2021/11/18 by Evan M. O’Dorney, O'Dorney, Evan M.
Mathematics · #11A15 #11E76 #11G20 #11R16 #11R54 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2111.09784
openalex publication_date 2021/11/18 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
The Ohno-Nakagawa (O-N) reflection theorem is an unexpectedly simple identity\nrelating the number of \GL2 \ℤ-classes of binary cubic forms\n(equivalently, cubic rings) of two different discriminants D, -27D; it\ngeneralizes cubic reciprocity and the Scholz reflection theorem. In this paper,\nwe present a new approach to this theorem using Fourier analysis on the adelic\ncohomology H1( mathbbAK, M) of a finite Galois module, modeled after the\ncelebrated Fourier analysis on mathbbAK used in Tate's thesis. This\nmethod reduces reflection theorems of O-N type to local identities. We\nestablish reflection theorems of O-N type for cubic forms and rings over\narbitrary number fields, and also for quadratic forms counting by a peculiar\ninvariant a(b2 - 4ac). We also find relations for the number of forms over\n\ℤ[1/N] and for forms of highly non-squarefree discriminant\n(discriminant reduction).\n In a sequel to this paper, we will deal with reflection theorems for quartic\nrings, 2\× 3\× 3 symmetric boxes, and binary quartic forms. In these\ncases the local step is much more involved.\n