Mod-$p$ Galois representations not arising from abelian varieties

TL;DR

This paper demonstrates that for most primes and dimensions, there exist Galois representations with cyclotomic similitude character that do not originate from any abelian variety over ield.

Contribution

It generalizes known results by showing the existence of Galois representations not arising from abelian varieties for all primes and higher dimensions, except specific small cases.

Findings

Existence of non-abelian Galois representations for most primes and dimensions.

Counterexamples for all primes and dimensions ield, except (2,2), (2,3), (3,2).

Extension of previous results beyond small primes and dimensions.

Abstract

It is known that any Galois representation $\rho : G_{\mathbb{Q}} \rightarrow \mathrm{GL}(2,\mathbb{F}_p)$ with determinant equal to the mod-$p$ cyclotomic character, arises from the $p$-torsion of an elliptic curve over $\mathbb{Q}$, if and only if $p \leq 5$. In dimension $g = 2$, when $p \le 3$, it is again known that any Galois representation valued in $\mathrm{GSp}(4,\mathbb{F}_p)$ with cyclotomic similitude character arises from an abelian surface. In this paper, we study this question for all primes $p$ and dimensions $g \ge 2$. When $g \ge 2$ and $(g,p) \neq (2,2)$, $(2,3)$, $(3,2)$, we prove the existence of a Galois representation over $\mathbb{Q}$ valued in $\mathrm{GSp}(2g,\mathbb{F}_p)$ with cyclotomic similitude character, that cannot arise as the $p$-torsion representation of any $g$-dimensional abelian variety over $\mathbb{Q}$.