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

Abstract

We classify compact Riemann surfaces of genus \(g\), where \(g-1\) is a prime \(p\), which have a group of automorphisms of order \(\rho(g-1)\) for some integer \(\rho\ge 1\), and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for \(\rho>6\), and of the first and third authors for \(\rho=\) 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.