Determination of elliptic curves by their adjoint $p$-adic $L$-functions

Abstract

Fix $p$ an odd prime. Let $E$ be an elliptic curve over $\mathbb{Q}$ with semistable reduction at $p$. We show that the adjoint $p$-adic $L$-function of $E$ evaluated at infinitely many integers prime to $p$ completely determines up to a quadratic twist the isogeny class of $E$. To do this, we prove a result on the determination of isobaric representations of $\text{GL}(3,{\mathbb{A}}_{\mathbb{Q}})$ by certain $L$-values of $p$-power twists.