Papers › On the local-global principle for isogenies of abelian surfaces
On the local-global principle for isogenies of abelian surfaces
Davide Lombardo, Matteo Verzobio
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Let ℓ be a prime number. We classify the subgroups G of Sp₄(𝔽_ℓ) and GSp₄(𝔽_ℓ) that act irreducibly on 𝔽_ℓ⁴, but such that every element of G fixes an 𝔽_ℓ-vector subspace of dimension 1. We use this classification to prove that the local-global principle for isogenies of degree ℓ between abelian surfaces over number fields holds in many cases -- in particular, whenever the abelian surface has non-trivial endomorphisms and ℓ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes ℓ for which some abelian surface A/ℚ fails the local-global principle for isogenies of degree ℓ.
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