the functions qn (z). (s)
In the vector formalism where the analogues of the functions qn are given by q(s) n1 ,...,nk (x) =
∂ n1 +···+nk x1 e1 + · · · + xk ek (s) (s) , nk q0 (x) where q0 (x) = − 2 n1 ∂x1 · · · ∂xk |x1 + · · · + x2k |(k+1−s)/2
one obtains the following analogue of Theorem 1.40: Theorem 1.45. Let s be an odd integer with s < k. Let α ∈ {1, . . . , k} and n ∈ INk0 \{0}. Then for all x ∈ IRk \{0} we have n (s) (s) qn+τ (α) (x) = (1.53) |j|! qn−j (x) j 0≤j≤n k−1−s )[x−1 ek ]jk × · · · × [x−1 e1 ]j1 · (x−1 eα ) · (−1)|j|+1 ( 2 k+1−s (eα x−1 ) · [e1 x−1 ]j1 × · · · × [ek x−1 ]jk . + 2
44
Chapter 1. Function theory in hypercomplex spaces
The proof can be done in analogy to the proof of Theorem 1.40. The additional symmetry simplifies several computations. Similarly, one obtains the follow(s) ing formula for the functions qn for the cases where s is an even positive integer with s < k: Theorem 1.46. Let s be an odd integer with s < k. Let α ∈ {1, . . . , k} and n ∈ INk0 \{0}. Then for all x ∈ IRk \{0} we have n (s) (s) (1.54) qn+τ (α) (x) = |j|! qn−j (x) j 0≤j≤n
k − s [x−1 ek ]jk × · · · × [x−1 e1 ]j1 · (x−1 eα ) · (−1)|j|+1 2 + (eα x−1 ) · [e1 x−1 ]j1 × · · · × [ek x−1 ]jk .
Remark. These representation formulas hold also in its form for the analogues of (s) qm in the more general framework of arbitrary real and complex Minkowski type spaces of signature (p, q). However, one gets a different form of estimate due to the infinite extension of the singularity cone. For details, see [89] where we developed an estimate for the complexified case. (s)
Next we describe a closed representation formula for the functions qn in Ak+1 that was developed in [23] for the monogenic case s = 1. The formula that (s) will be derived is based on the fact that the functions qn (z) are radial symmetric functions of degree −(k − s + 1) whenever s is even, and that further to this the special relation 1 Dz qn(s) (z) (1.55) qn(s−1) (z) = − k−s+1 (s)
holds. The whole problem of finding a closed formula for the functions qn is thus reduced to finding a closed description for partial derivatives of functions in the space that have a radial symmetry. To meet this end one needs the following lemma: Lemma 1.47. Let f (r) be a C ∞ function of a single real variable r. Then
d dr
n
f (r) =
0≤2p≤n
(2r)n−2p n! (n − 2p)!p!
d d(r2 )
n−p f (r).
(1.56)
Proof. By applying the chain rule from classical one real variable calculus, one obtains directly that d 1 d d(r 2 ) f (r) = 2r dr f (r), which gives
d dr
f (r) = 2r
d d(r 2 )
f (r).
1.6. The generalized negative power functions
45
Iteration leads in the second step to
d 2 dr
f (r) = 2
d d(r 2 )
f (r) + 4r2
d d(r 2 )
2
f (r).
Arbitrary iterations lead finally to a formula of the form
d n dr
f (r) =
∞
cn,s (r)
s=0
d d(r 2 )
s f (r),
(1.57)
in which however only a finite number of terms do not vanish. It is thus only a finite sum. The main task is reduced to determining the functions cn,q (r) explictly. 2 To do so we apply the following trick: insert the particular function f (r) = ear into (1.57). A direct computation leads to ∞ d n ar2 2 s e = c (r)a (1.58) ear . n,s dr s=0
The functions cn,s (r) will now be determined by a comparison of both sides of the previous equation. Next we multiply both sides of equation (1.58) by bn /n! and sum over n = 0, 1, 2, . . ., which then leads to ∞ n=0
bn n!
d dr
n
∞ ∞
2
ear =
n=0
s=0
2 n cn,s (r)as ear bn! .
The left-hand side of this equation is nothing else than the Taylor expansion of 2 the function h(b) = ea(r+b) at the point b = 0, so that this equation simplifies to ) * ∞ 2 as bn a(r+b)2 = cn,s (r) n! (1.59) ear , e n,s=0
which in turn simplifies further to eab(2r+b) =
∞
s n
n,s=0
cn,s (r) a n!b .
(1.60)
Now we rewrite eab(2r+b) in terms of its Taylor expansion around r = 0, and we get ∞ q q ∞ s n a b (2r+b)q = cn,s (r) a n!b . (1.61) q! q=0
n,s=0
In view of the binomial formula (b + 2r)q = the previous equation in the form 0≤p≤q<∞
aq bp+q (2r)q−p (q−p)!p!
=
q
q p=0 p
∞ n,s=0
bp (2r)q−p , we can rewrite
s n
cn,s (r) a n!b .
(1.62)
46
Chapter 1. Function theory in hypercomplex spaces
Next we compare the terms with the same powers of a and b on both sides of the equation. This leads to q = s, p + q = n and hence q − p = n − 2p. Consequently the inequality 0 ≤ p ≤ q < ∞ is then equivalent to 0 ≤ p ≤ n − q < ∞ which further is equivalent to 0 ≤ 2p ≤ n < ∞. Summarizing, (1.62) can thus be written equivalently as ∞ s n as bn (2r)n−2p cn,s (r) a n!b . (1.63) (n−2p)!p! n,s=0
0≤2p≤n<∞
Finally, we obtain the formula ∞
cn,s (r)as =
s=0
0≤2p≤n
(2r)n−2p n! n−p . (n−2p)!p! a
Inserting this expression into (1.57) leads then immediately to the assertion ( 1.56). This lemma allows us now easily to develop a formula for the partial derivatives of radial symmetric functions. \{0} Lemma 1.48. Let f ( z ) be a C ∞ radial symmetric function and m ∈ INk+1 0 be an arbitrary multi-index. Then
∂n (f ( z )) =
0≤2p≤n
(2z)n−2p n! (n − 2p)!p!
d d( z 2 )
|n|−|p| f ( z ),
where z n−p stands for xn0 0 −p0 xn1 1 −p1 · · · · · xnk k −pk . Proof. To show the assertion, one applies the previous lemma to each coordinate x0 , . . . , xk for the n0 , . . . , nk -th derivative, respectively. This leads then directly to the formula nk −pk (2z)n−2p n! ∂ n0 −p0 ∂ ∂ |n| (f ( z )) = · · · f ( z ). ∂z n (n − 2p)!p! ∂(x20 ) ∂(x2k ) 0≤2p≤n
Let us put f ( z ) = g( z 2 ) = g(x20 + . . . + x2k ). From the radial symmetry it follows then that all derivatives with respect to x2i (i = 0, . . . , k) coincide with the derivative with respect to z 2 , so that one finally obtains the formula
∂ ∂(x20 )
n0 −p0
···
∂ ∂(x2k )
nk −pk f ( z ) =
d d( z 2 ) (s)
|n|−|p| f ( z ).
Applying now Lemma 1.48 to the functions qn (z) with s ∈ 2IN, leads then (s) immediately to the following formula for the functions qn where s is an even positive integer with s < k + 1:
1.6. The generalized negative power functions
47
\{0} be an arbitrary Theorem 1.49. Let s < k + 1 be an even integer and n ∈ INk+1 0 multi-index. Then
k − s + 1 (2z)n−2p n! 1 (−1)|n|−|p| . qn(s) (z) = (n − 2p)!p! 2 |n|−|p| z k−s+1+2|n|−2|p| 0≤2p≤n
(1.64) Remark. This formula allows us directly to conclude that qn(s) (z) 0 ≡ z=x0
if and only if n = (n0 , . . . , nk ) is a multi-index from INk+1 with n1 , . . . , nk ∈ 2IN0 . 0 Applying (1.55) to formula (1.64) leads to a closed representation formula for (s) the functions qn (z) for odd s for the cases where k > s. By a direct computation we obtain the following formula generalizing Theorem 6 from [23]: Theorem 1.50. For odd k ∈ IN with k ≥ s + 1 one has the closed representation formula (2z)n−2p n! k−s 1 1 (−1)|n|−|p| qn(s) (z) = k−s+2|n|−2|p| k−s (n − 2p)!p! 2
z |n|−|p| 0≤2p≤n k n − 2p k − s + 2|n| − 2|p| j j + (1.65) · ej . z x j j=0 In the monogenic case s = 1, this representation formula is thus valid in its form for dimensions greater than 3. Theorem 1.50 allows us to conclude that also for all odd s with s > k, Sc qn(s) (z) ≡ 0 z=x0
if and only if n = (n0 , . . . , nk ) is a multi-index from INk+1 with n1 , . . . , nk ∈ 2IN0 . 0
Chapter 2
Clifford-analytic Eisenstein series associated to translation groups 2.1
Multiperiodic Mittag-Leffler series
The classical meromorphic Eisenstein series for translation groups in complex analysis are given as Mittag-Leffler series of the negative power functions 1/(z + ω)m summed over a one- or two-dimensional lattice. They provide important building blocks for the construction of the trigonometric functions, the elliptic functions and the elliptic modular forms. In the higher dimensional Clifford analysis setting where we consider nullsolutions to D[s] or Ds in Ak+1 or IRk , respectively, the negative power functions (s) are generalized by the functions qm (z + ω), as explained in the previous chapter. Generalizations of the classical Eisenstein series to the context considered here (s) are, roughly speaking, thus given by summations of the expressions qm (T z ) over the corresponding translation group T . The following definition provides a more precise formulation: Definition 2.1 (s-monogenic Eisenstein series associated to translation groups). Let p ∈ IN with 1 ≤ p ≤ k + 1 and let s ∈ IN with 1 ≤ s ≤ k. Assume that ω1 , . . . , ωp are IR-linear independent paravectors from Ak+1 . Then the s-monogenic translative Eisenstein series associated with the p-dimensional lattice with |m| ≥ Ωp = Zω1 + · · · + Zωp are defined for all multi-indices m ∈ INk+1 0 max{0, p − k + s} by (s) (p) qm (z + ω). (2.1) m,s (z; Ωp ) = ω∈Ωp
50
Chapter 2. Clifford-analytic Eisenstein series for translation groups
In the cases where p−k−1+s ≥ 0 we have additionally for multi-indices m ∈ INk+1 0 with |m| = p − k − 1 + s:
(s) (s) (s) q (z; Ω ) = q (z) + (z + ω) − q (ω) . (2.2) (p) p m,s m m m ω∈Ωp \{0}
In the cases where p − k − 2 + s ≥ 0 one defines, for multi-indices m ∈ INk+1 with 0 |m| = p − k − 2 + s, (p)
m,s;a,b (z; Ωp ) := +
(s) (s) qm (z − a) − qm (z − b) (s) (s) (z − a + ω) − qm (z − b + ω) qm ω∈Ωp \{0}
(2.3)
(s) (s) − qm (ω − a) + qm (ω − b) ,
where a, b are paravectors from Ak+1 \Ωp with a ≡ b mod Ωp . Notice that all these series have only point singularities. This stems from the fact that we are working here in real Euclidean spaces which are endowed with a definite scalar product. As shown in [89, 90, 93] one can also introduce these function series in the more general context of real and complex Minkowski type spaces IRp,q and Cp,q endowed with a non-degenerate scalar product of arbitrary signature (p, q) with p + q = k + 1. However, in these cases one has to put restrictions on the generators of the lattice. In those quadratic spaces IRp,q where p and q are both different from zero, one does not always obtain a convergent series when considering a summa(s) tion of the associated expressions qm (z + ω) over any arbitrary lattice in IRp,q , independently how large |m| is chosen. This behavior stems from the fact that these expressions do not only have singularities in isolated points but in cones. Therefore, restrictions on the periodicity group have to be made in these cases. One obtains a convergent series (under the same conditions for |m| as mentioned above), when the lattice Ω is additionally completely contained in IRp or in IRq . This shows that the configuration (p, q) = (0, k +1) or (p, q) = (k +1, 0) is a special one. In these particular cases one obtains an even s-monogenic Eisenstein series associated to periodicity lattices of the codimension 0. In the general case working in IRp,q , one gets in the best case an Eisenstein series associated to a translation group with max{p, q} linear independent generators. In the complexified case working in Cp,q one has again a more balanced relationship between the corresponding spaces. In Cp,q one can always define convergent Eisenstein series to a group that has k + 1 translation generators. In Cp,q the period lattice needs to be contained in the linear IRk+1 -like domain manifold eiφ (IRp + iIRq ) where φ ∈ [0, 2π) is an arbitrary real parameter. Under this condition we get the same convergence conditions for |m| as mentioned above. Thus, in
2.1. Multiperiodic Mittag-Leffler series
51
the complexified case one can always introduce (independently from the signature) a p-fold periodic s-monogenic Eisenstein series, for 1 ≤ p ≤ k + 1. One obtains a divergent series when p > k + 1. In the cases p < k +1 these function series provide us with building blocks for k-fold periodic generalizations of classical trigonometric functions to the Clifford analysis setting (cf. [84, 85]). In the case p = k + 1, one obtains generalizations of the elliptic functions to the Clifford analysis setting. Notice that one cannot define (k + 1)-fold periodic s-monogenic Eisenstein series within the setting of IRp,q whenever p, q are both different from zero. (k+1)
The particular 1-monogenic function series m,1 , where the summation is extended over a period lattice of codimension 0, appears first in a paper by A. C. Dixon [40] within the particular framework of the three dimensional real Euclidean space IR0,3 , some decades later in works of R. Fueter [61, 62, 63, 64] in the setting of real quaternions, and in the beginning of the 1980s in J. Ryan’s first paper [133] in the setting of the (k + 1)-dimensional real Euclidean vector space. Some of their basic properties were studied in these works. In [128] T. Qian considered a class of functions defined on particular sector domains and studied Fourier multipliers and Lipschitz perturbations of the k(k) torus in IRk+1 within a cylinder. The particular function 0 associated with the k+1 turns out to be part of that special k-dimensional orthonormal lattice in IR function class. In [84, 85] we started a more extensive and systematic study of 1-monogenic Eisenstein series associated to arbitrarily dimensional translation groups in the Euclidean space IR0,k+1 within a more general framework, particularly under function theoretical and number theoretical motivation. Subsequently their complexification to C0,k+1 has been studied in [89]. Polymonogenic Eisenstein and Poincar´e series associated to general discrete subgroups of Vahlen type groups in IR0,k+1 have been introduced in [88] and their complexifications to C0,k+1 in [89]. The function series treated in [88] contain the series (p) (p) 0,s as special subseries. Explicit analogues of the series 0,s on the unit ball in IR0,k+1 and on the Lie ball in C0,k+1 also were developed in [88] and [90], respec(p) tively. In [96] the series 0,s with p ≤ min{k + 2 − s, k + 1} were used explicitly to develop Cauchy–Green formulas on conformally flat cylinders and tori which arise from factoring out IR0,k+1 by a translation group. For arbitrary multi-indices (p) m ∈ INk+1 and s < k + 1 the series m,s were introduced in [92] within the 0 framework of real Euclidean spaces. The family of all s-monogenic Eisenstein series associated to translation groups in turn contains the holomorphic Cliffordian Weierstraß functions from [67, 105], which are solutions of the Fueter–Sce equation, as very special and important subcases. Finally, in [93] we proceeded also to introduce s-monogenic generalizations of Eisenstein series within the framework of finite dimensional real and complex vector spaces.
52
Chapter 2. Clifford-analytic Eisenstein series for translation groups
We restrict ourselves mainly to a treatment of the theory within the context of finite dimensional real Euclidean vector spaces. For extensions of this theory to complexified Clifford analysis, we refer to our papers [89] and [90], and, more generally, for extensions to the framework of real and Minkowski type spaces of arbitrary signature to [93]. (p)
The Eisenstein series m,s (z, Ωp ) provide an extremely useful tool for the construction of large classes of s-monogenic p-fold periodic functions in Ak+1 . As we shall see in Chapter 3, they turn out to serve furthermore as building blocks for automorphic forms for much larger arithmetic groups which contain translation groups as subgroups. This gives them a similar key role as their two-dimensional counterparts from classical complex analysis have. We begin by giving a detailed convergence proof for these function series under the conditions mentioned in Definition 2.1. Proposition 2.2. Under the conditions for p, s, m stated in Definition 2.1, the s-monogenic generalized translative Eisenstein series converge normally in Ak+1 \Ωp . Proof. Let us first consider those cases where m is a multi-index from INk+1 with 0 |m| ≥ max{0, p − k + s}. To show that under these conditions the series
(s)
qm (z + ω)
(2.4)
ω∈Ωp
is normally convergent in Ak+1 let us take an arbitrary compact subset K ⊂ Ak+1 . Next one takes a real R > 0 so that the compact ball B(0, R) covers K completely. Let z ∈ B(0, R). For the qualitative convergence analysis we may drop without loss of generality a finite number of terms of the series. Let us thus restrict ourselves to summing those lattice points ω that satisfy
ω > (k + 1)R ≥ (k + 1) z .
(2.5)
(s) g(z) = qm (z + ω)
(2.6)
The function is left s-monogenic in 0 ≤ z < (k+1)R. Hence it is real analytic in B(0, (k+1)R) and is thus represented in the interior of this ball by its Taylor series which turns out to have the form (s)
qm (z + ω) =
∞ |l|=0
1 l0 l! x0
(s)
· · · · · xlkk qm+l (ω).
(2.7)
As a consequence of the estimate proved in Proposition 1.44 we obtain in particular ∞ (s) qm (z + ω) ≤
l=0 l=l0 +···+lk
l |m|+k−s + Rl l! 1 (l0 + · · · + lk + γ) l0 !···l . ω k+|m|+1−s l k ! w γ=1
2.1. Multiperiodic Mittag-Leffler series
53
An application of the multinomial formula l! l=l0 +···+lk
leads finally to (s) qm (z + ω) ≤
l0 !···lk !
= (k + 1)l
l ∞ |m|+k−s + 1 (l + γ) (k+1)R ω k+|m|+1−s ω γ=1
l=0
=
(|m|+k−s)! 1 (k+1)R k+|m|+1−s ω k+|m|+1−s 1− ω
≤
C ω k+|m|+1−s
with a real positive constant C. Due to G. Eisenstein’s lemma (cf. [41]) a series of the form
m1 ω1 + · · · + mt ωt −(t+α) (2.8) (m1 ,...,mt )∈Zt \{0}
is convergent if and only if α ≥ 1. Hence, the series (2.4) converges normally for |m| ≥ max{0, p − k + s}. is a multi-index Let us now turn to the second type of cases, where m ∈ INk+1 0 with |m| = p − k − 1 + s while we suppose that p − k − 1 + s ≥ 0. In this context, where we have to show that the expression (s) (p) (s) (s) qm (z + ω) − qm (ω) (2.9) m,s (z; Ωp ) = qm (z) + ω∈Ωp \{0}
is normally convergent in Ak+1 \Ωp , we can provide a similar argument. Again we suppose that z ∈ B(0, R) and restrict our study to those lattice points with
ω > (k + 1)R ≥ (k + 1) z . Instead of considering the function (2.6), we expand the function (s) (s) (z + ω) − qm (ω), g1 (z) = qm
|m| = p − k − 1 + s,
into a Taylor series in B(0, R ) where 0 < R < R. We obtain g1 (z) =
∞ |l|=1
1 l0 l! x0
(s)
· · · · · xlkk qm+l (ω).
(2.10)
Notice that in contrast to (2.7), this Taylor series starts only with |l| ≥ 1. A similar procedure, using the estimate from Proposition 1.44 and the multinomial formula, leads then finally to (s) (s) qm (z + ω) − qm (ω)
≤
∞ |m|+k−s l + 1 (l + γ) (k+1)R ω ω k−s+1+|m|
l=1
=
γ=1
l R (l + γ) (k+1)R ω ω k−s+2+|m|
∞ |m|+k−s+1 + l=0
γ=2
54
Chapter 2. Clifford-analytic Eisenstein series for translation groups
Hence, there is a real C > 0 such that (s) (s) qm (z + ω) − qm (ω) ≤ C R ω∈Ωp \{0}
ω∈Ωp \{0}
1 ω k−s+2+|m|
=CR
ω∈Ωp \{0}
1 ω p+1
which is convergent under the given conditions due to Eisenstein’s lemma (2.8). The cases where |m| = p−k−2+s (under the assumption that p−k−2+s ≥ 0) require already a bit more technical treatment. Again let K ⊂ Ak+1 be an arbitrary compact subset and choose R > 0 so that the closed ball B(0, R) covers completely K. Let z ∈ B(0, R). Now we restrict our consideration, without loss of generality, to those lattice points that satisfy (k+1)a <1− 1 <1− 1 . and (k+1)b (2.11) ω k+1 ω k+1 Further, we take only those lattice elements which furthermore satisfy
ω > (k + 1)2 R.
(2.12)
Now we consider the function (s) (s) (s) (s) (z − a + ω) − qm (z − b + ω) − qm (ω − a) + qm (ω − b) g2 (z) = qm
which is s-monogenic in B(0, (k + 1)R). Hence, it is represented in the interior of this ball by its Taylor series which reads g2 (z) =
∞ |l|=1
1 l0 l! x0
(s) (s) · · · · · xlkk qm+l (ω − a) − qm+l (ω − b) .
(2.13)
We thus obtain that
g2 (z) ≤
∞ |l|=1
1 l0 l! x0
(s) (s) (s) (s) · · · · · xlkk qm+l (ω − a) − qm+l (ω) + qm+l (ω − b) − qm+l (ω) . (2.14)
The crucial step is now to consider the function (s)
(s)
h(a) := qm+l (ω − a) − qm+l (ω)
(2.15)
which is s-monogenic in the variable a in 0 ≤ a < γ < ω . In particular, it is real analytic in B(0, γ) and can be thus represented in B(0, γ) by its Taylor series. Using the same argument as in the previous part of the proof, we obtain
h(a) ≤
(|m|+|l|+k−s+1)! (k+1)a (k+1)a k+2−s+|m|+|l| (1− ω )k+2−s+|m|+|l| ω
were we used condition (2.11). Again from (2.11) we infer that a
h(a) ≤ (|m| + |l| + k − s + 1)!(k + 1)k+3+|m|+|l|−s ω k+|m|+|l|+s−2 .
(2.16)
2.1. Multiperiodic Mittag-Leffler series Now we estimate ∞
g2 (z) ≤
55
(|m| + l0 + · · · + lk + k + 1 − s)!(k + 1)k+3+|m|+l−s
l=1 l=l0 +···+lk
·
a + b Rl . ω |m|+l0 +···+lk +k+2−s l0 !···lk !
After having applied the multinomial formula and an index shift we thus obtain
g(z)
≤
r ∞ |m|+k+2−s + (r + γ) (k+1)R (k + 1)k+3+|m|−s ω a + b k+|m|+3−s R ω
r=0
≤
γ=2
L ω a + b k+|m|+3−s R.
In view of |m| = p − k − 2 + s we can conclude from Eisenstein’s lemma (2.8) that the series 1 ω∈Ωp \{0}
ω k+|m|+3−s
converges absolutely under this condition, which completes the proof of this proposition. (p)
The series m,s (z, Ωp ) (p < k + 1) admit the construction of higher dimensional paravector valued s-monogenic variants of many elementary trigonometric functions, as explained in [84, 85, 89, 93] for the several types of higher dimensional spaces. Definition 2.3. Let p < k+1 and let Ωp = Zω1 +· · ·+Zωp be a p-dimensional lattice in Ak+1 . Let β stand for a multi-index of a length |β| = max{0, p − k − 1 + s}. The s-monogenic p-fold periodic generalizations of the cotangent, tangent, cosecant and secant function associated to Ωp are then defined by (p)
cotp,s,β (z)
:= β,s (z), := −
tanp,s,β (z)
cotp,s,β (z + v/2),
v∈Vp (2)\{0}
cscp,s,β (z)
:=
secp,s,β (z)
:=
1 cotp,s,β (z/2) − cotp,s,β (z), 2k−s cscp,s,β (z + v/2). v∈Vp (2)\{0}
ω
ω +···+ω
Here Vp (2) := {0, ω21 , . . . , 2p , . . . , 1 2 p } stands for the canonical system of representatives of the quotient module Ωp /2Ωp . In the case where β = 0 we write for simplicity cotp,s , tanp,s , secp,s and cscp,s . (p)
By means of the series m,s with |m| = β + 1 one can furthermore construct s-monogenic p-fold periodic generalizations of the cosecant-squared and the secantsquared function, etc..
56
Chapter 2. Clifford-analytic Eisenstein series for translation groups
Remark. The hypercomplex generalizations of the classical trigonometric functions with singularities given in Definition 2.3 are different from those that are considered in [66, 159] and Section 2 of [105] which are radial periodic generalizations. In [26] explicit relations between the classical complex-analytic trigonometric functions and the particular 1-monogenic higher dimensional one-fold periodic generalizations that have purely real periods are established. Without loss of generality, let us consider in this special context as one-dimensional periodicity “lattice” Zπ. Then the associated one-fold periodic monogenic cotangent cot1,1 (z, Zπ) function has the form z + nπ .
z + nπ k+1 n∈Z
Similarly, csc1,1 (z, Zπ) =
(−1)n
n∈Z
z + nπ
z + nπ k+1
while the one-fold periodic monogenic cosecant-squared function is given in the hypercomplex case by
∂ z + nπ − . csc21,1 (z, Zπ) = ∂x0 z + nπ k+1 n∈Z
Notice that z is the only paravector expression appearing here. The same holds for the corresponding formulas for the tangent, the secant and the secant squared functions since they stem from these functions just by applying a shift of ±π/2 in the argument z. We shall see that this has an important consequence. To proceed, consider the well-known transformation (cf. e.g., [56, 146, 66]) T : Ak+1 \{0} → C
z → ζ := Sc(z) + i V ec(z) .
The fact that z is the only paravector appearing in all these expressions admits actually description of the hypercomplex one-fold periodic trigonometric functions with real periods in terms of the associated complex variable ζ. To be more explicit, it is possible to recover the generalized one-fold periodic hypercomplex trigonometric functions with real periods, say f (z), from its associated complex transform T (f (z)) by means of f (z) = ReT (f (z)) +
V ec(z) ImT (f (z))
V ec(z)
in this special situation. Especially in the even-dimensional case, i.e., k = 3, 5, · · · one obtains a relatively simple relation. For simplicity let us introduce the abbreviations: cd,p
=
(−1)d/2−1 (d/2−2+p)! , (d/2−2)!p!(d/2−p−1)!(ζ−ζ)p+d/2−1
c∗d,p
=
(−1)d/2 (d/2−1+p)! , (d/2−1)!p!(d/2−p−2)!(ζ−ζ)p+d/2
p = 0, . . . , d/2 − 1, p = 0, . . . , d/2 − 2.
2.1. Multiperiodic Mittag-Leffler series
57
Theorem 2.4. d/2−p−2 d/2−p−1 d/2−2 ∂ cot(ζ) + c∗d,p cot(ζ), ∂ζ p=0 p=0 (2.17) d/2−p−1 d/2−p−2 d/2−1 d/2−2 2 ∂ ∂ T csc1,1 (z, Zπ) = cd,p csc2 (ζ) + c∗d,p csc2 (ζ), ∂ζ ∂ζ p=0 p=0 (2.18) and d/2−p−1 d/2−p−2 d/2−1 d/2−2 ∂ ∂ ∗ T (csc1,1 (z, Zπ)) = cd,p csc(ζ) + cd,p csc(ζ). ∂ζ ∂ζ p=0 p=0 (2.19)
d/2−1
T (cot1,1 (z, Zπ)) =
cd,p
∂ ∂ζ
The odd-dimensional case involves far more complicated formulas. For complete details, we refer the reader to [26]. The functions (k+1)
℘r+τ (i),s (z) := r+τ (i),s (z) i = 1, . . . , k + 1
(2.20)
with |r| = s − 1 provide natural s-monogenic generalizations of where r ∈ INk+1 0 the Weierstraß ℘-function. Their partial derivatives generalize the derivatives of the Weierstraß ℘-function. All the functions from (2.20) are generated by partial differentiation from one single s-monogenic primitive ζr,s so that one has ∂ζr,s (z) (k+1) = r+τ (j),s (z), ∂xj
j = 0, . . . , k. (k+1)
Because of the (k + 1)-fold periodicity of r+τ (j),s for j = 0, . . . , k, there are [s]
paravector constants ηh ∈ Ak+1 with [s]
ζr,s (z + ωh ) = ζr,s (z) + ηh ,
h = 0, . . . , k,
(2.21)
where the ωh are the primitive periods of the period lattice. In the 1-monogenic setting, the function ζ0,1 (z) is the monogenic generalized Weierstraß ζ-function, i.e., ζ(z) = q0 (z) +
ω∈Ωk+1 \{0}
q0 (z + ω) − q0 (ω) +
k
qτ (j) (ω)Vτ (j) (z) ,
j=1
which has been introduced by R. Fueter for the quaternionic case in [63, 64] and for arbitrary dimensions by J. Ryan in [133]. In the setting s = k, k odd (Fueter– Sce equation) the associated function ζr,s (z) is one of the generalized holomorphic Cliffordian Weierstraß ζ-functions discussed in [105] and [106].
58
Chapter 2. Clifford-analytic Eisenstein series for translation groups
By applying Mittag–Leffler’s theorem one can deduce the following structural result. The following theorem (compare to [84, 87]) points out the central role of this type of Eisenstein series in the study of s-monogenic p-fold periodic functions with singularities in higher dimensional Euclidean spaces. Theorem 2.5 (Representation theorem). Let p ∈ {1, . . . , k + 1} and s ∈ {1, . . . , k}. Let α be a non-negative integer with α ≥ max{0, p − k − 2 + s}. Let further Ωp be a p-dimensional lattice in Ak+1 . Suppose that f is a left s-meromorphic function in the whole space Ak+1 with values in Cl0k (IR). Let us assume that f has only isolated singularities and that there is only a finite number of singularities in each period cell. Let P be an arbitrary but fixed period cell with no singularities on its boundary and denote the singularities within P by a1 , . . . , al . Let us assume that the singularity order of ai with (1 ≤ i ≤ l) is N (ai ), that N (ai ) − (k + 1 − s) ≥ α for all i ∈ {1, . . . , l} and that the principal parts of f are given by
N (ai )−(k+1−s) (s) (i) qm (z − ai )bm .
|m|=α
Then there is a function g : Ak+1 → Cl0,k (IR) which is left s-monogenic in the whole space Ak+1 with f (z) =
l N (ai )−(k+1−s) i=1
(i) (p) m,s (z − ai )bm + g(z).
(2.22)
|m|=α (p)
In the cases |m| = α where α = p − k − 2 + s we take the function m,s;ai ,b with an arbitrarily chosen b ≡ ai mod Ωp and b ∈ Ωp . As mentioned in [93] this theorem can be adapted to the context of Minkowski type spaces endowed with a non-degenerate scalar product of arbitrary signature. We shall briefly see that we get an even stronger result exclusively for the class of s-monogenic (k+1)-fold periodic functions in real Euclidean space. The specialty of this function class within the particular framework of real Euclidean spaces becomes clear in terms of the generalizations of the classical Liouville theorems which we give in Chapter 2.3.
2.2
Some results on the zeroes of the generalized cotangent and tangent
In the classical complex case, all the zeroes of the complex cotangent lie on the line of the one-dimensional period lattice situated exactly in the middle between two poles. For the tangent the same holds. Since the tangent stems from the cotangent by a half-period shift in the argument, the zero set of the classical tangent function coincides thus with the pole set of the cotangent and vice versa.
2.2. Some results on the zeroes of the generalized cotangent and tangent
59
Let us now turn to the monogenic and more generally to the s-monogenic hypercomplex generalizations (s < k + 1) of the cotangent and tangent functions which are given by (p) (p) cotp,s,β (z, Ωp ) = β,s (z, Ωp ) and tanp,s,β (z, Ωp ) = − cotp,s,β (z + v/2) v∈Vp
where Ωp denotes a p-dimensional lattice with p < k + 1 in Ak+1 and where β stands for a multi-index of a length |β| = max{0, p − k − 1 + s}. By construction cotp,s,β (z, Ωp ) has isolated poles precisely in the points of the period lattice Ωp and tanp,s,β (z, Ωp ) in the points in between, i.e., in 12 Ωp \Ωp . In the case where s+|β| is odd, both functions are odd functions. In these cases it follows then immediately from the periodicity with respect to Ωp that cotp,s,β (z, Ωp ) has zeroes in 12 Ωp \Ωp and that tanp,s,β (z, Ωp ) has zeroes in the period lattice of Ωp . Important questions that arise immediately are whether these zeroes are isolated or not and whether there are more zeroes. Concerning the second question we were not yet able to give a fully satisfactory answer. In our joint paper with T. Hempfling [74] a partial answer that points in this direction has been given within the framework of 1-monogenic generalized elliptic functions. We will discuss these results in the following section. Concerning the first question we managed in [74] to give a partial answer for the one-fold periodic 1-monogenic generalizations of the cotangent and the tangent. In the following theorem we formulate this result in the more general context of the one-fold periodic s-monogenic generalizations of the cotangent and the tangent for s odd. Notice that we have β = 0 for p = 1. Proposition 2.6. Let s < k + 1 be an odd integer. Further, let j = 0, 1, . . . , k and let a ∈ Ak+1 ∪ {∞}. Then all a-points of cot1,s,0 (z, 2Zej ) and tan1,s,0 (z, 2Zej ) are isolated. Proof. Both functions have indeed only isolated poles which turn out to be all ∞-points in the sense of the remark following Definition 1.29. Let us now focus exclusively on a-points where a = ∞. One method to show the assertion is to compute explicitly the Jacobian J of cot1,s,0 (z, 2Zej ). Since the cotangent series is normally convergent, one obtains that (s) J[cot1,s,0 (z, 2Zej )] = Jq0 (x + 2mej ). m∈Z
By a direct computation one obtains under the condition that s is odd that the (s) Jacobian matrix of the function q0 has the form
x 2 − µx20 −µx0 x1 −µx0 x2 ... −µx0 xk µx0 x1 µx21 − x 2 µx1 x2 ... µx1 xk . . . . 1 .. .. .. .. µx x · 1 2 k+4−s
k 2 . . 2 . . xi .. .. .. .. µxk−1 xk i=0 µx0 xk µx1 xk ... µxk−1 xk µx2k − x 2
60
Chapter 2. Clifford-analytic Eisenstein series for translation groups
where we put µ = (k + 2 − s). Inserting a particular point of the form z = (xj + 2mj )ej where we assume that z ∈ 2Zej in this matrix provides significant simplifications due to the fact that xi = 0 for all i = j. In the case where j = 0 we obtain −(k + 1 − s)|x0 + 2m|s−k−2 0 0 ... 0 s−k−2 0 −|x0 + 2m| 0 ... 0 .. . . . . . . . . . . 0 . .. .. .. .. . . . . 0 s−k−2 0 0 . . . 0 −|x0 + 2m| In the cases where j = 0 it simplifies to a diagonal matrix of the form −diag(|xj + 2m|−µ , . . . , |xj + 2m|−µ , −(k + 1 − s)|xj + 2m|−µ , |xj + 2m|−µ , . . . , |xj + 2m|−µ ) where the entry (k + 1 − s)(xj + 2m)−µ appears precisely at the j-th position. A summation of these matrices over the one-dimensional lattice 2Zej gives actually a well-defined invertible matrix for all z ∈ IRej \2Zej . Hence, all a-points of cot1,s,0 (z, 2Zej ) that lie on the xj axis are isolated. Since tan1,s,0 (z; 2Zej ) stems from cot1,s,0 (z, 2Zej ) by a applying a shift in the argument in the direction of the xj -axis, it follows directly that also the a-points of tan1,s,0 (z; 2Zej ) that lie on the xj axis are all isolated. In particular the zeroes lying in between the poles are isolated zeroes.
2.3
Liouville type theorems for generalized elliptic functions
In classical function theory of one complex variable the elliptic functions play a very special role in the theory of value distribution. Their value distribution is namely completely described by the classical Liouville theorems (see for instance [112] or [82]). Some analogies can also be established in the context of (k + 1)-fold periodic Clifford-analytic functions for higher dimensions. In what follows let us denote the set of left (right) s-monogenic p-fold periodic functions in Ak+1 (IKr,q ), i.e., the set of left s-meromorphic (right s-meromorphic) p-fold periodic functions (s) (s) in Ak+1 (IKr,q ) by E(l) (Ωp ; Ak+1 (IKr,q )) (resp. E(l) (Ωp ; Ak+1 (IKr,q ))), where Ωp denotes the associated period lattice. In the real Euclidean case IK = IR and (s) (s) (s) r = 0 we simply write E(l) (Ωp ) or E(r) (Ωp ). Notice that E(l) (Ωp ; Ak+1 (IKr,q )) (s)
and E(r) (Ωp ; Ak+1 (IKr,q )) have the algebraic structure of a right or left differential module, respectively. One observes at once that we have a special situation in real Euclidean spaces, IK = IR and r = 0. As mentioned in Chapter 2.1, one can construct non-trivial
2.3. Liouville type theorems for generalized elliptic functions
61
normally convergent translative Eisenstein series to periodicity lattices that have real codimension zero. Notice that the fundamental domain F of a (k + 1)-fold periodic function in the real Euclidean space Ak+1 is then a bounded (k + 1)dimensional parallelepiped with the vertices ω0 , ω1 , . . . , ωk , . . . , ω0 + · · · + ωk . In what follows we call a parallelepiped that can be transformed by a translation into F a period parallelepiped of f . In the setting of the real Euclidean higher dimensional spaces one has in analogy to the complex case: (s)
Theorem 2.7 (Generalization of the first Liouville theorem). Let f ∈ E(l) (Ωk+1 ). If f is left- or right s-monogenic in the whole Euclidean space Ak+1 , then f is a constant. For 1-monogenic functions this theorem has already been proved by R. Fueter for the quaternionic case (cf. [61]) and later by J. Ryan for IR0,k (cf. [133]). Within the framework of the quaternionic Fueter–Sce equation a proof has been given by G. Laville and I. Ramadanoff in [105]. It can directly be extended to the class of s-monogenic functions, since an arbitrary s-monogenic function that is bounded in Ak+1 reduces similarly to the 1-monogenic case (cf. [33]) to a constant function as shown for instance in [11]. As a consequence of the compactness of the period cell of a (k + 1)-fold periodic function in Ak+1 the theorem thus follows readily. Remarks. In the particular context of real Euclidean spaces, the function g(z) from Theorem 2.5 reduces to a constant. Notice that one cannot transfer this compactness argument so directly for the other spaces. This is due to the fact, that in all the other cases, the associated fundamental domain is not finite. Theorem 2.7 reveals that every non-trivial generalized s-monogenic elliptic function in Ak+1 has singularities. It may have isolated and non-isolated singularities. In what follows we mainly restrict ourselves to the class of elliptic functions that have only isolated poles. In the complex case, the second Liouville theorem tells us that the sum of the residues in each period cell of any complex-analytic doubly-periodic function vanishes identically. For the class of 1-monogenic (k + 1)fold periodic functions a similar statement can also be established: Theorem 2.8 (Generalization of the second Liouville theorem). Let us assume that (1) f ∈ E(l) (Ωk+1 ) has only isolated singularities. Then in any period parallelepiped P with no singularities on the boundary res(f ; c) = 0. (2.23) c∈P
This statement has been proved for the quaternionic case by R. Fueter in [61] and for arbitrarily higher dimensional real Euclidean spaces in [84, 74]. In the
62
Chapter 2. Clifford-analytic Eisenstein series for translation groups
case of dealing with non-isolated singularities, the same result can be established. In this more general setting the notion of the residue is then understood in the sense of Leray–Norguet residues Res(f ; c) as in [61], [124] or [38]. As a direct consequence one may hence conclude that a non-trivial generalized elliptic function must have at least one pole of order (k + 1) or two poles of order k in one period parallelepiped. In classical complex analysis one has furthermore that the sum of the orders of the a-points per period cell of an elliptic function vanishes, which shows a balanced relationship between the sum of the orders of the poles and the sum of the orders of the a-points with a = ∞. This statement is often called the third Liouville theorem on elliptic functions. The question whether there is also an analogy of the third Liouville theorem in the Clifford analysis setting remained open for a long time. In [74] a first positive result in this direction has been developed by means of the generalized argument principle from Chapter 1.5. (1)
Theorem 2.9 (Generalization of the third Liouville theorem). Let f ∈ E(l) (Ωk+1 ) that takes only values in Ak+1 . Furthermore, let us suppose that f has only isolated singularities. Let us further assume that f has only isolated a-points (a ∈ Ak+1 ). Let P be a period parallelepiped that has no poles and no a-points on its boundary. Let us denote the poles that lie inside of P by b1 , . . . , bµ . Let δ1 , . . . , δµ > 0 be sufficiently small so that B(bi , δi )\{bi } contains no poles and no a-points. Then
ord(f − a; c) = −
c∈P \{b1 ,...,bµ }
where p(f − a; bi ) :=
1
µ
p(f − a; bi )
(2.24)
i=1
Ak+1
q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]
∂B(bi ,δi )
which is a finite expression. Proof. The set of poles and a-points are both discrete which follows by the assumption. Therefore, one can choose a period parallelepiped P in such a way that no poles and no a-points lie on its surfaces. Notice that under the given conditions there are at most a finite number of a-points in the interior of each period parallelepiped. A (k + 1)-dimensional parallelepiped has (2k + 2) hypersurfaces. Let us denote them by Bν , Bν , ν = 1, . . . , k + 1 where Bν + ων = Bν . These surfaces should be oriented in such a way that the normal vectors point to the outside. The surfaces Bj are then oriented in the opposite way to Bj . Since P is a period parallelepiped, the surfaces Bi and Bi (i = 1, . . . , k + 1) correspond to each other with respect to the range of values of f . In the sequel we denote the c-points of P by y 1 , . . . , y l . For each j ∈ {1, . . . , l} there exists an εj > 0 and mutually disjoint open balls B(y j , εj ) with ∂P ∩ B(y j , εj ) = ∅ for j = 1, . . . , l.
2.3. Liouville type theorems for generalized elliptic functions l
Notice that in P \
63
B(y j , εj ) there are no a-points. Now we compute:
j=1
ord(f − a; c) +
c∈P \{b1 ,...,bµ }
= = =
1
Ak+1 1 Ak+1 1 Ak+1
q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)] +
j=1 ∂B(y j ,εj )
,
k+1
,
q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]
ν=1 Bν
k+1
,
, ,
Bν 1 Ak+1
k+1
−
ν=1
+
q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]
, Bν
,
Bν
=
q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]
q0 (f (z + ων ) − a)[(Jf )∗ (z)] ∗ [dσ(z)]
ν=1 Bν
+ =
p(f − a; bi )
i=1
∂P
Bν 1 Ak+1
µ
q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]
+ =
p(f − a; bi )
i=1
,
l
µ
q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]
q0 (f (z) − a)[(Jf )∗ (z)] ∗ [dσ(z)]
0.
In view of the periodicity of f , the differential dσ(f (z)) = [(Jf )∗ (z)] ∗ [dσ(z)] is invariant under translations of the form z → z + ων . From Theorem 2.9 one may readily deduce (1)
Corollary 2.10. Let f ∈ El (Ωk+1 ) taking only values in Ak+1 . If f has an isolated zero at c ∈ Ak+1 or f (c) = 0, then ord(f ; c + ω) = ord(f ; c)
f or all ω ∈ Ωk+1 .
(2.25)
One can say more: Proposition 2.11. Let f be a non-constant meromorphic function that takes only values in Ak+1 and assume additionally that f has at most isolated poles. Let S stand for the set of poles. Let Ωk+1 be a (k + 1)-dimensional lattice in Ak+1 . We further assume that one can associate with each ω ∈ Ωk+1 a positive real α = α(ω) such that f (z + ω) = α(ω)f (z) (2.26) for all z ∈ Ak+1 \S. If f has an isolated zero at c (or f (c) = 0), then f has an isolated zero at c + ω for all ω ∈ Ωk+1 or f (c + ω) = 0 for all ω ∈ Ωk+1 ,
64
Chapter 2. Clifford-analytic Eisenstein series for translation groups
respectively. In this case, f or all ω ∈ Ωk+1 .
ord(f ; c + ω) = ord(f ; c)
Provided all zeroes of f are isolated, ord(f ; c) + p(f ; b) = 0 c∈P \S
(2.27)
(2.28)
b∈S
for any period parallelepiped P . Proof. Since α(ω) is a positive real for any arbitrary ω ∈ Ωk+1 , it follows immediately from Proposition 1.30 that c + ω is an isolated zero if and only if c is an isolated zero. Similarly, from f (c) = 0 follows that f (c + ω) = 0 for all ω ∈ Ωk+1 . Next let us consider , q0 (f (z))[(Jf )∗ (z)] ∗ [dσ(z)] Ak+1 ord(f ; c + ω) = ∂B(c+ω,ε)
,
=
q0 (f (z))dσ(f (z))
∂B(c+ω,ε)
=
,
q0 (f (y + ω))dσ(f (y + ω))
∂B(c,ε)
=
,
α(ω) q (f (y))α(ω)k dσ(f (y)) α(ω)k+1 0
∂B(c,ε)
=
,
q0 (f (y))[(Jf )∗ (y)] ∗ [dσ(y)]
∂B(c,ε)
= Ak+1 ord(f ; c) where the parameter ε > 0 has to be chosen sufficiently small. With the same argument one can show (2.28). One simply has to apply the same procedure as in the proof of the third Liouville theorem. The third Liouville theorem gives thus a first insight in the basic value distribution of the generalized elliptic functions from Clifford analysis that satisfy some special conditions. In particular it revealed that at least under some special conditions there is also a certain balance relation between a-points (a = ∞) and poles in a period cell of a monogenic generalized elliptic function. These statements should be regarded as a promising starting point for more sophisticated versions concerning the cases where manifolds of zeroes and singularities appear. Remarks. As mentioned in [93], generalizations of Theorem 2.8 and Theorem 2.9 can be established in the context of Ak+1 -like linear domain manifolds within the framework of complexified (k + 1)-fold periodic functions satisfying in Ck+1 the complexified Cauchy–Riemann equation. However, additional restrictions have to be put. In the context where the cones of the singularities have only point intersections with the linear domain manifold one can obtain a generalization of
2.4. Series expansions, divisor sums and Dirichlet series
65
Theorem 2.8. This is a consequence of the generalized Cauchy integral formula on IRk -like domain manifolds which was proved in [134]. If the manifolds of c-points have additionally only point intersections with the linear domain manifold, then one obtains also a generalization of Theorem 2.9. These Liouville type theorems indicate a special role of the (k + 1)-fold periodic functions in Ak+1 or its complexification.
2.4
Series expansions, divisor sums and Dirichlet series (1)
(2)
In the classical complex case the function series n and n are intimately related to the Riemann zeta function and to Eisenstein series of the type (3). These functions appear explicitly in their Laurent expansion. If 0 < r < 1, then the (1) Laurent expansion of 1 reads exactly ∞
(1)
1 (z) = π cot(πz) =
1 − 2ζ(2n)z 2n−1 , z n=1
and similarly also for m ≥ 1, (1) m (z)
2n − 1 1 m = + (−1) 2ζ(2n)z 2n−m . m−1 z 2n≥m
Concerning the second series let us assume without loss of generality that the twodimensional lattice Ω2 has the form Z+Zτ with Im(τ ) > 0. In 0 < r < min{1, |τ |} (2) the Laurent expansion of the associated Weierstraß ℘-function 2 turns out to be ∞ 1 (2) 2 (z, Ω2 ) = ℘(z, Ω2 ) = 2 + (2n − 1)G2n (Ω2 )z 2n−2 , z n=2 with Gn (Ω2 ) := (c,d)∈Z×Z\{(0,0)} (cτ +d)−n which converges absolutely for n ∈ IN with n > 2 and does not vanish whenever n is even. The Weierstraß ℘-function serves thus as generating function for the Eisenstein series (3). Next we turn to treat higher dimensional monogenic and s-monogenic analogies. As one can verify by a direct computation the Taylor coefficients of the function (s) |m| ≥ max{0, p − k − 1 + s} (p) m,s (z) − qm (z), in a sufficiently small neighborhood of the origin have the form (s) qm (ω),
(2.29)
ω∈Ωp \{0}
involving parameters m and s with |m|+s even and sufficiently large so that (2.29) converges absolutely. This is also true in the more general context of arbitrary real and complex Minkowski type spaces.
66
Chapter 2. Clifford-analytic Eisenstein series for translation groups
Since the series (2.29) with |m| + s even are the Laurent coefficients of the (p) s-meromorphic series m,s (z), there are actually indices m with |m|+s ≡ 0 mod 2, (s) for which the associated series (2.29) does not vanish. If ω∈Ωp \{0} qm (ω) van(p)
(s)
ished for all m, then one would obtain that r,s (z) ≡ qr (z) within a whole ball which is a contradiction. Notice that the series vanish identically whenever (s) |m| + s ≡ 1 mod 2 which is due to the fact that the functions qm are odd in these cases. (s) In the cases where p ≤ k −s and where s is even, the series ω∈Ωp \{0} q0 (ω) converges and coincides in these cases with the Epstein zeta function (gtr W tr W g)−s (2.30) ζW tr W (s) = g∈Zp \{0}
where W is the p × (k + 1) matrix which maps the generators of the lattice Lp = Ze0 + Ze1 + · · · + Zep−1 onto the generators of Ωp , i.e., W := (ω1 ω2 . . . ωp ). In the case s = 1 the series (1) n∈IN qm (n) coincide in the two-dimensional case with the classical Riemann zeta function up to a constant. The series (2.29) are thus generalizations of the classical Riemann zeta function and the Epstein zeta function which appear in a natural way in connection with Clifford-analytic multiperiodic functions. We shall see that these variants of Riemann zeta type functions play a rather central role in all that follows. In order to introduce also a non-vanishing Riemann zeta type function for the cases where |m| + s is odd, we decompose the lattice Ωp := Zω1 + · · · + Zωp into a positive and a negative part (cf. [84]). The positive part of Ωp is defined by Ω+ p
:=
INω1 + Zω2 + Zω3 + · · · + Zωp
∪ ...
INω2 + Zω3 + · · · + Zωp
∪
INωp .
Consequently, the negative part of the lattice Ωp is defined by + Ω− p := Ωp \{0} \Ωp so that
− − and Ω+ z ∈ Ω+ p ⇔ −z ∈ Ωp p ∪ Ωp ∪ {0} = Ωp .
In particular, for k = 1 and ω1 = 1 one has: Ω+ 1 = IN Now we can introduce:
Ω− 1 = −IN Ω1 = IN ∪ −IN ∪ {0} = Z.
2.4. Series expansions, divisor sums and Dirichlet series
67
Definition 2.12 (Generalized Riemann zeta function of Clifford analysis in Ak+1 ). be a multiLet p, s ∈ IN with 1 ≤ p ≤ k + 1 and 1 ≤ s ≤ k. Let further l ∈ INk+1 0 index with |l| ≥ max{0, p − k + s}. Then the generalized Riemann zeta function of Clifford analysis in Ak+1 is defined by (s) Ω ql (ω). (2.31) ζMp (l, s) := ω∈Ω+ p
To prove the convergence one applies first the estimate (1.50) that we devel(s) oped for the functions ql both for s odd and s even and after that Eisenstein’s lemma. This shows that the series (2.31) converge indeed absolutely under the given conditions. In the cases where we have additionally |l| + s ≡ 0 mod 2 we obtain the relation Ω (s) ql (ω). 2ζMp (l, s) = ω∈Ωp \{0} (s)
From the estimates on the functions qn we can readily derive that |l|−1 Ω
ζMp (l, s) ≤
-
(k + 1 − s + µ) ζW tr W
µ=0
1 2
(k + 1 − s + |l|)
where ζW tr W is again the Epstein zeta function associated to the matrix W tr W . The closed representation formulas that we developed in the second part of Sec(s) tion 1.6 for the functions ql enable us next to describe the generalized Riemann zeta functions explicitly in terms of finite sums of variants of Dirichlet series with polynomial coefficients of the form P (g)(gtr S g)−t . (2.32) δ(P ( · ), S, t) := g∈Zp \{0}
Here P denotes a real-valued polynomial in g1 , . . . , gp , S a positive definite matrix and t a real or complex parameter. Notice that a series of the form (2.32) is convergent if Re(t) − deg(P ) > (p/2). If P ≡ 1, then δ(1, S, t) coincides with the Epstein zeta function associated to the matrix S, i.e., ζS (t). We also consider the positive part of the Dirichlet series, which is defined by P (g)(gtr S g)−t . (2.33) D(P ( · ), S, t) := g∈Zp+
D(P ( · ), S, t) provides a canonical generalization of the classical Dirichlet series D(P ( · ), t) := P (n)n−t n∈IN
when putting p = 1 and S = (1). We observe that δ(P ( · ), S, t) = 2D(P ( · ), S, t) if P is an even function and that δ(P ( · ), S, t) = 0 if P is odd.
68
Chapter 2. Clifford-analytic Eisenstein series for translation groups
In the case where s is an even positive integer, we get directly by Theorem 1.49 that Ω D(a(k, n, p)ω n−2p , W tr W, k − s + 1 + 2|n| − 2|p|) ζMp (n, s) = 0≤2p≤n
with a(k, n, p) = (−1)|n|−|p|
k − s 2
|n|−|p| (n
n! . − 2p)!p!
(2.34)
Notice, ω n−2p means ω0n0 −2p0 · · · ωknk −2pk . In the case where |n| is even we have furthermore 1 Ω ζMp (n, s) = δ(a(k, n, p)ω n−2p , W tr W, k − s + 1 + 2|n| − 2|p|). 2 0≤2p≤n
In the case where s is odd with s < k we can use Theorem 1.50. Due to the more (s) complicated structure of the functions qn for these cases, we will get slightly more complicated representation formulas for the associated generalized Riemann zeta type functions in terms of these variants of Dirichlet series. To proceed in this direction we first observe that one can write ω n−2p
ω k−s+2|n|−2|p|
k k nj − 2pj ω n−2p−τ (j) · ej = (nj − 2pj ) ej ωj
ω k−s+2|n|−2|p| j=0 j=0
(2.35)
and also ω n−2n+τ (0) 1 ω n−2n+τ (j) = − ej .
ω k−s+2|n|−2|p| ω
ω k−s+2+2|n|−2|p| j=1 ω k−s+2+2|n|−2|p| k
ω n−2p
(2.36)
Inserting these relations into the representation formula (1.65) leads to qn(s) (ω) 1 = k−s
0≤2p≤n
.
a(k, n, p) ·
ω n−2p+τ (0)
ω k−s+2+2|n|−2|p|
−
ω n−2p+τ (j)
j=1
ω k−s+2+2|n|−2|p|
3 ω + bj (k, n, p) ej ,
ω k−s+2|n|−2|p| j=0 k
k
ej
n−2p−τ (j)
(2.37) after having put bj (k, n, p) = (−1)|n|−|p|
k − s 2
|n|−|p|
(nj − 2pj ) j = 0, 1, . . . , k.
For the sake of readability let us further introduce the real-valued polynomials Aj (ω) := ω n−2p+τ (j)
Bj (ω) := ω n−2p−τ (j)
(2.38)
2.4. Series expansions, divisor sums and Dirichlet series
69
so that we arrive at the compact formula Ω
ζMp (n, s) . a(k, n, p)A0 (ω) 1 b0 (k, n, p)B0 (ω) = + k−s
ω k−s+2+2|n|−2|p|
ω k−s+2|n|−2|p| + 0≤2p≤n +
k j=1
4
ω∈Ωp
5 3 bj (k, n, p)Bj (ω) a(k, n, p)Aj (ω) − ej .
ω k−s+2|n|−2|p|
ω k−s+2+2|n|−2|p| + +
ω∈Ωp
ω∈Ωp
From this formula we finally obtain also for the cases s < k where s is odd the following explicit representation formula for paravector components of the generalized Riemann zeta function in terms of finite sums of real-valued Dirichlet series with explicitly determined polynomial coefficients: 1 Ω D(a(k, n, p)A0 (W · ), W tr W, k − s + 2 + 2|n| − 2|p|) ζMp (n, s) = k−s 0≤2p≤n
+D(b0 (k, n, p)B0 (W · )W tr W, k − s + 2|n| − 2|p|) k D(bj (k, n, p)Bj (W · ), W tr W, k − s + 2|n| − 2|p|) + j=1
−D(a(k, n, p)Aj (W · ), W tr W, k − s + 2 + 2|n| − 2|p|)ej
.
In the case |n| ≡ 1 mod 2 one obtains 1 Ω δ(a(k, n, p)A0 (W · ), W tr W, k − s + 2 + 2|n| − 2|p|) ζMp (n) = 2(k − s) 0≤2p≤n
+δ(b0 (k, n, p)B0 (W · )W tr W, k − s + 2|n| − 2|p|) k + δ(bj (k, n, p)Bj (W · ), W tr W, k − s + 2|n| − 2|p|) j=1
−δ(a(k, n, p)Aj (W · ), W tr W, k − s + 2 + 2|n| − 2|p|)ej
.
Next we turn to generalizations of the Eisenstein series (3) in this framework. Suppose that Ωk+1 = Zω0 + · · · + Zωk is an arbitrary (k + 1)-dimensional lattice in Ak+1 . By a rotation, we can transform this lattice into a special lattice of the form Ω∗k+1 = Zτ + Ωk where Ωk is completely contained in IRk , i.e., Sc(Ωk ) = 0 and where Sc(τ ) > 0. The associated Riemann zeta functions 1 Ω∗ (s) ζMk+1 (m, s) = qm (ατ + ω) (2.39) 2 (α,ω)∈Z×Ωk \{(0,0)}
70
Chapter 2. Clifford-analytic Eisenstein series for translation groups
are precisely the Laurent coefficients of the s-monogenic generalized Weierstraß functions associated to Ω∗k+1 . They generalize to the higher dimensional case the expression (c,d)∈Z×Z\{(0,0)} (cτ +d)−n . Regarding τ as a hypercomplex variable of the half-space H + (Ak+1 ) leads then to s-monogenic generalizations of the classical Eisenstein series of type (3) to Clifford analysis in this sense. More generally as in our previous works [84, 86, 88] we introduce here: Definition 2.13. Let p ∈ {1, . . . , k} and let Ωp = Zω1 +· · ·+Zωp be a p-dimensional lattice in Ak+1 which is contained in spanIR {e1 , . . . , ek }. Let s < k + 1 be a with |m| + s ≡ 0 mod 2 and |m| ≥ positive integer. For a multi-index m ∈ INk+1 0 max{0, p + s + 1 − k} we introduce the following type of s-monogenic Eisenstein series on the right half-space: (s) G(p) qm (αz + ω) z ∈ H r (Ak+1 ). (2.40) m,s (z) := (α,ω)∈Z×Ωp \{(0,0)}
For the monogenic case s = 1 we introduced these types of Eisenstein series in [84] for p = k. The convergence proof from [84, 86] can easily be adapted to the more general setting by using the more general estimate for the s-monogenic (s) functions qn (z) from Proposition 1.44. The classical Eisenstein series (3) have a very interesting Fourier expansion, from the number theoretic point of view. It is directly related to the Riemann zeta function and representation numbers of sums of divisors [49]. (p)
In what follows let us abbreviate the 1-monogenic Eisenstein series Gm,1 (p)
by Gm for simplicity. In [84, 86] we determined the Fourier expansion of the (k) 1-monogenic Eisenstein series Gm where we considered without qualitative loss of generality the orthonormal lattice Lk = Ze1 + · · · + Zek and, in view of the Cauchy–Riemann system, multi-indices with m0 = 0. Theorem 2.14 (Fourier expansion). Let m = (0, m1 , . . . , mk ) ∈ INk+1 be a multi0 (k) index with |m| ≡ 1 mod 2, |m| ≥ 3. Then the Eisenstein series Gm,1 associated with the orthonormal lattice Lk in IRk have a Fourier expansion on the right halfspace, s 2πi s,x −2π s x0 Lk |m| )e G(k) σm (s)(1 + i e m (z) = 2ζM (m) + Ak+1 (2πi)
s k s∈Z \{0}
(2.41) where σm (s) =
rm
(2.42)
r|s
and where r|s means that there is an α ∈ IN such that αr = s. For the detailed proof we refer to [84, 86]. The basic idea of the proof is to (k) expand first the subseries m,1 (z) (|m| ≥ 2) on H r (Ak+1 ) into a Fourier series of
2.4. Series expansions, divisor sums and Dirichlet series the form
71
αf (r, x0 )e2πi r,x .
r∈Zk
By a direct computation one obtains αf (r, x0 ) = qm (z + m) dx1 · · · dxk = 0. m∈Zk
[0,1]k
For r = 0 one applies the partial integration method successively (integrating qm up until we obtain q0 and differentiating the exponential terms). This leads finally to αf (r, x0 ) = (2πi)|m| rm
q0 (z)e−2πi r,x dx1 · · · dxk .
IRk
The value of the remaining integral is well known. See for example [154, 108, 23]. It can be evaluated by applying the residue theorem for monogenic functions, Ak+1
r −2π r x0 q0 (z)e−2πi r,x dx1 · · · dxk = 1+i e 2
r IRk
so that one finally obtains for x0 > 0 that (k) m (z) =
Ak+1 2
r −2π r x0 2πi r,x e (2πi)|m| rm 1 + i e .
r
(2.43)
r∈Zk \{0}
(k)
Rearrangement arguments allow us next to rewrite Gm (z) in the form G(k) m (z)
= 2ζ
Lk
(m) + 2
∞
m (αz)
(2.44)
α=1 m∈Zk
where we used the notation introduced above. Applying (2.43) to (2.44) yields Gm (z) = 2ζ Lk (m) + 2(2πi)|m| (k)
Ak+1 2
∞ α=1 r∈Zk
αr e2πi αr,x e−2π αr x0 rm 1 + i αr
which finally may be expressed in the form (2.41). Here we have indeed a nice analogy between the form of the Fourier expansion of the classical Eisenstein series (3) and the structure of the Fourier expansion of the higher dimensional variant defined in (2.40). The ordinary Riemann zeta function is replaced in the higher dimensional monogenic context by the generalized paravector valued variant (2.31). As we have seen previously, their paravector
72
Chapter 2. Clifford-analytic Eisenstein series for translation groups
component can in turn be expressed in terms of a finite sum of scalar-valued Dirichlet series with polynomial coefficients, generalizing the Epstein zeta function. The Fourier coefficients αf (r) for r = 0 of the complex Eisenstein series σm (s) =
rm−1
(2.45)
r|s
where r|s means that there is an α ∈ IN such that αr = l are thus generalized by the expression (2.42) which can further be expressed in terms of (2.45): σm (s) = sm σ−|m| (gcd(s1 , . . . , sm−1 )).
(2.46)
As a consequence of the monogenicity, the monogenic plane wave function appears as a natural generalization of the classical exponential function in the sense of Cauchy–Kowalewski extension. Notice further that infinitely many Fourier coef(k) ficients do not vanish. Hence the series Gm (z) are definitely non-trivial functions whenever |m| is odd. In Chapter 3 we will observe furthermore, that they provide elementary building blocks for the generation of families of Clifford-analytic modular forms for larger discrete groups, in particular, also for the special hypercomplex modular groups Γp . This in turn provides us with an analogy to the classical complex case. (k) However, the proper series Gm (z) are not yet modular forms for Γk . Only their set of singularities Qe1 + · · · + Qek is totally invariant under the action of Γk . This in fact provides a difference from the complex case.
2.5
The integer multiplication of the Clifford-analytic Eisenstein series
One classical result from complex analysis (cf. e.g., [130]) states that the classical cotangent function π cot(πz) is completely characterized by the property of being 1 at each point m ∈ Z an odd and meromorphic function with principal parts z−m that satisfies additionally the duplication formula 1 2f (2z) = f (z) + f (z + ). 2
(2.47)
The doubly periodic Weierstraß ℘-function also satisfies an analogous duplication formula (cf. e.g., [82]): v (2.48) 4℘(2z) = ℘(z + ). 2 v∈V(2)
2.5. The integer multiplication of the Clifford-analytic Eisenstein series
73
Here in the summation, the symbol V(2) stands again for the canonical system of representatives in Ω/2Ω, as introduced in Definition 2.3. Conversely, every mero1 morphic function with principal parts (z−ω) 2 at each ω ∈ Ω that satisfies a functional equation of type (2.48) coincides with the ℘-function up to a constant. From (2.48) one can further directly derive that the sum of the integer division values gives ℘(v/2) = 0. 0=v∈V(2)
In [84] and [85] we deduced analogous duplication formulas for the monogenic p-fold periodic generalizations of the cotangent and the monogenic (k + 1)-fold periodic Weierstraß ℘τ (i) -functions and showed that these generalized duplication formulas permit an analogous characterization of these functions as in the complex case. In our recent paper [92] we developed more general multiplication formulas for a larger class of s-monogenic translative Eisenstein series. In this section and the following two we present the results from our recent paper [92]. In this section and in Section 2.6 we describe the multiplication of the smonogenic Eisenstein series with general integer multiplicators. In Section 2.7 we (p) will treat the hypercomplex multiplication of the series m,s . Throughout this section and the next one Ωp = Zω1 + · · · + Zωp stands always for an arbitrary p-dimensional lattice with 1 ≤ p ≤ k + 1 in Ak+1 . To study the integer multiplication of the s-monogenic Eisenstein series means to describe the relationship between the monogenic and polymonogenic Eisenstein series of an arbitrary lattice Ωp and those of a p-dimensional sublattice Ω∗p ⊂ Ωp that has the form Ω∗p = nΩp with a positive integer n ≥ 2. By Vp (n) := {m1 ω1 + · · · + mp ωp ; m1 , . . . , mp ∈ IN0 , 0 ≤ m1 , . . . , mp < n} we denote the canonical system of representatives of Ωp /nΩp . For the cases where m ∈ INk+1 is a multi-index with |m| ≥ max{0, p − k + s} 0 (p) we can immediately infer by a direct rearrangement that the series m,s satisfy the following integer multiplication formulas: nk+1+|m|−s (p) m,s (nz) =
v∈Vp (n)
(p) m,s (z + v/n).
(2.49)
74
Chapter 2. Clifford-analytic Eisenstein series for translation groups
Simply consider,
k+1+|m|−s (p) (p) m,s (nz) m,s (z + v/n) − n
v∈Vp (n)
=
(s) qm (z + v/n + ω) − nk+1+|m|−s
v∈Vp (n) ω∈Ωp
k+1+|m|−s
= n
(s) qm (nz + ω)
ω∈Ωp (s) qm (nz
+ v + nω)
v∈Vp (n) ω∈Ωp
− nk+1+|m|−s
(s) qm (nz + ω)
ω∈Ωp
k+1+|m|−s
= n
(s) qm (nz + v + ω )
v∈Vp (n) ω ∈nΩp
− n
k+1+|m|−s
(s) qm (nz + v + ω ) = 0,
v∈Vp (n) ω ∈nΩp (s)
where we used furthermore that qm is IR-homogeneous of degree s − |m| − k − 1. The following two statements permit us to establish an analogous statement is a multi-index with for the cases with p − k − 1 + s ≥ 0 where m ∈ INk+1 0 |m| = p − k − 1 + s and where s + |m| is odd. Proposition q2.15. Let Ωp = Zω1 + · · · + Zωp ⊂ Ak+1 be a p-dimensional lattice. Let v = i=1 αi ωi with 0 = αi < n where 1 ≤ q ≤ p. Write further v = q k+1 be a multi-index i=1 (n − αi )ωi . Let s ∈ IN with s < k + 1 and let m ∈ IN0 such that |m| + s is odd and that furthermore |m| ≥ p − k − 1 + s. Then (s) (s) (s) (s) qm qm (nω + v) − qm (nω) + (nω + v ) − qm (nω) ω∈Ωp \{0} ω∈Ωp \{0} (2.50) = −qm (v) − qm (v ). (s)
(s)
(s)
Proof. Both series converge absolutely. The functions qm are odd, under the condition that |m| + s is odd. Hence we can rewrite the left-hand side of (2.50) in the way q q q (s) (s) qm (nω + αi ωi ) − qm (n(ω − ωi ) + αi ωi ) . ω∈Ωp \{0}
i=1
i=1
i=1
This series in turn can be split into the following two parts: q q q q (s) (s) ωi + αi ωi ) − qm (n(r − 1) ωi + αi ωi ) (2.51) qm (nr r∈Z\{0}
+
ω∈Ωp \Z(
q i=1
i=1 (s)
qm (nω + ωi )
i=1 q i=1
i=1 (s)
αi ωi ) − qm (n(ω −
q i=1
ωi ) +
i=1 q i=1
αi ωi ) . (2.52)
2.5. The integer multiplication of the Clifford-analytic Eisenstein series
75
Here, the prime behind Ωp means that the origin is omitted in the summation. q (s) From lim qm (z) = 0 follows that (2.52) equals zero. ω ∈ Ωp \Z( i=1 ωi ) implies z→∞ q q (s) namely that also ω − i=1 ωi ∈ Ωp \Z( ωi ). In view of the oddness of qm , the i=1
sum in (2.51) simplifies to q q q q (s) (s) qm (nr ωi + αi ωi ) − qm (n(r − 1) ωi + αi ωi ) r∈IN (s)
i=1 q
−qm (nr =
i=1 q
ωi −
(s)
i=1 q
αi ωi ) + qm (n(r + 1)
i=1 i=1 q q (s) (s) −qm ( αi ωi ) − qm ( (n i=1 i=1
i=1 q
ωi −
i=1
αi ωi )
i=1
− αi )ωi ).
This proposition permits us to establish next Lemma 2.16. Let Ωp := Zω1 + · · · + Zωp be a p-dimensional lattice in Ak+1 where 1 ≤ p ≤ k + 1. Let further Vp (n) be the canonical system of representatives of be Ωp /nΩp where n ≥ 2 is an integer. Let s ∈ IN with s < k + 1 and let m ∈ INk+1 0 a multi-index such that |m| + s is odd and that furthermore |m| ≥ p − k − 1 + s. Then (s) (s) (s) [qm (nω + v) − qm (nω)] = − qm (v). (2.53) v∈Vp (n)\{0} ω∈Ωp \{0}
v∈Vp (n)\{0}
Remark. Lemma 2.16 provides us with a generalization of the particular relation (1) (1) (1) q0 (2ω + v) − q0 (2ω) = −q0 (v), (2.54) ω∈Ωk \{0}
which we established earlier in [84, 85] for proving the periodicity of the particular (k) function 0,1 . In equation (2.54) v is supposed to stand for an arbitrary element from Vk (2)\{0}. Note that (2.54) is not satisfied if one replaces 2 by an arbitrary different integer n > 2. Lemma 2.16 permits us next to establish be Theorem 2.17. Let p, s ≤ k and assume that p − k − 1 + s ≥ 0. Let m ∈ INk+1 0 a multi-index with |m| = p − k − 1 + s. Suppose further that s + |m| is odd. Then (p) (2.49) is also satisfied by the associated series m,s . Remark. In the monogenic case this theorem deals exclusively with the configuration p = k and m = 0. (p)
Proof of Theorem 2.17. In the context considered here the series m,s has the more complicated form (s) (p) (s) (s) (2.55) qm (z + ω) − qm (ω) m,s (z; Ωp ) = qm (z) + ω∈Ωp \{0}
76
Chapter 2. Clifford-analytic Eisenstein series for translation groups
in order to have convergence. In the 1-monogenic case we are dealing with the particular configuration p = k and m = 0. The expression
(p)
(p)
m,s (z + v/n) − nk+1+|m|−s m,s (nz)
(2.56)
v∈Vp (n)
can hence be rewritten in the form (s) qm (z + v/n) +
(s)
(s)
[qm (z + ω + v/n) − qm (ω)]
v∈Vp (n) ω∈Ωp \{0}
v∈Vp (n) (s) −nk+1+|m|−s qm (nz)
= nk+1+|m|−s
− nk+1+|m|−s
+n
(s)
(s)
[qm (nz + nω + v) − qm (nω)]
v∈Vp (n) ω∈Ωp \{0}
− nk+1+|m|−s
+n
(s)
(s)
[qm (nz + ω) − qm (ω)]
ω∈Ωp \{0}
(s)
qm (nz + v)
v∈Vp (n)\{0} k+1+|m|−s
(s)
(s)
(s) −nk+1+|m|−s qm (nz)
= nk+1+|m|−s
(s)
[qm (nz + ω) − qm (ω)]
ω∈Ωp \{0}
qm (nz + v)
v∈Vp (n) k+1+|m|−s
(s)
(s)
[qm (nz + nω + v) − qm (nω)]
v∈Vp (n) ω∈Ωp \{0}
−n
k+1+|m|−s
(s)
−nk+1+|m|−s
(s)
[qm (nz + nω) − qm (nω)]
ω∈Ωp \{0}
(s)
(s)
[qm (nz + nω + v) − qm (nω + v)]
v∈Vp (n)\{0} ω∈Ωp
k+1+|m|−s
= n
(s)
qm (nz + v)
v∈Vp (n)\{0}
k+1+|m|−s
+n
(s)
(s)
[qm (nz + nω + v) − qm (nω)]
v∈Vp (n)\{0} ω∈Ωp \{0}
−nk+1+|m|−s
v∈Vp (n)\{0}
−nk+1+|m|−s = nk+1+|m|−s
(s)
(s)
[qm (nz + v) − qm (v)]
(s)
(s)
[qm (nz + nω + v) − qm (nω + v)]
v∈Vp \{0} ω∈Ωp \{0}
v∈Vp (n)\{0}
(s)
qm (v) +
(s) (s) (qm (nω + v) − qm (nω)) .
v∈Vp (n)\{0} ω∈Ωp \{0}
(2.57) Since |m| + s is odd, Lemma 2.16 can be applied to (2.57). This allows us to establish the multiplication formula. Remark. Let us assume that p ≤ k + 1, s ≤ k, p − k − 1 + s ≥ 0 and that m is a multi-index from INk+1 with |m| = p − k − 1 + s. Whenever s + |m| is even, 0
2.5. The integer multiplication of the Clifford-analytic Eisenstein series
77
we cannot apply Lemma 2.16 in the previous line of the proof of Theorem 2.17. Nevertheless, the modulus of the expression within the brackets of (2.57) must be finite. This allows us to conclude at least that also in these cases there is a paravector constant C ∈ Ak+1 with (p) (2.58) nk+1+|m|−s (p) m,s (nz) = m,s (z + v/n) + C. v∈Vp (n)
In the special case where p = k + 1 and where m is a multi-index from INk+1 0 with |m| = s, we are dealing with s-monogenic generalizations of the Weierstraß ℘-function. In this special context one can use a very elegant method, involving the s-monogenic generalizations of the Weierstraß ζ-function, in order to show that C = 0. What follows is an extension in several directions of R. Fueter’s calculations and methods that were presented in [63] pp. 236–245 exclusively in the context of the particular problem of the multiplication of the four-fold periodic quaternionic 1-monogenic ℘-function. We show more generally: Theorem 2.18. Let Ωk+1 be a (k + 1)-dimensional lattice in Ak+1 . Let n ≥ 2 be a positive integer and let Vk+1 (n) be the canonical system of representatives of Ωk+1 /nΩk+1 . Then the s-monogenic generalizations of the Weierstraß ℘-function satisfy all the integer multiplication formulae nk+1 (k+1) (nz) = (k+1) (z + v/n) (2.59) s,s s,s v∈Vk+1 (n)
where s is a multi-index of length |s| = s. Proof. Let us take an arbitrary (k+1)-fold periodic s-monogenic translative Eisen(k+1) stein series s,s with |s| = s. As a consequence of the simple fact that |s| ≥ 1 there is an i ∈ {0, . . . , k} so that si > 0. Furthermore, there is a multi-index r ∈ IN0 k+1 of length s − 1 with r + τ (i) = s. Let us now take the k + 1many s-monogenic Eisenstein series associated to the multi-indices r + τ (j) where (k+1) j = 0, 1, . . . , k. The original Eisenstein series s,s is among them. In view of (k+1) |r| ≥ 0, all the Eisenstein series r+τ (j),s j = 0, 1, . . . , k can be generated by partial differentiation from one global s-monogenic quasi-periodic primitive, namely the s-monogenic Weiertraß ζ-functions ζr,s (see Section 2.1). From (2.58) follows that all the k + 1 partial derivatives of (k+1) (k+1) Ej (z) := nk+1 r+τ (j),s (nz) − r+τ (j),s (z + v/n) (2.60) v∈Vk+1 (n)
vanish identically. Therefore, E0 , E1 , . . . , Ek must be constants. This statement could be established alternatively by applying generalizations of the classical Liouville’s theorem to the class of s-monogenic functions, cf. [11, 105]. After this a comparison of the Laurent coefficients provides us with this statement.
78
Chapter 2. Clifford-analytic Eisenstein series for translation groups (k+1)
Since all the functions r+τ (j),s can be generated by partial differentiation from the function ζr,s , one may now integrate (2.60), so that we get
nk ζr,s (nz) =
ζr,s (z + v/n) +
k
Ej xj + E,
(2.61)
j=0
v∈Vk+1 (n)
where E denotes a further independent Clifford constant. If we next apply on both sides of (2.61) a shift of the argument of the form z → z + ωh (h = 0, . . . , k), then we arrive at
[s] nk ζr,s (nz)+nηh =
[s]
ζr,s (z+v/n)+ηh
k + Ej (xj +ωhj )+E. (2.62) j=0
v∈Vk+1 (n)
k
Here, and in the sequel we use the notation ωh = j=0 ωhj ej . In view of (2.61), (2.62) implies k
[s] k+1 [s] ηh = ηh + Ej ωhj . (2.63) n j=1
v∈Vk+1 (n) k+1
Since Vk+1 (n) = n
, the equation (2.63) simplifies to k
Ej ωhj = 0.
(2.64)
j=1
Let us write further
ω00 ω10 det W = det ··· ωk0
··· ··· ···
ω0k ω1k ··· ωkk
(2.65)
and Θhj for the adjoint determinant associated with the element ωhj . Then k
ωhi Θhl = δil · det W.
(2.66)
h=0
From (2.64) follows in particular that k h=0
Θhi
k
Ej ωhj
= 0.
(2.67)
j=1
Applying (2.66) to (2.67) leads to the system Ei det W = 0 for all i = 0, . . . , k. The k + 1 primitive periods ωh (h = 0, . . . , k) of the (k + 1)-fold periodic series (k+1) r+τ (j),s are all linearly independent. For this reason det W = 0. Hence, Ej = 0 for all j = 0, 1, . . . , k. The integer multiplication formulas for the s-monogenic generalized ℘-functions are hereby established.
2.6. Characterization theorems
79
For the remaining cases where p, s, m are chosen so that s + |m| is an even positive integer where |m| = p − k − 1 + s with p < k + 1, this argument cannot be adapted so directly. In order to show that Ej = 0 for all j = 0, . . . , k we used that the period matrix is an invertible matrix having full rank k + 1. Theorem 2.18 permits us now directly to show that the sum of the integer division values of the s-monogenic generalizations of the Weierstraß ℘-function vanishes. We only have to apply the following limit argument: (k+1) (v/n) = lim (k+1) (z + v/n) s,s s,s z→0
v∈Vp (n)\{0}
=
v∈Vp (n)\{0}
lim nk+1+|m|−s (k+1) (nz) − (k+1) (z) = 0. s,s s,s
z→0
The limit calculus in the line above is allowed since all expressions that are involved are s-monogenic in a sufficiently small annular domain around zero. The limit value vanishes since the Laurent expansion of (z) = qs(s) (z) + qs(s) (z + ω) − qs(s) (ω) (k+1) s,s ω∈Ωk+1 \{0} (s)
has the form qs (z) + O(z) in a neighborhood around the origin. This result provides a generalization of the classical result that the sum over the integer division values of the holomorphic Weierstraß ℘-function vanishes (see e.g., [82] p. 83, [103]). One can say more. Whenever |m| + s is an even positive integer so that (s) qm (z + ω) ω∈Ωp
converges, then the sum over the integer division values equals the value of one of those generalized Riemann zeta functions defined in Section 2.4, namely (s) 2ζ Ωp (m, s) = qm (ω). ω∈Ωp \{0} (p)
If |m| + s is odd, then the sum over the integer division values of the series m,s (p) gives zero, which is a consequence of the oddness of m,s in the cases where |m| + s is odd.
2.6
Characterization theorems
For a number of choices of parameters p, m, s we have shown in the previous section (p) that the s-monogenic Eisenstein series m,s are solutions of a functional equation of the form (2.49). In this section we conversely want to study now the general class of s-monogenic Clifford-valued functions that solve such kinds of functional
80
Chapter 2. Clifford-analytic Eisenstein series for translation groups
equations. In particular we are interested in understanding under which additional conditions one obtains a characterization of the class of s-monogenic Eisenstein series in terms of these types of functional equations. In our recent paper [92] we established two theorems in this direction. First we proved: Theorem 2.19. Let s < k + 1, p ≤ k + 1 and let Ωp = Zω1 + · · · + Zωp be be a multi-index with |m| ≥ a p-dimensional lattice in Ak+1 . Let m ∈ INk+1 0 max{0, p − k − 1 + s}. Assume further that g : Ak+1 → Cl0k is a left or right (s) s-monogenic function with principal parts qm (z − ω) at each ω ∈ Ωp . If g satisfies nk+1+|m|−s g(nz) = g(z + v/n), (2.68) v∈Vp (n) (p)
then there is a C ∈ Cl0k such that g(z) = m,s (z) + C for all z ∈ Ak+1 \Ωk+1 . (p)
Proof. To show the assertion consider h(z) = g(z) − m,s (z) which is a left or right s-monogenic function, respectively, in the whole space Ak+1 . For all configurations with |m| = s with |m| ≥ max{0, p − k + s}, as well as for multi-indices m ∈ INk+1 0 in the case p = k + 1, and also for all configurations with p − k − 1 + s ≥ 0 where m is a multi-index with |m| = p − k − 1 + s where |m| + s is additionally odd, we have managed to establish the integer multiplication formulas for the function (p) series m,s in the previous section. For all these cases we know then for sure that also the function h satisfies nk+1+|m|−s h(nz) = h(z + v/n), h(0) = h0 . (2.69) v∈Vp (n)
Let us now first concentrate on these cases and let us assume that h ≡ h0 . Further, put β := ω1 + · · · + ωp . Since the maximum principle holds for the class of smonogenic functions, we may conclude that there is an element c ∈ ∂B(0, nβ) such that
h(z) < h(c) for all z ∈ B(0, nβ). In view of
p n−1 β < 2β αi ωi )/n ≤ β + (c + n i=1
for all 0 ≤ αi < n, we obtain nk+1+|m|−s h(c)
=
v∈Vp (n)
<
h(
c+v ) ≤ n
n h(c) ≤ n p
h(
v∈Vp (n)
k+1−s+|m|
c+v ) n
h(c) ,
(p)
and we have arrived at a contradiction. Hence, g(z) ≡ m,s (z) + C where C = h0 . We further observe that the constant C vanishes at least when |m| = p − k − 1 + s
2.6. Characterization theorems
81
in view of Vp = np . In the particular cases with |m| = p − k − 1 + s we obtain then (s) )(z). C = lim (g − qm z→0
We are now left to prove the assertion for those cases where p, s ≤ k with with |m| = p − k − 1 + s p − k − 1 + s ≥ 0 and where m is a multi-index from INk+1 0 such that |m| + s is even. For these cases consider now the first partial derivatives ∂h for i = 0, . . . , k. hi (z) := ∂x i (p)
For any i ∈ {0, 1, . . . , k} we know for sure that the function series m+τ (i),s satisfies the integer multiplication formula (2.49). Hence the functions hi satisfy nk+2+|m|−s hi (nz) = hi (z + v/n), hi (0) = hi0 . v∈Vp (n)
Applying similar arguments as before leads to hi ≡ hi0 for all i = 0, . . . , k. In view of Vp = np = nk+1+|m|−s , all the constants hi0 have to vanish. Hence, the function h is a constant and the proof is hereby completed. Remark. This theorem provides a generalization of results that we obtained earlier in [84, 85] for the particular cases s = 1, n = 2, m = 0, p ≤ k and s = 1, n = 2, |m| = 1, p = k + 1. Let us proceed one step further. The classical Herglotz lemma (see [46, 130]) tells us that a complex-valued function that is analytic in a sufficiently large disc of the complex plane and that satisfies the cotangent duplication formula is a constant. The following theorem provides a generalization to the context of smonogenic functions. |m|
For simplicity let us use the notation gm (z) = ∂∂zm g(z) for a C |m| -function g, where m ∈ INk+1 stands for a multi-index of length |m|. We further use the 0 symbol M (gm , r) := max gm (z) where r denotes a positive real. z ≤r
Theorem 2.20 (Generalized Herglotz lemma). Let Ωp = Zω1 + · · · + Zωp be a nondegenerate p-dimensional lattice in Ak+1 , where the paravectors ω1 , . . . , ωp denote its primitive periods. Let further β := ω1 + · · · + ωp and let r be a positive real with r > β. Suppose that f : B(0, r) → Cl0k is a left (right) s-monogenic function in the whole ball B(0, r) that has a continuous extension to ∂B(0, r) and that there is furthermore another C |m| -function χ : B(0, r) → Cl0k (necessarily s-monogenic in B(0, r)) such that for a fixed real δ > 0 holds nδ f (nz) = f (z + v/n) + χ(z), (2.70) v∈Vp (n) ◦
whenever the points z, nz, z + v/n for all v ∈ Vp (n) lie in the ball B (0, r). If χ satisfies the growth condition M (χm , r) < (nδ+|m| − np )M (fm , r),
(2.71)
82
Chapter 2. Clifford-analytic Eisenstein series for translation groups
then f must be an s-monogenic polynomial of a total degree that is less than or equal to |m|. Proof. Let us take an arbitrary left s-monogenic function f that satisfies (2.70). |m| If one applies ∂∂zm on both sides of equation (2.70), then one gets nδ+|m| fm (nz) =
fm (z + v/n) + χm (z).
(2.72)
v∈Vp (n)
Let t be a positive real number within β < t < r. If z ∈ B(0, t), then (z + v)/n ∈ B(0, t) for all v ∈ Vp (n). If we replace z by z/n in (2.72), then we obtain nδ+|m| fm (z) ≤ fm ((z + v)/n) + χm (z/n) v∈Vp (n)
≤ np M (fm , t) + M (χm , t) < np M (fm , t) + nδ+|m| M (fm , t) − np M (fm , t) = nδ+|m| M (fm , t). In view of M (fm , t) < M (fm , t), we have thus arrived at a contradiction. Therewith length |m|. Hence f must be an s-monogenic fore, fr = 0 for all r ∈ INk+1 0 polynomial of a total degree being less than or equal to |m|. This theorem generalizes a result from our earlier paper [87] in which a special version of this lemma was proved exclusively for the monogenic case, and, exclusively for the configuration p ≤ k, m = 0, n = 2 and χ ≡ 0. Now we can draw the following conclusion: A non-constant s-monogenic function that satisfies a functional equation of the form (2.49) must have singularities. Whenever the singularities are all isolated and distributed in the form of a lattice and have furthermore all the same order and principal parts, then this function (p) coincides at least up to a constant with m,s . A question that arises in a natural way in this context is if there are also non-constant functions that satisfy these multiplication formulas which do not (p) coincide with m,s . Let us finish this section by giving one example. For a multik+1 with |m| ≥ max{0, p + s + 1 − k} consider the s-monogenic index m ∈ IN0 functions (s) qm (αz + ω) Sc(z) > 0 G(p) m,s (z) = (α,ω)∈Z×Ωp \{(0,0)}
where Ωp is a p-dimensional lattice in spanIR {e1 , . . . , ek }, as introduced in Section 2.4. Since the series is normally convergent one can directly conclude by rearrangement arguments in order to show that nk+1−s|m| G(p) G(p) m,s (nz) = m,s (z + v/n). v∈Vp (n)
2.7. Lattices with hypercomplex multiplication
83
Notice that these functions are only s-monogenic in the right half-space H r (Ak+1 ). They have non-isolated singularities in each point of the dense set Qω1 + · · · + Qωp and are thus not meromorphic in the whole space Ak+1 .
2.7
Lattices with hypercomplex multiplication
In the previous two sections we discussed integer multiplication for the s-monogenic Eisenstein series. The treatment could be performed for all arbitrary lattices in Ak+1 without putting any restrictions on the generators of the lattice. This was due to the fact that we have nΩ ⊂ Ω for all n ∈ Z, independently from the form of the lattice. In this section we present more general multiplication formulas for the s-monogenic Eisenstein series that involve more generally hypercomplex multipliers from Ak+1 , in particular, paravector multipliers stemming from the proper lattice. However, in order to meet this end, it will be necessary to put number theoretical conditions on the generators of the lattice. Also the results from this section stem from our recent paper [92]. Before we go into detail let us first recall the basic results for the classical complex case and give some historical background information on this problem. Without loss of generality let us assume as usual that a √ complex lattice Ω has the form Ω = Z + Zτ with Im(τ ) > 0. Whenever τ ∈ Z[ −D] with a positive square-free integer D, then the relation λΩ ⊂ Ω is also satisfied for some complex numbers from C\Z, namely exactly by all the elements of the lattice and no more. This special property has been known for quite a long time. For lattices with this property the terminology lattices with complex multiplication has been established. For more details about the classical theory, see for instance [82, 103]. Now suppose that f (z) is an elliptic function associated with such a lattice. Then fλ (z) := f (λz) is again an elliptic function associated with the same lattice for all λ ∈ Ω. As a consequence of this one obtains an explicit connection between fλ and f in terms of a functional equation. In particular one gets a rather simple functional equation between ℘λ and ℘ which provides a generalization of (2.48) for complex multipliers. From this particular functional equation one can deduce an explicit formula for the trace of the complex division values of the Weierstraß ℘-function, which is the expression ω ℘( ). λ ω λ ∈P \Ω
In the previous line, P stands for a period parallelepiped of the associated translation group. The complex division values of the Weierstraß ℘-function and its trace are closely related to important fundamental problems from algebraic number theory. Putting g2 = 60 ω∈Ω\{0} ω −4 and g3 = 140 ω∈Ω\{0} ω −6 , then all complex division values of the associated normalized doubly periodic Weierstraß √ 2 g3 −D] and function P(z) := g3g−27g 2 ℘(z) lie in abelian Galois field extensions of Q[ 2
3
84
Chapter 2. Clifford-analytic Eisenstein series for translation groups
√ its trace is an element from Q[ −D]. This property provides an analogy to the number theoretical behavior of that of the division values of the exponential function which lie in abelian Galois field extensions of Q. For this reason, the classical elliptic functions play a fundamental role in class field theory. This is illuminated in detail in some of R. Fueter’s early works, see for instance [52], [53]. To give also a reference to one of the modern textbooks dealing with this problem, we refer for instance to [149] among others. Hilbert’s famous twelfth problem from [75] which deals with the problem of the construction of abelian Galois field extensions to an arbitrarily given algebraic number field provided a relatively strong motivation to ask for higher dimensional analogies of the concept of complex multiplication of the elliptic functions. In the first decades of the twentieth century, E. Hecke dedicated himself to work on generalizations of the concept of complex multiplication to the setting of several complex variables using Abelian functions (see e.g. [70, 71]). However, in the context of several complex variables theory, the Riemann condition imposes an additional restriction to the choice of the periods of the Abelian functions. The Riemann condition is a quadratic relation that the generators of the period lattices have to satisfy. It causes important restrictions for the computations of the class fields (see [70, 71] and also [63]). R. Fueter outlined at the end of his mathematical career in [62, 63] an ansatz for a quaternionic multiplication within the context of the monogenic four-fold periodic quaternionic generalizations of the Weierstraß ℘-function and the monogenic quasi-periodic quaternionic Weierstraß ζ-function. If one chooses the four primitive quaternionic periods for the period lattice Ω from a maximal integral domain that lies in a quaternionic Brandt algebra (cf. [14, 57]), then all λ, µ ∈ Ω satisfy λΩ ⊆ Ω, Ωµ ⊆ Ω, and hence also λΩµ ⊆ Ω. In view of the non-commutativity of the quaternions it is indeed strongly motivated to consider multipliers from the left and from the right. This approach involves in particular lattices where the four primitive periods are quaternions whose real components are elements from biquadratic number fields. If f (z) is a four-fold periodic monogenic function that is associated to such a particular quaternionic lattice Ω, then the function fλ,µ (z) := µf (λzµ)λ is again a four-fold periodic monogenic function associated to the same Ω for all λ, µ ∈ Ω. According to Theorem 2.5 every four-fold periodic 1-monogenic elliptic function with non-essential singularities can be represented on the other hand in terms of a (4) finite sum of the elementary four-fold periodic 1-monogenic function series 0,1;a,b (4)
and m,1 with |m| ≥ 1. If Ω is a lattice with quaternionic multiplication, then fλ,µ can in turn be represented in terms of the same elementary generalized Weierstraß functions (associated to the same lattice) as f . In analogy to the complex case, this leads to important functional equations. In particular, one obtains explicit multiplication formulas for the quaternionic monogenic versions of the Weierstraß ℘-function — involving quaternionic multipliers. These formulas in turn lead to closed formulas
2.7. Lattices with hypercomplex multiplication
85
for the trace of the division values of the quaternionic versions of the ℘-function (4) (i.e., ℘τ (i),1 = τ (i),1 ) from which R. Fueter drew the conclusion that the sum of the quaternionic division values λ−1 ωµ−1 ∈P \Ω ℘τ (i),1 (λ−1 ωµ−1 ) is a quaternion whose real components all lie in a field that is generated by the real components of the involved primitive periods ω1 , ω2 , ω3 , ω4 (which stem in the simplest nontrivial case from a biquadratic number field) and by the real components of the quasi-periodicity constants ηh of the generalized quaternionic monogenic Weierstraß ζ-function ζ(z, Ω) which are ηh := 2ζ(ωh /2, Ω), h = 1, 2, 3, 4. A deeper and more concrete analysis has not been provided by R. Fueter. Nevertheless, his ideas might be rather promising: While the periods of the Abelian functions have to satisfy the Riemann condition, the periods of the quaternionic monogenic elliptic functions can be chosen absolutely free within the Brandt algebra. Every arbitrary ideal from an integral domain of a Brandt algebra provides us with four periods. One only has to take care in the sense of choosing four IR-linear independent periods ω1 , . . . , ω4 . It shall be noticed that the Abelian functions of two complex variables can be reconstructed from particular candidates of four-fold periodic monogenic elliptic functions, as mentioned for instance in [64]. Unfortunately the idea to link special monogenic functions with problems from class field theory or with general problems of number theoretical origin has been neglected after R. Fueter’s death. In this section we present extensions of R. Fueter’s ideas from [62] and [63] to the setting of the arbitrary finite dimensional Euclidean spaces Ak+1 and to more general function classes. From the very beginning it is not obvious at all whether complex and quaternionic multiplication can be generalized to arbitrary higher dimensional settings within the context of Clifford-analytic multiperiodic functions which are defined in Ak+1 . Note that both C and IH are closed under multiplication. However, if λ, ω, µ are more generally paravectors from Ak+1 , then neither products of the form λω, nor ωµ, nor λωµ remain in Ak+1 in general. It was possible to define Zorders in rational Clifford algebras Cl0k for general k (see e.g., [44, 45]); however, their restriction to the paravector space Ak+1 is not closed under multiplication within Ak+1 . It is thus not immediately so evident how the concept of complex and quaternionic multiplication of the Weierstraß functions can be adapted to the setting of the Clifford-analytic Weierstraß functions defined in the paravector space Ak+1 . In our recent paper [92] we illustrated a meaningful way to do so. First we started with the following definition: Definition 2.21 (Lattices with paravector multiplication). Let 1 ≤ p ≤ k + 1 and Ωp = Zω1 + · · · + Zωp be a p-dimensional lattice in Ak+1 . Then we say that Ωp has paravector multiplication if there are paravectors λ, µ ∈ Ak+1 where at least λ or at least µ is an element from Ak+1 \Z such that λΩp µ ⊂ Ωp .
(2.73)
86
Chapter 2. Clifford-analytic Eisenstein series for translation groups
The simplest examples of lattices in Ak+1 that satisfy this condition are rectangular lattices: e1 (Zα0 +Zα1 e1 +· · ·+Zαk ek )e1 = Zα0 +Zα1 e1 +· · ·+Zαk ek ; α0 , . . . , αk ∈ IR\{0}. (2.74) However, a special focus shall be put on those lattices that satisfy (2.73) in particular for multipliers λ, µ that are elements from the proper lattice. The following theorem provides us with a number of non-trivial examples of lattices in Ak+1 that have a paravector multiplication where the multipliers stem from the proper lattice. Theorem 2.22. Let p be an arbitrary but fixed integer with 0 ≤ p ≤ k. Let further Ωp+1 := Z + Zω1 + · · · + Zωp be a lattice in Ak+1 where the periods ω1 , . . . , ωp satisfy all N (ωi ) ∈ Z,
S(ωi ) ∈ Z,
2 ωi , ωj ∈ Z
∀i, j ∈ {1, . . . , p}.
(2.75)
Then Ωp+1 has paravector multiplication. In particular, it is stable under the conjugation anti-automorphism (i.e., Ωp+1 = Ωp+1 ) and each η ∈ Ωp+1 and α ∈ Z satisfies (2.76) α ηΩp+1 η ⊂ Ωp+1 . Conversely, if Ωp+1 = Z + Zω1 + · · · + Zωp is a (p + 1)-dimensional lattice that has the property that αηΩp+1 η ⊆ Ωp+1 is satisfied for any α ∈ Z and any arbitrary η ∈ Ωp+1 , then the primitive periods satisfy all (2.75). p Proof. Let ω ∈ Ωp+1 . Then there are α0 , . . . , αp ∈ Z so that ω = α0 + j=1 αj ωj . p Furthermore, ω = S(ω) − ω. Since S(ω) = 2α0 + j=1 αj S(ωj ) ∈ Z, we can conclude directly that ω ∈ Ωp+1 . The lattice Ωp+1 is indeed stable under the conjugation anti-automorphism. Next let us prove that every η ∈ Ωp+1 satisfies ηωη ∈ Ωp+1 for any arbitrary ω ∈ Ωp+1 from which (2.76) then readily follows. One obtains ηωη = S(ηω)η − N (η)ω = 2 η, ω η − N (η)ω.
(2.77)
Observe that both η, ω ∈ Ωp+1 . Therefore, there are integers α0 , . . . , αp , β0 , . . ., βp ∈ Z with p p 2 η, ω = 2 α0 + αj ωj , β0 + βj ωj = j=1
j=1
0≤i,j≤p
αi βj · 2 ωi , ωj
∈Z
where ω0 := e0 and ω1 , . . . , ωp are the primitive periods of Ωp+1 . In view of (2.75), one has S(ηω) ∈ Z for any ω, η ∈ Ωp+1 . Putting ω = η in the above calculation yields N (η) = η, η ∈ Z. Consequently, we can write ηωη = γη − δω with integers γ, δ ∈ Z. Hence, ηωη ∈ Ωp+1 .
2.7. Lattices with hypercomplex multiplication
87
Let us now suppose that we are dealing with a lattice Ωp+1 := Z + Zω1 + · · · + Zωp that satisfies αηΩp+1 η ⊂ Ωp+1 for all α ∈ Z and for all η ∈ Ωp+1 . A lattice with these properties satisfies then in particular ωi ωj ωi ∈ Ωp+1 for the primitive periods. The relation (2.77) implies that ωi ωj ωi = (S(ωi )S(ωj ) − 2 ωi , ωj )ωi − N (ωi )ωj . The product ωi ωj ωi has thus the form aωi + bωj . From ωi ωj ωi ∈ Ωp+1 follows that both elements a and b have to be integers. Hence N (ωi ) ∈ Z for all i = 0, 1, . . . , p. Further, (2.78) S(ωi )S(ωj ) − 2 ωi , ωj ∈ Z for all i, j ∈ {0, . . . , p}. Inserting in particular ωj = 1 into (2.78), implies conse quently that S(ωi ) ∈ Z. Hence, 2 ωi , ωj must also be an integer. Notice once more that the multipliers do not form anymore a ring within Ak+1 for general k. In general, the multiplication of the multipliers is only closed within the whole Clifford algebra over Ak+1 , where we have zero-divisors. Nevertheless, we have the impression that the lack of the closure of multiplication within Ak+1 does not really lead to a serious obstacle. Next we give a number of important examples of paravector lattices that have the properties from Theorem 2.22: Proposition 2.23. Let Ωp+1 be a p + 1-dimensional lattice in Ak+1 of the special form Ωp+1 = Z + Zω1 + · · · + Zωp where the primitive periods ω1 , . . . , ωp have the form ωi =
k
√ αij mj ej
(2.79)
j=0
where αij are all integers, m0 = 1 and m1 , . . . , mk are positive integers which may be square-free. Then Ωp+1 has paravector multiplication and for all α ∈ Z and all η ∈ Ωp+1 we have αηΩp+1 η ⊆ Ωp+1 . Proof. Consider simply N (ω0 ) = N (e0 ) = 1 and for i = 1, . . . , p: N (ωi ) =
k
√ (αij mj )2 = αi2j mj ∈ Z. k
j=0
j=0
Furthermore, S(ω0 ) = 2, and for i = 1, . . . , p we have S(ωi ) = 2αi0 ∈ Z. Obviously, ω0 , ωj = ωj , ω0 ∈ Z for all j = 0, . . . , p. For general i, l ∈ {1, . . . , p} we have 2 ωi , ωl =
k j=0
αij αlj mj ∈ Z.
88
Chapter 2. Clifford-analytic Eisenstein series for translation groups
Notice that the condition αij ∈ Z is only a sufficient condition. Indeed, it is not difficult to work out examples where the parameters αij stem from Q\Z, as given next in (2.82). In 2n -dimensional Euclidean spaces the following types of lattices play a particular role within the theory of paravector multiplication. In the four-dimensional case, i.e., n = 2, consider lattices with periods of the form (see also [63]) √ √ √ √ ωi = αi0 + αi1 m1 e1 + αi2 m2 e2 + αi3 m1 m2 e3
(2.80)
where m1 and m2 are two distinct positive square-free integers and where the elements αij are chosen in a way that (2.75) is satisfied. In the particular case where all the parameters αij are integers, the conditions (2.75) are always satisfied — independently from the choice of the parameters αij . In the eight-dimensional case consider lattices with primitive generators of the form ωi
√ √ √ = αi0 + αi1 m1 e1 + αi2 m2 e2 + αi3 m3 e3 √ √ √ √ √ √ +αi4 m1 m2 e4 + αi5 m1 m3 e5 + αi6 m2 m3 e6 √ √ √ +αi7 m1 m2 m3 e7
(2.81)
and similar lattices of dimensions 2n with n > 3 — claiming the analogous conditions on the elements m1 , m2 , . . . and αij . If one treats the case n = 2 with quaternions, i.e., by identifying the elements ei with the quaternionic imaginary units, e1 = i, e2 = j, e3 = k = ij, then the primitive periods of the quaternionic lattice (2.80) generate an ideal within an integral quaternionic Brandt algebra, as R. Fueter mentioned in [63]. The multiplication of two primitive periods from a quaternionic lattice of the form (2.80) produces again a quaternion that has the form √ √ √ √ βi0 + βi1 m1 e1 + βi2 m2 e2 + βi3 m1 m2 e3 with parameters βij from Q. If in particular all αij ∈ Z, then all the elements βij are again integers. If the elements αij from (2.80) are properly chosen so that the ideal generated by the primitive periods is furthermore a maximal integral domain in the underlying quaternionic Brandt algebra, then the associated lattice is closed under multiplication. Then λΩ ⊆ Ω, Ωµ ⊆ Ω, and thus λΩµ ⊆ Ω for all elements λ, µ ∈ Ω. Choosing two non-real linearly independent primitive periods ω1 and ω2 of the form (2.80) that satisfy moreover ω1 ω2 = ω2 ω1 , then, following [57], [1, ω1 , ω2 , ω1 ω2 ] is a basis for a maximal integral domain whenever N (ω1 ω2 − ω2 ω1 ) = 4| det W |.
2.7. Lattices with hypercomplex multiplication
89
Here W stands for the period matrix of the primitive periods (1, ω1 , ω2 , ω1 ω2 ). In this case the associated lattice Ω = Z + Zω1 + Zω2 + Zω1 ω2 is closed under multiplication. To give one explicit example for a non-rectangular quaternionic lattice that is closed under multiplication, take for instance as generating periods ω0 = 1,
ω3 = − 21 2 +
√
3 2 e1 , √ √ 7 3 5 2 e1 + 2 e2
ω1 =
1 2
+
√ √ ω2 = 7 3e1 + 5e2 , √
+
15 2 e3
(2.82)
= ω1 ω2 .
By a direct computation one can readily show that the conditions (2.75) are all satisfied, that ω1 ω2 − ω2 ω1 = 0 and that additionally N (ω1 ω2 − ω2 ω1 ) = 4| det W |. These periods generate actually a maximal integral domain within an integral quaternionic Brandt algebra. The associated lattice is hence closed under multiplication. In the eight-dimensional case, n = 3, one can embed the lattice (2.81) into the octonions O. In this sense simply identify the elements e1 , . . . , e7 with the octononic imaginary units, i.e. e1 = i, e2 = j, e3 = w, e4 = k, e5 = iw, e6 = jw, e7 = kw. Here i, j, k are the quaternionic imaginary units, while w is a further independent octonionic imaginary unit that satisfies w2 = −1, as introduced in Chapter 1.1. The multiplication of two primitive periods from an octonionic lattice that has the form (2.81) results again in an octonion of the form √ √ √ √ √ γi0 + γi1 m1 e1 + γi2 m2 e2 + γi3 m3 e3 + γi4 m1 m2 e4 √ √ √ √ √ √ √ +γi5 m1 m3 e5 + γi6 m2 m3 e6 + γi7 m1 m2 m3 e7 , where the coefficients γij are all elements from Q. In the particular case where all the elements αij are integers, the coefficients γij are integers, too. In analogy to the quaternionic case, the primitive periods generate an ideal of integral elements within the octonions. In this context, the notions “ideal” and “integral domain” shall be understood in a wider sense, having no longer the associativity. For more details about the general theory of ideals in octonions, see for instance [28]. If we choose the elements αij from (2.81) properly so that the ideal that is generated by the primitive periods is moreover a maximal integral domain, then we have in analogy to the quaternionic case that λΩ ⊆ Ω, Ωµ ⊆ Ω, and hence (λΩ)µ ⊆ Ω and λ(Ωµ) ⊆ Ω for all elements λ, µ ∈ Ω. Such a relation does in general not exist in the context of working in the eight-dimensional paravector space A8 . At this point we wish to point out once more that the octonions do not form a Clifford algebra, since they are non-associative. However, as mentioned in Chapter 1.1, they still form a normed division algebra. Higher dimensional Cayley–Dickson algebras where n > 3 do not form normed algebras anymore. Therefore, the notion of Brandt algebras cannot be transferred so directly to the setting of higher dimensional Cayley–Dickson algebras that extend the octonions
90
Chapter 2. Clifford-analytic Eisenstein series for translation groups
without imposing further restrictions. The special cases where the dimension of the underlying Euclidean space is either 2, 4 or 8 should therefore be regarded as very special subcases within the theory of hypercomplex multiplication. Now let us assume that Ω is an arbitrary lattice with hypercomplex multiplication of the form that λΩµ ⊂ Ω for some paravectors from Ak+1 , quaternions or octonions respectively. Under this condition we have the following situation: If f satisfies f (z + ω) = f (z) for all ω ∈ Ωp , then the function fλ,µ (z) := µf (λzµ)λ satisfies also fλ,µ (z + ω) = fλ,µ (z) for all ω from the same lattice. If f : Ak+1 → Cl0k is an arbitrary left and right s-monogenic function and if λ is an arbitrary paravector from Ak+1 , then in general the expression fλ,µ (z) remains only left and right s-monogenic, if µ = αλ with a real α ∈ IR. This provides a strong function theoretic motivation to regard exactly those lattices that are described in Theorem 2.22 as the canonical ones within the theory of paravector multiplication of the Clifford analytic multiperiodic functions defined in Ak+1 for arbitrary dimensions. We observe here an important connection point between Clifford analysis and number theory. Notice once more the particular role of the quaternionic and octonionic case: If f is left and right quaternionic (octonionic) s-monogenic then fλ,µ (z) is also left and right s-monogenic, for all quaternions (octonions) λ and µ. This special property is a consequence of the fact that both the quaternions and the octonions do still form normed algebras. Let us now describe in detail the paravector multiplication of the monogenic elliptic functions in arbitrary dimensions in the Clifford analysis setting. What follows provides an extension of R. Fueter’s result for the quaternionic case. The octonionic case is also treated implicitly within the following calculations. In this (p) sense, notice that the Eisenstein series m,s can be introduced directly into the octonionic setting. One simply has to substitute the elements e0 , e1 , . . . , e7 by the eight units of the octonions. The series that we subsequently obtain — having l f = 0 formally the same form as their Clifford counterparts — satisfy then DO where DO means the octonionic Cauchy–Riemann operator as used in [68, 114, 109, 110] and elsewhere. In view of the relation (ab)b = b(ba) = a b 2 = a(bb) = a(bb) for all a, b ∈ O, as a consequence of Theorem 4.1 from [164], all the following calculations can be directly adapted to the octonionic case putting the brackets in accordance to the octonionic lattice multiplications that are considered, either of the form (λΩ)µ or λ(Ωµ). To proceed, recall that every Ωk+1 -periodic monogenic function f that has at most unessential singularities can be represented in terms of a finite sum of the (k+1) associated to this lattice. monogenic k + 1-fold periodic Eisenstein series m This is the statement of Theorem 2.5. If Ωk+1 is a lattice that satisfies (2.75), then
2.7. Lattices with hypercomplex multiplication
91
fαλ,λ can consequently be represented in terms of a finite sum of the same se(k+1) ries m associated with this lattice. This leads to explicit relationships between (k+1) (k+1) αλm (αλzλ)λ and m (z) for arbitrary λ ∈ Ωk+1 and α ∈ Z. In particular we obtain an interesting generalization of the multiplication formula for the generalized ℘-functions from Theorem 2.18 for paravector multipliers. The following argumentation is more complicated than in Section 2.5 due (s) to the fact that the functions qm are only IR-homogeneous and not Ak+1 -homogeneous functions. Now consider the generalized monogenic Weierstraß ζ-function ζ(z, Ωk+1 ) := ζ0,1 (z, Ωk+1 ) where we assume that Ωk+1 is a lattice whose primitive periods satisfy the conditions (2.75), as for example, lattices of the form (2.79), (2.80), (2.81). Next let us take an arbitrary non-zero λ of such a lattice and an arbitrary non-zero positive integer α. Then αλzλ ∈ Ak+1 \Ωk+1 for every z ∈ Ak+1 \Ωk+1 and αλωλ ∈ Ωk+1 for every ω ∈ Ωk+1 . Hence ζαλ,λ (z) := αλζ(αλzλ, Ωk+1 )λ is a well-defined quasi (k + 1)-fold periodic function with respect to Ωk+1 . It is left and right monogenic in Ak+1 \ α1 λ−1 Ωk+1 λ−1 . Notice that the proper expression ζ(αλzλ, Ωk+1 ) is not monogenic anymore whenever λ ∈ IR. However, the factor λ on the left-hand side of this expression provides the left monogenicity, and the other one standing on the right-hand side, provides the right monogenicity. Notice that ζαλ,λ (z) has only point singularities of order k at the points 1 −1 −1 λ ωλ , ω ∈ Ωk+1 . α √ Each period parallelepiped contains {µ} = N ( αλ)k+1 -many point singularities (all of order k). The set of singularities that are contained in the fundamental period parallelepiped (including the origin) will be denoted by V in all that follows. Next we expand the function ζαλ,λ into a Laurent series in a sufficiently small neighborhood around zero: µ :=
αλζ(αλzλ)λ = αλq0 (αλzλ)λ + φ(z) = αλ = q0 (z)
αλzλ λ + φ(z)
αλzλ k+1
1
k−1 + φ(z). αN (λ)
Here φ stands for a function that is monogenic in a neighborhood of the origin. On the other hand, ζ(z + µ) = q0 (z) + Ψ(z) µ∈V
in a sufficiently small neighborhood around the origin, where Ψ stands analogously for a function that is monogenic in that neighborhood. The function 1 f (z) := αλζ(αλzλ, Ωk+1 )λ − ζ(z + µ, Ωk+1 ) (2.83) k−1 [αN (λ)] µ∈V
92
Chapter 2. Clifford-analytic Eisenstein series for translation groups
is thus left and right monogenic at the origin. By analogous arguments one concludes that f is also right and left monogenic at each µ ∈ V. Let i ∈ {1, . . . , k} and let us for simplicity denote the partial derivatives of f with respect to the variable xi by fτ (i) . From the quasi-periodicity property of the generalized Weierstraß ζ-function with respect to Ωk+1 , the k functions ∂ 1 fτ (i) (z) := αλ[ ζ(αλzλ, Ωk+1 )]λ − ℘τ (i) (z + µ, Ωk+1 ) (2.84) ∂xi [αN (λ)]k−1 µ∈V
turn out to be all (k + 1)-fold periodic with respect to Ωk+1 . Since f has no singularities in each period parallelepiped, the function fτ (i) has no singularities in each period cell, either. Thus, fτ (i) (z) ≡ Ei is constant for all i = 1, . . . , k. In view of the monogenicity of ζ we hence obtain: αλζ(αλzλ)λ =
k 1 ζ(z + µ) + Ej Vτ (j) (z) + E. [αN (λ)]k−1 j=1
(2.85)
µ∈V
Applying again a shift of the form (z → z + ωh ), h = 0, . . . , k on the argument of both sides of the previous equation yields αλζ(αλzλ + αλωh λ)λ =
k 1 ζ(z + µ + ω ) + Ej Vτ (j) (z + ωh ) + E h [αN (λ)]k−1 j=1 µ∈V
(2.86) for all h = 0, 1, . . . , k. The special property that Ωk+1 is a lattice with hypercomplex multiplication allows us to conclude that there are integers nhj ∈ Z with αλωh λ =
k
nhj ωj .
(2.87)
j=0
Notice here, that we do not obtain such a relation for general lattices. Applying (2.21) successively yields ζ(αλzλ + αλωh λ) = ζ(αλzλ) +
k
nhj ηj .
j=0 [1]
For simplicity we write here ηj for ηj . Next combine this relation with (2.86). After this one gets in view of (2.85): αλ
k
nhj ηj λ =
j=0
k
1 + η Ej Vτ (j) (ωh ). h [αN (λ)]k−1 j=1 µ∈V
√ From V = N ( αλ)k+1 follows that the previous equation simplifies to αλ
k
j=0
k √ nhj ηj λ − N ( αλ)2 ηh = Ej (ωhj − ej ωh0 ) j=1
(2.88)
2.7. Lattices with hypercomplex multiplication
93
k where we write again ωh = j=0 ωhj ej . Let us use the notation det W for the determinant of the period matrix W , as defined in (2.65), and Θhj for the adjoint determinant associated with the element ωhj . Since (2.88) is satisfied, the following equation is also satisfied: k k k k k
√ Θhi αλ nhj ηj λ − N ( αλ)2 Θhi ηh = Ej (Θhi ωhj − ej Θhi ωh0 ). j=0
h=0
h=0
h=0 j=1
(2.89) Applying (2.66) to (2.89), yields for all i = 1, . . . , k:
k k k √ Θhi αλ nhj ηj λ − N ( αλ)2 Θhi ηh Ei =
h=0
j=0
h=0
det W
.
(2.90)
In view of the linear independence of the primitive periods, we have det W = 0, so that we actually may divide by det W . Notice further that the elements ηj cannot all vanish, otherwise the ζ-function would be (k + 1)-fold periodic. This, however, would be a contradiction to Theorem 2.8 from Section 2.3. Hence the values Ej need not all vanish. In the context considered in this section, we get thus a difference to the integer multiplication formulas that we described in Section 2.5. Taking the limit z → 0 on both sides of equation (2.84) provides us with an explicit formula for the trace of hypercomplex division values of the generalized monogenic ℘-functions: Theorem 2.24. Let i ∈ {1, . . . , k}. Let Ωk+1 be a lattice whose primitive periods satisfy (2.75). Further let α ∈ IN, λ ∈ Ωk+1 \{0} and V = {µ ∈ F | µ = 1 −1 ωλ−1 , ω ∈ Ωk+1 } where F stands for the fundamental period parallelepiped. αλ Then
℘τ (j) (µ) =
µ∈V\{0}
k k k
√ [αN (λ)]k−1 nhj ηj λ − N ( αλ)2 Θhi ηh . Θhi αλ det W j=0 h=0
h=0
(2.91) The trace of the hypercomplex division points of the generalized monogenic ℘functions is a paravector from Ak+1 whose real components all lie in the field that is generated by the number field of the real components of the primitive periods ωh and by the real components of the quasi-periodicity constants ηh of the monogenic Weierstraß ζ-function. If we have a lattice of the form (2.79) with m1 , . . . , mk being all mutually distinct and square-free, then the real components of the primitive periods ωh √ √ generate the field Q[ m1 , . . . , mk ]. The quaternionic or octonionic components of the periods of the special lattices (2.80) or (2.81), respectively, generate a biquadratic or triquadratic number field, respectively. To obtain an exhaustive number theoretical description of the trace of the hypercomplex division values of the
94
Chapter 2. Clifford-analytic Eisenstein series for translation groups
generalized ℘-functions, a deeper number theoretical analysis of the constants ηh will be required. Notice further, that we obtain by applying on both sides of (2.86) differentiation arguments that αλ
(k+1) 1 ∂ |m|+|r| ζ(αλzλ, Ωk+1 )λ − m+r (z + µ, Ωk+1 ) = 0. m+r k−1 ∂x [αN (λ)] µ∈V
A similar limit argument as presented previously provides us with formulas for the (k+1) trace of the division values of the series m . In particular, we obtain (k+1) (µ) = 0 m µ∈V\{0}
for all multi-indices m where |m| is an even positive integer. Next let us treat the harmonic case. In this context keep in mind that the (2) series ω∈Ωk+1 qm (z + ω) only converges normally up from |m| ≥ 3 (instead for |m| ≥ 2 for s = 1). The harmonic analogues of the Weierstraß ℘-function in (k+1) this setting are given by the series m,2 where m is a multi-index of length 2. (2)
(1)
From Dqτ (i) (z) = (1 − k)qτ (i) (z) follows that for each m = τ (i) + τ (j) with i, j ∈ {0, 1, . . . , k}, k
(k+1)
(k+1)
τ (i)+τ (j),2 (z)ej = (1 − k)τ (i),1 (z).
j=0 (k+1)
The functions τ (i)+τ (j),2 which are scalar-valued coincide thus up to the real constant ±(1 − k) with the real components of the monogenic paravector-valued (k+1) series τ (i),1 (z). This admits an immediate transfer of the results obtained in the monogenic case to the harmonic case. This direct argument can only partially be adapted to the setting D[s] f = 0 where s > 2. With the formula D
[β] (s) qm (z)
(s−β) = C(k)qm (z),
where C(k) stands for a real constant that depends only on k, one can reduce the analysis of a number of s-monogenic variants of associated Weierstraß ℘-functions to the study of the monogenic ones. However, the relations become more complicated the larger s becomes. This relation cannot be used directly to draw conclusions for arbitrary choices of m, as for instance for the index 3τ (i) (i = 1, . . . , k) in (s) the setting s = 3, as one may easily verify. The inhomogeneity of qm with respect to Ak+1 for all |m| ≥ 1 is a significant obstacle for a direct derivation of multiplication formulas for all configurations of s and m. To discuss a further interesting
2.7. Lattices with hypercomplex multiplication
95
positive example consider in Ak+1 with k ≡ 1 mod 2 the special holomorphic Cliffordian ℘-functions (cf. [105]) (k+1)
[k]
(k−1,0,...,0)+τ (i) (z) =: ℘τ (i) (z)
i = 0, . . . , k
where (k − 1, 0, . . . , 0) is a multi-index whose first entry refers to the differentiation with respect to the x0 -direction. These functions stem from one global holomorphic Cliffordian primitive, which is here for simplicity denoted by ζ [k] (z) = ζ(k−1,0,...,0),k (z). We have ∂ [k] [k] ζ (z) = ℘τ (i) (z). ∂xi
(2.92)
Consequently, there are paravector constants βh with ζ [k] (z + ωh ) = ζ [k] (z) + βh for h = 0, . . . , k. If we deal with a lattice that satisfies the conditions from (2.75), then we can proceed in a similar way as in the proof for Theorem 2.24 in order to establish an explicit formula for the trace of their hypercomplex division points, however, involving one further restriction: [k]
Theorem 2.25. Let i ∈ {0, 1, . . . , k}, k be an odd positive integer, and ℘τ (i) (z) be an arbitrary function of those from (2.92). Let Ωk+1 be a lattice whose primitive periods satisfy (2.75). For all λ ∈ Ωk+1 with Sc(λ) = 0 we have [k] ℘τ (i) (µ) µ∈V\{0}
=
k−1 k
− N (λ) det W
Θhi λ
k
k nhj βj λ − (−1)k−1 N (λ)2 Θhi βh .
j=0
h=0
h=0
(2.93) Proof. Notice first that the Laurent expansion of the particular function ζ [k] (z) has the special form ζ [k] (z)
= =
∂ k−1 (k) q0 (z) + Φ(z) ∂xk−1 0
(−1)k−1 (k − 1)!z −k + Φ(z).
Under the condition that Sc(λ) = 0 we have the special relation (λzλ)−k = (−1)k−1 λ−1 z −k λ−1 Hence, λζ [k] (λzλ)λ = (k − 1)!
1 . N (λ)k−1
z −k + λΦ(λzλ)λ, N (λ)k−1
96
Chapter 2. Clifford-analytic Eisenstein series for translation groups
where the function Ψ(z) := λΦ(λzλ)λ is left and right holomorphic Cliffordian in an open neighborhood around zero. The function f (z) = λζ [k] (λzλ)λ −
(−1)k−1 [k] ζ (z + µ) N (λ)k−1
(2.94)
µ∈V
is left and right holomorphic Cliffordian at zero and also at the other points µ ∈ V. Its k + 1 partial derivatives fτ (i) (z) = λ[
∂ (−1)k−1 [k] {ζ [k] (λzλ)}]λ − ℘τ (i) (z + µ) i = 0, . . . , k, (2.95) ∂xi N (λ)k−1 µ∈V
are then (k + 1)-fold periodic holomorphic Cliffordian functions in the whole space Ak+1 . From the first Liouville theorem for holomorphic Cliffordian functions (cf. [105]) follows that fτ (i) (z) ≡ Ei and hence λζ [k] (λzλ)λ =
(−1)k−1 [k] ζ (z + µ) + E0 x0 + E1 x1 + · · · + Ek xk + E, N (λ)k−1 µ∈V
involving a further independent Clifford constant E0 , since f is not monogenic. Similar arguments as in the proof of Theorem 2.24 can now be applied so that one finally arrives at the stated result. Remarks. Qualitatively, the formula (2.93) has from the formal point of view a similar structure as (2.91). However, notice that we put an additional restriction on the multipliers λ, namely that Sc(λ) = 0. Note in this context that the particular lattices (2.79), (2.80) and (2.81) contain infinitely many lattice points with no scalar part. In general the elements βh do not coincide with the quasi-periodicity constants ηh from the monogenic ζ-function. Applying to all these cases a differentiation argument provides us consequently with formulas for the sum of the hypercomplex division values of the partial derivatives of these functions. (s)
As a consequence of the inhomogeneity of qm (z) with respect to Ak+1 for |m| ≥ 1, this argument cannot be adapted that directly to the functions (k+1) (k+1) (0,k−1,...,0)+τ (i) (z), (0,0,k−1,0,...,0)+τ (i) (z), etc.. In those cases the Laurent expansion of the associated primitive ζ˜[k] reads ζ˜[k] (z) = (−1)k−1 z −1 (ej z −1 )k−1 + Φ(z) and hence λζ˜[k] (λzλ)λ = (−1)k−1 λ(λ−1 z −1 λ−1 )(ej λ−1 z −1 λ−1 )k−1 λ + λΦ(λzλ)λ. Whenever j = 0 it is not possible simply to shift all the λ terms to the left or to the right-hand side of terms which only contain powers of z.
2.8. Bergman kernels of rectangular domains
97
We conclude this section by dedicating a few summarizing words to the hypercomplex multiplication of the translative Eisenstein series associated to a period lattice of dimension p < k + 1. If p < k + 1 − s, then the function series (s) qm (z + ω) ω∈Ωp
is already convergent up from m = 0. The same holds for the expression (s) (s) (s) q0 (z) + q0 (z + ω) − q0 (ω) ω∈Ωp \{0}
in the case p = k +1−s, as mentioned previously. In all these cases one can exploit (s) the particular Ak+1 -homogeneity of the functions q0 which we do not have for the cases with |m| ≥ 1, as mentioned a few lines above. In the 1-monogenic case, in which the series ω∈Ωp q0 (z + ω) is convergent for all p ≤ k − 1, this allows us readily to establish (p)
λ0 (λzλ)λ
=
ω∈Ωp
=
µ∈V
1 q0 (z + λ−1 ωλ−1 ) N (λ)k−1
1 (p) (z + µ) N (λ)k−1 0
for all λ ∈ Ωp .
(2.96)
A similar formula will be obtained for the case s = 2 with p ≤ k − 2, for the case s = 3 with p ≤ k − 3, etc. In the case s = 1 and p = k one can establish an analogous formula, using a generalized version of Lemma 2.16 for the multi-index m = 0. The cases s = 2n + 1, p = k − 2n can be treated in a similar way, since (s) q0 is odd whenever s is odd. In the cases where s = 2n, p = k − 2n + 1 it is not (s) allowed to apply Lemma 2.16, since in these functions q0 are even. However, one may immediately infer a similar multiplication formula up to a constant. These formulas give rise to explicit formulas for the sum of the hypercomplex division (p) values of the associated series m,s . Differentiation arguments and an analogous limit argument lead further to formulas for the trace of the hypercomplex division (p) values of m+r,s for |r| ≥ 1. The trace vanishes if |m| + |r| + s is odd. The remaining cases are more difficult to treat in view of the inhomogeneity (s) of the functions qm . The elegant argumentation with Liouville’s theorem cannot be applied for p < k + 1, either.
2.8
Bergman kernels of rectangular domains
In classical complex analysis, holomorphic automorphic forms are intimately related to reproducing kernel functions from the classical Bergman and Hardy spaces associated to their fundamental domains.
98
Chapter 2. Clifford-analytic Eisenstein series for translation groups
We briefly recall: The classical Bergman space B 2 (G, C) is the space of functions that are holomorphic and square-integrable over a domain G ⊂ C. The closure of the space of functions that are holomorphic over G with continuous extension to the boundary being square-integrable over the boundary ∂G is called the Hardy space and is denoted by H2 (∂G, C). Both function spaces are Hilbert spaces with a continuous point evaluation from which the existence of a uniquely defined reproducing kernel function follows. The kernel function is called the Bergman kernel in the first case and the Szeg¨o kernel in the second one. The Bergman kero kernel SG (z, w) are both functions in two complex nel BG (z, w) and the Szeg¨ variables which are holomorphic in the first variable and anti-holomorphic in the second one. They satisfy BG (z, w)f (w)dVw f (z) = G
and
SG (z, w)f (z)dSw
f (z) = ∂G
for any function f ∈ B 2 (G, C) or f ∈ H2 (∂G, C), respectively, where dV stands for the volume measure and dS for the Lebesgue surface measure, respectively. For details about the classical theory of these function spaces, see for instance the textbook [6]. In contrast to the behavior of the Cauchy kernel, the Bergman and the Szeg¨o kernel depend on the domain. With each domain a different Bergman and Szeg¨o kernel function is associated. The determination of explicit formulas for these reproducing kernel functions is very difficult in general. In the classical case when we deal with a simply connected domain G = C the associated kernel functions are directly related to the Riemann mapping function f , which is the function that maps G conformally onto the upper half-plane. One has BG (z, w) = −
1 f (z)f (w) , π (f (z) − f (w))2
(2.97)
and furthermore 2 (z, w) = SG
1 BG (z, w). 4π
(2.98)
For domains for which one knows the Riemann mapping function f explicitly, one thus obtains immediately explicit and closed formulas for the Bergman and the Szeg¨o kernel associated to G, respectively. The vertical strip S := {z = x + iy ∈ C | 0 < x < π} is mapped by the conformal map f (z) = exp(iz) onto the upper half-plane. An
2.8. Bergman kernels of rectangular domains
99
application of (2.97) leads thus immediately to BS (z, w)
= −
1 f (z)f (w) π (f (z) − f (w))2
exp(i(z − w)) 1 π (exp(iz) − exp(−iw))2 1 1 1 1
= − = 2 2 π (exp(i(z + w)/2) − exp(−i(z + w)/2)) 4π sin z+w 2 = −
=
1 (1) (z + w; 2πZ). π 2
By the same method one obtains for a rectangular domain of the form R = {z = x + iy ∈ C | 0 < x < 1, 0 < y < 1} that BR (z, w) =
1
℘(z + w; 2Z + 2iZ) − ℘(z − w; 2Z + 2iZ) . π
The modular function j(z) =
(60G4
60G4 (z) − 27(140G6 (z))2
(z))3
maps the hyperbolic triangle F = {z = x + iy ∈ C | 0 < x <
1 2 , x + y 2 > 1} 2
onto the upper half-plane. Again we can conclude consequently that BF (z, w) = −
1 j (z)j (w) . π (j(z) − j(w))2
An application of (2.98) provides then explicit formulas for the associated Szeg¨ o kernel functions. The explicit knowledge of the Bergman and the Szeg¨o kernel for a domain gives rise to the solution of an optimization and approximation problem: The Bergman operator, defined by [IBf ](z) = BG (z, w)f (z)dV G
where f stands now for an arbitrary L2 (G, C)-function is namely an orthogonal o projector projector from L2 (G, C) to B 2 (G, C). Similarly, the Szeg¨ SG (z, w)f (z)dS [Sf ](z) = ∂G
100
Chapter 2. Clifford-analytic Eisenstein series for translation groups
provides an orthogonal projection from L2 (∂G, C) into H2 (∂G, C). In contrast to this the Cauchy transform induces only an orthogonal projection if and only if G is the unit disk. In the 1970s R. Delanghe et al. started to study Hilbert spaces with continuous point evaluation that consist of Clifford-valued functions that are monogenic and square-integrable over a domain G ⊂ Ak+1 . These Hilbert spaces provide a canonical generalization of the classical Bergman spaces to the context of Clifford analysis. Similarly, Hardy spaces of Clifford-valued monogenic functions with boundary values in L2 (∂G, Cl0k (IR)) were introduced. See [35, 12] and the monograph [13]. The general theory of function spaces of monogenic functions being square-integrable over a domain in the Euclidean space or over its boundary respectively, has been extended by a number of authors including for example D. Constales (cf. e.g. [21, 22], M. Shapiro and N. Vasilevski (see for instance [150, 151, 163]), J. Cnops (cf. e.g., [29]), D. Calderbank [18] among many others. Meanwhile its study has become one of the main topics of Clifford analysis. Similarly, as in the complex case, one is interested in having closed and explicit formulas for the reproducing kernel functions for some special domains. Explicit formulas for the Bergman and the Szeg¨ o kernel for the unit ball can be found for instance in [13]. However, in contrast to the planar case, there is no direct higher dimensional analogue of the Riemann mapping theorem in spaces of real dimension n ≥ 3, due to the fact that the set of conformal mappings in the space is restricted to the set of M¨ obius transformations (Theorem 1.9). Only if G and G∗ are domains that can be mapped by a M¨ obius transformation onto each other, then there is an isometry between the associated Hardy spaces H2 (∂G, Cl0k (IR)) and H2 (∂G∗ , Cl0k (IR)) (cf. e.g., [21, 29, 83]). Based on the knowledge of the Szeg¨ o kernel of the unit ball, J. Cnops derived in [29] an explicit formula for the Szeg¨ o kernel of the upper halfspace of IRk . In contrast to the classical complex case there is no isometry between the corresponding Bergman spaces. In general there is no analogue of (2.98) in the higher dimensional case, either. However, within the special framework G = H + (IRk ) a particular analogue of (2.98) can be established in the form BH (x, w) = −2
∂ SH (x, w), ∂wk
(2.99)
as proved [29]. With this formula J. Cnops managed to establish a closed formula for the Bergman kernel of the upper half-space. The determination of explicit formulas for these kernel functions in the higher dimensional case is thus even more difficult than in the planar case. This explains why not many formulas are developed so far. D. Calderbank managed in 1996 to work out closed representation formulas for the kernel functions for annular domains, see [18]. J. Peetre and P. Sj¨ olin derived in [126] a non-explicit formula
2.8. Bergman kernels of rectangular domains
101
for the Szeg¨ o kernel for the special strip domain S0 = {z ∈ Ak+1 | 0 < x0 < d} in terms of plane wave integrals. In view of the complex case it is natural to ask whether perhaps the higher dimensional analogues of the translative Eisenstein series give rise to closed and explicit formulas for the kernel function associated to higher dimensional strip and rectangular domains. In the papers [25] and [24], both jointly written with D. Constales, this conjecture has been affirmed. It turned out that one can indeed express the Bergman reproducing kernel function of a rectangular domain that is (p) bounded in p-directions in terms of a finite sum of the Eisenstein series τ (j) . In this section we summarize the main results from [25]. In the following section we will summarize the first main result from [24] where we showed that the Szeg¨ o kernel function of the strip domain S0 is explictly given by the one-fold periodic monogenic cosecant function csc1,1 (z). We start by introducing the basic setting and some special notation. Without loss of generality we consider rectangular domains in Ak+1 that are described by Rk1 ,k2 := {z ∈ Ak+1 | 0 < xj < dj , j = 0, . . . , k1 − 1, xj > 0, j = k1 , . . . , k2 − 1} where its first k1 sides (0 ≤ k1 ≤ k + 1) are assumed to be each of finite length d0 , . . . , dk1 −1 , and its sides in the following k2 − k1 dimensions (k1 ≤ k2 ≤ k + 1) are semi-infinite and its sides in the remaining directionsare infinite in both direck tions. Let K2 := {0, 1, . . . , k2 − 1}. Suppose that w = j=0 wj ej is an arbitrary paravector. Then one associates to any subset A ⊆ K2 the paravector wA whose components are defined by (wA )j = (−1)j∈A wj where (−1)j∈A = 1 if j ∈ A and (−1)j∈A = −1 if j ∈ A. Next let us abbreviate the Cauchy kernel q0 (w − z) by K(z, w) and let us use the notation K (z, wA ) = {K(z, wA )}Dw which shall be understood in the distributional sense. Let us first treat the case k1 < k + 1. To meet our ends we first need some preparative lemmas: Lemma 2.26. For all A ⊆ K2 , (wA )Dw
w − z ( w − z 2 )Dw A
A
2
= k + 1 − 2|A|, = w − z , = 2(w − z A ),
(2.100) f or all z, w ∈ Ak+1 , f or all z, w ∈ Ak+1 .
A 2
(2.101) (2.102)
With this lemma one can show next (z, w) satisfy Lemma 2.27. Let A ⊆ K2 . Then the distributions KA
K∅ (z, w) = δ(w − z), KA (z, w) =
(2.103)
1 (k + 1 − 2|A|) w − z − (k + 1)(w − z)(w − z ) , (2.104) Ak+1
wA − z k+3 A
2
A
A
102
Chapter 2. Clifford-analytic Eisenstein series for translation groups
where we assume z = wA in (2.104). Furthermore KA (z, w) = O
1
z k+1
,
z → +∞,
(2.105)
uniformly for w in a given compact set and (z, w). KA (w, z) = KA
(2.106)
The expression K (z, wA ) is left monogenic in the first argument z and right conjugate monogenic in w, except at z = wA . In what follows let us use the abbreviation K π (z, w) :=
1 (k1 ) (w − z) Ak+1 0
for the periodization of the Cauchy kernel with respect to the rectangular lattice 2Zd0 + · · · + 2Zdk1 −1 . The Theodorescu transform associated to K π (z, wA ) is then given by the integral ((K π (z, wA ))Dw )f (w) dw0 · · · dwk ,
TA f (z) =
(2.107)
w∈R
where Dw is understood again in the distributional sense. Since the periodization of δ(z − w) has only one point z = w of its support belonging to R, one obtains T∅ f (z) = f (z)
(2.108)
as a consequence of (2.103). With these tools in hand one can prove the following important proposition: Proposition 2.28. Let 0 ≤ k1 < k. Let z ∈ R := Rk1 ,k2 and let U be an open neighborhood of R, the closure of R. If f : U → Cl0k (IR) is left monogenic in U , then (−1)|A| TA f (z) = 0. (2.109) A⊆K2
Proof. From (2.107) we have A⊆K2
(−1)|A| TA f (z) =
,
(−1)|A| (K π (z, wA ))Dw f (w) dw0 · · · dwk .
w∈R A⊆K2
Next one applies Stokes’ theorem in the distributional sense on this expression.
2.8. Bergman kernels of rectangular domains
103
This leads to
(−1)|A| TA f (z) =
k 1 −1
,
j=0 w∈R,wj =dj
A⊆K2
−
k 1 −1
,
(−1)|A| K π (z, wA ) dσw f (w)
A⊆K2
j=0 w∈R,wj =0 A⊆K2
−
k 2 −1
,
(−1)|A| K π (z, wA ) dσw f (w) (−1)|A| K π (z, wA ) dσw f (w)
j=k1 w∈R,wj =0 A⊆K2
,
−
w∈R
(−1)|A| K π (z, wA ) Df (w) dw0 · · · dwk .
A⊆K2 =0 =0
(2.110) When wj = 0, we have trivially that wA = wA∆{j} . When wj = dj , we have wA = wA∆{j} ± 2dj . Since K π (z + 2dj ej , w) = K π (z, w) one can consequently replace K π (z, wA ) by K π (z, wA∆{j} ) in all the boundary integrals that appear in (2.110). Next, let B = A∆{j}. Summing over all A ⊆ K2 and all j = 0, . . . , k2 is equivalent to summing over all B ⊆ K2 and all j = 0, . . . , k2 , replacing A by B∆{j}. However, we have (−1)|A| = −(−1)|B| as a consequence of |A| = |B| ± 1, so that by applying Stokes’ theorem in the other direction, the right-hand side of (2.110) simplifies precisely to − B⊆K2 (−1)|B| TB f (z),
so that the assertion follows. This proposition gives rise to the following result for the cases k1 < k + 1.
Proposition 2.29. Let k1 < k + 1. Then the Bergman kernel of R := Rk1 ,k2 has the form B(z, w) =
(n0 ,...,nk1 −1 )∈Zk1
A⊆K2 A=∅
(−1)|A|+1 KA (z + 2n0 d0 e0 + · · · + 2nk1 −1 dk1 −1 ek1 −1 , w) . (2.111)
Proof. The square integrability over R may be concluded by (2.104). None of the singularities z = wA , (A = ∅), lie in R, and the functions decrease fast enough to be square-integrable over the unbounded dimensions of R. The monogenicity in z follows simply by Weierstraß’ convergence theorem. Property (2.106) implies that (2.111) has the required conjugate symmetry in z and w. We are thus left to verify the reproducing property of B(z, w). In the case where f is monogenic in a neighborhood of R, the reproducing property follows (z, w), at once from (2.109). This follows immediately from the definition of KA
104
Chapter 2. Clifford-analytic Eisenstein series for translation groups
when the term A = ∅ is separated from the others and (2.108) is used to simplify it. Let us now suppose that f is more generally an arbitrary element from L2 (R, Cl0k (IR)). Then consider the functions
ε fε (z) = f (1 − ε)z + (d0 e0 + · · · + dk1 −1 ek1 −1 ) + ε(ek1 + · · · + ek2 −1 ) , ε > 0. 2 We observe that the function f can be approximated as closely as desired in the space L2 (R, Cl0k (IR))) by fε , ε → 0+ , which is a left monogenic function in a neighborhood of R and hence reproduced by the expression in (2.111). Taking the limit as ε → 0+ proves finally the reproducing property of the more general function f . The proposition follows from the uniqueness of the Bergman kernel. Finally let us turn to the case k1 = k + 1. This case is more difficult to handle, since there is no meromorphic (k + 1)-fold periodic function that has only one point singularity of order k in a period cell. Therefore we cannot define a useful analogue of the auxiliary function K π in the setting k1 = k + 1. For this reason a more technical argument is required. The crucial point for the following argumentation is that (2.105) can be strengthened to
1 , z → ∞. (2.112) (−1)|A|+1 KA (z, w) + O
z k+2 A⊆K2 ,A=∅
For the detailed proof, see Lemma 2 from [25]. From (2.112) follows that also
B(z, w) = (−1)|A|+1 KA (z + 2n0 d0 e0 + · · · + 2nk dk ek , w) (n0 ,...,nk−1 )∈Zk
A⊆K2 A=∅
(2.113) is a normally convergent series and one can prove: Proposition 2.30. In the case k1 = k + 1, B(z, w) as defined in (2.113), is the Bergman reproducing kernel of the rectangular domain R := Rk+1,0 . Proof. The qualitative idea of the proof is rather similar to that of Proposition 2.29. However, one has to take more care in the part of the proof concerning the reproducing property. To this end consider for the following partial sums of the series (2.113):
T6N (z, w) = (−1)|A| KA (z + 2n0 d0 e0 + . . . + 2nk dk ek , w) −N
(2.114) where N ∈ IN. Next, assume first again that f is a function that is monogenic in a neighborhood of R and consider , T6N (z, w)f (w) dw0 · · · dwk w∈R
=
−N
(−1)|A|
,
((K(z + (
w∈R
k
j=0
2nj dj ej ), wA ))Dw )f (w)dw0 · · · dwk .
2.8. Bergman kernels of rectangular domains
105
If one applies Stokes’ theorem as in Proposition 2.29, then one obtains k k
, (−1)|A| K(z + ( 2nj dj ej ), wA ) dσw f (w) j=0 w∈R,wj =dj j=0 −N
The treatment of every j is similar. Therefore, restrict the conditions to j = 0. In the integral with w0 = 0 one may change w0 to −w0 in the integrand. This amounts to replacing wA by wA∆{0} . In the integral with w0 = d0 , a replacement of w0 by −w0 amounts to subtracting 2d0 from wA if 0 ∈ A, so that , K(z + 2n0 d0 e0 + · · · + 2nk dk ek , wA ) dσw f (w) −N
w∈R w0 =d0
,
−N
w∈R w0 =d0
=
K(z − 2N d0 e0 + · · · + 2nk dk ek , wA∆{0} )
−K(z + 2(N − 1)d0 e0 + · · · + 2nk dk ek , wA∆{0} ) dσw f (w) , K(z + 2n0 d0 e0 + · · · + 2nk dk ek , wA∆{0} ) dσw f (w);
+
−N
w∈R w0 =d0
whereas if 0 ∈ A it amounts to adding 2d0 to wA , then , K(z + 2n0 d0 e0 + · · · + 2nk dk ek , wA ) dσw f (w) −N
w∈R w0 =d0
,
−N
w∈R w0 =d0
=
− K(z − 2(N − 1)d0 e0 + · · · + 2nk dk ek , wA∆{0} )
+K(z + 2N d0 e0 + · · · + 2nk dk ek , wA∆{0} ) dσw f (w) , K(z + 2n0 d0 e0 + · · · + 2nk dk ek , wA∆{0} ) dσw f (w).
+
−N
w∈R w0 =d0
As a consequence, one obtains
(−1)|A|
−N
+
−N
=
,
−N
w∈R w0 =d0
,
K(z + (
2nj dj ej ), wA ) dσw f (w)
j=0
w∈R w0 =d0
(−1)|A∆{0}|
k
,
K(z + (
w∈R w0 =d0
k
2nj dj ej ), wA∆{0} ) dσw f (w)
j=0
ZN (z, w) dσw f (w), (2.115)
106
Chapter 2. Clifford-analytic Eisenstein series for translation groups
where ZN (z, w) stands as an abbreviation for the positive and negative contributions at both “ends” of the sum, i.e., n0 = N and n0 = N − 1 at the upper, n0 = −N and n0 = −(N − 1) at the lower end. Since there are as many subsets A ⊂ K2 that contain the element 0 as subsets of K2 that do not contain the element 0, one may conclude that the sum of the coefficients associated to K(z + 2N d0 + · · · ) and to K(z + 2(N − 1)d0 + · · · ) (i.e., the upper end) vanishes. The asymptotic estimate K(z + 2N d0 + · · · ) = O(1/N k ) (uniformly in z, w ∈ R and independently of n1 ,. . ., nk ) can hence be strengthened to O(1/N k+1 ) for the sum at the upper end. A similar argumentation can be established concerning the lower end, so that one finally obtains that
1 . ZN (z, w) = O N k+1 The summation over the k indices n1 , . . . , nk produces a coefficient with the asymptotic behavior O(1/N ) in the integral, tending to zero for N → +∞. The function f is bounded since it is monogenic in a neighborhood of R. It follows that the right-hand side of (2.115) tends to zero for N → +∞. Both terms on the left-hand side of (2.115) coincide due to the fact that summing over the set A ⊆ K2 gives the same result as summing over the set A∆{0}, for all A ⊆ K2 . Summing the left-hand side of (2.115) for all j = 1, 2, . . . , k (instead of performing this procedure only for j = 0) leads finally to , T6N (z, w)f (w) dw0 · · · dwk = 0. lim (2.116) N →∞ w∈D
The rest of the proof can be done in analogy to the proof of Proposition 2.29, however, relying on (2.116) instead of (2.109). From Proposition 2.29 and Proposition 2.30 one can deduce now the main result from [25]: Theorem 2.31. Let k1 , k2 be arbitrary integers with 1 ≤ k1 ≤ k + 1 and k1 ≤ k2 ≤ k + 1. Then the Bergman kernel of the associated rectangular domain Rk1 ,k2 is given by BR (z, w) =
1 Ak+1
k
A⊆K2 ,A=∅ j=0
1 (−1)|A|+1 (−1)j∈A τ (j) (wA − z)ej .
(k )
(2.117)
Proof. In the cases where k1 < k + 1 the two summations in (2.111) can be exchanged due to the absolute convergence of each series KA (z + 2n0 d0 e0 + · · · + 2nk1 −1 dk1 −1 ek1 −1 , w) (n0 ,...,nk1 −1 )∈Zk1
and (2.117) follows readily. In the case k1 = k + 1 such a rearrangement is not allowed. However, since K∅ (z, w) is zero as a function, one may equivalently consider
2.8. Bergman kernels of rectangular domains
107
the sum over all A ⊆ K2 in (2.117) and we may equally well consider
(−1)|A|+1 (−1)j∈A Kτ (j) ( z + 2n0 d0 e0 + · · · (n0 ,...,nk1 −1 )∈Zk1
+2nk1 −1 dk1 −1 ek1 −1 , wA )ej .
A⊆K2
The relations,
and
(−1)|A|+1 =
k
A⊆K2 ,j∈A
p=0
k
(−1)|A|+1 =
A⊆K2 ,j∈A
(−1)p+1
(−1)p+1
p=0
k p
k p
= −(1 + (−1))k = 0
= −(1 + (−1))k = 0
lead to k A⊆K2 j=0
(−1)|A|+1 (−1)j∈A Kτ (j) (2n0 d0 e0 + · · · + 2nk1 −1 dk1 −1 ek1 −1 , 0) = 0.
Subtracting this result from the general term in (2.113) (except for (n0 , n1 , . . . , nk ) (k1 ) = 0) the definition of τ (j) is recovered so that (2.117) follows, except for the presence of an A = ∅ term. This term however is seen to vanish identically as a function. (p)
The fact that the functions τ (j) (wA − z) are reproducing kernel functions of Hilbert spaces allows us to derive some interesting estimates on them, just as a consequence of Cauchy–Schwarz inequality. In particular one gets an upper bound estimate of the first partial derivative of the simply-periodic cotangent from above in terms of the classical real-analytic Eisenstein series [41]. Proposition 2.32. Assume that z, w ∈ Ak+1 so that 2n < Sc(z + w) < 2(n + 1) for some n ∈ Z. Then ∂ (1) k 7 (2.118) ∂x0 0 (z + w, 2Z) ≤ 2k+1 εk+1 (x0 )εk+1 (w0 ) where εk (x) = (x + m)−k for x ∈ IR\Z and k ≥ 2. m∈Z
Proof. From Theorem 2.31 it follows readily that the Bergman kernel function of the rectangular strip S0 := {z ∈ Ak+1 | 0 < x0 < 2} can be rewritten in the form B(z, w)S0 = −
∂ (1) 2 (z + w, 2Z). Ak+1 ∂z0 0
The Cauchy–Schwarz inequality tells us that
Φ(B(·, z), B(·, w)) 2 ≤ Φ(B(·, z), B(·, z))Φ(B(·, w), B(·, w)),
(2.119)
108
Chapter 2. Clifford-analytic Eisenstein series for translation groups
where Φ(·, ·) is the scalar product defined on the Bergman space B2 (S0 , Cl0k (IR)), Φ(f, g) =
f (z)g(z)dV. S0
From the reproducing property of the Bergman kernel one can readily deduce that Φ(BS0 (·, z), BS0 (·, w)) = BS0 (z, w), and therefore
BS0 (z, w) 2 ≤ BS0 (z, z)BS0 (w, w).
(2.120)
Next we compute Ak+1 BS0 (z, z)
=
2
(k + 1)(2x0 + 2n)(2x0 + 2n)
2x0 + 2n k+3
n∈Z
=
k 1 2k (x0 + n)k+1
=
k εk+1 (x0 ). 2k
−
1
2x0 + 2n k+1
n∈Z
Applying finally (2.120) leads to the assertion.
2.9
Szeg¨ o kernels of strip domains
In the classical planar case the Szeg¨o kernel of a strip domain having without loss of generality the form 0 < x < π is explicitly S(z, w) =
1 csc(z + ω). 2π
In [24] it has been proved that in the higher dimensional case, one can express the Szeg¨o kernel of a strip domain being bounded in one direction in terms of the monogenic one-fold generalization of the cosecant function in the sense of Definition 2.3. Theorem 2.33 (cf. [24]). The Szeg¨ o kernel for the strip domain S0 := {z ∈ Ak+1 | 0 < x0 < d} is given explicitly by SS0 (z, w) =
1 Ak+1
csc1,1 (z + w, 2dZ).
(2.121)
Proof. Let us first assume again that f is monogenic in a neighborhood of S0 . Let
2.9. Szeg¨o kernels of strip domains
109
us write again for simplicity K(z, w) = q0 (w − z). We obtain * ) ∞ (−1)n+1 K(z + 2dn, −w) f (w) dS w∈∂S0
n=−∞
)
= w∈∂S0 ,w0 =0
w∈∂S0 ,w0 =d
* n+1
(−1)
n=−∞
)
+
∞
∞
K(z + 2dn, −w) f (w) dS *
n+1
(−1)
K(z + 2dn, −w) f (w) dS.
n=−∞
Note that w = −w in the first integral and w = 2d − w in the second one. In view of these relations one can reformulate the integral as a consequence of the (2d)-antiperiodicity as follows: * ) ∞ n+1 (−1) K(z + 2dn, w) f (w) dS w∈∂S0 ,w0 =0
n=−∞
)
∞
− w∈∂S0 ,w0 =d
* (−1)n+1 K(z + 2dn, w) f (w) dS
n=−∞
in which form the surface integral * ) ∞ − (−1)n+1 K(z + 2dn, w) dσw f (w) w∈∂S0
n=−∞
can be recognized. If z ∈ S0 , then the only term in which a singularity appears is the term that is associated with n = 0. As a consequence of Cauchy’s integral formula the original integral simplifies in fact to f (z). For a function f that is monogenic in a neighborhood of S0 the reproducing property is hereby proven. To show the assertion for more general f from L2 (∂S0 ), we stretch and translate the coordinates slightly by defining for instance
ε fε (z) = f (1 − ε)z + 2 for arbitrarily small ε > 0. We observe that f can be approximated as dense as desired in the space L2 (∂S0 ) by the function fε by letting ε tend to 0. The function fε however, is a left monogenic function in a neighborhood of S0 . Therefore, 1 fε (z) = csc1,1 (z + w, 2dZ)fε (w)dSw . Ak+1 ∂S0
Considering ε → 0 yields the reproduction property of f . Since the Szeg¨o kernel is uniquely defined, the theorem is hereby proven.
110
Chapter 2. Clifford-analytic Eisenstein series for translation groups
Remark. Theorem 2.33 provides a useful complement to [126] in which a non-explicit representation formula for the Szeg¨ o kernel of S0 in terms of plane wave integrals has been given. Furthermore, it provides a further justification for defining higher dimensional generalizations of the cosecant in the sense of Definition 2.3. As mentioned in the previous section there is no explicit relation between the Bergman and the Szeg¨ o kernel of an arbitrary domain in the higher dimensional case in general. However, for some special domains, these kernels are explicitly related as for instance in the half-space case in terms of the formula (2.99). In [24] explicit relations in the case of a strip domain were established. To proceed in this direction, recall that the Bergman kernel of S0 is explictly BS0 (z, w) = −
∂ (cot1,1 (z + w, 2dZ)) , Ak+1 ∂w0 2
according to Theorem 2.31. Next recall the definition of the one-fold periodic monogenic cosecant function (Definition 2.3) csc(1) (z, 2dZ) =
1 cot(1) (z/2, 2dZ) − cot(1) (z, 2dZ) 2k−1
and apply on the expression cot(1) (z, 2dZ) the duplication formula for the cotangent from Section 2.5 (putting n = 2), i.e., 2k cot(1) (2z, 2dZ) = cot(1) (z, 2dZ) + cot(1) (z + d, 2dZ). This leads to csc1,1 (z, 2dZ) =
1 1 z z+d , 2dZ). cot1,1 ( ) − k cot1,1 ( k 2 2 2 2
In view of (2.121) we obtain 2k SS0 (z, w) =
1 Ak+1
cot1,1 (
1 z+w z+w , 2dZ) − , 2dZ). cot1,1 ( 2 Ak+1 2
Performing partial differentation with respect to w0 leads finally to the following explicit relationship between the Bergman kernel and the Szeg¨ o kernel S0 :
z w z+d w+d ∂ , − BS0 , BS0 (SS0 (z, w)) . (2.122) = −2k+2 2 2 2 2 ∂w0 Formula (2.122) provides thus a nice application of the multiplication formulas described earlier in Section 2.5.
2.10
Boundary value problems on conformally flat cylinders and tori
The results outlined in this section were obtained in collaboration with John Ryan (University of Arkansas) and stem from our joint paper [96] to which we refer the interested reader for more details.
2.10. Boundary value problems on conformally flat cylinders and tori
111
In classical complex analysis, automorphic functions give rise to extend function theory of one complex variable from the complex plane to the more general setting of Riemann surfaces. Due to the conformal invariance of the generalized Cauchy–Riemann operator (Theorem 1.28) it is suggestive to regard conformally flat manifolds as natural generalizations of complex Riemann surfaces in the context of Clifford analysis. Conformally flat manifolds are exactly those manifolds which have an atlas whose charts are M¨ obius transformations. On the other hand they are characterized to be those Riemannian manifolds that have the property that their Weyl tensor vanishes (see e.g., [143]). The general study of these manifolds has become one of the mainstream research areas of mathematics. They have been studied under a number of different aspects independently from Clifford analysis, as for instance in [20, 148] among others. From the viewpoint of solving boundary value problems it is interesting to develop Clifford and harmonic analysis techniques on these manifolds. In [162, 113] for instance such techniques were developed for spheres and hyperbolae, which are two very simple examples of conformally flat manifolds. In [145] similar techniques are developed on manifolds that can be constructed by gluing a finite number of n-spheres together. As mentioned in [148], more examples of conformally flat manifolds can be constructed by factoring out a simply connected domain G ⊂ Ak+1 by a Kleinian group G of the M¨ obius group that has the property that G acts discontinuously on G. This gives rise to construction of particular conformally flat cylinders of infinite extension and conformally flat tori by factoring out Ak+1 by a translation group Tp . In the cases p < k + 1 this construction provides conformally flat cylinders which shall be denoted by Cp in what follows. In the remaining case p = k + 1, one obtains a conformally flat k + 1-torus Tk+1 . Since the universal covering of these manifolds is the Euclidean space Ak+1 , there is a well-defined projection map Πp : Ak+1 → Cp (p < k + 1) and Πk+1 : Ak+1 → Tk+1 . If U ⊂ Ak+1 is p-fold periodic with respect to a lattice Ωp ⊂ Ak+1 , then U = Πp (U ) is a subset of Cp , so that each z ∈ U induces an element z = Πp (z) ∈ Cp . This map induces further a Cauchy–Riemann operator on Cp , simply viz D = Πp (D). Functions [s] that are in the kernel of D are simply called cylindrical (toroidal) s-monogenic. Provided f : U → Cl0k (IR) is p-fold periodic with respect to Ωp , then Πp induces a well-defined function f : U → Cl0k (IR) defined by f (Π−1 p (z )) for each z ∈ U . In the cases where p ≤ k the translation invariant monogenic Eisenstein series (p) (p) (p) 0,1 (z − y) induce global Cauchy kernel functions viz 0,s (z − y ) =: 0,s (z , y ) on the associated flat cylinders Cp = Ak+1 /Tp , as a consequence of being the p-fold (1) periodic periodizations of the Euclidean Cauchy kernel function q0 (z − y). They give rise to the following global Cauchy integral formula on the cylinders Cp with p ≤ k, proved in [96]:
112
Chapter 2. Clifford-analytic Eisenstein series for translation groups
Theorem 2.34. Let G ⊂ Cp be a domain and let S ⊂ G be a compact differentiable k-dimensional oriented manifold with boundary. Let g : G → Cl0,k (IR) be ◦
cylindrical left monogenic, i.e., D g (z ) = 0 for all z ∈ G . Then for each y ∈S one has 1 (p) 0,1 (z , y )dσ(z) g (z ) (2.123) g (y ) = Ak+1 ∂S
where dσ (z ) is the oriented surface measure on ∂S . More generally, we obtain Theorem 2.35. Let G ⊂ Cp be a domain and let Γ ⊂ G be a k − 1 dimensional nullhomologous cycle in G’. Let g : G → Cl0,k (IR) be cylindrical left monogenic for all z ∈ G . Then for each y ∈ G \Γ , 1 (p) 0,1 (z , y )dσ(z) g (z ) (2.124) wΓ (y )g (y ) = Ak+1 Γ
where wΓ (y ) denotes the winding number of Γ at y and dσ (z ) the oriented surface measure on Γ . For the cases where p ≤ k + 1 − s the translation invariant s-monogenic (p) (p) Eisenstein series 0,s (z − y) induce global Green kernel functions viz 0,s (z , y ) on the associated flat cylinders Cp = Ak+1 /Tp . For details, see [96]. For instance one obtains for Clifford-valued harmonic functions on the cylinders Cp with p ≤ k − 1 the following Green integral formula: Theorem 2.36. Let G ⊂ Cp be a domain and let S ⊂ G be a compact orientable k-dimensional differentiable manifold. Let g : G → Cl0,k (IR) be cylindrical har◦
monic, i.e., ∆ g (z ) = 0 for all z ∈ G . Then for each y ∈S one has 1 (p) 0,1 (z , y )dσ(z) g (z ) g (y ) = Ak+1 ∂S 1−k (p) + 0,2 (z , y )dσ(z) g (z ). Ak+1 ∂S
Similar boundary value problems within the context of the torus, i.e., p = k + 1 are more difficult to treat. As a consequence of the second Liouville theorem it is not possible to construct a (k + 1)-fold periodic monogenic function with the property of having only one isolated pole of order k in each period cell. However, (k+1) the 1-monogenic translative Eisenstein series 0;a,b which has two poles of order k in each period cell gives rise to a local Cauchy integral formula on the k + 1-torus Tk+1 . More precisely, following [96]:
2.11. Order theory and argument principles on cylinders and tori
113
Theorem 2.37. Let G ⊂ Tk+1 be a domain and S ⊂ G be a compact orientable k-dimensional differentiable manifold. Let g : G → Cl0,k (IR) be a toroidal left monogenic function. Furthermore suppose that b = Πk+1 (b) does not belong to the closure of Sa := Πk+1 (S − a). Then for each y ∈ int(S ), 1 (k+1) (z , ya )dσ (z )G (z ) (2.125) g (y ) = Ak+1 ∂Sa 0;a,b where ya = Πk+1 (y − a) and G(z) = g(z + a). (p)
More generally for s ≥ 2, the s-monogenic series 0,s;a,b with p = k + 2 − s provide under the same conditions as in Theorem 2.37 a local Green formula for s-monogenic functions on the cylinders Ck+2−s . Remarks. These integral formulas provide powerful tools for a detailed study of Hardy spaces that arise in this context, including explicit Plemelj projection formulas and Kerzman–Stein formulas. For more details, we refer the interested reader to [96]. One can easily prove similar results within the more general framework of working in complex Minkowski spaces of signature (p, q), when putting additional restrictions on the extension and the location of the period lattice, so that the associated Eisenstein series converge.
2.11
Order theory and argument principles on cylinders and tori (p)
The monogenic translative Eisenstein series 0,1 lead in the particular case s = 1 to global Cauchy integral formulas on the cylinders Cp for all 1 ≤ p ≤ k, and (k+1) 0,1;a,b lead to a local Cauchy integral formula on the torus Tk+1 . For simplicity we shall also write Ck+1 for Tk+1 in this section. In [93] we have shown that these function series can furthermore be used to set up explicit argument principles and generalized versions of Rouch´e’s theorem for monogenic functions on conformally flat cylinders and tori. To proceed in this direction we first introduce in analogy to Definition 1.31 the notion of the order of an isolated a-point of a cylindrical or toroidal monogenic function. Hereinafter B (c , ε) stands for the projection of the ball B(c, ε) to the cylinder Cp or torus Tk+1 , respectively. Definition 2.38. Let p ∈ {1, . . . , k + 1}. Suppose that G ⊆ Cp is an open set and that f : G → Cp is cylindrical (toroidal) left monogenic and that c ∈ G is an isolated zero point on Cp so that there is an ε > 0 such that B (c , ε) ⊂ G and f |B (c ,ε)\{c } = 0. Then the integer 1 (p) ord(f ; c ) := 0 (z )dσ (z ) (2.126) Ak+1 f (∂B (c ,ε))
114
Chapter 2. Clifford-analytic Eisenstein series for translation groups
is called the order of f at c . In the case p = k + 1 one takes the function 0;c,b and one has to take care that ε is furthermore chosen so small that b ∈ B (c , ε). (k+1)
In order to show that ord (f ; c ) is an integer, we use now the generalized Cauchy integral formulas on cylinders and tori. The rest of the argumentation is then similar to the Euclidean case treated earlier in Section 1.5. In the cases where p < k + 1, one takes analogously to Section 1.5 g ≡ 1 and replaces y by f (c ) and ∂S by f (∂B (c , ε)). This leads consequently to 1 (p) 0 (z , f (c ))dσ (z ) = wf (∂B (c ,ε)) (f (c )).
Ak+1 f (∂B (c,ε))
=0
=0
The value ord(f , c ) = wf (∂B (c ,ε)) (0) is thus the integer counting how often the image of B under f around the isolated zero wraps around zero. For the case p = k + 1 one uses the Cauchy integral formula for the torus: put also g ≡ 1, replace yc by c and ∂Sc by f (∂B (c , ε)). In this context we need additionally to choose ε so small that b ∈ B (c , ε). Similarly to the Euclidean case, one can replace the projection of the ball in Definition 2.38 by a nullhomologous (k − 1)-dimensional cycle parameterizing a (k − 1)-dimensional surface of a k-dimensional simply connected domain inside of G ⊂ Cp which contains the isolated zero c in its interior and no further zeroes neither in its interior nor on its boundary. Let us now assume that G is a domain. Since f : G → Cp is continuously differentiable, we can then again apply the general transformation rule for differential forms from [165], so that we get for the oriented surface differential on the cylinder or on the torus, respectively, dσ (f (z )) = [(Jf )∗ (z )] ∗ [dσ (z )]
(2.127)
where Jf denotes the Jacobi matrix of f (z ) and (Jf )∗ its adjoint. The symbol ∗ between the brackets indicates again the matrix-vector multiplication. This allows us to rewrite (2.126) in the following form. Theorem 2.39. Let G ⊆ Cp be a domain. Let f : G → Cp be cylindrical (toroidal) monogenic in G and c ∈ G an isolated point. Then, under the same conditions for B (c , ε) ⊂ G as in Definition 2.38, 1 (p) ord(f ; c ) = (f (z ))[(Jf )∗ (z )] ∗ [dσ (z )]. (2.128) Ak+1 ∂B (c ,ε) 0 (k+1)
For the case p = k + 1 one takes again 0,1;c,b . By means of formula (2.128) we can then establish in a similar way as in the Euclidean case treated earlier in Chapter 1.5 (Theorem 1.34) an explicit argument principle on conformally flat cylinders and tori. In this context notice that
2.11. Order theory and argument principles on cylinders and tori
115
whenever Γ is a (k − 1)-dimensional cycle on Cp which has at most a countable number of points z at which det Jf (z ) = 0, then we may rewrite the expression [(Jf )(z)] ∗ [dσ(z)] in the form det Jf (z)[((Jf )−1 ) (z )] ∗ [dσ (z )] in analogy to (1.42). We also obtain a generalization of Rouch´e’s theorem within this context. To this end let us first introduce a norm on Cp as follows:
z Cp := min { z + ω }
z ∈ Cp .
ω∈Ωp
In terms of this norm we can show: Theorem 2.40. Let G ⊂ Cp be a domain and let Γ be a nullhomologous k − 1dimensional cycle parameterizing a k − 1-dimensional surface of a k-dimensional simply connected domain E ⊂ G . Suppose f , g : G → Cp are cylindrical (toroidal) monogenic functions having only a finite number of zeroes in E and no zeroes on the boundary ∂E . If
f (z ) − g (z ) Cp < f (z ) Cp then
c ∈E
ord(f ; c ) =
for all z ∈ ∂E ,
ord(g ; c ).
(2.129) (2.130)
c ∈E
The proof can be done in analogy to the proof of Theorem 1.35. One simply has to take instead of the Euclidean Cauchy kernel the monogenic translative (p) (k+1) Eisenstein series 0 for p < k + 1 and 0;c,b for the case p = k + 1.
Chapter 3
Clifford-analytic Modular Forms In the previous chapter we discussed s-monogenic Eisenstein series related to the translation groups T (Ωp ). This chapter is devoted to the construction of smonogenic automorphic forms for different, in particular, for larger arithmetic subgroups of the Vahlen group. What follows provides additionally a significant extension and a rectification of an attempt outlined by G. Z¨ oll in 1987 in [165]. In [165] on p.95 G. Z¨ oll was concerned to give one example of a 1-monogenic function f that shows an invariance behavior under a discrete group G < SV (Ak+1 ) that is different from the full translation group Tk+1 and that acts discontinuously on a domain G ⊂ Ak+1 . To this end he proposed to take a left monogenic function f˜ that is bounded on G and then to sum the expression (1) q0 (cz + d)f˜(M z )
(3.1)
over all M¨ obius transformations induced by all matrices from a group G that acts on G. However, as G. Z¨ oll pointed out, significant restrictions to the choices of the admissible groups for G have to be made in order to ensure that a summation of expressions of the form (3.1) over the whole group G converges. To meet this end he suggested to restrict consideration to those groups that have the property that the cardinality of the set a b M= ∈Gc=0 c d is only finite. This is a significant restriction, since all the hypercomplex generaliza(1) tions of the modular group are ruled out by this condition. Furthermore, since q0 is an odd function, one has to be very careful in the sense that such a construction does not only provide the zero function. Unfortunately, the only concrete example that G. Z¨ oll gave explicitly in his thesis using this attempt coincides identically with the zero function, as we have explained in [88].
118
Chapter 3. Clifford-analytic Modular Forms
In [88, 89] we managed to show that this approach provides actually some non-trivial examples of monogenic functions that satisfy (1)
f (z) = q0 (cz + d)f (M z )
for all M ∈ G.
However, the choice of G has to be restricted to quite small groups which are not significantly larger than the translation group, as for instance the transversion group Vp which is conjugated to the translation group Tp , as mentioned in Chapter 1.2. For more details, we refer the reader to our paper [89]. To get s-monogenic automorphic forms for larger groups, in particular for hypercomplex generalizations of the modular group this attempt has thus to be modified in several directions. In this chapter we present several construction theorems that meet these ends. In the first section of this chapter we construct Eisenstein series that show an invariance behavior under particular translation and rotation groups. In the second section we construct s-monogenic Clifford-valued modular forms on the upper halfspace that are quasi-invariant under the group action of the special hypercomplex modular groups Γp = Γp (IRk ) or their associated principal congruence subgroups of level n denoted by Γp [n] which were introduced at the end of Chapter 1.2. In the third section we consider generalizations on Cartesian products of several half-spaces which provide us with generalizations of Hilbert modular forms within the framework of Clifford analysis in several vector variables and their regularity concepts. All results will be formulated for real Euclidean spaces. In Section 3.4 we outline briefly how these results extend to the more general context of arbitrary real and complex Minkowski spaces IRp,q and Cp,q . Finally, in the last section of this chapter we outline some applications and possible perspectives for a further development in the future.
3.1
Rotation and translation invariant Eisenstein series
All the function series that were treated in Chapter 2 are totally invariant under the action of discrete translation groups in Ak+1 . In this chapter we shall see that a number of them provide us with useful building blocks for the construction of automorphic forms for larger discrete subgroups of Vahlen type groups. In this section we start with the construction of some examples of Cliffordvalued monogenic and polymonogenic functions that are invariant under a translation group and additionally also under a finite rotation group. In order to meet this end it is suggestive to sum a translation invariant s-monogenic Eisenstein (p) series m,s over a rotation group. However, we shall see that we have to put restrictions on the period lattice of the Eisenstein series and also on the generators of the rotation group. To proceed in this direction let us first suppose that u1 , . . . , ut are some IR-linear independent unit paravectors in Ak+1 satisfying uN j = 1 for a positive
3.1. Rotation and translation invariant Eisenstein series
119
integer N ∈ IN (j = 1, . . . , t). We write ∗ u∗ ut 0 0 u1 ,...,ut 1 . = R := R ,..., 0 u−1 0 u−1 t 1 More generally, let us also consider the following modified rotation type groups, χ u∗ χt u∗t 0 0 1 1 ˜ R := ,..., 0 u−1 0 u−1 t 1 where χ1 , . . . , χt are supposed to beelements from {−1, 1}. All matrices M ∈ R or a 0 ˜ are thus diagonal matrices M ∈R whose entries a and d are products 0 d of the paravectors uj and of their inverse, or, up to a minus sign products of uj and their inverse elements, respectively. is a Suppose that s ∈ {1, . . . , k}, p ∈ {1, . . . , k + 1} and that m ∈ INk+1 0 multi-index with |m| ≥ max{0, p − k − 1 + s}. Then the associated translative (p) Eisenstein series m,s converges normally in Ak+1 \Ωp . Under the same conditions, the series (Ω ,R) −1 d−1 (p) ; Ωp )a, (3.2) Rm,sp (z) := m,s (azd M ∈R
or more generally, ˜ (Ω ,R)
Rm,sp
(z) :=
−1 d−1 (p) ; Ωp )a m,s (azd
(3.3)
˜ M ∈R
will also converge normally to a well-defined s-monogenic function. Notice that this type of function series can at most have singularities in a discrete set of points. However, for general translation lattices and for a general choice of the elements u1 , . . . , ut this series does not turn out to have the desired group invariance. For getting the translation and the rotation invariance we require that the paravectors u1 , . . . , ut and the primitive periods ω1 , . . . , ωp satisfy all u∗i ωj ui ∈ Ωp for all i ∈ {1, . . . , t} and all j ∈ {1, . . . , p}. We thus need to restrict our study to lattices with hypercomplex multiplication in the more general sense of Definition 2.21. Two of the simplest types of examples of admissible translation and rotation groups are thus realized by 1. T (L0p ) := Te0 , . . . , Tep for 0 ≤ p ≤ k + 1 and e 0 A ˜= R A ⊆ P (1, . . . , k) . −1 0 eA ˜ u1 ,...,ut with χj = −1 for all j ∈ {1, . . . , t} where all elements 2. T (Ωp ) and R ω1 , . . . , ωp and u1 , . . . , ut are supposed to satisfy especially ωi , uj = 0, Sc(ωi ) = Sc(uj ) = 0
for all i ∈ {1, . . . , p}, j ∈ {1, . . . , t}.
120
Chapter 3. Clifford-analytic Modular Forms
Let us denote the group generated by the matrices described in the first example by M1 and those described in the second example by M2 . For clarity let us denote ˜ M and the subgroup of rotations in M2 the subgroup of rotations in M1 by R 1 ˜ by RM2 in all that follows now. We show Proposition 3.1. Suppose that INk+1 is a multi-index with m1 , . . . , mk ∈ 2IN0 and 0 that p, m and s are properly chosen such that |m| ≥ max{0, p − k − 1 + s}. Then the function series a 0 Mi −1 (p) ˜M , Rm,s (z) := d m,s (M z )a M= ∈R i ∈ {1, 2}, i 0 d ˜M M ∈R i
do not vanish identically and provide non-trivial examples of s-monogenic automorphic forms with respect to M1 or M2 , respectively, that transform a b Mi −1 Mi Rm,s (z) = d Rm,s (M z )a for all M = ∈ Mi . 0 d Proof. The function series associated to the first group M1 has the explicit form M1 Rm,s (z) = eA (p) m,s (eA zeA ; L0p )eA , A⊆P (1,...,t)
where L0p stands for the orthonormal lattice Z + Ze1 + · · · + Zep in the paravector (p) space. Let us first treat those cases where s is even. Then the series m,s are scalar-valued. Thus, we obtain in particular Mi t+1 (p) Rm,s (z)(x0 ) = eA eA (p) m,s (x0 ; L0p ). m,s (x0 ; L0p ) = 2 A⊆P (1,...,t)
In a sufficiently small neighborhood around the origin, we have (s) (p) m,s (x0 ) = qm (x0 ) + O(1). (s)
As pointed out at the end of Chapter 1.6, we know that qm (x0 ) ≡ 0 for indices with (s) m1 , . . . , mk ∈ 2IN0 . Since qm (x0 ) is IR-homogeneous of degree −(k + 1 + |m| − s) we have (s) (x0 ) = +∞. lim + qm x0 →0
Hence, Mi lim Rm,s (x0 ) = +∞,
x0 →0+
Mi and the non-triviality of the particular function series Rm,s is shown for the cases where s is even.
3.1. Rotation and translation invariant Eisenstein series
121
In the cases where s is odd we consider 1
(p) Mi Sc{Rm,s (x0 )} = eA ((p) m,s (x0 ; L0p ) + m,s (x0 ; L0p ))eA 2 A⊆P (1,...,t) t+1 = Sc{(p) Sc{(p) m,s (x0 ; L0p )} = 2 m,s (x0 ; L0p )}. A⊆P (1,...,t) (s)
For odd s we have that Sc{qm (x0 )} ≡ 0 if and only if m1 , . . . , mk ∈ 2IN0 which follows from Theorem 1.50, as also mentioned in Section 1.6. Therefore, we obtain in these cases Mi (x0 )} = +∞ lim Sc{Rm,s x0 →0+
which proves the non-triviality of these function series for the cases where s is Mi odd. As a consequence of this, the functions Rm,s (z) provide us with non-trivial examples of s-monogenic automorphic forms transforming in the form Mi Mi Rm,s (z) = eA Rm,s (eA zeA )eA
for all
A ⊆ P (1, . . . , t)
and Mi Mi Rm,s (z) = Rm,s (z + ω) for all ω ∈ L0p .
For the treatment of the second concrete example, we can apply exactly the same u 0 ˜ M have the form argumentation. Notice that all matrices from R 2 0 u−1 where u is a product built with the vectors u1 , . . . , ut and with their inverse elements. Under the given condition we obtain, similarly to the case treated before when s is even, Mi (x0 ) Rm,s
=
u(p) m,s (ux0 u)u
˜M M ∈R 2
=
t+1 (p) (p) m,s (x0 ). m,s (x0 ) = 2
˜M M ∈R 2
For odd s we obtain (p,R) Sc{Rm,s (x0 )}
=
˜M M ∈R 2
=
˜M M ∈R 2
=
˜M M ∈R 2
1 (p) (p) um,s (ux0 u)u + um,s (ux0 u) u 2 1 (p) (p) u(m,s (x0 ) + m,s (x0 ))u 2 t+1 Sc{(p) Sc{(p) m,s (x0 )} = 2 m,s (x0 )}.
122
Chapter 3. Clifford-analytic Modular Forms
Also these function series lead thus to non-trivial s-monogenic automorphic forms transforming in the following way under translations and rotations: u 0 (p,R) (p,R) ˜M , for all M = ∈R Rm,s (z) = uRm,s (uzu)u 2 0 u−1 (p,R) Rm,s (z)
(p,R) = Rm,s (z + ω)
for all ω ∈ Ωp .
This can be readily verified by a direct rearrangement argument.
Notice that m1 , m2 , . . . , mk ∈ 2IN0 is just a sufficient condition for nontriviality. Let us call these types of function series s-monogenic translation and rotation invariant Eisenstein series in general. The two particular types of examples treated concretely in the previous proposition are the most simple ones within the family of s-monogenic automorphic forms that show an invariance under a discrete translation and rotation group. The associated lattices are the most simple ones among those that have hypercomplex multiplication. The set of admissible discrete translation and rotation groups is much larger: Every period lattice that has a paravector multiplication of the special form described in Theorem 2.22 and that contains furthermore unit paravectors from Ak+1 with non-vanishing vector parts provides us with generators for admissible translation and rotation groups. While the special prototype of the translation invariant monogenic Eisenstein (p) series, namely τ (j) , give rise to Bergman kernel functions of strip and rectangular domains, special variants of the function series of type (3.2) lead to Bergman kernel functions of certain wedge shaped domains with one or several rectangular restrictions. This is outlined briefly at the end of Section 6 in the forthcoming paper [27], jointly written with D. Constales. To go a bit more into detail we first recall Theorem 3 from [27]: Theorem 3.2. The Bergman reproducing kernel of the space of the left monogenic square-integrable functions in the wedge shaped domain πek W = z ∈ Ak+1 | Sc(α1/2 zα1/2 ) < 0, α = exp( ) n has the form BW (z, w)
=
n−1 1 1 Dz Dw p −p k−1 (k − 1)Ak+1 j=0
zα + α w
−
(3.4)
n−1 1 1 Dw . Dz p −p k−1 (k − 1)Ak+1 j=1
zα − α w
For the detailed proof, see [27]. Next let us rewrite this formula in a slightly different way. First let us introduce exp( πek ) 0 n . Rn = 0 exp(− πenk )
3.1. Rotation and translation invariant Eisenstein series Now we can rewrite formula (3.4) in the form 1 (2) BW (z, w) = Dz q0 (uzu + w)Dw 2(k − 1)Ak+1 M ∈Rn 1 (2) − Dz q0 (uzu − w)Dw . 2(k − 1)Ak+1
123
(3.5)
M ∈Rn \{0}
From this formula it is easy to deduce a formula for the Bergman kernel associated to a wedge-shaped domain of the form ˜ p := {z ∈ W | 0 < xj < dj , j = 1, · · · , p} W ˜ p is nothing else than the where we assume here first that p < k. Notice that W 2 Cartesian product of a wedge-shaped domain in IR and a p − 1-dimensional rectangular domain from IRk−1 , while W is the Cartesian product of a two dimensional wedge-shaped domain with the whole space IRk−1 . The problem is hereby reduced to make the transfer from IRk−1 to a rectangular domain in IRk−1 that satisfies 0 < xj < dj for 1 = 2, . . . , p. However, as we have seen in Chapter 2.8 this can be achieved by simply performing reflections and periodizations over the double of the thickness dj in the directions ej for 1 = 2, . . . , p, respectively. One simply has to lift the formulas from Section 2.8 for the Bergman kernel function of a rectangular domain in IRk−1 that is bounded in p-directions, to a Cartesian product of that rectangular domain with the planar fractional wedge-shaped domain for which we have formula (3.5). We finally obtain, as outlined in [27]: Theorem 3.3. Let p < k and n ≥ 2 be integers. Then the Bergman reproducing ˜ p , Cl0k (IR)) has the form kernel of B 2 (W BW ˜ p (z, w) =
1 2(k − 1)Ak+1
1 − 2(k − 1)Ak+1
(−1)|A|+1 Dz
A⊂{1,...,p}
A⊂{1,...,p}
(−1)|A|+1 Dz
(p) 0,2 (uzu − wA ; Lp ) Dw
M ∈Rn
(p) 0,2 (uzu + wA ; Lp ) Dw .
M ∈Rn \diag(1,1)
Summarizing: For a number of rotation and translation groups we can construct non-trivial s-meromorphic Eisenstein series by simply summing the attached (p) translative s-monogenic Eisenstein series m,s over the rotation group and to applying some constant weight factors from the left and from the right in order to ensure the monogenicity and to avoid in particular that the series vanishes. Unfortunately, it is highly more difficult to also include an invariance behavior under inversions. In [88, 89, 90] we proved several construction theorems that meet this end, in which the multiperiodic functions from the previous chapter can again be used as generating building blocks. This shall be treated in the following three sections.
124
3.2
Chapter 3. Clifford-analytic Modular Forms
Clifford-analytic modular forms in one hypercomplex variable
After having constructed s-monogenic automorphic forms that show an invariance behavior under particular translation and rotation groups, we want now to construct Clifford-valued s-monogenic functions that show an invariance behavior under the modular group Γp or at least under their principal congruence subgroups Γp [n]. Here again we can use some special prototypes of the s-monogenic translation invariant Eisenstein series from Chapter 2 as useful building blocks. For the reasons mentioned at the end of Chapter 1.4 we will work in this section and in the following ones in the vector formalism, since it admits a simpler treatment. To proceed in the proposed direction we first recall that from the conformal invariance formula for the iterated Dirac operators Ds (Theorem 1.28) it follows that if f˜ : H + (IRk ) → Cl0k (IR) is a left monogenic function, then (s) Dxl q0 (cx + d)∗ f˜(M x ) = 0 for every positive integer l with l ≥ s. The crucial idea is to begin with an appropriate monogenic starting function f˜ from Clifford analysis that is totally invariant under the action of the translation group Tp . Then a summation of expressions of the form (s) (f˜|s M )(x) := q0 (cx + d)∗ f˜(M x )
over a complete set of representatives of right cosets in Γp modulo Tp , will lead to several classes of non-trivial Clifford-valued modular forms with respect to Γp within the function classes Ker Dl for all positive integers l satisfying l ≥ s whenever s is an even positive integer. However, for the cases where s is odd, such a construction would vanish identically. To satisfy f (x) = (f |s M )(x) for all M ∈ Γp implies namely in particular, that f satisfies this relation for the (s)
negative identity matrix which is included in Γp . If s is odd, then q0 (x)∗ is odd, too. Therefore, −1 0 −I = f (x) = (f |s (−I))(x) = −f (x), 0 −1 for all x ∈ H + (IRk ) whenever s is odd. However, if n ∈ IN with n ≥ 3, then this construction theorem provides also for the case where s is odd non-trivial Cliffordmodular forms for the congruence groups Γp [n]. This will now be explained in detail.
3.2. Modular forms in one hypercomplex variable
125
In the sequel the notation M : Tp [n]\Γp [n] means that M runs through a system of representatives Rp [n] of the right cosets of Γp [n] with respect to Tp [n], i.e., 8 Tp [n]M = Γp [n] and Tp [n]M = Tp [n]N M ∈Rp [n]
for M, N ∈ Rp [n] with M = N . Notice that Γp = Γp [1]. The following theorem provides a generalization of Theorem 3 from our paper [88]: Theorem 3.4 (Construction theorem). Let k ∈ IN, s ∈ IN, s < k, p ∈ {1, . . . , k − 1} and p < k − s − 1. Let further n ∈ IN and assume that n ≥ 3 whenever s is odd. Let f˜ : H + (IRk ) → Cl0k (IR) be a bounded and left monogenic function on H + (IRk ) that is totally invariant under the translation group Tp . Then f (x) = (f˜|s M )(x) x ∈ H + (IRk ) (3.6) M :Tp [n]\Γp [n]
is a Clifford-valued function which is bounded in any compact subset of H + (IRk ). Moreover, it satisfies on the whole upper half-space Dl f (x) = 0 for all l ≥ s and f (x) = (f |s M )(x) for all M ∈ Γp [n]. Proof. First we show that the series (3.6) is normally convergent on the upper halfspace H + (IRk ). According to the assumption f˜ is a bounded function on H + (IRk ). For all p ∈ {1, . . . , k − 1} the group Γp [n] acts on H + (IRk ). Hence one can readily infer that there is a positive real L such that
f˜(M x ) ≤ L for all x ∈ H + (IRk ) and for all M ∈ Γp [n]. Therefore, it suffices to show the normal convergence of
cτ + d −α (3.7) M :Tp [n]\Γp [n]
∗ ∗ on H (IR ) for α > p + 1. To this end consider a matrix ∈ Γp [n]. If c d c = 0, then c is invertible and c−1 d ∈ IRp . For ε > 0 consider the vertical strip Vε (H + (IRk )) := τ = (x, xk ) ∈ H + (IRk )x ∈ IRk−1 : x ≤ 1ε , xk > ε . (3.8) +
k
The crucial point is to show that one can find for every ε > 0 a real ρ > 0 so that ∗ ∗ ∈ Γp [n] (3.9)
cτ + d ≥ ρ cek + d ∀τ ∈ Vε (H + (IRk )) and c d is satisfied. The estimate (3.9) is satisfied trivially for (c, d) = (0, 0). In the sequel let us thus assume that (c, d) = (0, 0). We observe that
cτ + d ≥ ρ cek + d
⇔
˜ ≥ ρ,
˜ cτ + d
126
Chapter 3. Clifford-analytic Modular Forms
˜ = 1. which satisfy via construction ˜ cek + d ˜ = ˜ ˜ ≥ ˜ ˜ Hence, for τ := x + xk ek one gets: ˜ cτ + d cxk ek + c˜x + d c(εek + x) + d . ˜ x ≤ 1 , ˜ ˜ 2 = 1} is compact. The function The set K := {(x, c˜, d)|, cek + d ε ≥0 ˜ → ˜ ˜ is continuous. Hence, it takes its Φ : K → IR , (x, c˜, d) c(εek + x) + d ˜ = d ˜ = 1 > 0. minimum value on K. If c = 0, then Φ(x, c˜, d) ˜ = 0, then εek + x = −˜ ˜ In view of c−1 d. Next consider c = 0. If Φ(x, c˜, d) c−1 d˜ ∈ IRk we hence obtain that (εek + x) ∈ IRk−1 . This implies x ∈ IRk−1 and −˜ ˜ > 0 and (3.9) is proved. ε = 0, which provides a contradiction. Hence Φ(x, c˜, d)
with c˜ :=
c cek +d
and d˜ :=
d cek +d
The series
cek + d −α
(3.10)
M :Tp [n]\Γp [n]
has the same convergence abscissa as the series
cep+1 + d −α
(3.11)
M :Tp \Γp
since the elements ep+1 , . . . , ek all play an equal role satisfying the same calculation rules in relation with the elements c, d ∈ spanIR {1, e1 , . . . , ep , . . . , e1...p }. According to [45], the series (3.11) has precisely the convergence abscissa p + 1. Hence, the series converges normally on H + (IRk ). By the Weierstraß’ convergence theorem we may next conclude that f satisfies Dl f = 0 for all l ≥ s in H + (IRk ), since f˜ is monogenic in H + (IRk ). In order to show that f is an automorphic form with respect to Γp [n] let us take an arbitrary matrix N ∈ Γp [n]. In view of the homogeneity of the weight factors q0 , i.e., (s) (s) (s) q0 (ab) = q0 (b)q0 (a) we obtain for even and odd s:
f (N x ) =
=
(s)
q0
M :Tp [n]\Γp [n]
=
cM (aN x + bN )(cN x + dN )−1 + dM
) (s) q0
M :Tp [n]\Γp [n]
*∗ *∗
cN x+dN cN x+dN 2
·f˜(M N x ) −1 (s) = q0 (cN x + dN )∗ =
(s) q0 (cN x
f˜(M N x )
cM (aN x+bN )(cN x+dN )+dM (cN x+dN )(cN x+dN ) cN x+dN 2
∗
) (s) q0
M :Tp [n]\Γp [n]
=
(s)
q0 (cM N x + dM )∗ f˜(M N x )
M :Tp [n]\Γp [n]
+ dN )∗
−1
(s)
q0
f˜(M N x )
(cM aN + dM cN )x + cM bN + dM dN
∗
(s)
q0 (cM N x + dM N )∗ f˜(M N x )
M :Tp [n]\Γp [n]
f (x).
The last step follows by rearrangement which is allowed since f˜ is invariant under Tp [n].
3.2. Modular forms in one hypercomplex variable
127
The simplest non-trivial example of a monogenic automorphic form with respect to Γp [n] with p < k − 2 for n being a positive integer with n ≥ 3 are the following Γp [n]-Eisenstein series. Proposition 3.5. Suppose that k, p, n ∈ IN with k ≥ 4, p < k − 2 and n ≥ 3. Then the Eisenstein type series (p,n) (1) G1 (x) = q0 (cx + d)∗ (3.12) M :Tp [n]\Γp [n]
are convergent and represent a non-trivial Clifford-valued monogenic automorphic form with respect to Γp [n] on the upper half-space H + (IRk ). Proof. The convergence and regularity properties follow from Theorem 3.4; insert (p,n) do not vanish identically on the upper there f˜ ≡ 1. To show that the series G1 half-space under the condition n ≥ 3, consider the following limit: (p,n) (1) lim G1 (x) = lim q0 (cx + d)∗ xk →∞
M :Tp [n]\Γp [n]
xk →∞
=
(1)
q0 (d)∗ = 1.
M :Tp [n]\Γp [n],cM =0
More generally, let us next assume that s is an odd positive integer. Then we can show in complete analogy that the series (s) Gs(p,n) (x) = q0 (cx + d)∗ , (3.13) M :Tp [n]\Γp [n]
which converge normally on H + (IRk ) under the condition p < k − s − 1, provide us with non-vanishing Clifford-valued examples of automorphic forms with respect to Γp [n] in Ker Dl for all l ≥ s whenever n ≥ 3. In the case where s is an even (p,n) integer, the associated series Gs (x) do not vanish in the cases where n = 1 and (p,n) n = 2, either. Notice that the series Gs (x) are only scalar-valued when s is even. In the case n = 1 they read 1 Gs(p,1) (x) =
cx + d k−s M :Tp \Γp
and have a similar form as the non-analytic Eisenstein series considered in [45] and the non-analytic quaternionic series considered in [54] and [55]. Notice that lim Gs(p,1) (x) = lim cx + d s−k xk →∞
M :Tp \Γp
=
xk →∞
M :Tp \Γp ,cM =0
d s−k = 2p+1 .
128
Chapter 3. Clifford-analytic Modular Forms
By means of the Eisenstein series introduced in Section 2.4 we can generate a class of non-trivial vector-valued automorphic forms for the group Γp in Ker Dl for l ≥ s where s is even. We show: Proposition 3.6. Let k ∈ IN, s ∈ 2 IN, s < k, p ∈ {1, . . . , k − 1}, p < k − s − 1 and suppose that m = (m1 , . . . , mk−1 ) ∈ INk−1 is a multi-index where |m| ≥ 3 is an odd 0 Lk−1 ˜ m (x) := G(k−1) number such that ζM (m, 1) = 0. Then, putting G (x + ek ; Lk−1 ), m the series p ˜ m |s M )(x) (x) = (G (3.14) Es,m M :Tp \Γp
does not vanish and represents a non-trivial vector-valued automorphic form with respect to Γp of weight (k − s) on the upper half-space in the classes Ker Dl for every positive integer l with l ≥ s. (s)
˜ m |s M )(x) stands for q (cx + d)∗ Gm (M x + ek ). FollowRemark. For clarity, (G 0 L for which ζMk−1(m,1) = 0, ing Chapter 2.4 there are actually multi-indices m ∈ INk−1 0 Lk−1 (k−1) since ζM (m, 1) are the Laurent coefficients of m . Proof. We use the notation x = x + xk ek with x ∈ IRk−1 . If x ∈ H + (IRk ), then M x ∈ H + (IRk ). Hence, M x + ek ∈ {x ∈ H + (IRk )|xk ≥ 1}. We can always find a lattice point ω ∈ Lk−1 = Ze1 + · · · + Zek−1 such that M x + ek + ω is contained in the vertical strip V k1 (H + (IRk )). In Section 2.4 we (k−1)
have shown that Gm1 ,...,mk−1 (M x + ek + ω) is bounded on V k1 (H + (IRk )); let us write
G(k−1) m1 ,...,mk−1 (M x + ek + ω) ≤ N, (k−1)
with a properly chosen N ∈ IR. Since Gm1 ,...,mk−1 is invariant under the action of the translation group Tk−1 , we hence obtain that for every x ∈ H + (IRk ): (k−1)
G(k−1) m1 ,...,mk−1 (M x + ek ) = Gm1 ,...,mk−1 (M x + ek + ω) ≤ N. p Therefore, Es,m (x) is in fact a well-defined function on H + (IRk ).
Now we consider
(s) (k−1) lim q0 (cx + d)∗ Gm1 ,...,mk−1 (M x + ek ) M :Tp \Γp xk →∞ (k−1) −1
d s−k lim Gm1 ,...,mk−1 (a[x + ek xk ]d−1 + bd
+ek ) x →∞ k M :Tp \Γp ,cM =0 ∈Lp (k−1) 2p+1 lim Gm1 ,...,mk−1 (x) xk →∞ (1) 2p+1 qm1 ,...,mk−1 (ω). ω∈Lk−1 \{0}
p lim Es,m (x) =
xk →∞
= = =
The last expression is different from zero, and the proof is finished.
3.2. Modular forms in one hypercomplex variable
129
To discuss a concrete example let us consider the class of Clifford-valued functions in Ker D∆4 defined in IR10 (Fueter–Sce solutions). Let us take the particular group Γ2 which acts discontinuously on H + (IR10 ). If we take an appropriate multiL index m ∈ IN90 of odd length |m| ≥ 3 for which ζMk−1 (m; 1) = 0, then the families 2 2 2 of function series E2,m , E4,m and E6,m provide us with non-trivial examples of vector-valued hypercomplex modular forms for the group Γ2 within the function class of holomorphic Cliffordian functions when expressing them in the paravector formalism. (k−1) by a general (k − 1)-fold periodic function f˜ ∈ Ker D∆4 , If we substituted Gm ˜ then in general (f |2α M )(x) would not be anymore an element of Ker D∆4 , due to the fact that α has to be chosen from the set {1, 2, 3} in order to preserve the convergence of the associated function series. (k−1)
In the case where n ∈ IN with n ≥ 3 the series Gm (x) serve furthermore to construct further non-trivial examples of Clifford-valued monogenic automorphic forms functions, and more generally, non-vanishing examples of automorphic forms in the classes Ker Dl where l ≥ s for odd s. Under the same convergence conditions as mentioned previously the series p,n (x) = Es,m
(s)
(k−1)
q0 (cx + d)∗ Gm,1 (M x + ek )
(3.15)
M :Tp [n]\Γp [n]
do not vanish even for odd s whenever n ≥ 3, since in these cases
p,n lim Es,m (x) =
xk →∞
(s)
q0 (d)∗
M :Tp [n]\Γp [n],cM =0
=
qm1 ,...,mk−1 (ω)
ω∈Lk−1 \{0}
L
2ζMk−1 (m; 1)
which does not vanish for properly chosen m. Behind the background of constructing cusp forms it is interesting to look for candidates of s-monogenic automorphic forms that satisfy in particular lim f (x) = 0.
xk →∞
(3.16)
To this end it is suggestive to insert in (3.6) for f˜ the monogenic plane wave exponential function 1 r 2πi x,r −2π r xk )e e f˜(x) = E(2πr, x) := (iek + 2
r and to consider
130
Chapter 3. Clifford-analytic Modular Forms
Definition 3.7. Let k ∈ IN, s ∈ IN with s < k and p ∈ {1, . . . , k − 1} such that k − s > p + 1. Let n ∈ IN and assume additionally in the case where s is odd that n ≥ 3. For an arbitrary parameter r ∈ Lk−1 \{0} define (s) q0 (cx + d)∗ E(r, M x ), x ∈ H + (IRk ). (3.17) Qp,n s,r (x) := M :Tp [n]\Γp [n]
These series, whose form generalizes that of the classical prototype for a Poincar´e series, might be possible candidates for cusp forms in Ker Dl for l ≥ h. In particular they satisfy (s) lim q0 (cx + d)∗ E(r, M x ) lim Qp,n s,r (x) = xk →∞
M :Tp [n]\Γp [n]
xk →∞
=
M :Tp [n]\Γp [n],cM =0
2 (s) lim q (d)∗ e−2π r xk / d e2πi
xk →∞ 0
= 0,
where we put y := (axd−1 + bd−1 ). Unfortunately, we were so far not able to provide an analytic proof for the non-vanishing behavior of these series. At the moment, it is not clear whether one can use similar methods as one uses in the classical complex case in order to show that these types of Poincar´e series do not vanish. Notice further that in the cases where s is even, the associated functions (x) have a very similar form to the non-analytic Poincar´e series considered Qp,n s,r in [45] which are eigenfunctions to the Laplace–Beltrami operator for special nonvanishing eigenvalues. Suppose again that k ∈ IN, s ∈ IN with s < k and p ∈ {1, . . . , k − 1} such that k − s > p + 1. Again, let n ∈ IN and assume additionally in the case where s is odd that n ≥ 3. Inserting in (3.6) for f˜ the translative monogenic Eisenstein (k−1) series m1 ,...,mk−1 (x + ek ) lead to convergent series of the form (s) p,n (x) := q0 (cx + d)∗ (p) x ∈ H + (IRk ). Es,m m1 ,...,mk−1 (M x + ek ) M :Tp [n]\Γp [n]
(3.18) In all the cases |m| ≥ 1 one can readily deduce that lim (k−1) m1 ,...,mk−1 (x + ek ) = 0.
xk →∞
p,n As a consequence of this also the series Es,m (x) with |m| ≥ 1 satisfy:
=
(s) (k−1) lim q0 (cx + d)∗ m1 ,...,mk−1 (M x + ek ) x →∞ k M :Tp [n]\Γp [n] (s) (k−1) q0 (d)∗ lim m1 ,...,mk−1 (a[x + ek xk ]d−1 + bd−1 + ek ) x →∞ k M :Tp [n]\Γp [n],cM =0
p,n lim Es,m (x) =
xk →∞
= 0.
3.3. Modular forms in several hypercomplex variables
131
p,n (x) we were not yet able to provide a non-triviality However, also for the series Es,m argument.
3.3
Clifford-analytic modular forms in two and several hypercomplex variables
In 1904 O. Blumenthal introduced a class of complex-valued automorphic forms that are defined on the m-fold Cartesian product of the upper half-plane (see [9]). These functions are holomorphic in each complex variable and show an invariance under the hyperabelian group which is the m-fold direct product of the classical modular group SL(2, Z). This class of automorphic forms is today commonly known under the designation Hilbert–Blumenthal modular forms, or often simply Hilbert modular forms for short. For the basic theory, see for instance [48]. Many authors extended the theory of Hilbert modular forms. Very recently researchers have become more and more interested in automorphic forms that are defined on m-fold Cartesian products of higher dimensional half-spaces. For example, O. Richter et al. considered very recently in [131] and [132] automorphic forms on m-fold Cartesian products of quaternionic half-spaces. The functions considered in these papers are not endowed with the regularity concepts of quaternionic analysis. In [90] we introduced also Clifford-analytic generalizations of Hilbert modular forms defined on Cartesian products of upper half-spaces in IRk that show an invariance behavior under direct products of a discrete group of Vahlen’s group that acts discontinuously on the respective half-space. In this section we summarize the main results from [90] and present some extensions. We begin by considering Clifford-valued functions in two vector variables that are left s-monogenic in the first variable and right t-monogenic in the second variable, both being defined on the Cartesian product H2+ (IRk ) := H + (IRk ) × H + (IRk ) that shows a quasi-invariant behavior under the direct product of the group Γp [n]× Γp [n]. This group acts discontinuously on each (x, y) ∈ (H2+ (IRk ) by (x, y) → (M x , N y ) for all M, N ∈ Γp [n]. Notice that we can write N y = (aN y + bN )(cN y + dN )−1 = (yc∗N + d∗N )−1 (ya∗N + b∗N ) =: N ∗ [y]. Functions of two vector variables that are left monogenic in the first one and right monogenic in the second one are called biregular functions (cf. [155, 156]). In what follows, functions of two vector variables that are left s-monogenic in
132
Chapter 3. Clifford-analytic Modular Forms
the first vector variable and t-monogenic in the second one will be called (s, t)biregular, or alternatively also polybiregular for short. As a direct consequence of Theorem 1.28 it follows that if f (x, y) is an (s, t)-biregular function on H2+ (IRk ), then the function F (x, y) = q0 (cM x + dM )∗ f (M x , N y )q0 (yc∗N + d∗N ) (s)
(t)
is for all (M, N ) ∈ Γp [n] × Γp [n] contained in the set of (s1 , t1 )-biregular functions for all (s1 , t1 ) with s1 > s and t1 > t. This background motivates the following definition: Definition 3.8. Let p, n, s, t ∈ IN with p ≤ k − 1, s, t ≤ k − 1, and assume that n ≥ 3 whenever s or t is an odd integer. An (s, t)-biregular function on H2+ (IRk ) is called an (s, t)-biregular automorphic form with respect to Γp [n] × Γp [n] if it satisfies for all (x, y) ∈ H2+ (IRk ), f (x, y) = q0 (cM x + dM )∗ f (M x , N y )q0 (yc∗N + d∗N ) (s)
(t)
(3.19)
for all M, N ∈ Γp [n]. The group action is not simultaneous in both variables. As a consequence of this, one obtains in the particular cases n = 1 and n = 2 only non-vanishing modular forms with this transformation behavior, if both s and t are even integers. Since J ∈ Γp , also the negative identity matrix −I is included in Γp . If s for instance is an odd integer, then the automorphy relation applied to the special matrices (M, N ) = (−I, I) leads immediately to f ≡ 0 on H2+ (IRk ). If n ≥ 3, then one gets also non-trivial examples when s or t or both of them are odd, as we shall see in what follows. For simplicity let us use the following notation: (f ||s,t (M, N ))(x, y) = q0 (cM x + dM )∗ f (M x , M y )q0 (yc∗M + d∗M ). (s)
(t)
We prove: Theorem 3.9. Let k, p, n, s, t ∈ IN with p ≤ k − 1 and s, t ≤ k − 1. Assume additionally that k ≥ 2, p < k − max{s, t} − 1 and that n ≥ 3 whenever s or t are odd. Let f˜ : H2+ (IRk ) → Cl0k (IR) be a bounded function with Dxs f˜(x, y) = 0 and f˜(x, y)Dyt = 0 in the whole domain H2+ (IRk ). Then the function series f (x, y) = (f˜||s,t (M, N ))(x, y) (3.20) M,N :Tp [n]\Γp [n]
converges normally on
H2+ (IRk )
and satisfies there
Dxs f (x, y) = f (x, y)Dyt = 0.
(3.21)
f (x, y) = (f ||s,t (M, N ))(x, y)
(3.22)
Moreover, for all M, N ∈ Γp [n].
3.3. Modular forms in several hypercomplex variables
133
Proof. Let K ⊂ H2+ (IRk ) be a compact set. As shown in the proof of Theorem 3.4 one can find a real ρ > 0 with
q0 (cM x + dM )∗ ≤ ρs−k cM ek + dM s−k (s)
for all
M ∈ Γp [n]
and for all x that belong to K. For every (x, y) ∈ K one thus obtains: (s) (t)
q0 (cM x + dM )∗ f˜(M x , N y )q0 (yc∗N + d∗N ) M,N :Tp [n]\Γp [n]
≤ C
M,N :Tp [n]\Γp [n]
1 1 ,
cM ek + dM k−s cN ek + dN k−t
(3.23)
where C > 0 is a real constant. Notice that the matrices M and N run independently from each other through systems of representatives of the right cosets of Γp [n] modulo Tp [n] in the summation process (3.23). Since the series
cM ek + dM −(k−α) M :Tp [n]\Γp [n]
converges under the condition p < k − α − 1, the series in (3.23) consequently converges normally for p < k − max{s, t} − 1. The regularity properties follow as an application of the Weierstraß’ convergence theorem. To show the automorphy behavior one takes two arbitrary matrices M1 , N1 ∈ Γp [n]. Let (x, y) ∈ H2+ (IRk ). Next we consider f (M1 x , N1 y ) (s) = q0 (cM M1 x + dM )∗ M,N
∈Tp [n]\Γp [n] · f˜ (aM M1 x + bM )(cM M1 x + dM )−1 , (aN N1 y + bN )(cN N1 y + dN )−1 (t) · q0 (N1∗ [y]cN + dN ) ∗ (s) (s) ∗ q0 (cM1 x + d−1 = M1 q0 (c(M ·M1 ) x + d(M ·M1 ) ) M,N
∈Tp [n]\Γp [n] · f˜ (a(M ·M1 ) x + b(M ·M1 ) )(c(M ·M1 ) x + d(M ·M1 ) )−1 , (a(N ·N1 ) y + b(N ·N1 ) )(c(N ·N1 ) y + d(N ·N1 ) )−1 (t) (t) · q0 (yc∗(N ·N1 ) + d∗(N ·N1 ) ) q0 (yc∗N1 + d∗N1 )−1 ∗
= q0 (cM1 x + dM1 )−1 · f (x, y) · q0 (yc∗N1 + d∗N1 )−1 . (s)
(t)
The previous line follows after having applied again a direct rearrangement argu ment, where we used in particular the invariance of f˜ under Tp [n] × Tp [n]. Next we give some non-trivial examples. Let us put f˜ ≡ 1 into (3.20). The function series (s) (t) f (x, y) = q0 (cM x + dM )∗ q0 (yc∗N + d∗N ) (3.24) M,N :Tp [n]\Γp [n]
134
Chapter 3. Clifford-analytic Modular Forms
converges whenever p < k − max{s, t} − 1. If n ≥ 3, then (s) (t) lim f (xk , yk ) = q0 (dM )∗ q0 (d∗N ) = 1, xk ,yk →∞
M,N :Tp [n]\Γp [n],(cM ,cN )=(0,0)
independently if s and t are even or odd numbers. The function defined in (3.24) provides the simplest example of a non-vanishing Hilbert modular form for Γp [n]× Γp [n] that is left s-monogenic in the first variable and right t-monogenic in the second one and f (x, y) = q0 (cM x + dM )∗ f (M x , N y )q0 (yc∗N + d∗N ) (s)
(t)
for all (M, N ) ∈ Γp [n] × Γp [n] and all (x, y) ∈ H2+ )(IRk ). If s and t are both even, then (3.24) does not vanish either for the cases n = 1 and n = 2. We obtain for instance for n = 1:
dM
dN = 2p+2 . lim f (xk , yk ) = xk ,yk →∞
M,N :Tp \Γp ,(cM ,cN )=(0,0)
Further examples of (s, t)-biregular automorphic forms can be generated by taking for f˜ a product of an s-monogenic translation invariant Eisenstein series in the variable x with a t-monogenic translation invariant Eisenstein series in the variable y, as for example (x + ek ; Lk−1 )G(k−1) (y + ek ; Lk−1 ). f˜(x, y) = G(k−1) m n Moreover, there is also the possibility of making the following multiplicative construction termwise, as e.g., (k−1) (1) Gm,n;1 (x, y) := qm (α(x + ek ) + ω)qn(1) (β(y + ek ) + η), (α,ω) (β,η) ∈Z×Lk−1 \{(0,0)} ∈Z×Lk−1 \{(0,0)}
which provides a further variant of a biregular generalization of the monogenic (k−1) Eisenstein series Gm;1 discussed in Chapter 2.4. By similar limit arguments as presented before, one can indeed show that inserting these examples for f˜ into (3.20) leads to non-vanishing (s, t) biregular Hilbert modular forms with respect to Γp [n] × Γp [n], claiming additionally in the cases where s or t is odd that n ≥ 3. Next we turn to a very particular variant of generalized Hilbert modular forms in the Clifford analysis setting. In contrast to the Hilbert modular forms treated so far, we want to consider now non-trivial (s, t)-biregular automorphic forms that are quasi-invariant under the simultaneous action of the group Γp [n] in both arguments. In other words we consider now (s, t)-biregular functions that show an invariance behavior under the action (x, y) → (M x , M y ),
M ∈ Γp [n],
3.3. Modular forms in several hypercomplex variables
135
in contrast to the action considered before which was of the form (x, y) → (M x , N y ) where M, N may be different matrices from Γp [n]. In order to exclude trivial examples, we only call an (s, t)-biregular function f (x, y) in H2+ (IRk ) a non-trivial automorphic form under the simultaneous action of Γp [n], if its restriction to the diagonal g(x) := f (x, x) is at least also a non-constant well-defined C ∞ automorphic form with respect to Γp [n]. In this sense we introduce Definition 3.10. Let p, n, s, t ∈ IN with p ≤ k − 1. An (s, t)-biregular function on H2+ (IRk ) is called a non-trivial automorphic form with respect to simultaneous actions of Γp [n] if it satisfies in each (x, y) ∈ H2+ (IRk ), f (x, y) = q0 (cM x + dM )∗ f (M x , M y )q0 (yc∗M + d∗M ) (s)
(t)
(3.25)
for all M ∈ Γp [n] and if additionally its restriction to the diagonal f (x, x) is at least a non-constant well-defined C ∞ -function that satisfies for all x ∈ H + (IRk ): f (x, x) = q0 (cM x + dM )∗ f (M x , M x )q0 (xc∗M + d∗M ) (s)
(t)
(3.26)
for all M ∈ Γp [n]. Remember that the Poincar´e summation ansatz of the form (3.6) only provided us with non-trivial polymonogenic automorphic forms for Γp [1] and Γp [2] in one variable whenever s is even. Similarly, the Poincar´e summation ansatz of the form (3.20) did not provide us with non-trivial (s, t)-biregular functions for Γp [1]×Γp [1] and Γp [2]×Γp [2] in the framework of Hilbert modular forms with nonsimultaneous group actions in those cases where s or t are odd. Furthermore, in both construction theorems Theorem 3.4 and Theorem 3.9 we had the convergence conditions p < k − s − 1 or p < k − max{s, t} − 1, respectively. In what follows next we shall see that applying a similar Poincar´e summation ansatz in the framework of Definition 3.10 will provide also non-trivial examples (in the sense of Definition 3.10) for Γp [1] and Γp [2] for the cases where s and t are both odd. Additionally we obtain better convergence conditions in this context, so that a similar Poincar´e summation ansatz provides us in particular also with non-trivial (s, t) biregular automorphic forms with respect to the simultaneous action of the full modular group Γk−1 , provided s and t are not too large. For simplicity let us use the following notation: (f ||s,t M )(x, y) = q0 (cM x + dM )∗ f (M x , M y )q0 (yc∗M + d∗M ). (s)
We prove the following theorem:
(t)
136
Chapter 3. Clifford-analytic Modular Forms
Theorem 3.11. Let p < min{k, 2k − (s + t) − 1} and let s + t ≡ 0 mod 2. Suppose that f˜ : H2+ (IRk ) → Cl0k (IR) is a bounded function that satisfies for all (x, y) ∈ H2+ (IRk ) the relation Dxs [f˜(x, y)] = [f˜(x, y)]Dyt = 0 and furthermore f˜(T x , T y ) = f˜(x, y) for all T ∈ Tp . Then f (x, y) := (f˜||s,t M )(x, y) (3.27) M :Tp \Γp
is left s-monogenic in the variable x and right t-monogenic in y and satisfies for all (x, y) ∈ H2+ (IRk ), f (x, y) = (f ||s,t M )(x, y)
for all M ∈ Γp .
(3.28)
Proof. Let K ⊂ H2+ (IRk ) be a compact subset. According to the proof of Theorem 3.4, there is a real ρ > 0 such that for all M ∈ Γp , (s)
q0 (cx + d) ≤ ρs−k cek + d s−k
(3.29)
for all x that belong to K. In view of the inequality abc ≤ C a
b
c which is satisfied for all Clifford numbers in Cl0k (IR) we obtain for every arbitrary pair (x, y) ∈ K the estimate
q0 (cx + d)∗ f˜(M x , M y )q0 (yc∗ + d∗ ) (s)
M :Tp \Γp
(t)
˜ s+t−2k ≤ Lρ
M :Tp \Γp
1 1 ≤L
cek + d k−s ek c∗ + d∗ k−t
M :Tp \Γp
1 .
cek + d 2k−s−t
˜ and L denote non-negative real constants. The series in the previous line is Here L absolutely convergent, whenever p < 2k − s − t − 1. From Weierstraß’ convergence theorem we conclude that f is (s, t)-biregular. To show the automorphy relation under simultaneous actions of Γp , just consider f (−x−1 , −y−1 ) (s) = q0 (−cx−1 + d)∗ M :Tp
\Γp ·f˜ (−ax−1 + b)(−cx−1 + d)−1 ), (−y−1 c∗ + d∗ )−1 (−y−1 a∗ + b∗ ) (t) ·q0 (−y−1 c∗ + d∗ ) (s) q0 ((c − dx)(−x−1 ))∗ = M :Tp \Γp
·f˜ (a − bx)x−1 x(c − dx)−1 , (c∗ − yd∗ )−1 yy−1 (a∗ − yb∗ ) (t) ·q0 ((−y−1 )(c∗ − yd∗ ))
3.3. Modular forms in several hypercomplex variables =
M :Tp \Γp
∗
(s)
137
(s)
q0 (−x−1 )−1 q0 (c − dx)∗
(t) (t) ·f˜ (a − bx)(c − dx)−1 , (c∗ − yd∗ )−1 (a∗ − yb∗ ) q0 (c∗ − yd∗ ) q0 (−y−1 )−1 ∗
= q0 (−x−1 )−1 f (x, y)q0 (−y−1 )−1 . (s)
(t)
Here we applied a simple rearrangement argument. By a similar rearrangement procedure one can show that the automorphy relation holds for simultaneous translations (x, y) → (x + ω, y + ω) where ω = ei for i = 1, . . . , p. Let therefore T be an arbitrary translation matrix from Tp . Let us consider without loss of generality T1 x = x + e1 . Then we obtain: f (T x , T y )
=
M :Tp \Γp
=
(s)
q0 (cM x + cM e1 + dM )∗ f˜((M T ) x , (M T ) y )
(t) · q0 (yc∗M + e1 c∗M + d∗M ) (s) q0 (cM T x + dM T )∗ f˜((M T ) x , (M T ) y )
M :Tp \Γp
(t) · q0 (yc∗M T + d∗M T ) ,
in view of c∗M T = c∗M and d∗M T = (cM e1 + dM )∗ = e1 c∗M + d∗M . The rest follows by rearrangement. The automorphy relation is thus shown for the generators of Γp . Remarks. Theorem 3.11 can easily be extended to the more general context of the congruence groups Γp [n] with n ≥ 2. To this end one simply starts with a generating function f˜ that is invariant under simultaneous actions of the group Tp [n]. Notice that we get convergence even for p = k−1 whenever s+t ≤ k−1. Even for the monogenic case s = t = 1, we obtain non-trivial Clifford-valued modular forms for the full modular group Γk−1 in the sense of Definition 3.10, provided we are in a space with dimension k ≥ 3. The simplest non-trivial examples in the sense of Definition 3.10 are the biregular Eisenstein series E(x, y)
=
(1||1,1 M )(x, y)
(3.30)
M :Tp \Γp
=
q0 (cM x + dM )∗ q0 (yc∗M + d∗M ), (1)
(1)
(x, y) ∈ H2+ (IRk ).
M :Tp \Γp
To prove that these series are actually non-trivial automorphic forms with respect
138
Chapter 3. Clifford-analytic Modular Forms
to the simultaneous action of Γp−1 , consider lim E(ek xk , ek xk ) =
xk →∞
(1) lim q (cek xk xk →∞ 0
M :Tp \Γp ,cM =0
(1)
=0
+
+ d)∗ q0 (ek xk c∗ + d∗ )
(1) (1) q0 (d)∗ q0 (d∗ )
M :Tp \Γp ,cM =0
=
2
eA eA = 2p+1 .
A⊆{1,...,p}
The restriction to the diagonal E(x, x) is thus actually a non-constant function. It is furthermore a C ∞ -function in the single vector variable x and satisfies (3.26). Also the series Gm can be used for the generation of classes of non-trivial examples for hypercomplex modular forms transforming in the sense of Definition 3.10; in this setting, even for the full modular group Γk−1 . To this end apply (3.27) on (x + ek ; Lk−1 )G(k−1) (y + ek ; Lk−1 ) (3.31) f˜(x, y) = G(k−1) m m which is bounded on H2+ (IRk ). To verify the non-triviality condition, consider lim
xk →∞
M :Tp \Γp
(1)
q0 (cek xk + d)∗ G(k−1) (M ek xk + ek ) m · G(k−1) (M ek xk + ek )q0 (ek yk c∗ + d∗ ) m L e∗A ζMk−1 (m, 1) 2 eA ∗ . (−8) · (1)
=
A⊆{1,...,p}
As explained in Chapter 2.4, there are at least some multi-indices m ∈ INk−1 0 L L with |m| ≡ 1 mod 2 for which ζMk−1 (m, 1) and hence ζMk−1 (m, 1) 2 does not vanish. For these multi-indices the limit under consideration is thus different from zero or infinity. Hence, the associated series M :Tp \Γp (f˜||s,t M )(x, y) give indeed non-constant C ∞ -functions on the diagonal and satisfy the automorphy relation. In the context of simultaneous group actions, also the following biregular (k−1) generalization of the series Gm , f˜(x, y) := qr(1) (α(x + ek ) + ω)qr(1) (y + ek ) + ω) |r| ≥ 3, (α,ω)∈Z×Lk−1 \{(0,0)}
(3.32) serve as generating functions for non-trivial (s, t)-biregular automorphic forms with respect to Γp [n] in the sense of Definition 3.10 for properly chosen r, as a similar limit argument shows. Notice that these functions cannot be used for the generation of (s, t)-biregular Hilbert modular forms transforming in the sense of Definition 3.8. The function defined in (3.32) is only invariant under simultaneous
3.3. Modular forms in several hypercomplex variables
139
actions of the group Tp . It is not invariant under the non-simultaneous group actions of Γp × Γp in both arguments. Remarks. Notice that restricting the functions constructed in Theorem 3.11 to the diagonal gives in general only a C ∞ -function in the single vector variable x. This is due to the fact that if f˜ is s-monogenic, then F (x) := q0 (cx + d)∗ f˜(M x )q0 (xc∗ + d∗ ) (s)
(t)
is in general not s-monogenic anymore. However, one should keep in mind that the higher we choose s, the larger becomes the chance to find adequate special generating functions f˜ such that the associated function F (x) remains in the set of s-monogenic functions. Within the class of the Fueter–Sce solutions (holomorphic Cliffordian functions), i.e., functions defined in the space IR ⊕ IR2m+1 that are in Ker D∆m , there is already a large class of special functions f˜ that belong to Ker D∆m which have the property that the transformation ((cz + d)∗ )−1 f˜(M z )(zc∗ + d∗ )−1 gives for every M ∈ SV (IR2m+1 ) again a function in Ker D∆m . The simplest examples are obtained by simply putting f˜ ≡ 1. Then F (z) := ((cz + d)∗ )−1 f˜(M z )(zc∗ + d∗ )−1 belongs to Ker D∆m for all M ∈ SV (IR2m+1 ). However, a series of the form ((cz + d)∗ )−1 f˜(M z )(zc∗ + d∗ )−1 (3.33) M :Tp \Γp
does not even converge normally for p = 1. Notice that the higher we choose s, the worse become the convergence conditions of the associated series (s) (t) q0 (cx + d)∗ f˜(M x )q0 (xc∗ + d∗ ). (3.34) M :Tp \Γp
The series (3.34) converges normally for p ≤ 2k − 2s − 1. In the holomorphic Cliffordian case (s = k − 1), this inequality turns out to read precisely p < 1. Note further, that if one substitutes the exponent −1 in (3.33) by larger ones in order to get convergence, then we will lose the automorphy behavior of the series. To illustrate this, just substitute simply z by z −1 in the above expression and observe that (cz −1 + d)−n = z −n (c + dz)−n whenever n > 1. To construct Hilbert modular forms with the property that its restriction to the diagonal is additionally s-monogenic, different ideas are therefore required. An interesting question in the context of non-trivial (s, t)-biregular Hilbert modular forms in the sense of Definition 3.8 and Definition 3.10 is to ask for biregular and polybiregular monogenic modular forms f that satisfy in particular lim f (x, y) = lim f (x, y) = 0.
xk →∞
yk →∞
(3.35)
140
Chapter 3. Clifford-analytic Modular Forms
To this end it is suggestive to insert for f˜ in (3.20) or in (3.27), respectively, the following variant of a biregular exponential function on H2+ (IRk ): Definition 3.12. For m, n ∈ IRk−1 \{0} and (x, y) ∈ H2+ (IRk ) define expm,n (x, y) := (iek +
m −xk m i x,m i y,n −yk n n )e ). e e e (iek +
m
n
This function is left monogenic in x and right monogenic in y on the whole domain H2+ (IRk ). Notice that left and right monogenicity is attained by the constant factors standing left and right from the exponential expressions. Notice that lim expm,n (x, y) = lim expm,n (x, y) = 0
xk →∞
yk →∞
for any parameter m and n from IRk−1 \{0}. In order to construct hypercomplex cusp forms with respect to Γp [n] × Γp [n] that are left s-monogenic in x and right t-monogenic in y the following construction is thus plausible (claiming that n ≥ 3 in the case where s or t is odd): Ps,t (x, y; m, n) = (expm,n ||s,t (M, N ))(x, y). (3.36) M,N :Tp [n]\Γp [n]
To construct (s, t)-biregular cusp forms that transform in the way of Definition 3.10, it is suggestive to consider Ps,t (x, y; m, n) = (expm,n ||s,t M )(x, y). (3.37) M :Tp [n]\Γp [n]
The function expm,n is bounded on H2+ (IRk ); thus the convergence abscissa of the series (3.36) is p < k − max{s, t} − 1 and that of (3.37) is p < 2k − (s + t) − 1. However, similarly to the one-variable case treated in the previous section, it is extremely difficult to find an analytic argument to show that these series do not vanish. At the moment it is not clear how to provide an analytic non-triviality proof, since it is not clear either how to extend the non-triviality arguments that are applied for cusp forms in the classical theory to the Clifford analysis setting. Note that by construction lim Ps,t (x, y; m, n) = lim Ps,t (x, y; m, n) = 0.
xk →∞
yk →∞
Inserting in (3.20) and (3.27) for f˜ products of the form (p)
(p) m,s (x + ek )n,t (y + ek ), or
(p)
G(p) m,s (x + ek )n,t (y + ek ),
3.3. Modular forms in several hypercomplex variables
141
lead also to functions with lim
xk ,yk →∞
f (xk , yk ) = 0.
However, also for these cases we have not yet managed to find a non-triviality argument of the associated function series. Next we proceed to give some examples of non-trivial hypercomplex monogenic and polymonogenic modular forms with respect to simultaneous actions of a discrete group on the Cartesian product of two unit balls. Let * ) √1 √1 ek − 2 2 . g= √1 − √12 ek 2 The associated Cayley transformation (x, y) → (g x , g y ) maps H2+ (IRk ) conformally onto the Cartesian product of the two unit balls, B 2 (0, 1) := B(0, 1) × B(0, 1) = {(x, y) ∈ IRk × IRk ; x < 1 and y < 1}. In order to construct monogenic and polymonogenic automorphic forms on B 2 (0,1) it is suggestive to make an ansatz of the form (f˜||h,l M )(x, y) (3.38) M :gG g −1 \gGg −1
where G ≤ G ≤ Γk−1 and where f˜ is a function that is bounded on B 2 (0, 1) and totally invariant under gG g −1 and furthermore left s-monogenic in the variable x and right t-monogenic in y. At this point one sees that the transfer to the unit ball is not as direct, as one may think at first. Notice that if f is monogenic in the vector variable x, then in general the expression f (M x ) will only remain monogenic, if M is a translation matrix. Putting G = Tk−1 (which would be plausible), then not every monogenic f˜ has the property that f˜((gG g −1 ) x ) remains monogenic. The same holds in the more general context of s-monogenic functions. The simplest nontrivial (poly-) biregular examples that we can construct for the whole modular group G := gΓp g −1 on B 2 (0, 1) and for which we can avoid this problem, are the following (poly-) biregular Eisenstein series on the Cartesian product of the two unit balls. In [91] we introduced: Definition 3.13. Let p < 2k−(s+t)−1 where s+t ≡ 0 mod 2 and let Gp := gΓp g −1 , Gp := gTp g −1 where g stands for the Cayley transformation. Then the (poly-) biregular Eisenstein series on the Cartesian product of the two unit balls B 2 (0, 1) are defined by H(x, y) := (1||s,t M )(x, y). (3.39) M :Gp \Gp
142
Chapter 3. Clifford-analytic Modular Forms
The function f˜(x, y) ≡ 1 is trivially left s-monogenic and right t-monogenic and totally invariant under Gp . Therefore, the limit function H(x, y) is (s, t)biregular. The series has the same convergence abscissa as its analogue on the half-space. Its transformation behavior H(x, y) = (H||s,t M )(x, y)
for all M ∈ Gp
can be verified in a similar way as we did for the previous examples. To show that the series is not a constant function on the diagonal whenever s + t ≡ 0 mod 2, one only has to evaluate the function at the point (0, 0). The origin is a (poly-) biregular point, because it lies inside the unit balls while all the singularities are concentrated on the boundary of B 2 (0, 1). The series is therefore normally convergent in any compact set around (0, 0) that is strictly contained in B 2 (0, 1). Thus, H(0, 0) =
q0 (d)∗ q0 (d∗ ) = (s)
M :Gp \Gp
(t)
d 2 = 0.
(3.40)
M :Gp \Gp
Hence we can conclude that H(x, x) is a non-constant C ∞ -function. It satisfies at each x ∈ B(0, 1), H(x, x) = (H||s,t M )(x, x) for all M ∈ Gp . After having constructed Clifford-analytic Hilbert modular forms in two vector variables it is natural to ask for generalizations to the framework of several vector variables. We conclude this section by exploring the possibility of extending the constructions made in Theorem 3.9 and Theorem 3.11 to the setting of more than two vector variables. Fundamentals for a general theory of monogenic functions in more than two vector variables has been established recently in [157]. To meet our ends it is suggestive to begin with a multiperiodic starting function f˜(x1 , . . . , xm ) that is left monogenic in the first r vector variables and right monogenic in the remaining m−r ones. Here and in all that follows we assume that 1 ≤ r ≤ m − 1. A first idea would be to make the following construction:
+ r (s ) [ q0 j (cMj xj + dMj )∗ ]f˜(M1 x1 , . . . , Mm xm ) M1 ,...,Mm :Tp \Γp j=1 m + (s ) · [ q0 j (xj c∗Mj + d∗Mj )] .
f (x1 , . . . , xm ) =
j=r+1
With similar arguments as in Theorem 3.9 one can infer that this function series + (IRk ) for p < k − max{s1 , . . . , sm } − 1. Under the is normally convergent on Hm additional assumption that all the numbers sj of the weight factors that stand in (s ) the middle, i.e., q0 j for j = 2, . . . , m − 1, are even, the limit function f shows the
3.3. Modular forms in several hypercomplex variables
143
automorphy relation f (x1 , . . . , xm ) r m + + (s ) (s ) =[ q0 j (cMj xj + dMj )∗ ]f (M1 x1 , . . . , Mm xm )[ q0 j (xj c∗Mj + d∗Mj )]. j=1
j=r+1
(3.41) However, if one (or more) of the elements s2 , . . . , sr−1 were odd, then we would have at least one Clifford-valued weight in the middle, which — when forming f (M1 x1 , . . . , Mm xm ) — cannot be factored out of the sum (3.3) neither to the left-hand side nor to the right-hand side. The same would already happen in the case m = 2 under the assumption that r = 0 or r = 2. In the case of two variables with additionally r = 1, one can factor one automorphy factor out of the sum to the left and the other one to the right. However, having more than two variables, then such a construction involves weight factors in the middle which can only be switched through to the right or to the left hand-side if they commute with the other expressions which however are Clifford valued in general. Additionally one has to keep in mind, that if f˜ is left monogenic in the vector variables x1 , . . . , xr and right monogenic in the remaining ones, then in general the function [
r -
j=1
(1) q0 (cM xj
+ dM
)∗ ]f˜(M x
1 , · · ·
, M xm )[
m -
q0 (xj c∗M + d∗M )] (1)
(3.42)
j=r+1
is not monogenic anymore in the vector variables x2 , . . . , xm−1 . This is a consequence of the fact that the set of left (right) monogenic functions is only a right (left) Clifford module. In the case 1 ≤ r ≤ m−1, the expression (3.42) is in general only left monogenic in the vector variable x1 and right monogenic in xm . In the following proposition we give one positive non-trivial example for a Clifford-analytic modular form that is left s1 -monogenic in the first vector variable, right sm -monogenic in the last vector variable and polyharmonic in the vector variables x2 , . . . , xm−1 : Proposition 3.14. Suppose that m ≥ 3 and that p, n, s1 , s2 , . . . , sm−1 , sm ∈ IN. Suppose additionally that s2 , . . . , sm−1 are all even integers and that p < k − max{s1 , . . . , sm } − 1 is satisfied. Let us assume furthermore that n ≥ 3 whenever s1 or sm is odd. Then the function Es1 ,s2 ,...,sm (x1 , . . . , xm )
m−2 + (sj ) (s ) q0 1 (cM1 x1 + dM1 )[ q0 (cMj xj + dMj )] = j=2 M1 ,...,Mm :Tp [n]\Γp [n] (s ) · q0 m (xm c∗Mm + d∗Mm )
(3.43)
144
Chapter 3. Clifford-analytic Modular Forms
+ (IRk ) (3.41) for does not vanish identically and satisfies for all (x1 , . . . , xm ) ∈ Hm all M1 , . . . , Mm ∈ Γp [n]
This example can be constructed from (3.3) by inserting f˜ ≡ 1. The nontriviality follows again by applying the standard limit argument. Restricting to simultaneous group actions in the sense of generalizing Definition 3.10, the previous example leads also to non-trivial functions in the case n = 1 and n = 2 in the case when more generally s1 + sm is even. The function f (x1 , x2 , . . . , xm−1 , xm ) =
(1)
q0 (cx1 + d)∗
M :Tp \Γp
m−1 -
(2) (1) q0 (cxj +d) q0 (xm c∗ +d∗ )
j=2
provides the simplest non-trivial example of a function that is left monogenic in x1 , right monogenic in xm , Euclidean harmonic in x2 , . . . , xm−1 and that transforms for all (x1 , . . . , xm ) as f (x1 , x2 , . . . , xm−1 , xm )
(1)
= q0 (cM x1 + dM )∗ f (M x1 , . . . , M xm ) m−1 -
·[
q0 (cM xj + dM )]q0 (xm c∗M + d∗M ) (2)
(1)
j=2
for all M ∈ Γp . As one can readily verify by the standard limit argument, the restriction of f to the diagonal gives a non-constant function being at least C ∞ that transforms in the way m−2 (1) (2) (1) q0 (xc∗ + d) f (x, . . . , x) = q0 (cx + d)∗ f (M x , . . . , M x ) q0 (cx + d) for all M ∈ Γp .
3.4
Some remarks on Clifford-analytic modular forms in real and complex Minkowski spaces
We conclude by briefly outlining how the theory of s-monogenic modular forms for Γd [n] extends to the more general context of arbitrary real or complex quadratic spaces with a non-degenerate bilinear form (p, q). A detailed discussion on this topic can be found in [89, 90, 93]. In the beginning of Chapter 2.1 we have al(d) ready outlined how the translative Eisenstein series m (z) extend to IKp,q and (d) which convergence conditions are required. Remember that the series Gm (z) and the families of s-monogenic modular forms associated to Γd [n] that we discussed previously are all defined on a proper upper half-space. In order to treat generalizations of them to the setting of arbitrary Minkowski type spaces IKp,q , one first needs to figure out what are the adequate analogues of the upper half-space in the Minkowski spaces for our needs.
3.4. Modular forms in Minkowski spaces
145
The structure of the null cones in the underlying Minkowski type spaces IKp,q motivates the consideration of the following circular half-cones in IKp,q . In the case IK = IR consider H p+ (IRp,q ) = {x ∈ IRp,q |
p j=1
and H
q+
p,q
(IR
) = {x ∈ IR
p,q
|
p
k
x2j >
x2j , x1 > 0}
j=p+1
x2j
j=1
<
k
x2j , xk > 0}.
j=p+1
Notice that in the case p = 0 one obtains H p+ (IR0,k ) = ∅ and H q+ (IR0,k ) = {x ∈ IR0,k | xk > 0}. In the case q = 0 one gets similarly that H q+ (IRk,0 ) = ∅ and H p+ (IRk,0 ) = {x ∈ IRk,0 | x1 > 0}. In what follows we assume without loss of generality that p < q and use for simplicity the notation H + (IRp,q ) for H q+ (IRp,q ). In the complexified case consider in the case p = 0 the following half-cone (compare with [3]) as its complexification: H + (C0,k ) = {Z = x + iy ∈ C0,k | x2k >
k−1
yj2 , xk > 0}.
j=1
In the case p = 1, q = k − 1 consider the classical Siegel half-space H + (C1,k−1 ) = {Z = x + iy ∈ C1,k−1 | yk2 >
k−1
yj2 , yk > 0}.
j=1
In the remaining complexified cases with p, q ≥ 2, p < q consider the domains +
p,q
H (C
) = {Z = x + iy ∈ C
p,q
|
yk2
>
p
yj2 +
j=1
k−1
x2j , yk > 0}
j=p+1
which of course are Siegel type domains as well. Similar domains are considered for p > q. One can verify by a direct calculation that these half-spaces have no intersection with the nullcone. As outlined in Section 1.2, M¨ obius transformations in IKp,q can be described in terms of the Vahlen groups SV (IKp,q ). The group SV (Cp,q ) contains the real Vahlen groups SV (IRr,s ) where r ≤ p, s ≤ q as subgroups. Each arbitrary discrete subgroup of SV (Cp,q ) is contained in a group that is isomorphic to ΓC (Ω2k ) where ΓC (Ω2k ) = J, Tω1 , · · · , Tω2k
146
Chapter 3. Clifford-analytic Modular Forms
where ω1 , . . . , ω2k are IR-linear independent vectors from Ck . Of course, the whole group ΓC (Ω2k ) does not act discontinuously for any arbitrary choice of lattice. Let us again restrict consideration to ω1 = e1 , . . . , ωk = ek , ωk+1 = ie1 , . . ., ω2k = iek which involves discontinuous actions. Turning first to the case IK = IR, one sees immediately that the group Γd (IRp,q ) = J, Tek−d , . . . , Tek−1 leaves H q+ (IRp,q ) invariant when d ≤ q − 1. One observes that the case where p = 0, q = k (or similarly p = k, q = 0) is special. Under this condition, H + (IRp,q ) is invariant under the modular group with the maximal number of translative generators. The situation is different in the complexified case where all the cases are equivalent. In the complexified case we always have an invariance under the inversion and a translation group generated by k linear independent translation matrices, independently from the choice of p and q. The space H q+ (C0,k ) is left invariant under Γk (C0,k ) = J, Te1 , . . . , Tek−1 , Tiek . Similarly, H q+ (C1,k−1 ) is left invariant under Γk (C1,k−1 ) = J, Te1 , . . . , Tek . Analogously, for p, q ≥ 2, the half-cone H + (Cp,q ) is invariant under Γk (Cp,q ) = J, Te1 , . . . , Tep , Tiep+1 , . . . , Tiek−1 , Tek . All these groups and subsequently also their principal congruence subgroups of any arbitrary level act discontinuously on the respective half-space. These halfspaces are then the adequate definition domains for an extension of the theory of s-monogenic modular forms related to the associated hypercomplex modular groups to the setting of arbitrary Minkowski type spaces. The conformal invariance formula from Theorem 1.28 extends in a natural way to the framework of circular half-cones and related groups considered here, simply by replacing the expression (s) q0 (cx + d) more generally by 1 s ≡ 0 mod 2, s < k = p + q,
(cx + d)(cx + d) (k−s)/2 (s) Q0 (cx + d) = cx + d s ≡ 1 mod 2, s < k = p + q,
(cx + d)(cx + d) (k−s+1)/2 for x ∈ IRp,q , or by (s) Q0 (cZ + d) =
1 [(cZ + d)(cZ + d)](k−s)/2 cZ + d [(cZ + d)(cZ + d)](k−s+1)/2
s ≡ 0 mod 2, s < k = p + q, s ≡ 1 mod 2, s < k = p + q,
3.5. Some Perspectives
147
for Z ∈ Cp,q whenever p + q = k ≡ 0 mod 2. It is crucial to observe that the (s) (s) expressions Q0 (cx + d) and Q0 (cZ + d) are again automorphy factors. In order to construct Clp,q (IKp,q )-valued modular forms on H q+ (IKp,q ) to Γd (IKp,q )[n] that are annihilated by DIsKp,q one only has to replace in Theorem 3.4 (s) (s) (s) the expression q0 (cx + d) by Q0 (cx + d) or Q0 (cZ + d), respectively, and to ˜ consider for f adequate starting functions that are invariant under the translation group contained in Γd (IKp,q )[n], say Td (IKp,q )[n]. In the case IK = IR one obtains the convergence condition d < k − s, d ≤ q. The convergence conditions for the complexified modular forms on H q+ (Cp,q ) with respect to Γd (IKp,q )[n] are always d < k − s, independently from the signature of the space. Under these conditions, the constructions (s) Q0 (cx + d)∗ f˜(M x ) x ∈ H + (IRp,q ) M :Γd (IRp,q )[n]\Td (IRp,q )[n]
and
p,q
M :Γd (C
(s)
Q0 (cZ + d)∗ f˜(M Z ) p,q
)[n]\Td (C
Z ∈ H + (Cp,q )
)[n]
provide us with non-trivial examples of Clp,q (IK)-valued modular forms with respect to Γd (IRpq )[n] in Ker DIsKp,q when inserting e.g. for f˜ the function f˜ ≡ 1 (d) (d) or f˜(x) = Gm (x + β, Td (IRp,q )), x ∈ H + (IRp,q ) resp. f˜ = Gm (Z + β, Td (Cp,q )), Z ∈ H + (Cp,q ) where β may be chosen arbitrarily from H + (IKp,q ). In [89, 90] we gave elementary convergence proofs. The convergence of the complexified series in the half-cone H q+ (Cp,q ) which is the complexified extension of a series that converges in a half-space from IRk can also by proved by applying the classical extension theorems from several complex variables theory on its restriction to the proper half-space of IRk whenever the centers of the singularity cones of the complexified series lie on IRk -like domain manifolds. In this case a function series that converges on a half-space of IRk normally to a real analytic function lifts naturally to a complex-analytic function on the complexification of the half-space under the same convergence conditions. See [152, 77] for the basic theory of the general extension theorems that are needed to meet this end. Applying the same modifications, we can also obtain extensions of Theorem 3.9 and Theorem 3.11 in the context of Cartesian products of the half-spaces H q+ (IKp,q ). For the analogue of Theorem 3.9 one needs again to claim that n ≥ 3 for odd s in order to get non-trivial examples. For the detailed elementary convergence proofs and particular examples see [90] and [93].
3.5
Some Perspectives
The constructions and examples developed in Chapter 3 extend the results from Chapter 2 from the context of translation groups to that of more general arith-
148
Chapter 3. Clifford-analytic Modular Forms
metic subgroups of the Vahlen group, including generalizations of the classical modular group SL(2, Z) and their principle congruence subgroups. In turn it is realistic to expect an amplification of the range of the particular applications of the translative s-monogenic Eisenstein series to a much more general context, using s-monogenic automorphic forms for more general discrete subgroups of the Vahlen group instead. This includes the hope to arrive at closed formulas for Bergman and Szeg¨o reproducing kernel functions of a number of much more general polyhedron type domains that include rectangular and strip domains as the simplest ones. Indeed, as mentioned at the end of Chapter 3.1, the Bergman kernel function of a domain that is composed by the Cartesian product of a two-dimensional fractional wedge-shaped domain in spanIR {1, ek } and a rectangular domain from spanIR {e1 , . . . , ep } turned out actually to be a variant of an Eisenstein type series which is related to the discrete rotation group generated by the matrix diag(exp( πenk ), exp(− πenk )) and to a translation group in spanIR {e1 , . . . , ep }. This result is a first step in the suggested direction and underlines some hope that in future analogous results could be established within a more general context. While the translation invariant s-monogenic Eisenstein series from Chapter 2 provide building blocks for s-monogenic functions that are defined on conformally flat cylinders and tori in IRk , the s-monogenic automorphic forms associated to more general discrete subgroups of the Vahlen group give rise to s-monogenic functions defined on more general conformally flat spin manifolds which in turn arise from factoring out a domain from IRk by the discrete group. A central problem that arises in this context is then to look for special representatives within the classes of s-monogenic automorphic forms that give rise to Cauchy or Green kernels which admit consequently the solution of important boundary value problems on these kinds of manifolds, including in particular the Dirichlet problem. In the recent paper [97] one step has been taken in this more general direction within the framework of some particular discrete subgroups of Vahlen’s group that are different from the translation group. With special automorphic forms associated to the groups G1 = {±1} and G2 = {2k } a Cauchy kernel could be constructed for the real projective spaces P IRk ∼ = S k /G1 and for S 1 × S k−1 ∼ = IRk \{0}/G2 . Furthermore, Cauchy and Green k kernels to IR \{0}/Vp , (1 ≤ p ≤ k) were constructed by means of adequate representatives of automorphic forms for the transversion group Vp . The transversion group is conjugated to the translation group Tp . While the Cauchy kernel associ(p) ated to the translation group, given by 0 , has no accumulation of singularities k within IR but at ∞, the Cauchy kernel function to the transversion group has a finite accumulation point of singularities, namely the origin. A combination of representation formulas for the Cauchy kernel derived for Tp , G1 and G2 lead then to further Cauchy kernel functions that are defined on manifolds that are constructed by factoring out a proper domain by a group that arises by forming the semi-direct product of Tp , G1 and G2 , respectively.
3.5. Some Perspectives
149
These kernel functions give then an immediate access to derive Plemelj projection formulas and explicit formulas for the Kerzman–Stein kernels providing important tools to study Hardy spaces on these kinds of manifolds. As mentioned in [93], also the argument principle which we described explicitly for the Euclidean space in Chapter 1.5 and for conformally flat cylinders and tori in Chapter 2.11 extends directly to the context of these manifolds when the kernel function is known. In this case one simply replaces q0 by the proper kernel functions. These results indicate a hope that it might be possible to carry out the techniques developed for these particular examples to the framework of more general discrete subgroups of the groups considered in Chapter 3.1 – Chapter 3.3. The associated s-monogenic automorphic forms would perhaps provide the building blocks for obtaining the appropriate kernel functions. Here, we see a great potential of applications of the new function classes that we constructed in this chapter. In conclusion, on the one hand, the theory of Clifford-analytic automorphic forms provides a certain potential to contribute to analytic number theory in the framework of arithmetic subgroups of the orthogonal group as a counterpart to holomorphic modular forms in several complex variables and to the non-analytic Maaß wave forms. On the other hand this theory opens the door to treat a number of problems of current interest from functional analysis, index theory, boundary value problems and spin geometry that arise in a natural way from harmonic analysis.
Bibliography [1] L. V. Ahlfors. M¨ obius transformations in Rn expressed through 2×2 matrices of Clifford numbers. Complex Variables 5 (1986), 215-224. [2] P. Alexandroff and H. Hopf. Topologie I. Chelsea, Bronx, New York, 1935. [3] V. Avanssian. Sur les fonctions harmoniques d’ ordre quelconque et leur prolongement analytique dans Cn . In: S´eminaire Pierre Lelong - Henri Skoda (Analyse) Ann´ees 1980/81 et Colloque de Wimereux, Mai 1981 ed. A. Dold and B. Eckmann, Springer Lecture Notes 919, Springer Berlin-HeidelbergNew York, 1982, pp. 192-281. [4] J. Baez. The Octonions. Bull. Amer. Math. Soc. 39 No. 2 (2001), 145-205. [5] K. Bakkesa, Swamy and N. Nagaraj. Conformality, Differentiability and regularity of quaternionic functions. Journal of the Indian Math. Soc. 47 (1983), 21-30. [6] S. Bell. The Cauchy Transform, Potential Theory and Conformal Mapping. CRC Press, Boca Raton, 1992. [7] A. Bitsadse. Grundlagen der Theorie der Analytischen Funktionen, Akademie-Verlag, Berlin, 1973. [8] W. Blaschke. Vorlesung u ¨ber Differentialgeometrie I, Springer Verlag, Berlin, 1924. ¨ [9] O. Blumenthal. Uber Modulfunktionen von mehreren Ver¨ anderlichen. Math. Ann. 56 (1903), 509-548 and 58 (1904), 497-527. [10] P. Boßhard. Die Cliffordschen Zahlen, ihre Algebra und ihre Funktionentheorie. PhD Thesis, Univ. Z¨ urich, 1940. [11] F. Brackx. Teorie van de (k)-monogene funkties. PhD Thesis Ghent State University, 1973-74. [12] F. Brackx and R. Delanghe. Hypercomplex function theory and Hilbert modules with reproducing kernel. Proc. London Math. Soc. 37 (1978), 545-576. [13] F. Brackx, R. Delanghe and F. Sommen. Clifford Analysis. Pitman Res. Notes 76, Boston-London-Melbourne, 1982.
152
Bibliography
[14] H. Brandt. Idealtheorie in Quaternionenalgebren. Math. Ann. 99 (1928), 129. [15] J.H. Bruinier and M. Kuss. Eisenstein series attached to lattices and modular forms on orthogonal groups. Preprint: http://kleene.wisc.edu/ bruinier/eis.ps [16] J.J. Buff. Characterization of analytic functions of a quaternion variable. Pi Mu Epsilon J. (1973), 387-392. [17] F. B¨ uhler. Die symplektische Struktur f¨ ur orthogonale Gruppen und Thetareihen als Modulformen, PhD Thesis, RWTH Aachen, 2000. [18] D. Calderbank. Dirac operators and Clifford analysis on manifolds with boundary. Max Plank Institute for Mathematics Bonn, Preprint number 96131, 1996. [19] S. Carlip. Quantum Gravity in (2+1) Dimensions. Cambridge Monographs on Mathematical Physics, Cambridge, 1998. [20] A. Chang, J. Qing and P. Yang. Compactification of a class of conformally flat 4-manifold. Invent. Math. 142 (2000), 65-93. [21] D. Constales. The relative position of L2 domains in complex and Clifford analysis. PhD Thesis, Ghent State University, 1989-1990. [22] D. Constales. The Bergman and Szeg¨ o kernels for separately monogenic functions. Z. Anal. Anwendungen 9 No. 2 (1990), 97-103. [23] D. Constales and R. S. Kraußhar. Representation formulas for the general derivatives of the fundamental solution of the Cauchy-Riemann operator in Clifford Analysis and Applications. Z. Anal. Anwendungen 21 No. 3 (2002), 579-597. [24] D. Constales and R.S. Kraußhar. Szeg¨ o and Polymonogenic Bergman Kernels for half-space and strip domains, and Single-periodic Functions in Clifford Analysis. Complex Variables 47 No. 4 (2002), 349-360. [25] D. Constales and R.S. Kraußhar. Bergman Kernels for rectangular domains and Multiperiodic Functions in Clifford Analysis. Mathematical Methods in the Applied Sciences 25 No. 16-18 (2002), 1509-1526. [26] D. Constales and R.S. Kraußhar. Closed formulas for singly-periodic monogenic cotangent, cosecant and cosecant-squared functions in Clifford Analysis. Journal of the London Mathematical Society 67 No. 2 (2003), 401 416. [27] D. Constales and R.S. Kraußhar. The Bergman Kernels for the half-ball and for fractional wedge-shaped domains in Clifford Analysis. Submitted for publication. [28] H.S.M. Coxeter. Integral Cayley Numbers. Duke Math. Journal 13 (1946), 561-578.
Bibliography
153
[29] J. Cnops. Hurwitz pairs and applications of M¨ obius transformations. Habilitation Thesis Ghent State University, 1993-1994. [30] J. Cnops. An Introduction to Dirac Operators on Manifolds. Progress in Mathematical Physics, Birkh¨ auser, Basel, 2002. [31] J. Cnops and H. Malonek. An introduction to Clifford Analysis. Textos de Matem´atica S´erie B, Departamento de Matem´atica da Universidade de Coimbra, 1995. [32] R. Delanghe. On regular-analytic functions with values in a Clifford algebra. Math. Ann. 185 (1970), 91-111. [33] R. Delanghe. On regular points and Liouville’s theorem for functions with values in a Clifford algebra. Simon Stevin 44 (1970-71), 55-66. [34] R. Delanghe. On the singularities of functions with values in a Clifford algebra. Math. Ann. 196 (1972), 293-319. [35] R. Delanghe. On Hilbert modules with reproducing kernel, Funct. theor. Meth. part. Differ. Equat., Proc. int. Symp., Darmstadt 1976, Lect. Notes Math. 561 (1976), 158-170. [36] R. Delanghe, R.S. Kraußhar and H.R. Malonek. Differentiability of functions with values in some real associative algebras: approaches to an old problem. Bulletin de la Soci´et´e Royale des Sciences de Li`ege 70 No. 4-6 (2001), 231249. [37] R. Delanghe, F. Sommen and V. Souˇcek. Clifford Algebra and Spinor Valued Functions. Kluwer, Dordrecht-Boston-London, 1992. [38] R. Delanghe, F. Sommen and V. Souˇcek. Residues in Clifford analysis. In: Begehr H. and Jeffrey A.(eds.), Partial differential equations with complex analysis, Pitman Res. Notes in Math. Ser. 262 (1992), 61-92. [39] T. Dern. Hermitesche Modulformen zweiten Grades. PhD Thesis RWTH Aachen, Wissenschaftsverlag Mainz, Aachen, 2001. [40] A. C. Dixon. On the Newtonian Potential. Quarterly Journal of Mathematics 35 (1904), 283-296. [41] G. Eisenstein. Genaue Untersuchung der unendlichen Doppelproducte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind, und der mit ihnen zusammenh¨ angenden Doppelreihen (als eine neue Begr¨ undung der Theorie der elliptischen Functionen, mit besonderer Ber¨ ucksichtigung ihrer Analogie zu den Kreisfunctionen). Crelle’s Journal 35 (1847), 153-274. [42] J. Elstrodt, F. Grunewald and J. Mennicke. Eisenstein Series on threedimensional hyperbolic space and imaginary quadratic number fields. J. Reine Angew. Math. 360 (1985), 160-213.
154
Bibliography
[43] J. Elstrodt, F. Grunewald and J. Mennicke. Vahlen’s Group of Clifford matrices and spin-groups. Math. Z. 196 (1987), 369-390. [44] J. Elstrodt, F. Grunewald and J. Mennicke. Arithmetic applications of the hyperbolic lattice point Theorem. Proc. London Math. Soc. III 57 No.2 (1988), 239-288. [45] J. Elstrodt, F. Grunewald and J. Mennicke. Kloosterman sums for Clifford algebras and a lower bound for the positive eigenvalues of the Laplacian for congruence subgroups acting on hyperbolic spaces. Invent. Math. 101 No.3 (1990), 641-668. [46] J. Elstrodt. Partialbruchentwicklung des Kotangens, Herglotz-Trick und die Weierstraßsche stetige, nirgends differenzierbare Funktion. Math. Semesterber. 45 No.2 (1998), 207-220. [47] S.-L. Eriksson-Bique and H. Leutwiler. Some integral formulas for hypermonogenic functions. Submitted to: Clifford Analysis and its Applications, Birkh¨ auser Basel. (Proceedingsvolume of the International Conference on Clifford Analysis and its Applications held at Macau (China), 15-18 August 2002). [48] E. Freitag. Hilbert Modular Forms. Springer, Berlin-Heidelberg-New York, 1990. [49] E. Freitag and R. Busam. Funktionentheorie, 2nd edition. Springer, BerlinHeidelberg, 1995. [50] E. Freitag and C. F. Hermann. Some modular varieties of low dimension. Adv. Math. 152 No.2 (2000), 203-287. ¨ [51] G. Frobenius. Uber lineare Substitutionen and bilineare Formen. J. Reine Angew. Math. 84 (1878), 1-63 [52] R. Fueter. Einige S¨ atze aus der Theorie der komplexen Multiplikation der elliptischen Funktionen. Comp. Rend. Congres Int. Math. Strasbourg 1920, 1920. [53] R. Fueter. Vorlesungen u ¨ber die singul¨ aren Moduln und die komplexe Multiplikation der elliptischen Funktionen. Teubner, Leipzig I-1924. ¨ [54] R. Fueter. Uber automorphe Funktionen in Bezug auf Gruppen, die in der Ebene uneigentlich diskontinuierlich sind. Crelle’s Journal 137 (1927), 66-77. [55] R. Fueter. Ueber automorphe Funktionen der Picard’schen Gruppe I. Comment. Math. Helv. 3 (1931) 42-68. [56] R. Fueter. Analytische Funktionen einer Quaternionenvariablen. Comment. Helv. Mat. 4 (1932), 9-20. [57] R. Fueter. Quaternionenringe. Comment. Math. Helv. 6 (1934), 199-222.
Bibliography
155
¨ [58] R. Fueter. Uber die analytische Darstellung der regul¨ aren Funktionen einer Quaternionenvariablen. Comment. Math. Helv. 8 (1935-36), 371-378. [59] R. Fueter. Die Theorie der regul¨ aren Funktionen einer Quaternionenvariablen. Compt´e Rendus du Congr´es Internactional de Math´ematiciens, Oslo, 1936, 75-91. [60] R. Fueter. Die Singularit¨ aten der eindeutigen regul¨ aren Funktionen einer Quaternionenvariablen I. Comment. Math. Helv. 9 (1936-37), 320-334. ¨ [61] R. Fueter. Uber vierfachperiodische Funktionen. Monatshefte Math. Phys. 48 (1939), 161-169. ¨ [62] R. Fueter. Uber die Quaternionenmultiplikation der vierfachperiodischen regul¨ aren Funktionen. Experientia 1 (1945), 57. [63] R. Fueter. Functions of a Hyper Complex Variable. Lecture notes written and supplemented by E. Bareiss, Math. Inst. Univ. Z¨ urich, Fall Semester 1948/49. [64] R. Fueter. Ueber Abelsche Funktionen von zwei Komplexen Variablen. Ann. Mat. Pura Appl., IV Ser. 28 (1949), 211-215. [65] V. Gritsenko. Arithmetic of quaternions and Eisenstein series. J. Sov. Math. 52 No.3 (1990); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 160 (1987), 82-90. [66] K. G¨ urlebeck and W. Spr¨ ossig. Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley & Sons, Chichester-New York, 1997. [67] F. G¨ ursey and H. Tze. Complex and Quaternionic Analyticity in Chiral and Gauge Theories I. Annals of Physics 128 (1980), 29-130. [68] F. G¨ ursey and H. Tze. On the role of division, Jordan and related algebras in particle physics. World Scientific, Singapore, 1996. [69] P. Hartman. On isometries and a theorem of Liouville. Math. Z. 69 (1958), 202-210. [70] E. Hecke. H¨ohere Modulfunktionen und ihre Anwendungen auf die Zahlentheorie. Math. Ann. 71 (1911), 1-37. ¨ [71] E. Hecke. Uber die Konstruktion relativ Abel’scher Zahlk¨ orper durch Modulfunktionen von zwei Variablen. Math. Ann. 74 (1913), 465-510. [72] T. Hempfling. Beitr¨ age zur Modifizierten Clifford-Analysis. PhD Thesis, University of Erlangen-N¨ urnberg, 1997. [73] T. Hempfling. Some remarks on zeroes of monogenic functions. Advances in Appl. Clifford Algebras 11(S2) (2001), 107-116. [74] T. Hempfling and R.S. Kraußhar. Order Theory for Isolated Points of Monogenic Functions. Archiv der Mathematik 80 (2003), 406-423.
156
Bibliography
[75] D. Hilbert. The Theory of Algebraic Number Fields (Zahlbericht 1896). Springer, Berlin-Heidelberg-New York, 1998. [76] L. H¨ ormander. An introduction to complex analysis in several variables. 3rd edition, North-Holland Mathematical Library, 1990. [77] L. K. Hua. Harmonic Analysis of Functions of Several Complex Variables and the Classical Domains. AMS Providence, Rhode Island, 1963. [78] G. Jank and L. Volkmann. Meromorphe Funktionen und Differentialgleichungen. Birkh¨ auser, Basel, 1985. [79] V. Iftimie. Fonctions hypercomplexes. Bull. Math. Soc. Sci. Math. Repub. Soc. Roum., Nouv. Ser. 9 (57), (1965) 279-332. [80] E. K¨ ahler. Die Poincar´e-Gruppe. Mathematica, Festschrift Ernst Mohr, Berlin, 1985, 117-144. [81] F. Klein and R. Fricke. Vorlesungen u ¨ber die Theorie der elliptischen Modulfunktionen I,II. Teubner, Leipzig 1890-1892. [82] M. Koecher and A. Krieg. Elliptische Funktionen und Modulformen. Springer, Berlin-Heidelberg, 1998. [83] R. S. Kraußhar. Conformal Mappings and Szeg¨ o Kernels in Quaternions. Diplomarbeit, Lehrstuhl II f¨ ur Mathematik, RWTH Aachen, 1998 [84] R. S. Kraußhar. Eisenstein Series in Clifford Analysis. Ph.D. Thesis RWTH Aachen, Aachener Beitr¨age zur Mathematik 28, Wissenschaftsverlag Mainz, Aachen, 2000. [85] R. S. Kraußhar. Monogenic multiperiodic functions in Clifford analysis. Complex Variables 46 No. 4 (2001), 337-368. [86] R. S. Kraußhar. On a new type of Eisenstein series in Clifford analysis. Z. Anal. Anwendungen 20 No. 4 (2001), 1007-1029. [87] R. S. Kraußhar. Generalizations of the complex analytic trigonometric functions to Clifford analysis by Eisenstein series. In: Functional-Analytic and complex methods, their interactions, and Applications to Partial Differential Equations (Proceedings of the International Graz Workshop Graz, Austria, 12-16 February 2001) eds. H. Florian, N. Ortner, F.J. Schnitzer, W. Tutschke, World Scientific, 2001, pp. 438-456. [88] R. S. Kraußhar. Automorphic Forms in Clifford Analysis. Complex Variables 47 No. 5 (2002), 417-440. [89] R. S. Kraußhar. Eisenstein Series in Complexified Clifford analysis. Computational Methods and Function Theory 2 No. 1 (2002), 29-65. [90] R. S. Kraußhar. Monogenic Modular Forms in Two and Several Real and Complex Vector Variables. Computational Methods and Function Theory 2 No. 2 (2002), 299-318.
Bibliography
157
[91] R. S. Kraußhar. Poincar´e series in Clifford analysis. In: Clifford Algebras: Applications to Mathematical Physics and Engineering, ed. Rafal Ablamowicz, Progress in Mathematical Physics, Birkh¨ auser, Boston, 2003, pp. 75-90. [92] R. S. Kraußhar. The Multiplication of the Clifford-analytic Eisenstein series. Journal of Number Theory 102 No. 2 (2003), 353-382. [93] R. S. Kraußhar. A Theory of Modular Forms in Clifford Analysis, their Applications and Perspectives. To appear in: Advances in Analysis and Geometry. New Developments Using Clifford Algebras, eds. T. Qian, T. Hempfling, A. MacIntosh, F. Sommen, Trends in Mathematics, Birkh¨ auser, Basel, 2004. Preprint: http://www.mathpreprints.com/math/Preprint/krauss/20021127/1/mac.pdf
[94] R. S. Kraußhar: Automorphic Forms and Functions in Clifford Analysis and Their Applications, Habilitationsschrift, RWTH Aachen, 2003. [95] R. S. Kraußhar and H. R. Malonek: A characterization of conformal mappings in R4 by a formal differentiability condition. Bulletin de la Soci´et´e Royale des Sciences de Li`ege 70 No. 1 (2001), 35-49. [96] R. S. Kraußhar and J. Ryan. Clifford and Harmonic Analysis on Cylinders and Tori. To appear in: Revista Matematica Iberoamericana. Preprint: http://www.mathpreprints.com/math/Preprint/krauss/20020801/3/kr-ryan5.pdf
[97] R. S. Kraußhar and J. Ryan. Some Conformally Flat Spin Manifolds, Dirac Operators and Automorphic Forms. Submitted for publication. Preprint: http://arXiv.org/abs/math.AP/0212086
[98] V. Kravchenko and M. Shapiro: Integral representations for spatial models of mathematical physics, Addison Wesley Longman, Harlow, 1996. [99] A. Krieg. Modular Forms on Half-Spaces of Quaternions. Springer Verlag, Berlin-Heidelberg, 1985. [100] A. Krieg. Eisenstein series on real, complex and quaternionic half-spaces. Pac. J. Math. 133 No.2 (1988), 315-354. [101] A. Krieg. Eisenstein-Series on the Four-Dimensional Hyperbolic Space. Journal of Number Theory 30 (1988), 177-197. [102] N.N. Krylov. Dokl. AN SSSR 55 No.9 (1947),799-800. [103] S. Lang. Elliptic functions (2nd edition). Graduate Texts in Mathematics 112, Springer, New York, 1987. [104] G. Laville and I. Ramadanoff. Holomorphic Cliffordian functions. Adv. Appl. Clifford Algebr. 8 No.2 (1998), 323-340. [105] G. Laville and I. Ramadanoff. Elliptic Cliffordian Functions. Complex Variables 45 No.4 (2001), 297-318.
158
Bibliography
[106] G. Laville and I. Ramadanoff. Jacobi Elliptic Cliffordian Functions. Complex Variables 47 No.9 (2002), 787-802. [107] H. Leutwiler. Modified Clifford analysis. Complex Variables 17 (1991), 153171. [108] C. Li, A. McIntosh and T. Qian. Clifford Algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces. Rev. Mat. Iberoamericana 10 (1994), 665-721. [109] X. Li, Z. Khai and L. Peng. The Laurent Series on the Octonions. Advances in Appl. Clifford Algebras 11(S2) (2001), 205-127. [110] X. Li, Z. Khai and L. Peng. Characterization of octonionic analytic functions. To appear in Complex Variables. [111] J. Liouville. Extension au cas de trois dimensions de la question du trac´e g´eographique. Application de l’ analyse a ` la g´eometrie, G. Monge, Paris (1850), 609-616 [112] J. Liouville. Lectures published by C.W. Borchardt. J. Reine Angew. Math. 88 (1880), 277-310. [113] H. Liu and J. Ryan. Clifford analysis techniques for spherical pde. Journal of Fourier Analysis and its Applications 8 No.6 (2002), 535-564. [114] P. Lounesto. Clifford algebras and spinors. 2nd edition. London Mathematical Society Lecture Note Series, 286, Cambridge University Press, Cambridge, 2001. [115] H. Maaß. Zur Theorie der automorphen Funktionen von n Ver¨anderlichen. Math. Ann. 117 (1940), 538-578. ¨ [116] H. Maaß. Uber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 121 (1949), 141-183. [117] H. Maaß. Automorphe Funktionen von mehreren Ver¨ anderlichen und Dirichletsche Reihen. Abh. Math. Sem. Univ. Hamb. 16 (1949), 53-104. [118] H. Malonek. Zum Holomorphiebegriff in h¨ oheren Dimensionen. Habilitation Thesis, Halle, 1987. [119] H. Malonek. A new hypercomplex structure of the Euclidean space IRm+1 and the concept of hypercomplex differentiability. Complex Variables 14 (1990), 25-33 [120] H. Malonek. Power series representation for monogenic functions in IRm+1 based on a permutational product. Complex Variables 15 (1990), 181-191. [121] A. S. Melijhzon. Because of monogenicity of quaternions (in russ.) Doklady Acad. Sc. USSR 59 (1948), 431-434.
Bibliography
159
[122] G. C. Moisil and N. Theodorescu. Fonctions holomorphes dans l’espace, Bul. Soc. Stiint. Cluij 6 (1931), 177-194. ¨ [123] W. Nef. Uber die singul¨aren Gebilde der regul¨ aren Funktionen einer Quaternionenvariablen. Comment. Math. Helv. 15 (1942-43), 144-174. [124] W. Nef. Die unwesentlichen Singularit¨ aten der regul¨ aren Funktionen einer Quaternionenvariablen. Comment. Math. Helv. 16 (1943-44), 284-304. [125] W. Nef. Funktionentheorie einer Klasse von hyperbolischen und ultrahyperbolischen Differentialgleichungen 2. Ordnung. Comm. Math. Helv. 17 (1944), 83-107. [126] J. Peetre and P. Sj¨ olin: Three-line theorems and Clifford analysis, Complex Variables 19 No. 3 (1992), 151-163. [127] H. Poincar´e. Oeuvres II, Gauthier-Villars, Paris, 1916-1954 [128] T. Qian. Singular integrals with monogenic kernels on the m-torus and their Lipschitz perturbations. In: Clifford Algebras in Analysis and Related Topics, J. Ryan (eds.), CRC Press, Boca Raton, 1996, 157-171. ¨ [129] H. Reich. Uber das Verh¨ altnis einer regul¨ aren Quaternionenfunktion in der N¨ ahe eines unwesentlich singul¨ aren Punktes. PhD Thesis Univ. Z¨ urich, 1948. [130] R. Remmert. Theory of Complex Functions. Springer, Berlin-Heidelberg, 1991. [131] O. Richter. Theta functions with harmonic coefficients over number fields. J. Number Theory 95 No.1 (2002), 101-121. [132] O. Richter and H. Skogman. Jacobi theta functions over number fields. To appear in Monatshefte f¨ ur Mathematik. [133] J. Ryan. Clifford Analysis with Generalized Elliptic and Quasi Elliptic Functions. Appl. Anal. 13 (1982), 151-171. [134] J. Ryan. Complexified Clifford Analysis. Complex Variables 1 (1982), 119149. [135] J. Ryan. Singularities and Laurent Expansions in Complex Clifford Analysis. Applicable Analysis 16 (1983) 33-49. [136] J. Ryan. Properties of isolated singularities of some functions taking values in a real Clifford algebra. Math. Proc. Cambr. Phil. Soc. 95 (1984), 277-298. [137] J. Ryan. Conformal Clifford manifolds arising in Clifford analysis. Proc. R. Ir. Acad 85A No.1 (1985), 1-23. [138] J. Ryan. Iterated Dirac operators and conformal transformations in IRn . In: Proceedings XVth International Conference on Differential Geometric Methods and Theoretical Physics, Clausthal 1986, World Scientific, Cleveland, 1987.
160
Bibliography
[139] J. Ryan. Iterated Dirac operators in C n . Z. Anal. Anwendungen 9 (1990), 385-401. [140] J. Ryan. Generalized Schwarzian derivatives for generalized fractional linear transformations. Ann. Pol. Math. 5 No.1 (1992), 29-44. [141] J. Ryan. Intertwining operators for iterated Dirac operators over Minkowskitype spaces. J. Math. Anal. Appl. 177 No.1, (1993) 1-23. [142] J. Ryan: Cauchy-Green type formulae in Clifford analysis. Trans. Am. Math. Soc. 347 No.4 (1995), 1331-1341. [143] J. Ryan. Dirac operators on spheres and hyperbolae. Bolletin de la Sociedad Matematica a Mexicana 3 (1996), 255-270. [144] J. Ryan. Basic Clifford Analysis. Cubo Matem´ atica Educacional 2 (2000), 226-256. [145] J. Ryan. Cauchy kernels for some Conformally Flat Manifolds. Preprint 2003. [146] M. Sce. Osservazione sulle serie di potenzi nei moduli quadratici. Lincei Rend. Sci. Fis. Mat. et Nat. 23 (1957), 220-225. [147] G. Scheffers. Verallgemeinerung der Grundlagen der gew¨ ohnlichen complexen Zahlen. Berichte kgl. S¨ achs. Ges. der Wiss. 52 (1893), pp.60 [148] R. Schoen and S.-T. Yau. Conformally flat manifolds, Kleinian groups and scalar curvature. Inventiones Mathematica 92 (1988), 47-71. [149] B. Schoeneberg. Elliptic Modular Functions. Die Grundlagen der mathematischen Wissenschaften 201, Springer, Berlin-Heidelberg-New York, 1974. [150] M. Shapiro and N. Vasilevski. On the Bergman kernel function in the Clifford analysis. Brackx, F. (ed.) et al., Clifford algebras and their applications in mathematical physics. Proceedings of the third conference held at Deinze, Belgium, 1993. Dordrecht: Kluwer Academic Publishers. Fundam. Theor. Phys. 55 (1993), 183-192. [151] M. Shapiro and N. Vasilevski. On the Bergmann kernel function in hyperholomorphic analysis, Acta Appl. Math. 46 No. 1 (1997), 1-27. [152] J. Siciak. Holomorphic continuation of harmonic functions. Ann. Polon. Math. 29 (1974), 67-73. [153] C.L. Siegel. Topics in Complex Function Theory Vol. III. Wiley, New YorkChichester, 1988. [154] F. Sommen. Some connections between Clifford and Complex Analysis. Complex Variables 1 (1982), 97-118. [155] F. Sommen. Plane waves, biregular functions and hypercomplex Fourier analysis. Rend. Circ. Mat. Palermo, II. Ser., Suppl. 9 (1985), 205-219.
Bibliography
161
[156] F. Sommen. Clifford analysis in two and several vector variables. Appl. Anal. 73 (1999), 225-253. [157] F. Sommen and G. Bernardes. Multivariable monogenic functions of higher spin. J. Nat. Geom. 18 No.1-2 (2000), 101-114. [158] V. Souˇcek. Complex-quaternionic analysis applied to spin-1/2 massless fields. Complex Variables 1 (1983), 327-346. [159] W. Spr¨ ossig. On operators and elementary functions in Clifford analysis. Z. Anal. Anwendungen 18 No. 2 (1999), 349-360. [160] A. Sudbery. Quaternionic analysis. Math. Proc. Camb. Phil. Soc. 85 (1979), 199-225. ¨ [161] K. Th. Vahlen. Uber Bewegungen und Complexe Zahlen. Math. Ann. 55 (1902), 585-593. [162] P. Van Lancker. Clifford analysis on the sphere. In: Clifford Algebras and their Applications in Mathematical Physics, eds. V. Dietrich et al, Kluwer, Dordrecht, 1998, 201-215. [163] N. Vasilevski and M. Shapiro. On Bergmann kernel functions in quaternion analysis. Russ. Math. 42, No. 2 (1998), 81-85. [164] J.P. Ward. Quaternions and Cayley Numbers. Mathematics and its Applications 403, Kluwer, Dordrecht, 1997. [165] G. Z¨ oll. Ein Residuenkalk¨ ul in der Clifford-Analysis und die M¨ obiustransformationen f¨ ur euklidische R¨ aume. PhD Thesis, Lehrstuhl II f¨ ur Mathematik RWTH Aachen, 1987.
List of Symbols Ak+1 Ak+1 Ak+1 (IKp,q ) B(z ∗ , R) B(z ∗ , R) B 2 (G, C), B 2 (G, Cl0k (IR)) C (C, IRk ), (C, Ak+1 ) Clp,q (IK) Cl+ p,q (IK) Cp D, DIKp,q D, DAk+1 (IKp,q ) Dhyp ∆ dσ (s) E(l) (Ωp ; Ak+1 (IKr,q )), E(l) (Ωp ) Γp (IRk ), Γp (Ak+1 ), Γp Γp [n] GV (IRk ), GV (Ak+1 ) IH H2 (G, C), H2 (G, Cl0k (IR)) H + (IRk ), H + (Ak+1 ) H r (Ak+1 ) I Im J IK L2 (·, ·) Lp M SV (Ak+1 ) IN IN0 O O Op Ωp ord(f ; c) Q
surface of (k + 1)-dim. unit ball, 18 set of paravectors in Cl0,k (IR), 3 set of paravectors in Clp,q (IK), 3 open ball with center z ∗ and radius R, 19 closed ball with center z ∗ and radius R, 19 Bergman spaces, 98 complex numbers, 2 Clifford group, 6 Clifford algebra over IKp,q , 3 even Clifford algebra, 3 cylinders, 111 Dirac operators, 16 Cauchy–Riemann operators, 16 hyperbolic Cauchy–Riemann operator, 24 Laplace operator in real Euclidean space, 23 oriented surface differential, 18 set of elliptic functions, 60 special hypercomplex modular groups, 9 principal congruence subgroups of Γp , 9 general Vahlen group 6 quaternions, 1 Hardy spaces, 98 upper half-spaces, 7 right half-space, 7 identity matrix, 9 imaginary part, 3 inversion matrix, 7 the real or complex number field, 2 L2 -spaces, 99 p-dimensional orthonormal lattice, 70 modified Vahlen group, 7 positive integers, 5 non-negative integers 5 octonions, Cayley numbers, 1 Landau symbol, 79,102 standard Z-order, 8 p-dimensional lattice, 9 order of f at c, 28 rational numbers, 8
164
List of Symbols IR Re res Res S S0 Sz ∗ Sc SV (IRk ), SV (Ak+1 ) Ta Tp Tk+1 T (Ωp ) τ (i) V Vp V ec wΓ Z
real numbers, 2 real part, 3 ordinary residue, 21 Leray–Norguet residue, 22 trace of a paravector, 4 null cone, 4 singularity cone with center z ∗ , 4 scalar part, 3 special Vahlen group 7 translation matrix, 7 translation group for Lp , 9 (k + 1)-torus, 111 translation group for Ωp , 9 a particular multi-index, 5 IRk or Ak+1 , 8 transversion group, 9 vector part, 3 winding number, 18 integers, 5
Index algebraic number fields, 84, 93 argument principle in Euclidean spaces, 33 on cylinders, 114 on tori, 114 automorphic forms, ix automorphic functions, ix Bergman kernel, 98 rectangular domains, 101, 106 wedge shaped domains, 122 Bergman projector, operator, 99 Bergman space complex, 98 hypercomplex, 100 biregular functions, 131 Brandt algebras, 84 maximal integral domains, 88 Cauchy integral formula in Euclidean spaces, 18 on cylinders, 111 on tori, 112 Cauchy integral theorem, 18 Cauchy kernel, 19 Cauchy–Riemann operator, 16 Cayley transformations, 141 class fields, x, 84 Clifford algebras, 2 Clifford conjugation, 3 Clifford main involution, 4 Clifford norm, 4 Clifford numbers, 3 Clifford reversion, 4 Clifford analysis, 16
Clifford group, 6 Clifford-analytic functions, 23 Clifford-analytic modular forms in one vector variable, 124 in two vector variables, 131 several vector variables, 142 complex conjugation, 4 complex multiplication, 83 conformal invariance formulae, 26 conformal manifolds, 111, 148 boundary value problems, 111 conformally flat cylinders, 111 conformally flat tori, 111 cusp forms one vector variable, 129 two vector variables, 140 Dirac operator, 16 Dirichlet series, 67 divisor sums, x, 72 domain manifolds, 28 Eisenstein series classical Eisenstein series, ix (p) Eisenstein series Gm,s Fourier expansion, 70 (p) Eisenstein series Gm,s , 70, 128, 134 for hypercomplex modular groups biregular Eisenstein series, 137 one vector variable, 127 two vector variables, 133 for translation groups, 49 translation + rotation invariant, 122
166
Index
elliptic functions, 57 hypercomplex division values, 93 order relations, 62 value distribution, 60 Epstein zeta function, 66, 67
meromorphic functions, 20 monogenic functions, 16 cylindrical monogenic, 111 toroidal monogenic, 111 multi-index notation, 5
Fueter polynomials, 19 Fueter–Sce solutions, 24
non-analytic automorphic forms, xi null cone, 4, 145
generalized negative powers, 19, 35 closed formulas, 47 recurrence formulas, 40, 42 generalized Riemann zeta functions, 67 Green formulas, 25 on cylinders, 112, 113
octonions, 1, 89, 90 octonionic lattices, 89 order of isolated a-points, 30, 114 orthogonal groups, xi
half-cones, 145 half-spaces, 7, 145 Hardy spaces complex, 98 hypercomplex, 100, 113 Herglotz lemma, 81 Hilbert modular forms, 131 several vector variables, 142 two vector variables, 132, 134 Hilbert’s twelfth problem, 84 holomorphic Cliffordian functions, 24 hypercomplex differentiability, 17 hypercomplex division values, 93 hypercomplex modular groups, 9, 124 congruence subgroups, 9, 124 hypercomplex multiplication, 83 hypermonogenic functions, 24 integer multiplication, 73 integer division values, 79 Laurent expansion translative Eisenstein series, 65 Laurent expansion theorem, 21 Liouville theorems, 61, 62 M¨obius transformations, 6 Maaß wave forms, xi
paravector, 3 norm, 4 trace, 4 paravector formalism, 16 paravector multiplication, 85 permutational product, 5 Poincar´e series Clifford-analytic, 130 complex-analytic, x polybiregular functions, 132 polymonogenic functions, 23 quantum gravity, x quaternionic differentiability, 11 quaternionic multiplication, 84 quaternionic division values, 85 quaternionic symplectic groups, xi quaternions, 1 rectangular domains, 101 regular points, 20 isolated a-points, 30 residue, 21 Leray–Norguet residues, 22 Riemann mapping function, 98 Riemann surfaces, x, 111 Rouch´e’s theorem in Euclidean space, 34 on cylinders, 115 s-monogenic functions, 23
Index Siegel half-space, 145 Siegel modular forms, xi Siegel type domains, 145 singular points, 20 essential singularities, 20 removable singularities, 20 unessential singularities, 20 singularity cones, 4, 50, 147 strip domains, 101 structural sets, 11 Szeg¨o kernel, 98 strip domains, 101, 108 Szeg¨o projector, 99 Taylor series expansion, 19 Theodorescu transform (periodic), 102 translation groups, 9 transversion group, 9, 148 trigonometric functions, 55 isolated zeroes, 59 Vahlen group, 6 arithmetic subgroups, 8, 124 vector formalism, 16 Weierstraß ℘-function, 57 Weierstraß ζ-function, 57 winding number, 18 Yang–Mills theory, x Z-order, 8, 85
167
