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arXiv · 0910.4695

Semi-direct Galois covers of the affine line

Linda Gruendken·Laura Hall-Seelig·Bo-Hae Im·Ekin Ozman·Rachel Pries·Katherine Stevenson
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Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be $Z/\ell Z$ semi-direct product $Z/pZ$ where $\ell$ is a prime distinct from $p$. In this paper, we study Galois covers $\psi:Z \to P^1_k$ ramified only over $\infty$ with Galois group $G$. We find the minimal genus of a curve $Z$ that admits such a cover and show that it depends only on $\ell$, $p$, and the order $a$ of $\ell$ modulo $p$. We also prove that the number of curves $Z$ of this minimal genus which admit such a cover is at most $(p-1)/a$.

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Linda Gruendken, Laura Hall-Seelig, Bo-Hae Im, Ekin Ozman, Rachel Pries, Katherine Stevenson. 2009-10-25. Semi-direct Galois covers of the affine line. https://arxiv.org/abs/0910.4695

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