Remarkable identities in the counting functions for cubic and quartic rings

Remarkable identities in the counting functions for cubic and quartic rings

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Evan O'Dorney , Princeton University
Fine Hall 110

Let h(D) be the number of cubic rings having discriminant D, and let h'(D) be the number of cubic rings having discriminant -27D where the traces of all elements are divisible by 3. (In each case, weight rings by the reciprocal of their number of automorphisms.) At first glance, there is no relation between these two quantities, nor was there expected to be until in 1997, Y. Ohno computed them up to |D| = 200 and found a stunning coincidence that generalizes both cubic reciprocity and the Scholz reflection principle. The next year, J. Nakagawa verified Ohno's conjecture. I will explain the main ideas in my streamlined version of Nakagawa's proof, and an extension to quartic rings still in progress.