期刊论文详细信息
JOURNAL OF NUMBER THEORY 卷:176
On a remarkable identity in class numbers of cubic rings
Article
O'Dorney, Evan1 
[1] Princeton Univ, Dept Math, Fine Hall,Washington Rd, Princeton, NJ 08544 USA
关键词: Shintani zeta functions;    Binary cubic forms;    Higher composition laws;   
DOI  :  10.1016/j.jnt.2016.12.002
来源: Elsevier
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【 摘 要 】

In 1997, Y. Ohno empirically stumbled on an astoundingly simple identity relating the number of cubic rings h(Delta) of a given discriminant A, over the integers, to the number of cubic rings (h) over cap(Delta) of discriminant -27 Delta in which every element has trace divisible by 3: (h) over cap(Delta) ={3h (Delta) if Delta > 0 h (Delta) if Delta < 0 where in each case, rings are weighted by the reciprocal of their number of automorphisms. This allows the functional equations governing the analytic continuation of the Shintani zeta functions (the Dirichlet series built from the functions h and <(h)over cap>) to be put in self-reflective form. In 1998, J. Nakagawa verified (1). We present a new proof of (1) that uses the main ingredients of Nakagawa's proof (binary cubic forms, recursions, and class field theory), as well as one of Bhargava's celebrated higher composition laws, while aiming to stay true to the stark elegance of the identity. (C) 2017 Elsevier Inc. All rights reserved.

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