On the effective version of Serre's open image theorem
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866914682184400896 |
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| author | Mayle, Jacob Wang, Tian |
| author_facet | Mayle, Jacob Wang, Tian |
| contents | Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod $\ell$ Galois representation $\overlineρ_{E, \ell}$ of $E$ is surjective for each prime number $\ell$ that is sufficiently large. Under the generalized Riemann hypothesis, we give an explicit upper bound on the largest prime $\ell$, linear in the logarithm of the conductor of $E$, such that $\overlineρ_{E, \ell}$ is nonsurjective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_08656 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the effective version of Serre's open image theorem Mayle, Jacob Wang, Tian Number Theory 11G05 (Primary) 11F80 (Secondary) Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod $\ell$ Galois representation $\overlineρ_{E, \ell}$ of $E$ is surjective for each prime number $\ell$ that is sufficiently large. Under the generalized Riemann hypothesis, we give an explicit upper bound on the largest prime $\ell$, linear in the logarithm of the conductor of $E$, such that $\overlineρ_{E, \ell}$ is nonsurjective. |
| title | On the effective version of Serre's open image theorem |
| topic | Number Theory 11G05 (Primary) 11F80 (Secondary) |
| url | https://arxiv.org/abs/2109.08656 |