论文标题
带有大型galois图像的超ellipriptic曲线
Superelliptic curves with large Galois images
论文作者
论文摘要
令$ r> 2 $和$ \ ell $为素数。在本文中，我们研究了$ y^r = f（x）$的mod $ \ ell $ galois表示形式，其中$ f $是monic且具有属于$ r $ r $ th Cyclotomic领域的系数。 我们提供$ f $系数（和程度）的条件，这些条件允许一个人验证mod $ \ ell $图像在（通常很小）有限的显式素数集中的外部很大。我们允许所有$ r $ $ r $ th Cyclotomic字段具有奇数类别的值。这似乎是尺寸大于两个而不的$ {\ rm gl} _2 $ -Type的Abelian品种的第一个显式结果，该品种允许地面场具有未引起的扩展。 在证明大图像结果时，我们对包含某些经典组的转向的最大亚组进行了分类，并描述了（在许多情况下）惯性组的图像。 确切的mod $ \ ell $图像由“内态特征”约束，这是一个概括CM字符的某些代数Hecke字符。当$ r = 3 $时，我们将整体描绘图像。据作者所知，这是文献中的第一个准确描述。 最后，我们提供了几个属属的范围为10到36。
Let $r>2$ and $\ell$ be primes. In this paper we study the mod $\ell$ Galois representations attached to curves of the form $y^r = f(x)$ where $f$ is monic and has coefficients belonging to the $r$-th cyclotomic field. We provide conditions on the coefficients (and degree) of $f$ which allow one to verify the mod $\ell$ image is large outside of a (typically small) finite explicit set of primes. We allow all values of $r$ for which the $r$-th cyclotomic field has odd class number. This appears to be the first explicit result for abelian varieties of dimension greater than two and not of ${\rm GL}_2$-type which allows the ground field to have unramified extensions. In proving the large image result we give a classification of the maximal subgroups containing transvections of certain classical groups and describe (in many cases) the images of inertia groups. The exact mod $\ell$ image is governed by the "endomorphism character", a certain algebraic Hecke character which generalises the CM character. When $r=3$, we depict the image in its entirety. To the author's knowledge, this is the first accurate description in the literature. Finally, we give several examples with genus ranging from 10 to 36. Applications to the Inverse Galois Problem are also included.
