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On the Davenport–Heilbronn theorems and second order terms

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Abstract

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.

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Notes

  1. We thank Franz Lemmermeyer for pointing out to us that the basic ideas of this correspondence were already essentially contained in the work of Levi [24] in 1914! See also the 1926 work of Delone [16].

  2. We follow here the convention that, for e≤0, we have a≡0 (mod p e) for any integer a.

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Acknowledgements

We thank Mohammad Bardestani, Karim Belabas, Andrew Granville, Piper Harris, Carl Pomerance, Peter Sarnak, Christopher Skinner, Frank Thorne, Ila Varma, Melanie Wood, and the anonymous referees for helpful comments on earlier versions of this manuscript. We are also grateful to Boris Alexeev and Sucharit Sarkar for helping us compute the precise values of the second main terms.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Manjul Bhargava

  2. Institute for Advanced Study, Princeton, NJ, 08540, USA

    Arul Shankar

  3. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Jacob Tsimerman

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  1. Manjul Bhargava
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  2. Arul Shankar
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Correspondence to Manjul Bhargava.

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Bhargava, M., Shankar, A. & Tsimerman, J. On the Davenport–Heilbronn theorems and second order terms. Invent. math. 193, 439–499 (2013). https://doi.org/10.1007/s00222-012-0433-0

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  • DOI: https://doi.org/10.1007/s00222-012-0433-0

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