Exceptional zero formulas for anticyclotomic p-adic L-functions

Abstract

In this note we define anticyclotomic p-adic measures attached to a modular elliptic curve E over a general number field F, a quadratic extension, and a set of places S of F above p. We study the exceptional zero phenomenon that arises when E has multiplicative reduction at some place in S. In this direction, we obtain p-adic Gross-Zagier formulas relating derivatives of the corresponding p-adic L-functions to the extended Mordell-Weil group of E. Our main result uses the recent construction of plectic points on elliptic curves due to Fornea and Gehrmann and generalizes their main result in [9]. We obtain a formula that computes the r-th derivative of the p-adic L-function, where r is the number of places in S where E has multiplicative reduction, in terms of plectic points and Tate periods of E.

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 682152). Moreover, the second author has been partially funded by the projects PID2021-124613OB-I00 and PID2022-137605NB-I00 from Ministerio de Ciencia e innovación. We would also like to thank Lennart Gehrmann and Marc Masdeu for all their helpful remarks and comments.