The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910914182119424 |
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| author | Gajović, Stevan Radičević, Lazar Verzobio, Matteo |
| author_facet | Gajović, Stevan Radičević, Lazar Verzobio, Matteo |
| contents | We determine the probability that a random Weierstrass equation with coefficients in the $p$-adic integers defines an elliptic curve with a non-trivial $3$-torsion point, or with a degree $3$ isogeny, defined over the field of $p$-adic numbers. We determine these densities by calculating the corresponding $p$-adic volume integrals and analyzing certain modular curves. Additionally, we explore the case of $\ell$-torsion for $\ell>3$ prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny Gajović, Stevan Radičević, Lazar Verzobio, Matteo Number Theory 11G07 We determine the probability that a random Weierstrass equation with coefficients in the $p$-adic integers defines an elliptic curve with a non-trivial $3$-torsion point, or with a degree $3$ isogeny, defined over the field of $p$-adic numbers. We determine these densities by calculating the corresponding $p$-adic volume integrals and analyzing certain modular curves. Additionally, we explore the case of $\ell$-torsion for $\ell>3$ prime. |
| title | The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny |
| topic | Number Theory 11G07 |
| url | https://arxiv.org/abs/2502.08583 |