대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제61권4호
  • /
  • Pages.907-915
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  • 2024
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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A NOTE ON REPRESENTATION NUMBERS OF QUADRATIC FORMS MODULO PRIME POWERS

  • Ran Xiong (School of Mathematics and Statistics Weinan Normal University)
  • 투고 : 2023.05.14
  • 심사 : 2024.02.29
  • 발행 : 2024.07.31
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초록

Let f be an integral quadratic form in k variables, F the Gram matrix corresponding to a ℤ-basis of ℤk. For r ∈ F-1ℤk, a rational number n with f(r) ≡ n mod ℤ and a positive integer c, set Nf(n, r; c) := #{x ∈ ℤk/cℤk : f(x + r) ≡ n mod c}. Siegel showed that for each prime p, there is a number w depending on r and n such that Nf(n, r; pν+1) = pk-1Nf(n, r; pν) holds for every integer ν > w and gave a rough estimation on the upper bound for such w. In this short note, we give a more explicit estimation on this bound than Siegel's.

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과제정보

The author would like to thank the referees for their valuable comments on this work. Also the author is greatly indebted to Nils-Peter Skoruppa, for teaching to the author basic knowledge of lattices, finite quadratic modules and related topics during 2015-2019.

참고문헌

  1. A. Ajouz, Hecke operators on Jacobi forms of lattice index and the relation to elliptic modular forms, Ph.D. thesis, University of Siegen, 2015.
  2. B. C. Berndt, R. J. Evans, and K. S. Williams, Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1998.
  3. J. H. Bruinier and M. Kuss, Eisenstein series attached to lattices and modular forms on orthogonal groups, Manuscripta Math. 106 (2001), no. 4, 443-459. https://doi.org/10.1007/s229-001-8027-1
  4. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, third edition, Grundlehren Math. Wiss. 290, Springer, New York, 1999. https://doi.org/10.1007/978-1-4757-6568-7
  5. C. L. Siegel, Uber die analytische Theorie der quadratischen Formen, Ann. of Math. (2) 36 (1935), no. 3, 527-606. https://doi.org/10.2307/1968644