Groups of automorphisms of Riemann surfaces and maps of genus p + 1 where p is prime

Izquierdo, Milagros, Jones, Gareth A. and Reyes-Carocca, Sebastián (2021) Groups of automorphisms of Riemann surfaces and maps of genus p + 1 where p is prime. Annales Fennici Mathematici, 46 (2), 839-867. (doi:10.5186/aasfm.2021.4649).

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Abstract

We classify compact Riemann surfaces of genus g, where g − 1 is a prime p, which have a group of automorphisms of order ρ(g − 1) for some integer ρ ≥ 1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ > 6, and of the first and third authors for ρ = 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p + 1, together with the non-orientable regular hypermaps of characteristic −p, with automorphism group of order divisible by the prime p; this extends results of Conder, Širáň and Tucker for maps.

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Accepted/In Press date: 30 October 2020

Published date: 4 August 2021

Additional Information: Funding Information: Acknowledgment. The authors are grateful to David Singerman for a number of helpful remarks concerning Fuchsian groups, and to an anonymous referee for suggesting some very useful improvements to the paper. The third author was partially supported by Fondecyt Grants 11180024 and 1190991. The first and third authors were partially supported by Redes Grant 170071. Publisher Copyright: © 2021,Annales Fennici Mathematici.All Rights Reserved.

Keywords: Automorphism group, Compact riemann surface, Dessin d’enfant, Finite group, Hypermap, Jacobian, Map

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