Computing Class Groups of Function Fields Using Stark Units

Abstract

Let $k$ be a fixed finite geometric extension of the rational function field $\mathbb{F}_q(t)$. Let $F/k$ be a finite abelian extension such that there is an $\Fq$-rational place $\infty$ in $k$ which splits in $F/k$ and let $\mathcal{O}_F$ denote the integral closure in $F$ of the ring of functions in $k$ that are regular outside $\infty$. 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 $\ell$ not dividing $q(q-1)[F:k]$, the structure of the $\ell$-part of the ideal class group $\p(\cO_F)$ of $\mathcal{O}_F$ 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 $\ell$ primary part of the divisor class group of a narrow ray class field for all primes $\ell$ not dividing $q(q-1)[F:k]$.