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Multiplicative property of representation numbers of ternary quadratic forms

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Abstract

Let f be a positive definite integral ternary quadratic form and \(\theta (z;f)=\sum _{n=0}^{\infty }a(n;f)q^n\) its theta function. For any fixed square-free positive integer t with \(a(t;f)\ne 0\), we define \(\rho (n;t,f):=a(tn^2;f)/a(t;f)\). For the case when \(f=x_1^2+x_2^2+x_3^2\) and \(t=1\), Hurwitz proved that \(\rho (n;t,f)\) is multiplicative and he gave its expression. Cooper and Lam proved four similar formulas and proposed a conjecture for some other cases. Using the results given in this paper, we can check the multiplicative property of \(\rho (n;t,f)\) for many cases. All cases in Cooper and Lam’s conjecture are included in ours.

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References

  1. Shimura, G.: On modular forms of half integral weight. Ann. Math. (2) 97, 440–481 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. Wang, X., Pei, D.: Modular Forms with Integral and Half-Integral Weights. Science Press Beijing, Springer, Beijing, Heidelberg (2012)

    Book  MATH  Google Scholar 

  3. Shaun Cooper and Heung Yeung Lam: On the Diophantine equation \(n^2=x^2+by^2+cz^2\). J. Number Theory 133(2), 719–737 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Guo, X., Peng, Y., Qin, H.: On the representation numbers of ternary quadratic forms and modular forms of weight 3/2. J. Number Theory 140, 235–266 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Rallis, S.: The Eichler commutation relation and the continuous spectrum of the Weil representation. In: Noncommutative Harmonic Analysis (Proc. Third Colloq., Marseille-Luminy, 1978). Lecture Notes in Mathematics, vol. 728, pp. 211–244. Springer, Berlin (1979)

  6. Schulze-Pillot, R.: Thetareihen positiv definiter quadratischer Formen. Invent. Math. 75(2), 283–299 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lu, W., Qin, H.: Eichler’s commutation relation and some other invariant subspaces of Hecke operators. To appear in Ramarujan J. doi:10.1007/s11139-016-9831-z

  8. van Asch, B., van der Blij, F.: Integral quadratic forms and Dirichlet series. Ramanujan J. 22(1), 1–10 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kim, K., Oh, B.-K.: The number of representations of squares by integral ternary quadratic forms. arXiv:1509.09111 (2015)

  10. Kim, K., Oh, B.-K.: The number of representations of squares by integral ternary quadratic forms (II). J. Number Theory 173, 210–229 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hsia, J.S., Jöchner, M., Shao, Y.Y.: A structure theorem for a pair of quadratic forms. Proc. Am. Math. Soc. 119(3), 731–734 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lu, W., Qin, H.: Eligible integers represented by positive ternary quadratic forms. Acta Arith. 179(1), 17–23 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 211189, China

    Wei Lu

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Hourong Qin

Authors
  1. Wei Lu
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  2. Hourong Qin
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Correspondence to Hourong Qin.

Additional information

This work was supported by NSFC (Nos. 11571163, 11171141, 11471154) and PAPD. The first author was Supported by the program B for Outstanding Ph.D. candidate of Nanjing University.

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Lu, W., Qin, H. Multiplicative property of representation numbers of ternary quadratic forms. manuscripta math. 156, 457–467 (2018). https://doi.org/10.1007/s00229-017-0964-1

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  • DOI: https://doi.org/10.1007/s00229-017-0964-1

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