Papers › Reflection theorems of Ohno-Nakagawa type for quartic rings and pairs of n-ary quadratic forms
Reflection theorems of Ohno-Nakagawa type for quartic rings and pairs of n-ary quadratic forms
Evan M. O'Dorney
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We prove a reflection theorem, conjectured by Nakagawa and Ohno, for the number of quartic rings, or pairs of ternary quadratic forms, with a given cubic resolvent. Over ℤ, our results are unconditional; we also allow the base to be the ring of integers of a general number field, conditional on some algebraic identities that are Monte Carlo verified. We also establish a reflection theorem for quartic $11111$-forms and $48441$-forms that relates them to the number of 3×3 symmetric matrices with given characteristic polynomial. Along the way, we find elegant new results on Igusa zeta functions of conics and the average value of a quadratic character over a box in a local field. We conjecture that a reflection theorem holds for pairs of n-ary quadratic forms for any odd n, and we prove this for odd cubefree discriminant. This furnishes a more satisfactory answer for a question raised by Cohen, Diaz y Diaz, and Olivier, namely whether there exist an infinite family of reflection theorems of Ohno-Nakagawa type.
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