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arXiv · 2006.12685

Superelliptic curves with many automorphisms and CM Jacobians

Andrew Obus·Tony Shaska
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Abstract

Let $\mathcal{C}$ be a smooth, projective, genus $g\geq 2$ curve, defined over $\mathbb{C}$. Then $\mathcal{C}$ has \emph{many automorphisms} if its corresponding moduli point $p \in \mathcal{M}_g$ has a neighborhood $U$ in the complex topology, such that all curves corresponding to points in $U \setminus \{p \}$ have strictly fewer automorphisms than $\mathcal{C}$. We compute completely the list of superelliptic curves having many automorphisms. For each of these curves, we determine whether its Jacobian has complex multiplication. As a consequence, we prove the converse of Streit's complex multiplication criterion for these curves.

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Andrew Obus, Tony Shaska. 2020-06-23. Superelliptic curves with many automorphisms and CM Jacobians. https://doi.org/10.1090/mcom/3639

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