Periods of generalized Fermat curves

2018/12/22 by Torres-Nova, Yerko · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1812.09436

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 ℤ_kⁿ, such that the quotient surface S/H is biholomorphic to the Riemann sphere ℂ 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.

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