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A $p$-adic $L$-function with canonical motivic periods for Hida families

In 2006, Fukaya and Kato formulated a general conjecture about $p$-adic $L$-functions for a large class of motives and derived a precise conjectural interpolation formula from the ETNC. We study this conjectural $p$-adic $L$-function in the setting of Hida families and compare it to the $p$-adic $L$-function constructed by Kitagawa. A priori it is not clear whether the interpolation formulas coincide, mainly because of the rather inexplicit definition of periods (via comparison isomorphisms) that appear in the general formula. Using $p$-adic Eichler-Shimura isomorphisms and their interpolation in Hida families, we compare these periods to the expressions in Kitagawa's formula coming from modular symbols. This allows in certain cases to modify Kitagawa's construction such that it produces a $p$-adic $L$-function compatible with Fukaya's and Kato's conjecture.