QUADRATIC CONGRUENCES FOR COHEN- EISENSTEIN SERIES.

The notion of quadratic congruences was introduced in the recently appeared paper [1]. In this note we present another, somewhat more conceptual proof of several results from loc. cit. Our method allows to refine the notion and to generalize the results quoted. Here we deal only with the quadratic congruences for Cohen- Eisenstein series. A similar phenomena exists for cusp forms of half-integral weight as well. However, as one can expect, in the case of Eisenstein series the argument is much simpler. In particular, we do not make use of other techniques then p-adic Mazur measure, whereas the consideration of cusp forms of half-integral weight involves much more sophisticated construction. Moreover, in the case of Cohen-Eisenstein series we are able to get the full and exhaustive result. For these reasons we dare to present the argument here. Our result deals with modular forms, but the argument essentially does not. One can think just about the congruences for Cohen numbers H(r,N). These are arithmetically interesting rational numbers defined below. Our proof relies on the construction of p-adic Mazur measure. We formulate the precise statement as proposition 1 in the text. After that we present a corollary (proposition 2) which is sufficient for our purposes. Let χ be a Dirichlet character modulo M> 1, and denote by L(s, χ) the associated L-series. L(s, χ) = n≥1 χ(n)n−s. The series converges for ℜ(s)> 1 and admits the analytic continuation over all s ∈ C. Its values at negative integers essentially coincide with generalized Bernoulli numbers. More precisely, one has for a positive integer r L(1 − r, χ) = −Br,