Teichmüller Curves Through Fermat Curves

Abstract

The billiard in a regular n-gon is known to give rise to a Teichmuller curve. For odd n, we compute the genus of this curve, a number field over which the curve may be defined and branched covering relations between certain pairs of these curves. If n is a power of a prime congruent to 3 or 5 modulo 8, the Teichmuller curve may be defined over the rationals.