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.