Ramification in Iwasawa modules en collaboration avec Chandrashekhar Khare

RAMIFICATION IN IWASAWA MODULES CHANDRASHEKHAR KHARE AND JEAN-PIERRE WINTENBERGER Abstract. We make a reciprocity conjecture that extends Iwasawa's analogy of direct limits of class groups along the cyclotomic tower of a totally real number field F to torsion points of Jacobians of curves over finite fields. The extension is to generalised class groups and generalised Jacobians. We state some “splitting conjectures” which are equivalent to Leopoldt's conjecture. 1. Introduction For a number field F , with ring of integers OF , we may define the class group of F to be Pic(OF ), i.e., the isomorphism classes of invertible sheaves on Spec(OF ). Iwasawa deepened this formal analogy between class groups of number fields and Jacobians. He considered X?∞, the inverse limit under norm maps of the minus parts under complex conjugation of the Sylow p- sugroups of the class groups of F (µpn), where F is a totally real number field, p a fixed (odd) prime, and n varying. Iwasawa viewed X?∞ ?Qp as a p-adic vector space, which he proved to be finite dimensional, equipped with the action of ?, a generator for the p-part of Gal(F (µp∞)/F ). He conjectured that the characteristic polynomial for this action should be the same as a certain p-adic L-function, at least when F = Q.

• zp


• iwasawa


• let x¯ ?


• galois group


• extension


• almost totally ramified


• adic tate


• f? ? ?


• extension class