Riquelme Faúndez, Edgardo. (2016) . Algorithms for l-sections on genus two curves over finite fields and applications. Universitat de Lleida. http://hdl.handle.net/10803/393881.

We study \ell-section algorithms for Jacobian of genus two over finite fields. We provide trisection (division by \ell=3) algorithms for Jacobians of genus 2 curves over finite fields \F_q of odd and even characteristic. In odd characteristic we obtain a symbolic trisection polynomial whose roots correspond (bijectively) to the set of trisections of the given divisor. We also construct a polynomial whose roots allow us to calculate the 3-torsion divisors. We show the relation between the rank of the 3-torsion subgroup and the factorization of this 3-torsion polynomial, and describe the factorization of the trisection polynomials in terms of the galois structure of the 3- torsion subgroup. We generalize these ideas and we determine the field of definition of an \ell-section with \ell \in {3, 5, 7}. In characteristic two for non-supersingular hyperelliptic curves we characterize the 3-torsion divisors and provide a polynomial whose roots correspond to the set of trisections of the given divisor. We also present a generalization of the known algorithms for the computation of the 2-Sylow subgroup to the case of the \ell-Sylow subgroup in general and we present explicit algorithms for the computation of the 3-Sylow subgroup. Finally we show some examples where we can obtain the central coefficients of the characteristic polynomial of the Frobenius endomorphism reduced modulo 3 using the generators obtained with the 3-Sylow algorithm.