Primes Dividing Invariants of CM Picard Curves

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.

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.…”

An inverse Jacobian algorithm for Picard curves

“…We get the sextic fields from [23, Table 3]. The authors of [13], working off an earlier version of this paper [16], give supporting evidence of the correctness of our examples. We have now confirmed the correctness of the models using the implementation of the algorithm in [5].…”

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.…”