Counting number fields with symplectic Galois group

Abstract

Let $n\ge 1$ and let $\ell$ be an odd prime. Let $G={\mathrm{PGSp}}_{2n}({\mathbb{F}}_{\ell })$ or ${\mathrm{GSp}}_{2n}({\mathbb{F}}_{\ell })$, and fix a faithful transitive permutation representation $\pi :G⟶S_d$. We study degree-$d$ number fields whose Galois closures have Galois group $G$ and whose associated permutation representation is $\pi$. For $\sigma \in S_d$, write $\mathrm{ind}(\sigma )$ for its permutation index, namely $\operatorname{ind}(σ) = d-\#\{\text{orbits of $σ$on$\{1,\ldots,d\}$}\}$. If $\tau$ denotes a symplectic transvection, or its image in the projective symplectic group, we prove that the number of such fields with absolute discriminant at most $X$ is bounded below by a constant multiple of $X^{1/(2n\mathrm{ind}(\pi (\tau )))}$. For the natural vector and projective actions, the exponents we obtain are asymptotically $1/(4n)$ of those predicted by the weak form of Malle's conjecture as $\ell \to \infty$. The fields are constructed from the mod-$\ell$ Galois representations attached to the Jacobians of a one-parameter family of hyperelliptic curves. The proof combines large symplectic monodromy for this family with a squarefree sieve.

arXiv commentv1: 17 pages