CONGRUENCES OF L-VALUES FOR CYCLIC EXTENSIONS

Abstract

Abstract. We study the consequences of Gross’s conjecture forcyclic extensions of degree l 2 where l is prime, and deduce that the L -values at s = 0 satisfy certain congruence relations. 1. IntroductionWe flrst review Gross’s conjecture brie°y. Let K=k be an abelianextension of global flelds with Galois group G . Let S be a flnite non-empty set of places of k which contains all archimedean places and allplaces ramifled in K , and let T be a flnite non-empty set of places of k which is disjoint from S . We choose T so that U S;T , the group of S -unitsin k which are congruent to 1 (mod v ) for all v 2 T , is a free abeliangroup of rank n = jSji 1.For a complex character ´ 2 G b = Hom( G; C ⁄ ), the associated modi-fled L -function is deflned as L S;T ( ´;s ) =Y v2T (1 i´ ( g v ) Nv 1 is )Y v62S (1 i´ ( g v ) Nv is ) i 1 ; where g v 2 G is the Frobenius element for v . The Stickelberger element µ G 2 C[ G ] is the unique element that satisfles ´ ( µ G ) = L S;T ( ´; 0)for all ´ 2 G b. In fact,