On the effective version of Serre’s open image theorem. (English) Zbl 07828047
The paper under review studies the surjectivity of the Galois representations
\[
\rho_{E,\ell} \colon \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{Aut}_\mathbb{Z}(E(\overline{\mathbb{Q}})[\ell]) \cong \mathrm{GL}_2(\mathbb{Z}/\ell \mathbb{Z})
\]
associated to elliptic curves \(E\) defined over \(\mathbb{Q}\), where \(\ell \in \mathbb{N}\) is a rational prime and \(E(\overline{\mathbb{Q}})[\ell]\) denotes the set of \(\ell\)-torsion points of the elliptic curve \(E\) defined over the field of algebraic numbers.
More precisely, Theorem 1.1 of the paper under review shows that, conditionally under the validity of the generalized Riemann hypothesis, given an elliptic curve \(E\) defined over the rational numbers, which does not have complex multiplication, and a prime \(\ell \in \mathbb{N}\) such that \[ \ell > 964 \log\left( 2 b_E \right) + 5760, \] the Galois representation \(\rho_{E,\ell}\) is surjective. Here, \(b_E\) denotes the product of all the rational primes \(p \geq 3\) at which \(E\) has bad reduction.
This can be seen as an explicit version of Serre’s celebrated open image theorem [J.-P. Serre, Invent. Math. 15, 259–331 (1972; Zbl 0235.14012)], which shows that for every elliptic curve \(E\) defined over a number field \(K\), if \(E\) does not have complex multiplication then there exists a constant \(c(E)\) such that for every prime \(\ell > c(E)\) the Galois representation \(\rho_{E,\ell}\) is surjective. As explained in the introduction of the paper under review, several explicit versions of this result already exist in the literature, proven either unconditionally or assuming the generalized Riemann hypothesis. Up to date, the best unconditional upper bound is due to D. Zywina [Bull. Lond. Math. Soc. 54, No. 6, 2404–2417 (2022; Zbl 1528.11043)], and is worse than linear in the constant \(b_E\) introduced above. On the other hand, E. Larson and D. Vaintrob [Bull. Lond. Math. Soc. 46, No. 1, 197–209 (2014; Zbl 1292.11067)] have proven a conditional and unexplicit upper bound on \(c(E)\) which is of the same order of the upper bound proven in Theorem 1.1 of the paper under review. Since Larson and Vaintrob’s proof cannot easily be made explicit, the authors of the paper under review use a different proof technique, which is based on a strategy set forth by Serre himself [J.-P. Serre, Publ. Math., Inst. Hautes Étud. Sci. 54, 123–201 (1981; Zbl 0496.12011)], and on an explicit version of Chebotarev’s density theorem which the authors prove in Section 2 of the paper under review, improving slightly a result of E. Bach and J. Sorenson [Math. Comput. 65, No. 216, 1717–1735 (1996; Zbl 0853.11077)].
To conclude, the paper under review provides an interesting new conditional upper bound for the constant \(c(E)\) associated to elliptic curves \(E\) defined over \(\mathbb{Q}\), which has the virtue of being completely explicit, and also of the same order of magnitude of the best results currently available in the literature.
More precisely, Theorem 1.1 of the paper under review shows that, conditionally under the validity of the generalized Riemann hypothesis, given an elliptic curve \(E\) defined over the rational numbers, which does not have complex multiplication, and a prime \(\ell \in \mathbb{N}\) such that \[ \ell > 964 \log\left( 2 b_E \right) + 5760, \] the Galois representation \(\rho_{E,\ell}\) is surjective. Here, \(b_E\) denotes the product of all the rational primes \(p \geq 3\) at which \(E\) has bad reduction.
This can be seen as an explicit version of Serre’s celebrated open image theorem [J.-P. Serre, Invent. Math. 15, 259–331 (1972; Zbl 0235.14012)], which shows that for every elliptic curve \(E\) defined over a number field \(K\), if \(E\) does not have complex multiplication then there exists a constant \(c(E)\) such that for every prime \(\ell > c(E)\) the Galois representation \(\rho_{E,\ell}\) is surjective. As explained in the introduction of the paper under review, several explicit versions of this result already exist in the literature, proven either unconditionally or assuming the generalized Riemann hypothesis. Up to date, the best unconditional upper bound is due to D. Zywina [Bull. Lond. Math. Soc. 54, No. 6, 2404–2417 (2022; Zbl 1528.11043)], and is worse than linear in the constant \(b_E\) introduced above. On the other hand, E. Larson and D. Vaintrob [Bull. Lond. Math. Soc. 46, No. 1, 197–209 (2014; Zbl 1292.11067)] have proven a conditional and unexplicit upper bound on \(c(E)\) which is of the same order of the upper bound proven in Theorem 1.1 of the paper under review. Since Larson and Vaintrob’s proof cannot easily be made explicit, the authors of the paper under review use a different proof technique, which is based on a strategy set forth by Serre himself [J.-P. Serre, Publ. Math., Inst. Hautes Étud. Sci. 54, 123–201 (1981; Zbl 0496.12011)], and on an explicit version of Chebotarev’s density theorem which the authors prove in Section 2 of the paper under review, improving slightly a result of E. Bach and J. Sorenson [Math. Comput. 65, No. 216, 1717–1735 (1996; Zbl 0853.11077)].
To conclude, the paper under review provides an interesting new conditional upper bound for the constant \(c(E)\) associated to elliptic curves \(E\) defined over \(\mathbb{Q}\), which has the virtue of being completely explicit, and also of the same order of magnitude of the best results currently available in the literature.
Reviewer: Riccardo Pengo (Hannover)
References:
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