Zeta functions enumerating subforms of quadratic forms
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929492586397696 |
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| author | Kim, Daejun Lee, Seok Hyeong Lee, Seungjai |
| author_facet | Kim, Daejun Lee, Seok Hyeong Lee, Seungjai |
| contents | In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas linking this Dirichlet series with Dirichlet series counting ideal classes of the imaginary quadratic field associated with the quadratic form. Utilizing the result, we provide explicit formulas of the Dirichlet series for several lattices, including square lattice and hexagonal lattice. Moreover, we investigate some analytic properties of this Dirichlet series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05625 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zeta functions enumerating subforms of quadratic forms Kim, Daejun Lee, Seok Hyeong Lee, Seungjai Number Theory 11E12, 11E16, 11H06, 11M41 In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas linking this Dirichlet series with Dirichlet series counting ideal classes of the imaginary quadratic field associated with the quadratic form. Utilizing the result, we provide explicit formulas of the Dirichlet series for several lattices, including square lattice and hexagonal lattice. Moreover, we investigate some analytic properties of this Dirichlet series. |
| title | Zeta functions enumerating subforms of quadratic forms |
| topic | Number Theory 11E12, 11E16, 11H06, 11M41 |
| url | https://arxiv.org/abs/2409.05625 |