Periods of generalized Fermat curves

Abstract

Let k,n≥2 be integers. A generalized Fermat curve of type (k,n) is a compact Riemann surface S that admits a subgroup of conformal automorphisms H≤Aut(S) isomorphic to Zkn, such that the quotient surface S/H is biholomorphic to the Riemann sphere Cˆ and has n+1 branch points, each one of order k. There exists a good algebraic model for these objects, which makes them easier to study. Using tools from algebraic topology and integration theory on Riemann surfaces, we find a set of generators for the first homology group of a generalized Fermat curve. Finally, with this information, we find a set of generators for the period lattice of the associated Jacobian variety.