Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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REPRESENTATION NUMBERS BY QUADRATIC FORMS OF CERTAIN LEVELS

  • Ick Sun Eum (Department of Mathematics Education Dongguk University WISE Campus)
  • Received : 2023.08.07
  • Accepted : 2024.05.27
  • Published : 2024.11.01
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Abstract

Let Q be an integral positive definite quadratic form of level N in 2k(≥ 4) variables. We assume that (-1)kN is a fundamental discriminant and the associated character χ of Q is primitive of conductor N. Under our assumption, we find the pairs (k, N) such that the dimension of spaces of cusp forms of weight k and level N with Nebentypus χ is one. Furthermore, we explicitly construct their bases by using Eisenstein series of lower weights. For the above pairs (k, N), we use these cusp forms to provide closed formulas for the representation numbers by quadratic forms of level N in 2k variables, which are expressed in terms of divisor functions and their convolution sums.

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Acknowledgement

This work was supported by the Dongguk University Research Fund of 2023 and the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(No. NRF-2020R1F1A1A01070647).

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