\(p\)-adic interpolation of real analytic Eisenstein series. (English) Zbl 0354.14007

Katz, Nicholas M.

Ann. Math. (2) 104, 459-571 (1976).

The correspondence between real analytic Eisenstein series on congruence subgroups of \(\mathrm{SL}_2(\mathbb Z)\) and Ramanujan series (the latter viewed as \(p\)-adic modular forms) is under investigation. This gives the possibility to construct the relatively complete theory of the \(p\)-adic \(L\)-functions (including the \(\Gamma\)-factor) attached to an imaginary quadratic field \(K_0\) in which \(p\) splits. These \(L\)-functions are obtained as the “Mellin transforms” of some \(p\)-adic measure in two variables whose moments are essentially the values of Ramanujan series \(\Phi_{k,l}\) on suitable “trivialized elliptic curves” with complex multiplication by \(K_0\). The theory developed by the author is discussed in correspondence with the results of Manin and Vishik concerning the \(p\)-adic interpolation of special values of Hecke \(L\)-series attached to größencharacter of type \(A_0\) of the field \(K_0\). One of the applications of the theory is the proof of the “second limit formula” of Kronecker for \(p\)-adic \(L\)-functions, generalising Leopold’s \(p\)-adic formula for the rational field to imaginary quadratic field.
The paper contains 10 chapters. The first chapter is devoted to the study of the Halphen-Fricke differential operator on analytic and \(C^\infty\) modular forms. lt was influenced by A. Weil’s fall 1974 lectures “Elliptic functions according to Eisenstein”. The second chapter reviews the interplay between the algebraic and analytic approaches to modular forms. The third chapter constructs real analytic Eisenstein series as special values of Epstein zeta-functions. Chapter four gives a mild generalisation of Damerell’s theorem [R. M. Damerell, Acta Arith. 17, 287–301 (1970; Zbl 0209.24603)]. The fifth chapter reviews the \(p\)-adic theory of modular forms. The last five chapters are devoted to the construction and over-detailed explication of the \(p\)-adic \(L\)-functions attached imaginary quadratic fields in which \(p\) splits.

Reviewer: V. A. Iskovskikh (Moskva)

MSC:

14G20	Local ground fields in algebraic geometry
11F12	Automorphic forms, one variable
11F85	\(p\)-adic theory, local fields
11E45	Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)

Keywords:

real analytic Eisenstein series; special values of Epstein zeta-functions; Damerell’s theorem; \(p\)-adic theory of modular forms; \(p\)-adic \(L\)-functions

Citations:

Zbl 0209.24603