A p-adic L-functionwith canonical motivic periodsfor families of modular forms

Abstract

We prove a version of the conjecture of Fukaya and Kato concerning the existence of p-adic L-functions for motives in the case of certain Hida families of modular forms and for commutative towers of fields, the novelty being the exact accordance with the conjectural interpolation formula. To do so, we first calculate the expressions in their conjectural formula as explicitly as possible and compare the result to the interpolation formula of the p-adic L-function constructed by Kitagawa. Here we use Eichler- Shimura isomorphisms to relate the periods appearing in Fukaya’s and Kato’s formula to the error terms appearing in Kitagawa’s formula, which are defined in terms of modular symbols. Under a technical hypothesis on the Hida family we show that the conjectural interpolation formula differs from Kitagawa’s one only by a unit in the Iwasawa algebra, so we find a p-adic L-function having the interpolation behaviour predicted by Fukaya and Kato (up to a non-constant sign).