Primes dividing invariants of CM Picard curves

TL;DR

This paper establishes a sharp, explicit bound on primes dividing invariants of genus 3 Picard curves with complex multiplication, based on a novel approach involving good reduction, enabling practical curve construction.

Contribution

It introduces a new method for bounding primes dividing invariants of CM Picard curves, improving sharpness and applicability over previous bounds.

Findings

Derived a sharper prime bound for invariants of CM Picard curves

Utilized a novel approach based on explicit good reduction

Enabled practical construction of Picard curves with CM

Abstract

We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof, and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.