Congruences modulo $23$ to $y^2=x^3-23$ are trivial

Abstract

We say that two elliptic curves $E$ and $F$ over $\mathbb{Q}$ are congruent modulo a prime $p$ if their $p$-torsion Galois modules (over the algebraic closure of $\mathbb{Q}$) are isomorphic. Such a congruence is called trivial if there is a rational isogeny between $E$ and $F$ with degree prime to $p$. A version of the Frey-Mazur conjecture states that any congruence modulo any prime $p \geq 19$ is trivial. Given an elliptic curve $E/\mathbb{Q}$ and a prime $p$, it is well-known that there is a twist of the classical modular curve $X(p)$ whose rational points describe the elliptic curves congruent to $E$ modulo $p$. In this article, we apply Mazur's strategy to determine the rational points of such a twisted modular curve under certain assumptions. This involves, among others, the determination of the previously unknown Tate module of its Jacobian and new instances of the Birch and Swinnerton--Dyer conjecture (for abelian varieties not of $\mathrm{GL}_2$-type). In particular, we determine an explicit bound on the conductor of any elliptic curve congruent modulo $p$ to $y^2=x^3-p$ when $p$ is prime and congruent to $5$ modulo $9$, and deduce that any congruence modulo $23$ to $y^2=x^3-23$ is trivial.