Luca Dall’Ava (Universität Mailand)

This talk aims to introduce a new balanced triple product p-adic L-function and discuss its application to the equivariant Birch & Swinnerton-Dyer conjecture; we state a conjecture in a rank-1 situation analogous to the Elliptic–Stark conjecture formulated by Darmon–Lauder–Rotger in rank-2 and prove it in the CM case; this work fits in the general framework studied by Darmon–Lauder–Rotger and Andreatta–Bertolini–Seveso–Venerucci. The main feature of this new p-adic L-function is to allow classical weight one modular forms in the chosen families. Its construction crucially relies on an extension of Chenevier’s p-adic Jacquet–Langlands for quaternionic modular forms with level structure given by Pizer orders. Time permitting, we will explain the intriguing technical difficulties one has to deal with constructing this morphism of eigenvarieties. That is joint work with Aleksander Horawa.