Computing class groups of function fields using stark units

Let k be a fixed finite geometric extension of the rational function field F-q(t). Let F/k be a finite abelian extension such that there is an F-q-rational place on in k which splits in F/(k) and let O-F denote the integral closure in F of the ring of functions in k that are regular outside co. We describe algorithms for computing the divisor class number and in certain cases for computing the structure of the divisor class group and discrete logarithms between Galois conjugate divisors in the divisor class group of F. The algorithms are efficient when F is a narrow ray class field or a small index subextension of a narrow ray class field. We prove that for all prime e not dividing q(q-1)[F : k], the structure of the e-part of the ideal class group Pic(O-F) of OF is determined by Kolyvagin derivative classes that are constructed out of Euler systems associated with Stark units. This leads to an algorithm to compute the structure of the e primary part of the divisor class group of a narrow ray class field for all primes e not dividing q(q 1)[F : k].