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On primes dividing the invariants of Picard curves Lorenzo Garcia, Elisa


In Bouw et al. 15 and Kilicer et al. 17, we give a bound for the primes of bad reduction of curves of genus 3 with CM. This bound does not directly translates into a bound for the primes appearing in the denominators of class polynomials of curves of genus 3 because the bad reduction locus for genus 3 has co-dimension 2. However, for special subfamilies in which the bad reduction locus has co-dimension 1, this bound do translate into a bound for primes in the denominators. The family of Picard curves is an example of such subfamily. The bound obtained in these works in huge and not computationally practical. In this new work we give a better set of invariants for Picard curves and a sharper bound for the corresponding denominators. We do it by studying not only primes of bad reduction but also primes of a very particular type of good reduction.

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