A note on large automorphism groups of compact Riemann surfaces

Abstract

Belolipetsky and Jones classified those compact Riemann surfaces of genus g admitting a large group of automorphisms of order λ(g − 1), for each λ > 6, under the assumption that g − 1 is a prime number. In this article we study the remaining large cases; namely, we classify Riemann surfaces admitting 5(g − 1) and 6(g − 1) automorphisms, with g − 1 a prime number. As a consequence, we obtain the classification of Riemann surfaces admitting a group of automorphisms of order 3(g − 1), with g − 1 a prime number. We also provide isogeny decompositions of their Jacobian varieties.

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

“…This theorem confirms and extends results obtained earlier by Belolipetzky and the second author [1] for ρ > 6, and more recently by the first and third authors [24,39] for ρ = 3, 4, 5, 6. There are similar but less uniform results for primes p ≤ 5 and for ρ = 1 and 2, discussed briefly in Sections 8 and 9 after the proof of Theorem 1.…”

“…The following result extends and confirms previous results in [24] and [39] and provides a complete treatment of isogeny decompositions for each surface S in Theorem 1.…”

“…(The primes p = 2, 3 and 5 are often omitted, since automorphism groups for very small genera g behave differently and are well-known: see [4,9], for example.) The results in [1] for ρ > 6 have recently been extended by the first and third authors [24,39] to the cases where ρ = 3, 4, 5 or 6 and p ≥ 7; for instance, they show that for ρ = 5 there are no new surfaces (the groups G with ρ = 5 are all subgroups of those appearing in [1] as full automorphism groups of surfaces S with ρ = 10), whereas for ρ = 6 they describe an infinite family of surfaces, two members of which also appear in [1] with larger automorphism groups; this family also realises the groups arising for ρ = 3.…”

“…Here we will build on these results, filling in gaps (such as for small primes where ρ > 6) to give a unified treatment and a complete classification of the groups G and surfaces S for all integers ρ ≥ 1. We also describe some group actions, such as those in case (v) of Theorem 1(a), which are only implicit in [24], where the emphasis is more on the surfaces than on the groups. For conciseness of exposition and proof, the main result, Theorem 1, is stated (below) only for integers ρ ≥ 3 and primes p ≥ 7.…”

“…where ω is a primitive rth root of 1 in the field F p of order p. This is a semidirect product of a ∼ = C p by b ∼ = C r , with the latter acting faithfully by conjugation on the former. Up to isomorphism this group, denoted by C p ⋊ r C r in [24], is independent of the choice of ω, and is the unique subgroup of order pr in the affine group AGL 1 (p) ∼ = G p,p−1 . For example, G p,2 is a dihedral group D p of order 2p.…”