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Elementary Dirichlet series and modular forms. (English) Zbl 1148.11002

Springer Monographs in Mathematics. New York, NY: Springer (ISBN 978-0-387-72473-7/hbk). viii, 147 p. (2007).
Two types of Dirichlet series are thoroughly discussed in the book under review, namely \[ D^r_{a,N}(s)= \sum_{0\neq n\in a+ N\mathbb Z} n^r|n|^{-r-s},\tag{1} \] where \(s\in\mathbb{C}\), \(r= 0\) or \(1\), \(a,N\in\mathbb Z\), \(N> 0\), and \[ L^r(s;\alpha,{\mathfrak b})= \sum_{0\neq\xi\in\alpha+{\mathfrak b}}\xi^{-r}|\xi|^{r- 2s},\tag{2} \] where \(s\in\mathbb C\), \(r> 0\) is an integer, \(K\subset\mathbb C\) is an imaginary quadratic number field, \(\alpha\in K\) and \({\mathfrak b}\) is a \(\mathbb Z\)-lattice in \(K\). The principal aim of the monograph is an investigation of the values of these series at certain integral values of \(s\). As a preparatory step the analytic continuations and the functional equations of (1) and (2) are proved, and the relations with Dirichlet \(L\)-functions and certain Hecke \(L\)-functions are deduced. A substantial part of the volume is devoted to the investigation of the relations of special values of (1) and (2) with various types of modular forms. The two leitmotives, Dirichlet series and modular forms, are treated here both in classical and rather unorthodox ways, and this yields known results alongside with new ones. As a typical example we mention that there are many new formulae for \(L(k,\chi)\). The most basic one of these expresses \(L(k,\chi)\) in terms of values of certain Euler type polynomials.
We indicate the contents of the various chapters. In Chapter I the necessary preliminaries on modular forms, Fourier analysis and Dirichlet series with functional equation are recalled. As a kind of appetizer the real subject matter of the book is taken up in the case of Dirichlet \(L\)-functions in Chapter II. There are numerous beautiful formulae, old (e.g. Dirichlet’s class number formula) and new ones, on the special values of these series at integral values of \(s\). As an application, new formulae for the quotient \(h_K/h_F\) of class numbers (\(K\) a certain cyclotomic field and \(F\) its maximal real subfield) are obtained.
The main subject of study in this book are Dirichlet series of type (2) and Hecke \(L\)-functions of \(K\). The motivation is drawn from the author’s theorem: Let \(k\) be an integer such that \(2-r\leq k\leq r\) and \(r-k\in 2\mathbb Z\). Then there exists a constant \(\gamma\) which depends only on \(K\) such that \(L^r(k/2; \alpha,{\mathfrak b})\) is equal to \(\pi^{(r+ k)/2}\gamma^r\) times an algebraic number. The main problem is to find a suitable \(\gamma\) such that the algebraic number can be computed. Typically, \(\gamma\) can be given in terms of a value \(\varphi(\tau)\) where \(\varphi\) is a (holomorphic or nearly holomorphic) modular form and \(\tau\in K\cap \mathbb H\) (\(\mathbb H\) = upper half-plane). A typical example of a nearly holomorphic modular form, which plays an important role in this work, is the well-known Eisenstein series \(E_2\).
Dirichlet series associated with an imaginary quadratic number field, their analytic continuation and functional equation, and nearly holomorphic modular forms and their behaviour under “invariant” differential operators are treated in Chapter III. An important role is played by Eisenstein series (of Maaß’ type); these are most carefully treated in Chapter IV. Topics include the Fourier expansion of Eisenstein series, their behaviour under differential operators, algebraicity properties of their Fourier coefficients, analytic continuation and functional equation of Eisenstein series, polynomial relations between Eisenstein series, modular forms of weight 2 and level 2 associated with the Weierstrass function \(\wp\), recurrence formulae for the critical values of certain Dirichlet series.
{Reviewer’s remark: The polynomial relation (10.8) as a relation between holomorphic modular forms and its application to the deduction of relations between divisor sums appears to be known. The reviewer got it to know in a course on modular forms given by Hans Petersson at the University of Münster in the summer semester 1962. The formula appears in print in the book: T. M. Apostol: Modular functions and Dirichlet series in number theory. New York etc.: Springer (1976; Zbl 0332.10017), p. 12 f., Theorem 1.13.}
Chapter V deals with critical values of Dirichlet series associated with imaginary quadratic number fields in terms of singular values of nearly holomorphic modular forms. Special attention is paid to the question whether such a form may be chosen in a canonical way. There is no clear-cut general answer, but once the values \(L^r(k/2;\alpha,{\mathfrak b})\) are determined for a suitable set of finitely many \((k,r)\), the values for further data of \((k,r)\) can be found by means of a recurrence formula. The basic result on the algebraicity of special values of Hecke \(L\)-functions reads as follows (see Theorem 13.6): Let \(r\) be a positive integer and \(k\in\mathbb Z\) be such that \(2-r\leq k\leq r\), \(r- k\in\mathbb Z\). Further, let \(\tau\in K\cap \mathbb H\) and \(h\) be a modular form on a congruence subgroup of \(\text{SL}_2(\mathbb Z)\) of weight \(r\) with Fourier coefficients in \(\overline{\mathbb Q}\) such that \(h(\tau)\neq 0\). Then \(L(k/2,\lambda)\) is an algebraic number times \(\pi^{(r+k)/2}h(\tau)\). There are many beautiful explicit examples. In certain cases, \(L(s,\lambda)\) is closely connected with an elliptic curve \(C\) defined over an algebraic number field with complex multiplication. In a special case, it is even equal to the zeta-function of \(C\). This aspect is pursued in some detail in Section 14. It turns out that \(L(1/2,\lambda)\) is related with a period of a holomorphic one-form on \(C\). The supplementary results in Chapter VI deal with isomorphism classes of Abelian varieties with complex multiplication and with an explicit determination of holomorphic differential operators on \(\mathbb H\) sending modular forms to modular forms.
This book may be regarded as a supplement to the author’s classic “Introduction to the arithmetic theory of automorphic functions.” Princeton, NJ: Princeton Univ. Press (1971; Zbl 0221.10029), reprint (1994; Zbl 0872.11023). It will be of great interest for everybody who is interested in modular forms and/or \(L\)-series. Since the preliminaries are modest, the monograph will be accessible to graduate students and will quickly lead them to frontiers of current research. The book is written in the well-known masterly style of the author, and prospective readers should bear in mind that mathematics ought to be studied with the masters.

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11F11 Holomorphic modular forms of integral weight
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G05 Elliptic curves over global fields
11G10 Abelian varieties of dimension \(> 1\)
11G15 Complex multiplication and moduli of abelian varieties
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
11M41 Other Dirichlet series and zeta functions
11R29 Class numbers, class groups, discriminants
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