Type Congruences Research Articles

Modular Invariance Characteristic Numbers and η Invariants

We extend the “miraculous cancellation” formulas of Alvarez–Gaumé, Witten and Kefeng Liu to a twisted version where an extra complex line bundle is involved. We apply our result to discuss intrinsic relations between the higher dimensional Rokhlin type congruences due to Ochanine, Finashin and Zhang. In particular, an analytic proof of the Finashin congruence is given.

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

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.

Q-Analogue of Riemann's ζ-function and q-Euler numbers

We construct a complex and a p-adic q-L-series which interpolate q-Bernoulli numbers and q-Euler numbers, respectively. As an application, we prove Kummer type congruences for q-Euler numbers.

Iwasawa invariants and Kummer conguruences

Let f( x, χ) be the Iwasawa power series for the p-adic L-function L p ( s, χ), where χ is an even nonprincipal character with conductor not divisible by p 2 (or by 8, when p = 2). The divisibility by p of the first p coefficients of f( x, χ) is characterized by Kummer type congruences of generalized Bernoulli numbers. Applications to Iwasawa invariants and class numbers of imaginary Abelian fields are discussed.