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The number of representations of integers by 4-dimensional strongly N-modular lattices

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In this article, we prove that a space of cusp forms of weight 2 with level N and real character \(\chi \) has dimension 1 if and only if \(\chi \) is trivial and N is in \(\{11,14,15,17,19,20,21,24,27,32,36,49\},\) and derive bases for spaces of cusp forms of weight 2 with trivial character and \(N\in \{17,19,21,49\}\). As applications, we provide formulas for the number of representations of integers by 4-dimensional strongly N-modular lattices for N in

$$\begin{aligned} \{11,14,15,17,19,20,21,24,27,32,36\}. \end{aligned}$$

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Authors and Affiliations

  1. Department of Mathematics, Dankook University, Cheonan, 31116, Korea

    Ho Yun Jung

  2. Department of Mathematics, Sungkyunkwan University, Suwon, 16419, Korea

    Chang Heon Kim, Kyoungmin Kim & Soonhak Kwon

  3. Department of Mathematics, Hannam University, Daejeon, 34430, Korea

    Kyoungmin Kim

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  1. Ho Yun Jung
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  2. Chang Heon Kim
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  3. Kyoungmin Kim
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  4. Soonhak Kwon
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Correspondence to Kyoungmin Kim.

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This work of H. Y. Jung was supported by the research fund of Dankook University in 2020. This work of C. H. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2016R1A5A1008055 and 2021R1A2C1003998). This work of K. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2016R1A5A1008055 and NRF-2018R1C1B6007778)

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Jung, H.Y., Kim, C.H., Kim, K. et al. The number of representations of integers by 4-dimensional strongly N-modular lattices. Ramanujan J 57, 1253–1275 (2022). https://doi.org/10.1007/s11139-021-00490-z

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  • DOI: https://doi.org/10.1007/s11139-021-00490-z

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