Teitelbaum's exceptional zero conjecture in the function field case

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 q(T) with split multiplicative reduction at two places and ∞, avoiding the construction of a -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.