Computing $L$-polynomials of Picard curves from Cartier–Manin matrices

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.