Divisibility of anticyclotomic L-functions and theta functions with complex multiplication

Abstract

The divisibility properties of Dirichlet L-functions in infinite families of characters have been studied by Iwasawa, Ferrero and Washington. The families considered by them are obtained by twisting an arbitrary Dirichlet character with all characters of p-power conductor for some prime p. One has to distinguish divisibility by p (the case considered by Iwasawa and FerreroWashington [FeW]) and by a prime � � p (considered by Washington [W1], [W2]). Ferrero and Washington proved the vanishing of the Iwasawa µ-invariant of any branch of the Kubota-Leopoldt p-adic L-function. This means that each of the power series, which p-adically interpolate the nontrivial L-values of twists of a fixed Dirichlet character by characters of p-power conductor, has some coefficient that is a p-adic unit. In the case � � p Washington [W2] obtained the following theorem on divisibility of L-values by � : given an integer n ≥ 1 and a Dirichlet character χ, for all but finitely many Dirichlet characters ψ of p-power conductor with χψ(−1) = (−1) n ,