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Let Ct(n) denote the number of t−core partitions of n, where a partition is a t−core if none of the hook numbers of the associated Ferrers-Young diagram are multiples of t. It is well known that t−cores describe the local-global behavior of representations of many finite groups, and are important for special cases of the Ramanujan congruences for the ordinary partition function. We show that these partitions also capture the structure of ideal class groups of suitable imaginary quadratic fields, which is a new perspective in the study of class groups. We show that if 8n + 5 is square-free, then C4(n) = 1 2 h(−32n − 20) where h(D) is the discriminant D class number. Moreover using the theory of abaci, we obtain an explicit map from the set of 4−cores to binary quadratic forms. As immediate corollaries, we obtain the Hirschhorn-Sellers Conjectures, and we find that a unique genus in the class group CL(−32n − 20) labels p−defect zero unipotent characters of suitable finite general linear groups GL(n, q).

4-core partitions and class numbers