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.