Exceptional zero formulas for anticyclotomic p-adic L-functions

TL;DR

This paper develops exceptional zero formulas for anticyclotomic p-adic L-functions of elliptic curves over number fields, linking derivatives of these functions to plectic points and Tate periods, especially when E has multiplicative reduction.

Contribution

It introduces a new framework for anticyclotomic p-adic measures and extends p-adic Gross-Zagier formulas using plectic points, generalizing recent constructions.

Findings

Derived formulas for derivatives of p-adic L-functions in terms of plectic points.

Established connections between exceptional zero phenomena and Tate periods.

Generalized previous results to broader settings with multiple places of multiplicative reduction.

Abstract

In this note we define anticyclotomic p-adic measures attached to a finite set of places S above p, a modular elliptic curve E over a general number field F and a quadratic extension K/F. 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. 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.