Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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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)
  • Received : 2023.05.14
  • Accepted : 2024.02.29
  • Published : 2024.07.31
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Abstract

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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Acknowledgement

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.

References

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