Author:
Lombardo Davide,Verzobio Matteo
Abstract
AbstractLet $$\ell $$
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
.
Publisher
Springer Science and Business Media LLC
Subject
General Physics and Astronomy,General Mathematics
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