Skip to main content
SpringerLink
Log in

Representations of finite number of quadratic forms with same rank

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Let m, n be positive integers with \(m\le n\). Let \(\kappa (m,n)\) be the largest integer k such that for any (positive definite and integral) quadratic forms \(f_1,\ldots ,f_k\) of rank m, there exists a quadratic form of rank n that represents \(f_i\) for any i with \(1\le i \le k\). In this article, we determine the number \(\kappa (m,n)\) for any integer m with \(1\le m\le 8\), except for the cases when \((m,n)=(3,5)\) and (4, 6). In the exceptional cases, it will be proved that \(1\le \kappa (3,5), \kappa (4,6)\le 2\). We also discuss some related topics.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bhargava, M.: On the Conway–Schneeberger fifteen theorem. Contemp. Math. 272, 27–37 (2000)

    Article  MathSciNet  Google Scholar 

  2. Cassels, J.K.S.: Rational Quadratic Forms. Academic Press, London (1978)

    MATH  Google Scholar 

  3. Conway, J.H.: Universal quadratic forms and the Fifteen Theorem. Contemp. Math. 272, 23–26 (2000)

    Article  MathSciNet  Google Scholar 

  4. Conway, J.H., Sloane, N.J.A.: Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 290. Springer, New York (1999)

    Google Scholar 

  5. Görtz, U.: Arithmetic intersection numbers. Astérisque 312, 15–24 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Gross, B.H., Keating, K.: On the intersection of modular correspondences. Invent. Math. 112, 225–245 (1993)

    Article  MathSciNet  Google Scholar 

  7. Kim, B.M., Kim, M.-H., Oh, B.-K.: 2-Universal positive definite integral quinary quadratic forms. Contemp. Math. 249, 51–62 (1999)

    Article  MathSciNet  Google Scholar 

  8. Kitaoka, Y.: Arithmetic of Quadratic Forms. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  9. Lee, I., Oh, B.-K., Yu, H.: A finiteness theorem for positive definite almost \(n\)-regular quadratic forms. J. Ramanujan Math. Soc. 35(1), 81–94 (2020)

    MathSciNet  MATH  Google Scholar 

  10. Oh, B.-K.: Universal \({\mathbb{Z}}\)-lattices of minimal rank. Proc. Am. Math. Soc. 128, 683–689 (1999)

    Article  MathSciNet  Google Scholar 

  11. O’Meara, O.T.: Introduction to Quadratic Forms. Springer, New York (1963)

    Book  Google Scholar 

  12. Ono, K., Soundararajan, K.: Ramanujan’s ternary quadratic form. Invent. Math. 130, 127–159 (1997)

    Article  MathSciNet  Google Scholar 

  13. Ramanujan, S.: On the expression of a number in the form \(ax^2+by^2+cz^2+dw^2\). Proc. Cambridge Phil. Soc. 19, 11–21 (1917)

    Google Scholar 

  14. van der Waerden, B.L.: Die Reduktionstheorie der positiven quadratischen Formen. Acta Math. 96, 265–309 (1956)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute of Mathematics, Seoul National University, Seoul, 08826, Korea

    Daejun Kim

  2. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Korea

    Byeong-Kweon Oh

Authors
  1. Daejun Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Byeong-Kweon Oh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daejun Kim.

Additional information

Publisher's Note

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

This work was supported by the National Research Foundation of Korea (NRF-2019R1A2C1086347).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, D., Oh, BK. Representations of finite number of quadratic forms with same rank. Ramanujan J 56, 631–644 (2021). https://doi.org/10.1007/s11139-020-00314-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-020-00314-6

Keywords

Mathematics Subject Classification

Search

Navigation