Local points on twists of X(p) with applications

2025/09/04 by Nuno Freitas, Diana Mocanu, Freitas, Nuno +1
Mathematics · #11G07. Secondary 14H10 #14G12 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 11G05

paper · pdf · doi:10.48550/arxiv.2509.04294

openalex publication_date 2025/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E/\mathbb Q be an elliptic curve and p ≥ 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_ℓ) is non-empty, for all primes ℓ≠ p; our result also includes ℓ=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 x³+y³=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.

Citations

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• Local and global densities for Weierstrass models of elliptic curves
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Related

• Revisiting the equation x²+y³=z^p
• On the symplectic type of isomorphims of the p-torsion of elliptic\n curves
• Global methods for the symplectic type of congruences between elliptic\n curves
• Congruences modulo 23 to y²=x³-23 are trivial
• Symplectic criteria for elliptic curves, revisited