 
Summary: The distribution of class groups of function fields
Jeffrey D. Achter
j.achter@colostate.edu
Abstract
For any sufficiently general family of curves over a finite field Fq and any elementary
abelian group H with relatively prime to q, we give an explicit formula for the propor
tion of curves C for which Jac(C)[ ](Fq) = H. In doing so, we prove a conjecture of Friedman
and Washington.
In 1983, Cohen and Lenstra introduced heuristics [5] to explain statistical observations about class
groups of imaginary quadratic fields. Their principle, although still unproven, remains an impor
tant source of guidance in number theory. A concrete application of their heuristics predicts that
an abelian group occurs as a class group of an imaginary quadratic field with frequency inversely
proportional to the size of its automorphism group.
Six years later Friedman and Washington [9] addressed the function field case. Fix a finite field Fq
and an abelian group H, where is an odd prime relatively prime to q. Friedman and Washing
ton conjecture that H occurs as the Sylow part of the divisor class group of function fields over
Fq with frequency inversely proportional to Aut(H). As evidence for this, they prove that the
uniform distribution of Frobenius automorphisms of curves of genus g in GL2g(Z ) would im
ply their conjecture. (Of course, autoduality of the Jacobian means that these Frobenius elements
are actually in GSp2g(Z ); still, [9] entertains the hope that this distinction is immaterial to the
