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

(With Sualeh Asif and Francesc Fité) This paper gives a fast and practical algorithm to compute L-polynomials of generic Picard curves over Q for all primes p of good reduction up to some integer N. We prove an interesting theoretical result to achieve this: for all but a density zero subset of primes, we can either compute the L-polynomial from the Cartier-Manin matrix or from this matrix and the splitting behavior of f and a polynomial psi_f related to the 2-torsion of the Jacobian. These show that surprisingly little work is needed to compute the L-polynomial at p from the L polynomial modulo p (which is found by the Cartier-Manin matrix).