Examples of abelian surfaces failing the local–global principle for isogenies

S., Banwait, Barinder

doi:10.1007/s40993-021-00283-9

Cited by 2 publications

(1 citation statement)

References 27 publications

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“…For example, when g = 1 and m is a positive integer, the articles [Sut12], [Ann14], [Vog20] study the problem of characterizing the existence of a degree m isogeny of A in terms of the existence of degree m isogenies for A p for almost all p ∈ Σ K . The analogous problem when g = 2 has been recently explored by [Ban21] and [LV22]. Rather than on the degree of the isogeny, in the present article we focus on its field of definition.…”

Section: Introductionmentioning

confidence: 98%

A local-global principle for polyquadratic twists of abelian surfaces

Fité¹,

Perucca²

2022

Preprint

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We say that two abelian varieties A and A ′ defined over a field F are polyquadratic twists if they are isogenous over a Galois extension of F whose Galois group has exponent dividing 2. Let A and A ′ be abelian varieties defined over a number field K of dimension g 1. In this article we prove that, if g 2, then A and A ′ are polyquadratic twists if and only if for almost all primes p of K their reductions modulo p are polyquadratic twists. We exhibit a counterexample to this local-global principle for g = 3. This work builds on a geometric analogue by Khare and Larsen, and on a similar criterion for quadratic twists established by Fité, relying itself on the works by Rajan and Ramakrishnan.

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Section: Introductionmentioning

confidence: 98%

A local-global principle for polyquadratic twists of abelian surfaces

Fité¹,

Perucca²

2022

Preprint

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On the local-global principle for isogenies of abelian surfaces

Lombardo,

Verzobio

2024

Sel. Math. New Ser.

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Let $$\ell $$ ℓ be a prime number. We classify the subgroups G of $${\text {Sp}}_4({\mathbb {F}}_\ell )$$ Sp 4 ( F ℓ ) and $${\text {GSp}}_4({\mathbb {F}}_\ell )$$ GSp 4 ( F ℓ ) that act irreducibly on $${\mathbb {F}}_\ell ^4$$ F ℓ 4 , but such that every element of G fixes an $${\mathbb {F}}_\ell $$ F ℓ -vector subspace of dimension 1. We use this classification to prove that a local-global principle for isogenies of degree $$\ell $$ ℓ between abelian surfaces over number fields holds in many cases—in particular, whenever the abelian surface has non-trivial endomorphisms and $$\ell $$ ℓ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes $$\ell $$ ℓ for which some abelian surface $$A/{\mathbb {Q}}$$ A / Q fails the local-global principle for isogenies of degree $$\ell $$ ℓ .

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Examples of abelian surfaces failing the local–global principle for isogenies

Cited by 2 publications

References 27 publications

A local-global principle for polyquadratic twists of abelian surfaces

A local-global principle for polyquadratic twists of abelian surfaces

On the local-global principle for isogenies of abelian surfaces

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