Picard curves over with good reduction away from 3

Abstract

Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over Q with good reduction away from 3, up to Q-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined and an application to a question of Ihara is discussed.

Picard Curves with Small Conductor

“…Proof. This is an adaption to Picard curves of the algorithm given by Smart for hyperelliptic curves, see [25] and [13]. The idea is that it suffices to determine the finite set of equivalence classes of binary forms of degree 4 over K whose discriminant is an S-unit (corresponding to the polynomial f (x)).…”

“…Let K = Q and S = {3}. Then there are precisely 63 isomorphism classes of Picard curves over Q with good reduction outside S. See [13]. For example, the curve Y : y 3 = f (x) = x 4 − 3x 3 − 24x 2 − x has good reduction outside S = {3} (the discriminant of f is ∆ ( f ) = 3 10 ).…”

“…The stable reductionȲ of Y at p = 3 is as in Case (c) of Theorem 3.2, the exponent conductor is f 3 = 10 (see [4], Appendix A1.1). This is the lowest value for the conductor which occurs for the curves in the list of [13]. The conductor exponents of all 63 Picard curves from [13] have been computed in [4], Appendix A1.2.…”

“…This is the lowest value for the conductor which occurs for the curves in the list of [13]. The conductor exponents of all 63 Picard curves from [13] have been computed in [4], Appendix A1.2. From this calculation it follows that the conductor exponent f 3 only takes the values f 3 = 10, 11, 12, 13, 15, 17, 19, 21. The upper bound on the conductor exponent from abelian varieties of genus 3 from [3], Theorem 6.2 yields f 3 ≤ 21.…”

Cyclic covers and Ihara’s question

“…This is enough to prove Theorem B. In §7, we study further the arithmetic of S and W s ; these two sets usually, but not always, define the same extension of k. In §8, we settle a question posed in [MR16] regarding Picard curves over Q. This also gives an example where Theorem B applies, but Theorem A does not.…”