Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the local-global principle for isogenies of abelian surfaces

Davide Lombardo, Matteo Verzobio

Abstract

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

On the local-global principle for isogenies of abelian surfaces

Abstract

Let be a prime number. We classify the subgroups of and that act irreducibly on , but such that every element of fixes an -vector subspace of dimension 1. We use this classification to prove that the local-global principle for isogenies of degree between abelian surfaces over number fields holds in many cases -- in particular, whenever the abelian surface has non-trivial endomorphisms and is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes for which some abelian surface fails the local-global principle for isogenies of degree .
Paper Structure (44 sections, 72 theorems, 50 equations, 2 tables)

This paper contains 44 sections, 72 theorems, 50 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Computer calculations.
  3. Notation
  4. Structure of the paper
  5. Preliminaries
  6. Endomorphism rings and algebraic monodromy groups
  7. Invariance under isogeny
  8. Group theory
  9. Subgroups of
  10. Hasse subgroups of
  11. Preliminary lemmas
  12. Handling the 'small' groups
  13. Further input from representation theory
  14. of type :
  15. Case in the normaliser of a split Cartan subgroup
  16. ...and 29 more sections

Key Result

Lemma 1.2

The pair $(A,\ell)$ is a weak counterexample if and only if the action of $G_\ell$ on $A[\ell]$ leaves no line invariant, but every $g \in G_\ell$ admits an $\mathbb{F}_\ell$-rational eigenvalue. Moreover, $(A, \ell)$ is a strong counterexample if and only if the action of $G_\ell$ on $A[\ell]$ is i

Theorems & Definitions (151)

  • Lemma 1.2
  • Corollary 1.3: Corollary \ref{['cor:final']}
  • Theorem 1.4: Theorem \ref{['thm:SemistableCase']}
  • Proposition 1.5: Proposition \ref{['prop:strongcounter']}
  • Definition 1.6
  • Remark 2.1
  • Conjecture 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 141 more