This chapter addresses the problem of computing in the group of lsuperscript k-torsion rational points in the Jacobian variety of algebraic curves over finite fields, with an application to computing modular representations. An algorithm in this chapter usually means a probabilistic Las Vegas algorithm. In some places it gives deterministic or probabilistic Monte Carlo algorithms, but this will be stated explicitly. The main reason for using probabilistic Turing machines is that there is a need to construct generating sets for the Picard group of curves over finite fields. Solving such a problem in the deterministic world is out of reach at this time. The unique goal is to prove, as quickly as possible, that the problems studied in this chapter can be solved in probabilistic polynomial time.
Some fundamental groups in the arithmetic of algebraic curves over finite fields
