Two abelian varieties A and B over a number field K are said to be strongly locally quadratic twists if they are quadratic twists at every completion of K. While it was known that this does not imply that A and B are quadratic twists over K, the only known counterexamples (necessarily of dimension <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3858_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>) are not geometrically simple. We show that, for every prime <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3858_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">$p\equiv 13 \pmod {24}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>13</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>24</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, there exists a pair of geometrically simple abelian varieties of dimension <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3858_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$p-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3858_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> that are strongly locally quadratic twists but not quadratic twists. The proof is based on Galois cohomology computations and class field theory.

Geometrically simple counterexamples to a local-global principle for quadratic twists

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2025年9月18日

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Emiliano Ambrosi,
Nirvana Coppola,
Francesc Fité

摘要

Two abelian varieties A and B over a number field K are said to be strongly locally quadratic twists if they are quadratic twists at every completion of K. While it was known that this does not imply that A and B are quadratic twists over K, the only known counterexamples (necessarily of dimension $\ge 4$ $\geq 4$ ) are not geometrically simple. We show that, for every prime $p\equiv 13 \pmod {24}$ $p \equiv 13 (mod 24)$ , there exists a pair of geometrically simple abelian varieties of dimension $p-1$ $p - 1$ over $\mathbb {Q}$ $Q$ that are strongly locally quadratic twists but not quadratic twists. The proof is based on Galois cohomology computations and class field theory.