p-ADIC L-FUNCTIONS FOR GALOIS DEFORMATIONS AND RELATED PROBLEMS ON PERIODS

Abstract. At the first half of this article, we present a conjecture (cf. Conjecture 1.10) to associate “the p-adic L-function ” to a family of Galois representation. In recent years, we have plenty of examples for families of Galois representations of the absolute Galois group of . However, because of lack of examples of explicit constructions, formulating a conjecture was difficult. We give conjectures in §1.4 based on our detailed study [O3] of the Iwasawa theory for the two-variable Hida deformation and various other examples coming from Hida theory on higher dimensional modular forms, Coleman theory, convolution product, etc. At the later half of this article, we explain the example of the two-variable Hida deformation by showing that it will be the first evidence (except the cyclotomic de-formations) which satisfies our generalized Iwasawa main conjecture (cf. Conjectur