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

TL;DR

This paper proves that the adjoint $p$-adic $L$-function of an elliptic curve over $Q$, evaluated at many integers, uniquely determines its isogeny class up to quadratic twist, using results on $ ext{GL}(3)$ representations.

Contribution

It establishes a new method to determine elliptic curves from their adjoint $p$-adic $L$-functions via $ ext{GL}(3)$ representation theory.

Findings

The adjoint $p$-adic $L$-function determines the isogeny class of elliptic curves.

A new link between $L$-values of $p$-power twists and isogeny classes.

Results on $ ext{GL}(3)$ representations aid in identifying elliptic curves.

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.