Local data of elliptic curves under quadratic twist

2025/01/06 by Alexander J. Barrios, Manami Roy, Barrios, Alexander J. +9 · 3 citations
Computer Science · Mathematics · #11G05 #11G07 #14H10 #14H52 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2501.03209

openalex publication_date 2025/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be the field of fractions of a complete discrete valuation ring with a perfect residue field. In this article, we investigate how the Tamagawa number of E/K changes under quadratic twist. To accomplish this, we introduce the notion of a strongly-minimal model for an elliptic curve E/K, which is a minimal Weierstrass model satisfying certain conditions that lead one to easily infer the local data of E/K. Our main results provide explicit conditions on the Weierstrass coefficients of a strongly-minimal model of E/K to determine the local data of a quadratic twist E^d/K. We note that when the residue field has characteristic 2, we only consider the special case K=ℚ₂. In this setting, we also determine the minimal discriminant valuation and conductor exponent of E and E^d from further conditions on the coefficients of a strongly-minimal model for E.

Cited by

• On the Birch and Swinnerton-Dyer formula modulo squares for certain quadratic twists of elliptic curves
• Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants
• Small Tamagawa numbers of elliptic curves with isogenies or torsion

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