The distribution of class groups of function elds

For any sufciently general family of curves over a nite eld 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 elds. 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 eld with frequency inversely proportional to the size of its automorphism group. Six years later Friedman and Washington [9] addressed the function eld case. Fix a nite eld 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 elds over Fq with frequency inversely proportional to jAut(H)j. As evidence for this, they prove that the uniform distribution of Frobenius automorphisms of curves of genus g in GL2g(Z`) would im