Local points on twists of $X(p)$ with applications

TL;DR

The paper analyzes local solvability for twists $X_E^-(p)$ of modular curves, giving a complete classification of $X_E^-(p)(\mathbb{Q}_\ell)$ for all $\ell\neq p$ and, in many cases, $\ell=p$ as well, depending on the reduction type and semistability defect of the base curve $E/\mathbb{Q}$. It develops a refined local theory based on symplectic vs anti-symplectic isomorphisms of $p$-torsion Galois modules, and introduces new elimination techniques in the modular method to tackle Diophantine equations, notably $x^3+y^3=5^\alpha z^p$. The first major contribution is a complete classification of local points, enabling construction of local-global obstructions and CM-based Hasse-principle counterexamples (including conditional infinite families via Frey–Mazur). A second key advance is a novel approach to the elimination stage of the modular method, paired with streamlined local symplectic criteria, which together yield concrete nonexistence results for specific prime signatures and Diophantine problems. These results have substantial implications for Diophantine geometry by broadening the toolkit for ruling out rational points on twists of modular curves and informing the search for counterexamples to the Hasse principle in CM settings.

Abstract

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. We give a complete classification of when $X_E^-(p)(\mathbb Q_\ell)$ is non-empty, for all primes $\ell\neq p$; our result also includes $\ell=p$ in most cases when $E$ is semistable at $p$. We give two different applications. First, we classify CM curves $E/\mathbb Q$ where the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for infinitely many $p$. Assuming the Frey--Mazur conjecture, we prove that for at least $60\%$ of rational elliptic curves $E$, the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for at least $50\%$ of primes $p$. Secondly, we introduce a new technique to the elimination stage of the modular method and apply it to show that $x^3+y^3=5^αz^p$ has no non-trivial primitive solutions for various primes $p$ satisfying $(α/p)=-1$. Moreover, as a by-product of our work, we simplify the assumptions of several local symplectic criteria due to the first author and Alain Kraus.

Theorems & Definitions (122)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Corollary 1.7
• Corollary 1.8
• Lemma 3.1
• proof
• Proposition 3.4