Point counting on Picard curves in large characteristic

We present an algorithm for computing the cardinality of the Jacobian of a random Picard curve over a finite field. If the underlying field is a prime field Fp, the algorithm has complexity O(p).

Quotients of Fermat curves and a Hecke character

AbstractWe explicitly identify infinitely many curves which are quotients of Fermat curves. We show that some of these have simple Jacobians with complex multiplication by a non-cyclotomic field. For a particular case we determine the local zeta functions with two independent methods. The first uses Jacobi sums and the second applies the general theory of complex multiplication, we verify that both methods give the same result.

Computing generators of the tame kernel of a global function field

Abstract The group K 2 of a curve C over a finite field is equal to the tame kernel of the corresponding function field. We describe two algorithms for computing generators of the tame kernel of a global function field. The first algorithm uses the transfer map and the fact that the l -torsion can easily be described if the ground field contains the l th roots of unity. The second method is inspired by an algorithm of Belabas and Gangl for computing generators of K 2 of the ring of integers in a number field. We finally give the generators of the tame kernel for some elliptic function fields.

On the number of prime divisors of the order of elliptic curves modulo p