Class group frequencies of real quadratic function fields: The degree 4 case

The distribution of ideal class groups of F-4(T, root M9T)) is examined for degree-four monic polynomials M is an element of F-q[T] when F-q is a fini te field of characteristic greater than 3 with q is an element of [20000,10 0000] or q is an element of [1020000, 1100000] and M is irreducible or has an irreducible cubic factor. Particular attention is paid to the distributi on of the p-Sylow part of the class group, and these results agree with tho se predicted using the Cohen-Lenstra heuristics to within about 1 part in 1 0000. An alternative set of conjectures specific to the cases under investi gation is in even sharper agreement.