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

TL;DR

The paper investigates when congruences modulo a prime $p$ between elliptic curves over $\mathbb{Q}$ are trivial, focusing on the curve $E_{23}: y^2=x^3-23$ and its $p=23$-congruences. It develops Mazur’s strategy for twisted modular curves $X_E^{\alpha}(p)$, and analyzes the Tate modules of their Jacobians in the non-split Cartan context, using Beilinson–Flach Euler systems to produce finiteness results for Selmer groups of suitable quotient abelian varieties. By constructing explicit quotients $J_{\Gamma,\omega}^{\alpha}$ and studying their $L$-values and BSD-type properties, the authors derive conductor bounds and conclude that any congruence modulo $23$ to $E_{23}$ is trivial. The approach interlaces local analysis of inertia at $p$, global automorphic representations, and rank-zero BSD phenomena to advance towards the Frey–Mazur and Serre uniformity questions. This yields concrete verifications of triviality for certain congruences and offers a general blueprint for ruling out nontrivial congruences via structured quotients of twisted Jacobians and Euler-system techniques.

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.