论文标题
由大型正交单构的家庭产生的Galois群体
Galois groups arising from families with big orthogonal monodromy
论文作者
论文摘要
我们研究了由大型正交单形成型的兼容家族引起的多项式的galois组。我们表明，鉴于功能方程和判别考虑对它们施加的约束，Galois组通常尽可能大。作为一个应用程序，我们考虑$ \ Mathbb {p} _ {\ Mathbb {f} _q} _q}^{2n+1} $至少$ 3 $的frobenius多项式群。我们还考虑了在功能字段上$ \ mathbb {f} _q（t）$上固定度曲线的固定度曲线的二次曲折的$ l $ functions。为了确定椭圆曲线设置中的典型Galois组，需要使用一些已知的桦木和Swinnerton-Dyer猜想。这扩展了Chavdarov，Katz和Jouve的工作。
We study the Galois groups of polynomials arising from a compatible family of representations with big orthogonal monodromy. We show that the Galois groups are usually as large as possible given the constraints imposed on them by a functional equation and discriminant considerations. As an application, we consider the Frobenius polynomials arising from the middle étale cohomology of hypersurfaces in $\mathbb{P}_{\mathbb{F}_q}^{2n+1}$ of degree at least $3$. We also consider the $L$-functions of quadratic twists of fixed degree of an elliptic curve over a function field $\mathbb{F}_q(t)$. To determine the typical Galois group in the elliptic curve setting requires using some known cases of the Birch and Swinnerton-Dyer conjecture. This extends and generalizes work of Chavdarov, Katz and Jouve.
