On primes dividing the invariants of Picard curves

Abstract: The j-invariants of elliptic curves with complex multiplication (CM) are algebraic integers. For invariants of genus g = 2 or 3, this is not the case, though suitably chosen invariants do have smooth denominators in many cases. Bounds on the primes in these denominators have been given for g=2 (Goren-Lauter) and some cases of g=3. For Picard curves of genus 3, we give a new approach based not on bad reduction of curves but on a very explicit type of good reduction. This approach simultaneously yields much sharper bounds and a simplification of the proof. This is a joint work with Marco Streng and Elisa Lorenzo García.