计算具有辛伽罗瓦群的数域

中文摘要 AI 辅助

设 $n\geq 1$，$\ell$ 为奇素数。令 $G=\mathrm{PGSp}_{2n}(\mathbb{F}_\ell)$ 或 $\mathrm{GSp}_{2n}(\mathbb{F}_\ell)$，并固定一个忠实的传递置换表示 $\pi:G\longrightarrow S_d$。我们研究次数为 $d$ 的数域，其伽罗瓦闭包具有伽罗瓦群 $G$，且其关联的置换表示为 $\pi$。对于 $\sigma\in S_d$，记 $\operatorname{ind}(\sigma)$ 为其置换指标，即 $\operatorname{ind}(\sigma) = d-\\#\{\text{$\sigma$ 在 $\{1,\ldots,d\}$ 上的轨道数}\}$。若 $\tau$ 表示一个辛换位（symplectic transvection），或其射影辛群中的像，我们证明绝对判别式至多为 $X$ 的此类数域的数量至少为 $X^{1/(2n\\,\operatorname{ind}(\pi(\tau)))}$ 的常数倍。对于自然向量作用和射影作用，当 $\ell\to\infty$ 时，我们得到的指数渐近为 Malle 猜想弱形式所预测值的 $1/(4n)$。这些数域由一参数超椭圆曲线族的雅可比簇所附带的模 $\ell$ 伽罗瓦表示构造而成。证明结合了该族的大辛幺模群（large symplectic monodromy）与无平方因子筛法。

英文摘要

Let $n\geq 1$ 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 $π:G\longrightarrow S_d$. We study degree-$d$ number fields whose Galois closures have Galois group $G$ and whose associated permutation representation is $π$. For $σ\in S_d$, write $\operatorname{ind}(σ)$ for its permutation index, namely $\operatorname{ind}(σ) = d-\#\{\text{orbits of $σ$ on $\{1,\ldots,d\}$}\}$. If $τ$ 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 $X^{1/(2n\,\operatorname{ind}(π(τ)))}$. 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.

发表机构

• Chennai Mathematical Institute（金奈数学研究所）

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