A proof of a conjecture on trace-zero forms and shapes of number fields

Abstract

In 2012 the first named author conjectured that totally real quartic fields of fundamental discriminant are determined by the isometry class of the integral trace zero form; such conjecture was based on computational evidence and the analog statement for cubic fields which was proved using Bhargava’s higher composition laws on cubes. Here, using Bhargava’s parametrization of quartic fields we prove the conjecture by generalizing the ideas used in the cubic case. Since at the moment, for arbitrary degrees, there is nothing like Bhargava’s parametrizations we cannot deal with degrees \(n > 5\) in a similar fashion. Nevertheless, using some of our previous work on trace forms we generalize this result to higher degrees. We show that if n is an integer bigger than 2 such that \((\mathbb {Z}/n\mathbb {Z})^{*}\) is a cyclic group, the shape is a complete invariant for degree n number fields that are totally real and have fundamental discriminant.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Bhargava, M., Varma, I., et al.: On the mean number of \(2\)-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields. Duke Math. J. 164(10), 1911–1933 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bhargava, M.: Higher composition laws II: on cubic analogues of Gauss composition. Ann. Math. 159, 865–886 (2004)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bhargava, M.: Higher composition laws III: the parametrization of quartic rings. Ann. Math. 159, 1329–1360 (2004)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bhargava, M.: Higher composition laws, Ph.D thesis, Princeton University (2001)

  5. 5.

    Bhargava, M., Harron, P.: The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields. Compositio Mathematica 152(6), 1111–1120 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Harron, P.: The equidistribution of lattice shapes of rings of Integers of cubic, quartic, and quintic Number Fields: an artist’s rendering, Ph.D thesis, Princeton Univ (2016)

  7. 7.

    Harron, R.: The shapes of pure cubic fields. Proc. Am. Math. Soc. 145(2), 509–524 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Harron, R.: Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent arXiv preprint arXiv:1907.07209 (2019)

  9. 9.

    Jones, J., Roberts, D.: A data base of number fields. LMS J. Comput. Math. 17(1), 595–618 (2014)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Mantilla-Soler, G.: On number fields with equivalent integral trace forms. Int. J. Number Theory 8–7, 1569–1580 (2012)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Mantilla-Soler, G.: Integral trace forms associated to cubic extensions. Algeb. Number Theory 4–6, 681–699 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Mantilla-Soler, G., Monsurrò, M.: The Shape of \({\mathbb{Z}/\ell {\mathbb{Z}}}\)-number fields With M. Monsurrò. Ramanujan J. 39(3), 451–463 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Mantilla-Soler, G., Rivera-Guaca, C.: An introduction to Casimir pairings and some arithmetic applications. arXiv: 1812.03133v3 (2019)

  14. 14.

    Maurer, D.: The Trace-Form of an algebraic number field. J. Number Theory 5, 379–384 (1973)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Neukirch, J.: Algebraische Zahlentheorie. Springer, Berlin (2007)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Guillermo Mantilla-Soler.

Ethics declarations

Acknowledgements

We would like to thank the referee for the careful reading of the paper, and for their helpful comments.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mantilla-Soler, G., Rivera-Guaca, C. A proof of a conjecture on trace-zero forms and shapes of number fields. Res. number theory 6, 35 (2020). https://doi.org/10.1007/s40993-020-00210-4

Download citation