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 $\equiv 1(\mathrm{mod}3)$ the matrix $A_p$ suffices and that for primes $\equiv 2(\mathrm{mod}3)$ 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.

arXiv commentMinor changes. To appear in Mathematics of Computation

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