Classifying compact Riemann surfaces by number of symmetries

2023/10/11 by Reyes-Carocca, Sebastián, Speziali, Pietro
#14H30 #14H37 #30F10 #30F35 #32G15 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.07520

Abstract

In this article we consider compact Riemann surfaces that are uniquely determined by the property of possessing a group of automorphisms of a prescribed order, strengthening uniqueness results proved by Nakagawa. More precisely, we deal with the cases in which such an order is 3g and 3g+3, where g is the genus. We prove that if g is odd (respectively g even and g \not ≡ 2 mod 3) then there exists a unique Riemann surface of genus g with a group of automorphisms of order 3g (respectively 3g+3). A similar conclusion can be derived in terms of orientably-regular hypermaps. In addition, we determine the full automorphism group of such Riemann surfaces and provide decompositions of their Jacobians.

Related

• Riemann surfaces of genus 1+q² with 3q² automorphisms
• Groups of automorphisms of Riemann surfaces and maps of genus p+1\n where p is prime
• An algorithm to compute automorphism groups of K3 surfaces and an application to singular K3 surfaces
• Riemann surfaces with a large abelian group of automorphisms
• On Riemann Surfaces of genus g with 4g automorphisms