L-PARTS OF DIVISOR CLASS GROUPS OF CYCLIC FUNCTION FIELDS OF DEGREE l

Abstract

Let l be a prime number and K be a cyclic extension of degree l of the rational function field 𝔽q(T) over a finite field of characteristic ≠ = l. Using class field theory we investigate the l-part of Pic 0(K), the group of divisor classes of degree 0 of K, considered as a Galois module. In particular we give deterministic algorithms that allow the computation of the so-called (σ - 1)-rank and the (σ - 1)2-rank of Pic 0(K), where σ denotes a generator of the Galois group of K/𝔽q(T). In the case l = 2 this yields the exact structure of the 2-torsion and the 4-torsion of Pic 0(K) for a hyperelliptic function field K (and hence of the 𝔽q-rational points on the Jacobian of the corresponding hyperelliptic curve over 𝔽q). In addition we develop similar results for l-parts of S-class groups, where S is a finite set of places of K. In many cases we are able to prove that our algorithms run in polynomial time.