An identity on class numbers of cubic rings

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• Evan O'Dorney (Cambridge)
• Tuesday 17 May 2016, 14:15-15:15
• MR13.

If you have a question about this talk, please contact Jack Thorne.

Let h(D) be the number of cubic rings (over Z) with discriminant D, and let h’(D) be the number of cubic rings with discriminant -27D such that the traces of all elements are multiples of 3, in each case weighting each ring by the reciprocal of its number of automorphisms. While studying the Dirichlet series associated to these two functions, Y. Ohno discovered in 1997 the pattern that h’(D) = h(D) (if D is negative) or h’(D) = 3h(D) (if D is positive): a highly unexpected generalization of the Scholz reflection principle that was verified by Nakagawa the following year. I will speak on an original proof of this identity that combines class field theory with one of Bhargava’s higher composition laws.

This talk is part of the Number Theory Seminar series.

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