Constructing abelian surfaces over rational numbers, reveals unique Galois group actions on torsion subgroups.
We construct infinitely many abelian surfaces A A defined over the rational numbers such that, for a prime ℓ ⩽ 7 7 , the ℓ -torsion subgroup of A A is not isomorphic as a Galois module to the ℓ -torsion subgroup of its dual A ∨ A^ . We do this by explicitly analyzing the action of the Galois group on the ℓ -adic Tate module and its reduction modulo ℓ .
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Frei et al. (2025) studied this question.
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