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

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We study the sequence of zeta functions Z ( C p , T ) Z(C_p,T) of a generic Picard curve C : y 3 = f ( x ) C:y^3=f(x) defined over Q \mathbb {Q} at primes p p of good reduction for C C . We define a degree 9 polynomial ψ f ∈ Q [ x ] \psi _f\in \mathbb {Q}[x] such that the splitting field of ψ f ( x 3 / 2 ) \psi _f(x^3/2) is the 2 2 -torsion field of the Jacobian of C C . We prove that, for all but a density zero subset of primes, the zeta function Z ( C p , T ) Z(C_p,T) is uniquely determined by the Cartier–Manin matrix A p A_p of C C modulo p p and the splitting behavior modulo p p of f f and ψ f \psi _f ; we also show that for primes ≡ 1 ( mod 3 ) \equiv 1 \pmod {3} the matrix A p A_p suffices and that for primes ≡ 2 ( mod 3 ) \equiv 2 \pmod {3} the genericity assumption on C 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 ) Z(C_p,T) for almost all primes p ≤ N p \le N using N log ⁡ ( N ) 3 + o ( 1 ) N\log (N)^{3+o(1)} bit operations. This is the first practical result of this type for curves of genus greater than 2.