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Descent properties for an abelian variety with extended Galois representation

Ludovic Felder

Abstract

Let $K$ be a field, $L$ a finite Galois extension of $K$, and $X$ an abelian variety defined over $L$. If $X$ is isogenous over $L$ to an abelian variety defined over $K$, then the $\ell$-adic Galois representations associated to $X$ extend to representations $\barρ_{\ell,X}:\mathrm{Gal}(\bar{L}/K)\to\mathrm{Aut}(V_\ell X)$ for every prime $\ell$. This paper aims to show that the converse is true for abelian varieties of Type I, with some supplementary conditions needed on the endomorphisms of $X$, when $L$ is either a number field or a function field of prime characteristic different from $2$.

Descent properties for an abelian variety with extended Galois representation

Abstract

Let be a field, a finite Galois extension of , and an abelian variety defined over . If is isogenous over to an abelian variety defined over , then the -adic Galois representations associated to extend to representations for every prime . This paper aims to show that the converse is true for abelian varieties of Type I, with some supplementary conditions needed on the endomorphisms of , when is either a number field or a function field of prime characteristic different from .
Paper Structure (7 sections, 17 theorems, 44 equations)

This paper contains 7 sections, 17 theorems, 44 equations.

Table of Contents

  1. Definitions
  2. The Galois representations associated to an abelian variety
  3. Galois conjugates of abelian varieties
  4. The action on the torsion points
  5. Extension of the representation
  6. A descent result
  7. Extension of the representation for all primes

Key Result

Proposition 1.3

If there exists a morphism of abelian varieties $g:Y\to X$ over $L$ such that $\phi=T_\ell g$ (resp. an element $g\in\mathrm{Hom}^0(Y,X)$ such that $\phi=V_\ell g$), then Thus ${}^{\sigma}T_\ell g=T_\ell {}^{\sigma}g=\pi_X(\tilde{\sigma})\circ T_\ell g\circ \pi_Y(\tilde{\sigma})^{-1}$ (resp. ${}^{\sigma}V_\ell g=V_\ell {}^{\sigma}g$) for any $\tilde{\sigma}\in\mathrm{Gal}(\bar{L}/K)$ extending $\

Theorems & Definitions (41)

  • Remark 1.1
  • Remark 1.2
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • proof
  • Remark 1.5
  • Proposition 2.1
  • proof : Proof
  • Proposition 2.2
  • ...and 31 more