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arXiv · 2501.03209

Local data of elliptic curves under quadratic twist

Alexander J. Barrios·Manami Roy·Nandita Sahajpal·Darwin Tallana·Bella Tobin·Hanneke Wiersema
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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 $K=\mathbb{Q}_{2}$. 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 $E$.

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Alexander J. Barrios, Manami Roy, Nandita Sahajpal, Darwin Tallana, Bella Tobin, Hanneke Wiersema. 2025-01-06. Local data of elliptic curves under quadratic twist. https://doi.org/10.1007/s40993-025-00650-w

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