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, \v Sirá\v n and Tucker for maps.

29 pages, 5 figures