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2-adic and 3-adic part of class numbers and properties of central values of L-functions

Let d be prime or the product of two primes. In this paper we study the connection between 2-parts and 3-parts of the class numbers h(-d) and h(-3d) and ray class groups of Q(sqrt(d)) unramified outside 2 (and 3). We obtain certain "reflection" theorems, and we reproduce the result of Williams on divisibility of h(-d) by 16 when d is prime (and we get a similar result when d is the product of two primes). The main ingredients of the proofs are congruences between L2(1 ; d) (and L3(1 ; d)) and h(-d)(and h(-3d)) modulo powers of 2 (and 3) which we prove using modular forms. We also obtain similar congruences for the central values of L-functions associated to Ramanujan Delta-function, and we relate them to the structure of 2-adic and 3-adic Galois representation attached to the Delta-function.Let d be prime or the product of two primes.

congruences between modular forms ; delta function ; Ray class groups ; special values of L-functions

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