Computing L-Polynomials of Picard curves from Cartier-Manin matrices

TL;DR

This paper develops a practical algorithm to compute the zeta functions of Picard curves at almost all primes, leveraging Cartier-Manin matrices and splitting field properties, advancing computational number theory for genus > 2 curves.

Contribution

It introduces a new deterministic algorithm for computing zeta functions of Picard curves at almost all primes, based on Cartier-Manin matrices and splitting field analysis.

Findings

The zeta function is determined by Cartier-Manin matrices and splitting behavior for most primes.

For primes ≡ 1 mod 3, Cartier-Manin matrices suffice; for ≡ 2 mod 3, no genericity assumption needed.

The algorithm operates in N log(N)^{3+o(1)} bit operations for primes ≤ N.

Abstract

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 $\psi_f\in \mathbb{Q}[x]$ 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 $N\log(N)^{3+o(1)}$ bit operations. This is the first practical result of this type for curves of genus greater than 2.