Papers › Computing L-Polynomials of Picard curves from Cartier-Manin matrices
Computing L-Polynomials of Picard curves from Cartier-Manin matrices
Sualeh Asif, Francesc Fité, Dylan Pentland
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We study the sequence of zeta functions Z(Cₚ,T) of a generic Picard curve C:y³=f(x) defined over ℚ at primes p of good reduction for C. We define a degree 9 polynomial ψ_f∈ℚ[x] such that the splitting field of ψ_f(x³/2) 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(Cₚ,T) is uniquely determined by the Cartier-Manin matrix Aₚ of C modulo p and the splitting behavior modulo p of f and ψ_f; we also show that for primes ≡1 3 the matrix Aₚ suffices and that for primes ≡2 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(Cₚ,T) for almost all primes p ≤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.
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