On Ramanujan congruences between special values of Hecke and Dirichlet L-functions

Abstract

Introduction. The Fourier coefficients of modular forms of even integer weight on the full modular group SL(2,Z) can by now be considered classical in number theory. Among their intriguing properties appears a remarkable congruence of Ramanujan τ(n) ≡ σ11(n) mod 691. Plenty of excellent works, such as [9], [10], [12] and [13], that appeared in the last two decades were devoted to various generalisations of this congruence. In the present paper we consider how this congruence and its higher weight analogues can be carried over to congruences between special values of L-functions of modular forms. The first two examples of this phenomenon appeared in the works of N. Koblitz [6] and [7]. In this paper we present a general construction which is based on the classical Rankin’s method. This construction yields Koblitz type congruences in an arbitrary weight, provided an additional hypothesis (see H1 below) is satisfied in that weight. Computer calculations show that H1 is satisfied for all weights less than or equal to 34. We give a heuristic argument that H1 is satisfied frequently. The philosophic foundation of the results of this type is explained in [6]. We will briefly recall it. If we deal with an arithmetic object to which one can associate a motivated L-function, then we can expect that congruences between objects induce, after removing some transcendental factors, congruences modulo the same ideals between the special values of their L-functions. This phenomenon becomes less striking if we take into account that these special values are up to a common factor algebraic numbers. The present construction can be considered as an attempt to prove some assertions along these lines when the “arithmetic objects” are modular forms and the “congruences” are Ramanujan type congruences.