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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2503.17944 |
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| _version_ | 1866910024087896064 |
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| author | Mondal, Koustav |
| author_facet | Mondal, Koustav |
| contents | In this paper, we analyze the theta series associated to the quadratic form $Q(\mathbf{x}) := x_1^2 + x_2^2 + x_3^2 + x_4^2$ with congruence conditions on $x_i$ modulo $2, 3, 4$, and $6$. By employing special operators on modular, non-holomorphic Eisenstein series of weight $2$, we construct a basis for the Eisenstein space for levels $2^k$ (with $k \le 7$), $3^{\ell}$ (with $\ell \le 3$), and $p$, where $p>3$ is an odd prime. Using the relation between the trace of Frobenius on an elliptic curve and the Fourier coefficients of the cusp-form part of the theta series corresponding to $Q$, we establish a relation between the number of integer solutions to the equation $Q(\mathbf{x}) = p$ and the number of $\mathbb{F}_p$-rational points on the associated elliptic curve under certain congruence conditions on $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17944 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relating elliptic curve point-counting and solutions of quadratic forms with congruence conditions Mondal, Koustav Number Theory 11: Number theory In this paper, we analyze the theta series associated to the quadratic form $Q(\mathbf{x}) := x_1^2 + x_2^2 + x_3^2 + x_4^2$ with congruence conditions on $x_i$ modulo $2, 3, 4$, and $6$. By employing special operators on modular, non-holomorphic Eisenstein series of weight $2$, we construct a basis for the Eisenstein space for levels $2^k$ (with $k \le 7$), $3^{\ell}$ (with $\ell \le 3$), and $p$, where $p>3$ is an odd prime. Using the relation between the trace of Frobenius on an elliptic curve and the Fourier coefficients of the cusp-form part of the theta series corresponding to $Q$, we establish a relation between the number of integer solutions to the equation $Q(\mathbf{x}) = p$ and the number of $\mathbb{F}_p$-rational points on the associated elliptic curve under certain congruence conditions on $p$. |
| title | Relating elliptic curve point-counting and solutions of quadratic forms with congruence conditions |
| topic | Number Theory 11: Number theory |
| url | https://arxiv.org/abs/2503.17944 |