On the pull-back method for p-adic regulator formulae

Constructing a p-adic L-function and proving the associated regulator formula have been a powerful tool for the Bloch–Kato conjecture. In the last decade, proofs of the regulator formulae for various p-adic L-functions

mostly belong to the so-called ‘push-forward’ situation, and use syntomic/finite polynomial cohomology for computation.

Recently, Marco Sangiovanni Vincentelli and Christopher Skinner constructed an Euler system for the adjoint of a modular form via the ‘pull-back’ method, which does not use the language of finite polynomial

cohomology.

Inspired by their work, A. Kazi, L. Marannino, and the speaker reconstructed the diagonal class for the triple product via pull-back. Moreover, we proved the regulator formula for the p-adic Asai L-function for

finite slope families.

The work is still in progress and this talk will mainly focus on the first part.