On the symplectic type of isomorphims of the p-torsion of elliptic\n curves

2016/07/05 by Nuno Freitas, Freitas, Nuno, Alain Kraus +1 · 6 citations
Mathematics · Medicine · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Spinal Hematomas and Complications

paper · pdf · doi:10.48550/arxiv.1607.01218

Abstract

Let p \≥ 3 be a prime. Let E/\ℚ and E'/\ℚ be elliptic\ncurves with isomorphic p-torsion modules E[p] and E'[p]. Assume further\nthat either (i) every G_\ℚ-modules isomorphism \φ : E[p] \→\nE'[p] admits a multiple \λ \⋅ \φ with \λ \∈\n mathbbF_p^\× preserving the Weil pairing; or (ii) no\nG_\ℚ-isomorphism \φ : E[p] \→ E'[p] preserves the Weil pairing.\nThis paper considers the problem of deciding if we are in case (i) or (ii).\n Our approach is to consider the problem locally at a prime \ℓ \≠ p.\nFirstly, we determine the primes \ℓ for which the local curves\nE/\ℚ_\ℓ and E'/\ℚ_\ℓ contain enough information to\ndecide between (i) or (ii). Secondly, we establish a collection of criteria, in\nterms of the standard invariants associated to minimal Weierstrass models of\nE/\ℚ_\ℓ and E'/\ℚ_\ℓ, to decide between (i) and (ii).\nWe show that our results give a complete solution to the problem by local\nmethods away from p.\n We apply our methods to show the non-existence of rational points on certain\nhyperelliptic curves of the form y² = x^p - \ℓ and y² = x^p - 2\ℓ\nwhere \ℓ is a prime; we also give incremental results on the Fermat\nequation x² + y³ = z^p. As a different application, we discuss variants of\na question raised by Mazur concerning the existence of symplectic isomorphisms\nbetween the p-torsion of two non-isogenous elliptic curves defined over\n\ℚ.\n

Citations

• Good Reduction of Abelian Varieties
• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques
• Modular Elliptic Curves and Fermat's Last Theorem

Cited by

• Revisiting the equation x²+y³=z^p
• Effective surjectivity of Galois representations of products of elliptic curves over function fields
• Symplectic criteria for elliptic curves, revisited
• Local points on twists of X(p) with applications
• Wild Galois representations: elliptic curves with wild cyclic reduction
• The Generalized Fermat Equation x² + y³ = z²⁵

Related

• Symplectic criteria for elliptic curves, revisited
• Local points on twists of X(p) with applications
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