I‑AIMS Institute for Artificial Intelligence
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Superelliptic curves with many automorphisms and CM Jacobians

Andrew Obus, Tanush Shaska

I-AIMS paper 2019-3 · Published in Math. Comp. 90 (2021), 2951–2975 · 2021

Let C be a smooth, projective, genus g ≥ 2 curve, defined over ℂ. Then C has many automorphisms if its corresponding moduli point 𝔭 ∈ ℳ_g has a neighborhood U in the complex topology, such that all curves corresponding to points in U ∖ {𝔭} have strictly fewer automorphisms than C. We compute completely the list of superelliptic curves C for which the superelliptic automorphism is normal in the automorphism group Aut(C) and 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.

MSC classes: Primary 14H37; Secondary 14H45, 14K22

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