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Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/214506
On a local-global principle for quadratic twists of abelian varieties
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Let $A$ and $A^{\prime}$ be abelian varieties defined over a number field $k$ of dimension $g \geq 1$. For $g \leq 3$, we show that the following local-global principle holds: $A$ and $A^{\prime}$ are quadratic twists of each other if and only if, for almost all primes $\mathfrak{p}$ of $k$ of good reduction for $A$ and $A^{\prime}$, the reductions $A_{\mathfrak{p}}$ and $A_{\mathfrak{p}}^{\prime}$ are quadratic twists of each other. This result is known when $g=1$, in which case it has appeared in works by Kings, Rajan, Ramakrishnan, and Serre. We provide an example that violates this local-global principle in dimension $g=4$.
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FITÉ NAYA, Francesc. On a local-global principle for quadratic twists of abelian varieties. Mathematische Annalen. 2022. Vol. 388, num. 769-794. ISSN 0025-5831. [consulted: 13 of June of 2026]. Available at: https://hdl.handle.net/2445/214506