群$\u200b\operatorname{SL}_2(\mathbb{F}_{13})$和$\operatorname{SL}_2(\mathbb{F}_{19})$是$\mathbb{Q}$上的伽罗瓦群
The groups $\operatorname{SL}_2(\mathbb{F}_{13})$ and $\operatorname{SL}_2(\mathbb{F}_{19})$ are Galois over $\mathbb{Q}$
- Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过构造对应次数的多项式、验证Böge准则并求解中心嵌入问题，结合二次扭与Hurwitz族、模方程工具，证明了两个特殊线性群是有理数域全实扩张的伽罗瓦群。
AI中文摘要：
本文证明了$\operatorname{SL}_2(\mathbb{F}_{13})$和$\operatorname{SL}_2(\mathbb{F}_{19})$是有理数域$\mathbb{Q}$的全实扩张的伽罗瓦群。对这两个素数$\ell$，我们各找到一个次数为$\ell+1$的多项式，其分裂域的伽罗瓦群为$\operatorname{PSL}_2(\mathbb{F}_\ell)$。这些域满足Böge准则，且对应的中心嵌入问题存在伽罗瓦群为$\operatorname{SL}_2(\mathbb{F}_\ell)$的真解，可通过二次扭将所得域选为全实域。14次多项式通过亏格1的14次覆盖Hurwitz族得到，20次多项式则借助Yang给出的志村曲线$X_6^*(1)$的模方程求得。
英文摘要：
In this paper, we show that $\operatorname{SL}_2(\mathbb{F}_{13})$ and $\operatorname{SL}_2(\mathbb{F}_{19})$ are Galois groups of totally real extensions of $\mathbb{Q}$. For each of these primes $\ell$, we find a polynomial of degree $\ell+1$ whose splitting field has Galois group $\operatorname{PSL}_2(\mathbb{F}_\ell)$. These fields satisfy Böge's criterion and the associated central embedding problem has a proper solution with Galois group $\operatorname{SL}_2(\mathbb{F}_\ell)$. One can choose the resulting fields totally real via a quadratic twist. The degree $14$ polynomial is found through a genus one Hurwitz family of degree 14 covers. The degree 20 polynomial is found using Yang's modular equations for the Shimura curve $X_6^*(1)$.
