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Mathematics of Computation

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Superelliptic curves with many automorphisms and CM Jacobians

Authors: Andrew Obus and Tanush Shaska
Journal: Math. Comp. 90 (2021), 2951-2975
MSC (2020): Primary 14H37; Secondary 14H45, 14K22
Published electronically: April 13, 2021
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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 many automorphisms if its corresponding moduli point $\mathfrak {p} \in \mathcal {M}_g$ has a neighborhood $U$ in the complex topology, such that all curves corresponding to points in $U \setminus \{\mathfrak {p} \}$ have strictly fewer automorphisms than $\mathcal {C}$. We compute completely the list of superelliptic curves $\mathcal {C}$ for which the superelliptic automorphism is normal in the automorphism group $\mathrm {Aut} (\mathcal {C})$ and $\mathcal {C}$ has 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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Additional Information

Andrew Obus
Affiliation: 1 Bernard Baruch Way, New York, New York 10010
MR Author ID: 890287
ORCID: 0000-0003-2358-4726

Tanush Shaska
Affiliation: Department of Mathematics, Oakland University, 146 Library Drive, 368 Mathematics Science Center, Rochester, Michigan 48309-4479
MR Author ID: 678224
ORCID: 0000-0002-2293-8230

Keywords: Complex multiplication, superelliptic curves.
Received by editor(s): June 30, 2020
Received by editor(s) in revised form: January 20, 2021
Published electronically: April 13, 2021
Additional Notes: The first author was supported by the National Science Foundation under DMS Grant No. 1900396. The second author was supported in part by the Research Institute of Science and Technology (Vlora, Albania) under Grant no. RISAT-2020-137
Article copyright: © Copyright 2021 American Mathematical Society