Primes dividing invariants of CM Picard curves

Plane quartics over $\mathbb {Q}$ with complex multiplication

“…Taking this even further allowed Lauter-Viray [35] to find out exactly which prime powers divide the discriminant. Similar bounds on primes dividing invariants have been obtained by Kılıçer-Lauter-Lorenzo-Newton-Ozman-Streng for hyperelliptic [24] and Picard [24,25] curves.…”

Constructing Picard curves with complex multiplication using the Chinese remainder theorem

“…If g 3 = 0, then C is a double cover of an elliptic curve (see [22,Lemma 2.1] and [22,Theorem 2.4]). Thus the invariants for a Picard curve C whose Jacobian is simple are always defined.…”

“…For genus 3 curves with CM by a sextic CM-field K, a bound on the primes of bad reduction in terms of a value depending on K was obtained in [8] and [21]. A bound on the primes occurring in the denominators of the above invariants of Picard curves was obtained in [22].…”

“…We work with the set of invariants for Picard curves j 1 , j 2 , j 3 defined in [22]. They are discussed in more detail in Section 3.…”

“…Now assume that C is a Picard curve and that v P (j i (C)) < 0 for some j i . After possibly extending M , we may assume that Jac(C) has good reduction modulo P. Then the reduction of Jac(C) modulo P has two non-trivial abelian subvarieties by[22, Lemma 2.1]. However, as p splits completely in K, we again obtain a contradiction.Remark 3.3.…”