\(p\)-adic interpolation of real analytic Eisenstein series
type: Item
instance of: scholarly article
title: \(p\)-adic interpolation of real analytic Eisenstein series
zbMATH Open document ID: 0354.14007
DOI: 10.2307/1970966
author: Nicholas M. Katz
published in: Annals of Mathematics. Second Series
publication date: 1976-01-01T00:00:00Z
review text: 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 [\textit{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 \…
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 11E45
zbMATH DE Number: 3550866
zbMATH Keywords: real analytic Eisenstein series
zbMATH Keywords: special values of Epstein zeta-functions
zbMATH Keywords: Damerell's theorem
zbMATH Keywords: \(p\)-adic theory of modular forms
zbMATH Keywords: \(p\)-adic \(L\)-functions
Wikidata QID: Q105611462
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1970966
OpenAlex ID: W2097966430
description: scientific article; zbMATH DE number 3550866