Courbes elliptiques, fonctions L, et tours cyclotomiques. (French) Zbl 0565.14007

Sémin. Théor. Nombres, Univ. Bordeaux I 1983-1984, Exp. No. 14, 10 p. (1984).
Let E be an elliptic curve defined over \({\mathbb{Q}}\), P a finite set of prime numbers, and L the maximal abelian extension of \({\mathbb{Q}}\) unramified outside of P and infinity. It has been conjectured that the group E(L) of L-rational points on E is finitely generated. Assume that E is a Weil curve with corresponding form f. If the Birch-Swinnerton-Dyer conjecture holds, then E(L) is finitely generated. For a prime number p, let \(\mu_{p^ k}\) be the group of \(p^ k\)-th roots of unity and \(\mu_{p^{\infty}}=\cup^{\infty}_{k=0}\mu_{p^ k}\). Define \(\eta_{BSD}(p)\) as the smallest \(k\geq 0\) such that \(E({\mathbb{Q}})\mu_{p^{\infty}}))=E({\mathbb{Q}}(\mu_{p^ k}))\) and \(\eta(p)\) as the smallest \(k\geq 0\) such that \(L(1,f,\chi)\neq 0\) for every primitive Dirichlet character \(\chi\) of conductor \(p^ j\) with \(j>k\). Under the assumption that the Birch-Swinnerton-Dyer conjecture holds, \(\eta_{BSD}(p)=\eta (p)\) for all sufficiently large p. The problem of bounding \(\eta(p)\) is discussed. This is shown to be related to the problem of making certain diophantine approximations effective. The author proves that \(\eta (p)=O(p/\log p)\).
Reviewer: L.D.Olson


14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11J68 Approximation to algebraic numbers
14K22 Complex multiplication and abelian varieties
11R18 Cyclotomic extensions
14G25 Global ground fields in algebraic geometry
14H52 Elliptic curves
14H45 Special algebraic curves and curves of low genus
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