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

Abstract

It is known that any Galois representation $\rho :G_{\mathbb{Q}}\to \mathrm{GL}(2,{\mathbb{F}}_p)$ 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 $p\le 5$. 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 $(g,p)\ne (2,2)$, $(2,3)$, $(3,2)$, 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 $\mathbb{Q}$.

arXiv comment15 pages. Minor updates