## Elementary theory of $$L$$-functions and Eisenstein series.(English)Zbl 0942.11024

London Mathematical Society Student Texts. 26. Cambridge: Cambridge University Press. ix, 386 p. (1993).
This monograph has meanwhile become a classic reference text in the theory of $$p$$-adic and classic modular forms and the study of arithmetic and $$p$$-adic $$L$$-functions. The text is an outgrowth of courses given at various places (UCLA, Hokkaido, Grenoble, Paris-Sud).
Chapter 1 is meant to supply the necessary results from algebraic number theory, linear and homological algebra and the theory of $$p$$-adic numbers.
Chapter 2 deals with basic analytic properties of abelian $$L$$-functions in the classical setting and Eisenstein series treating Euler’s method of computing $$\zeta$$-values, analytic continuation and functional equation, Hurwitz and Dirichlet $$L$$-functions, Shintani $$L$$-functions, $$L$$-functions of real and imaginary quadratic fields and Hecke $$L$$-functions.
In Chapter 3 ‘$$p$$-adic Hecke $$L$$-functions’ the author first constructs $$p$$-adic Dirichlet $$L$$-functions for $$\mathbb Q$$ using Euler’s method of computing the values $$L(n,\chi)$$ following N. M. Katz (1995). Then he generalizes the result for Hecke $$L$$-functions of totally real fields. Roughly speaking, a $$p$$-adic $$L$$-function is a $$p$$-adic analytic function whose values coincide with those of its complex counterpart at enough integer points to guarantee its uniqueness. The $$p$$-adic $$L$$-function discussed here were first constructed for $$\mathbb Q$$ by Kubota and Leopoldt as a continuous function and later shown to be $$p$$-adic analytic by Iwasawa. The result was generalized to totally real fields independently by Deligne and Ribet, Cassou-Noguès and Barsky. Katz’s method is an interpretation of the methods of Casson-Noguès and Barsky in terms of formal groups.
The chapter consists of the following subheadings: Interpolation series, Interpolation series in $$p$$-adic fields, $$p$$-adic measures on $$\mathbb Z_p$$, The $$p$$-adic measure of the Riemann zeta-function, $$p$$-adic Dirichlet $$L$$-functions, Group schemes and formal group schemes, Toroidal formal groups and $$p$$-adic measures, $$p$$-adic Shintani $$L$$-functions of totally real fields, $$p$$-adic Hecke $$L$$-functions of totally real fields.
Chapter 4 ‘Homological interpretation’ gives such an interpretation of the theory of the special values of Dirichlet $$L$$-functions over $$\mathbb Q$$ and reconstructs $$p$$-adic Dirichlet $$L$$-functions by a homological method (called the “modular symbol” method). The modular symbol method was introduced by Mazur in the context of modular forms on $$\mathrm{GL}(2)$$. Basic facts from cohomology theory used in this section are summarized in the Appendix at the end of this book.
Subheadings are: Cohomology groups on $$\mathbb G_m(\mathbb C)$$, Cohomological interpretation of Dirichlet $$L$$-values, $$p$$-adic measures and locally constant functions, Another construction of $$p$$-adic Dirichlet $$L$$-functions.
Chapter 5 ‘Elliptic modular forms and their $$L$$-functions’ treats Classical Eisenstein series of $$\mathrm{GL}(2)/\mathbb Q$$, Rationality of modular forms, Hecke operators, The Petersson inner product and the Rankin product, and Standard $$L$$-functions of holomorphic modular forms.
In Chapter 6 ‘Modular forms and cohomology groups’ the author proves the Eichler-Shimura isomorphism between the space of modular forms and the cohomology group on each modular curve. This fact was first proven by Shimura in 1959. He gives two proofs of this isomorphism. The first one is the original proof of Shimura based on the two dimension formulas. One is the formula for the space of cusp forms and the other is for the cohomology group. The other proof makes use of harmonic analysis on the modular curve. After studying the Hecke module structure of modular cohomology groups, he constructs the $$p$$-adic standard $$L$$-function of $$\mathrm{GL}(2)/\mathbb Q$$ following the method (so-called “$$p$$-adic Mellin transform”) of Mazur and Manin.
Subheadings are: Cohomology of modular groups, Eichler-Shimura isomorphisms, Hecke operators on cohomology groups, Algebraicity theorem for standard $$L$$-functions of $$\mathrm{GL}(2)$$, Mazur’s $$p$$-adic Mellin transforms.
Chapter 7 ‘Ordinary $$\Lambda$$-adic forms, two variable $$p$$-adic Rankin products and Galois representations’ presents some recent developments in the theory of $$p$$-adic modular forms, in particular, (i) $$p$$-adic analytic parametrization of classical modular forms, (ii) $$p$$-adic $$L$$-functions attached to each $$p$$-adically parametrized family of modular forms and (iii) Galois representations of these families of modular forms.
In detail the author treats $$p$$-adic families of Eisenstein series, the projection to the ordinary part, ordinary $$\Lambda$$-adic forms, two variable $$p$$-adic Rankin product, ordinary Galois representations into $$\mathrm{GL}_2 (\mathbb Z_p [[X]])$$, and examples of $$\Lambda$$-adic forms.
After giving a brief summary of the notion of adeles of number fields in Chapter 8 ‘Functional equations of Hecke $$L$$-functions’ he proves the functional equations of Hecke $$L$$-functions. The author starts with the adelic interpretation of algebraic number theory, and then treats Hecke characters as continuous idele characters, self-duality of local fields, Haar measures and the Poisson summation formula, and adelic Haar measures. Finally he gives the functional equations for Hecke $$L$$-functions via the method of Tate-Iwasawa.
In Chapter 9 ‘Adelic Eisenstein series and Rankin products’ the author first gives an adelic interpretation of modular forms and then computes the Fourier expansion of adelic Eisenstein series for $$\mathrm{GL}(2)$$ over $$\mathbb Q$$ obtaining the result that the Eisenstein series has analytic continuation with respect to the variable $$s$$. Using this fact, he shows the analytic continuability of the Rankin product and its functional equations.
Subheadings are: Modular forms on $$\mathrm{GL}_2(F_{\mathbb A})$$, Fourier expansion of Eisenstein series, Functional equation of Eisenstein series, Analytic continuation of Rankin products, Functional equations of Rankin products.
Chapter 10 ‘Three variable $$p$$-adic Rankin products’ is the most important one. Here, the author first proves Shimura’s algebraicity theorem for Rankin product $$L$$-functions $$L(s, \lambda\otimes \varphi)$$. Then he constructs three variable $$p$$-adic Rankin products extending the result obtained in Chapter 7. As for the algebraicity theorem, he follows the treatment of Shimura and also his own former papers. He only treats the case of $$\mathrm{GL}_2(\mathbb Q)$$. For further study of this type of algebraicity questions for the algebraic group $$\mathrm{GL}(2)$$ over general fields, it is refered to Shimura’s papers and to his own for totally real fields and for fields containing CM fields. As for the $$p$$-adic $$L$$-functions, he generalizes his earlier method. There is one more method of getting $$p$$-adic continuation of $$L(s, \lambda \otimes \varphi)$$ along the cyclotomic line (i.e. varying $$s$$) found by Panchishkin.
Subheadings are: Differential operators of Maass and Shimura, The algebraicity theorem of Rankin products, Two variable $$\Lambda$$-adic Eisenstein series, Three variable $$p$$-adic Rankin products, Relation to two variable $$p$$-adic Rankin products. In his ‘Concluding remarks’ he gives some valuable indications for further reading.
The presentation of this book is almost self-contained and very concise. The author succeeded in explaining the difficult subjects to provide the reader with a detailed insight into this theory. Exercises and examples are included throughout with answers to some of these. The reviewer totally agrees with the cover phrase ‘this book offers a unique introduction to a fascinating branch of mathematics’.

### MSC:

 11Fxx Discontinuous groups and automorphic forms 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11R42 Zeta functions and $$L$$-functions of number fields 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F33 Congruences for modular and $$p$$-adic modular forms 11F75 Cohomology of arithmetic groups 11F80 Galois representations 11F85 $$p$$-adic theory, local fields 11M35 Hurwitz and Lerch zeta functions 11M41 Other Dirichlet series and zeta functions 11M38 Zeta and $$L$$-functions in characteristic $$p$$