On the Davenport-Heilbronn theorems and second order terms

Manjul Bhargava, Arul Shankar, Jacob Tsimerman

Research output: Contribution to journalArticlepeer-review

51 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)439-499
Number of pages61
JournalInventiones Mathematicae
Issue number2
StatePublished - Aug 2013

All Science Journal Classification (ASJC) codes

  • General Mathematics


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