Binary quadratic forms of odd class number
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913706945806336 |
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| author | Akbary, Amir Totani, Yash |
| author_facet | Akbary, Amir Totani, Yash |
| contents | Let $-D$ be a fundamental discriminant. We express the number of representations of an integer by a positive definite binary quadratic form of discriminant $-D$ with an odd class number $h(-D)$ as a rational linear expression involving the Kronecker symbol $\left(\frac{-D}{.}\right)$ and the Fourier coefficients of certain cusp forms. We prove these cusp forms have eta quotient representations only if $D=23$. This provides, using theta functions, a generalization of a result of F. van der Blij from 1952 for binary quadratic forms of discriminant $-23$ to the case of forms of discriminant $-D$ with odd $h(-D)$. We also classify all the eta quotients of prime level $D$ which are half the difference of two theta functions of level $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00184 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Binary quadratic forms of odd class number Akbary, Amir Totani, Yash Number Theory 11E25, 11E45 Let $-D$ be a fundamental discriminant. We express the number of representations of an integer by a positive definite binary quadratic form of discriminant $-D$ with an odd class number $h(-D)$ as a rational linear expression involving the Kronecker symbol $\left(\frac{-D}{.}\right)$ and the Fourier coefficients of certain cusp forms. We prove these cusp forms have eta quotient representations only if $D=23$. This provides, using theta functions, a generalization of a result of F. van der Blij from 1952 for binary quadratic forms of discriminant $-23$ to the case of forms of discriminant $-D$ with odd $h(-D)$. We also classify all the eta quotients of prime level $D$ which are half the difference of two theta functions of level $D$. |
| title | Binary quadratic forms of odd class number |
| topic | Number Theory 11E25, 11E45 |
| url | https://arxiv.org/abs/2408.00184 |