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It is known that any Galois representation $\rho : G_{\mathbb{Q}} \rightarrow\mathrm{GL}(2,\mathbb{F}_p)$ with determinant equal to the mod-$p$ cyclotomiccharacter, 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 againknown that any Galois representation valued in $\mathrm{GSp}(4,\mathbb{F}_p)$with cyclotomic similitude character arises from an abelian surface. In thispaper, 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 aGalois representation over $\mathbb{Q}$ valued in$\mathrm{GSp}(2g,\mathbb{F}_p)$ with cyclotomic similitude character, thatcannot arise as the $p$-torsion representation of any $g$-dimensional abelianvariety over $\mathbb{Q}$.

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