Elliptic curves and spin
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912632806572032 |
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| author | Koymans, Peter Uttenthal, Peter Vang |
| author_facet | Koymans, Peter Uttenthal, Peter Vang |
| contents | In the early 2000s, Ramakrishna asked the question: For the elliptic curve $$ E: y^2 = x^3 - x, $$ what is the density of primes $p$ for which the Fourier coefficient $a_p(E)$ is a cube modulo $p$? As a generalization of this question, Weston--Zaurova formulated conjectures concerning the distribution of power residues of degree $m$ of the Fourier coefficients of elliptic curves $E/\mathbb{Q}$ with complex multiplication. In this paper, we prove their conjecture for cubic residues using the analytic theory of spin. Our proof works for all elliptic curves $E$ with complex multiplication. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15644 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elliptic curves and spin Koymans, Peter Uttenthal, Peter Vang Number Theory In the early 2000s, Ramakrishna asked the question: For the elliptic curve $$ E: y^2 = x^3 - x, $$ what is the density of primes $p$ for which the Fourier coefficient $a_p(E)$ is a cube modulo $p$? As a generalization of this question, Weston--Zaurova formulated conjectures concerning the distribution of power residues of degree $m$ of the Fourier coefficients of elliptic curves $E/\mathbb{Q}$ with complex multiplication. In this paper, we prove their conjecture for cubic residues using the analytic theory of spin. Our proof works for all elliptic curves $E$ with complex multiplication. |
| title | Elliptic curves and spin |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.15644 |