On Riemann surfaces of genus g with 4g–4 automorphisms

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.…”

“…Case (iv) is contained in case (i); they are new and indeed improve the decomposition found in [24,Theorem 2]. Case (vi) is new but implicit in [24,Theorem 1] since it is contained in case (ii) and case (viii) improves the decomposition determined in [39,Theorem 3]. Case (vii) is dealt with in [39,Theorem 3].…”

“…(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.…”

“…Case (vi) is new but implicit in [24,Theorem 1] since it is contained in case (ii) and case (viii) improves the decomposition determined in [39,Theorem 3]. Case (vii) is dealt with in [39,Theorem 3]. Cases (x), (xi) and (xii) are well-known; these Jacobians decompose as a product of elliptic curves (see, for example, [18] for the former case in a more general context).…”