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Sur la nature non-cyclotomique des points d’ordre fini des courbes elliptiques. (On the noncyclotomic nature of finite-order points of elliptic curves). With an appendix by E. Kowalski and P. Michel. (French) Zbl 1020.11041

Let \(S_{0,c}\) be the set of prime numbers \(p\) for which there is an elliptic curve \(E\) defined over \(\mathbb{Q}(\mu_p)\) with no potentially good reduction in characteristic \(p\) having a subgroup \(\mathbb{Q} (\mu_p)\)-rational, of order \(p\). In the paper under review, the author proves that the set \(S_{0,c}\) is finite. Furthermore, he proves that for the primes \(p\) of \(S_{0,c}\) there is a nonquadratic even character \(\chi: (\mathbb{Z}/p\mathbb{Z})^* \to\mathbb{C}^*\) such that the following proposition is not valid: “There is a cusp form \(f=\Sigma_n a_nq^n\) of weight 2 for \(\Gamma_0(p)\) such that the function \(L(f,\chi,s)\) extending the Dirichlet series \(\sum_{n\geq 1}a_n \chi(n)n^{-s}\) has not a zero at \(s=1\).” Moreover, the paper contains an appendix in which it is proved that if \(p\) is a prime \(>10^{25}\) and \(\chi\) a nonquadratic even character, then the above proposition is valid.

MSC:

11G05 Elliptic curves over global fields
14G05 Rational points
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11L40 Estimates on character sums
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