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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.18389 |
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| _version_ | 1866911694844854272 |
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| author | McLeman, Cam Rasmussen, Christopher |
| author_facet | McLeman, Cam Rasmussen, Christopher |
| contents | An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(μ_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism classes of heavenly elliptic curves is finite, even running over all primes $\ell$. We prove a complementary result, that for a fixed prime $\ell\geq 7$, there are only finitely many such classes, even running over all quadratic fields. This naturally raises the question of whether to expect a finiteness result when both $K$ and $\ell$ are allowed to vary. We demonstrate similarities in the behavior of heavenly elliptic curves and elliptic curves with complex multiplication, in terms of their Frobenius traces modulo $\ell$. We determine the complete list of heavenly elliptic curves defined over quadratic fields with complex multiplication and with irrational $j$-invariant (up to isomorphism). We include various extensions of our results to higher degree fields and higher-dimensional abelian varieties where possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18389 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Heavenly elliptic curves over quadratic fields McLeman, Cam Rasmussen, Christopher Number Theory 11G05, 11G10, 11G15 An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(μ_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism classes of heavenly elliptic curves is finite, even running over all primes $\ell$. We prove a complementary result, that for a fixed prime $\ell\geq 7$, there are only finitely many such classes, even running over all quadratic fields. This naturally raises the question of whether to expect a finiteness result when both $K$ and $\ell$ are allowed to vary. We demonstrate similarities in the behavior of heavenly elliptic curves and elliptic curves with complex multiplication, in terms of their Frobenius traces modulo $\ell$. We determine the complete list of heavenly elliptic curves defined over quadratic fields with complex multiplication and with irrational $j$-invariant (up to isomorphism). We include various extensions of our results to higher degree fields and higher-dimensional abelian varieties where possible. |
| title | Heavenly elliptic curves over quadratic fields |
| topic | Number Theory 11G05, 11G10, 11G15 |
| url | https://arxiv.org/abs/2410.18389 |