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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_650_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$E^{d}/K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>E</mi> <mi>d</mi> </msup> <mo stretchy="false">/</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation>. We note that when the residue field has characteristic 2, we only consider the special case <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_650_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$K=\mathbb {Q}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <msub> <mi mathvariant="double-struck">Q</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. In this setting, we also determine the minimal discriminant valuation and conductor exponent of E and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_650_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$E^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>E</mi> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> from further conditions on the coefficients of a strongly-minimal model for E.

Local data of elliptic curves under quadratic twist

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2025年8月7日

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Alexander J. Barrios,
Manami Roy,
Nandita Sahajpal,
Darwin Tallana,
Bella Tobin,
Hanneke Wiersema

摘要

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 $E^{d}/K$ $E^{d} / K$ . We note that when the residue field has characteristic 2, we only consider the special case $K=\mathbb {Q}_{2}$ $K = Q_{2}$ . In this setting, we also determine the minimal discriminant valuation and conductor exponent of E and $E^d$ $E^{d}$ from further conditions on the coefficients of a strongly-minimal model for E.