Teitelbaum's exceptional zero conjecture in the function field case

Abstract

The exceptional zero conjecture relates the first derivative of the $p$-adic $L$-function of a rational elliptic curve with split multiplicative reduction at $p$ to its complex $L$-function. Teitelbaum formulated an analogue of Mazur and Tate's refined (multiplicative) version of this conjecture for elliptic curves over the rational function field $\FQ(T)$ with split multiplicative reduction at two places $\fp$ and $\infty$, avoiding the construction of a $\fp$-adic $L$-function. This article proves Teitelbaum's conjecture up to roots of unity by developing Darmon's theory of double integrals over arbitrary function fields. A function field version of Darmon's period conjecture is also obtained.