On the Davenport-Heilbronn theorems and second order terms

• 成果类型: Article
• 署名作者: Bhargava, Manjul; Shankar, Arul; Tsimerman, Jacob
• 署名单位: Princeton University; Institute for Advanced Study - USA; Harvard University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-012-0433-0
• 发表日期: 2013
• 页码: 439-499
• 关键词: cubic fields; density; discriminants; coefficients; rings

摘要

We give simple proofs of the Davenport-Heilbronn theorems, which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3-torsion elements in the class groups of quadratic fields having bounded discriminant. We also establish second main terms for these theorems, thus proving a conjecture of Roberts. Our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings.