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The distribution of class groups of function fields Jeffrey D. Achter

Summary: The distribution of class groups of function fields
Jeffrey D. Achter
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


Source: Achter, Jeff - Department of Mathematics, Colorado State University


Collections: Environmental Sciences and Ecology; Mathematics