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Register a new userPicard curves over Q with good reduction away from 3
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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.
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We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant.
Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We give a bound on the primes $mathfrak{p}$ of $M$ such that the stable reduction of $C$ at $mathfrak{p}$ contains three irreducible components of genus $1$.
We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $mathbb{Q}$ at primes $p$ of good reduction for $C$. We define a degree 9 polynomial $psi_fin mathbb{Q}[x]$ such that the splitting field of $psi_f(x^3/2)$ is the $2$-torsion field of the Jacobian of $C$. We prove that, for all but a density zero subset of primes, the zeta function $Z(C_p,T)$ is uniquely determined by the Cartier-Manin matrix $A_p$ of $C$ modulo $p$ and the splitting behavior modulo $p$ of $f$ and $psi_f$; we also show that for primes $equiv 1 pmod{3}$ the matrix $A_p$ suffices and that for primes $equiv 2 pmod{3}$ the genericity assumption on $C$ is unnecessary. An element of the proof, which may be of independent interest, is the determination of the density of the set of primes of ordinary reduction for a generic Picard curve. By combining this with recent work of Sutherland, we obtain a practical deterministic algorithm that computes $Z(C_p,T)$ for almost all primes $p le N$ using $Nlog(N)^{3+o(1)}$ bit operations. This is the first practical result of this type for curves of genus greater than 2.
We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so-called special Picard curves over $mathbb{Q}$ with good reduction outside 2 and 3, and use this to determine the smallest possible conductor a special Picard curve may have. We also collect a database of Picard curves over $mathbb{Q}$ of small conductor.
An elliptic curve $E$ over $mathbb{Q}$ is said to be good if $N_{E}^{6}<max!left{ leftvert c_{4}^{3}rightvert ,c_{6}^{2}right} $ where $N_{E}$ is the conductor of $E$ and $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of $E$. In this article, we generalize Massers Theorem on the existence of infinitely many good elliptic curves with full $2$-torsion. Specifically, we prove via constructive methods that for each of the fifteen torsion subgroups $T$ allowed by Mazurs Torsion Theorem, there are infinitely many good elliptic curves $E$ with $E!left(mathbb{Q}right) _{text{tors}}cong T$.
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