TAILIEUCHUNG - Đề tài " Divisibility of anticyclotomic L-functions and theta functions with complex multiplication "

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. . | Annals of Mathematics Divisibility of anticyclotomic L-functions and theta functions with complex multiplication By Tobias Finis Annals of Mathematics 163 2006 767 807 Divisibility of anticyclotomic L-functions and theta functions with complex multiplication By Tobias Finis 1. Introduction 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 Ferrero-Washington FeW and by a prime I p considered by Washington W1 W2 . Ferrero and Washington proved the vanishing of the Iwasawa -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 t p Washington W2 obtained the following theorem on divisibility of L-values by F. given an integer n 1 and a Dirichlet character X for all but finitely many Dirichlet characters ý of p-power conductor with X -1 -I n vr L 1 - n xf 0. Here Vi- denotes the Oadic valuation of an element in C and we apply Vi- to algebraic numbers in C after fixing embeddings itt o C and ý o C. By the class number formula these theorems are related to divisibility properties of class numbers in the cyclotomic Zp-extension of an abelian number field. One obtains the following qualitative picture let F be an abelian number field and F Fo its cyclotomic Zp-extension with unique intermediate extensions Fn F of degree pn. The vanishing of the -invariant of F F implies by a well-known result of Iwasawa that the p-part of the class number hn of Fn grows linearly with n for n x . Washington s theorem allows to control divisibility

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