Rank of elliptic curves and exponential sums
type: Item
instance of: scholarly article
title: Rank of elliptic curves and exponential sums
zbMATH Open document ID: 0796.14022
DOI: 10.1007/BF01404006
author: Jacek Pomykała
author: Étienne Fouvry
published in: Monatshefte für Mathematik
publication date: 1994-02-06T00:00:00Z
full work available at URL: https://eudml.org/doc/178619
review text: It is usually supposed true that the existence of elliptic curves \(E\) defined over \(\mathbb{Q}\) with high rank is rare. So there is a natural question of giving an average bound for the number of such elliptic curves. This was done by the first author unconditionally for the curves \(E(0,k)\) with Weierstrass equation \(y^ 2=x^ 3+k\), where \(k\) is a nonzero integer [\textit{E. Fouvry}, Sur le comportement en moyenne du rang des courbes \(y^ 2=x^ 3+k\), in Sém. Théorie Nombres, Paris 1990- 91, Prog. Math. 61-84 (1993)]. More precisely \[ \sum_{| k | \leq X} (\sqrt 3)^{\text{rk} \bigl( E(0,k) \bigr)} = O(X), \] for \(X \geq 1\), which shows that for a strong type of average the rank of the elliptic curves of this family is bounded. This result involves an argument of 3- descent which cannot be done so easily when one considers the elliptic curves \(E(a,b)\) with equation \(y^ 2 = x^ 3 + ax + b\), with \(a,b \in \mathbb{Z}\) and \(4a^ 3 + 27b^ 2 \neq 0\). To deal with these curves the authors suppose that they satisfy the conjectures of Birch and Swinnerton-Dyer and Weil-Taniyama and that their Hasse-Weil \(L\)-function \(L(E,s)\) satisfies the generalized Riemann's hypothesis. In this set-up there is a result due to \textit{A. Brumer} [Invent. Math. 109, No. 3, 445-472 (1992; Zbl 0783.14019)] where under the above hypotheses he shows that \[ \sum_{a,b} \text{rank} \bigl( E(a,b) \bigr) \leq \bigl( 2,3 + o(1) \bigr)\;(4T^{5/6})\;\zeta (10), \] as \(T \to \infty\), where the sum is taken over the \(a\) and \(b\)'s such that if \(n^ 4 | a\), \(n^ 6 | b\), then \(n = \pm 1\), \(| a | \leq T^{1/3}\), \(| b | \leq T^{1/2}\) and \(\Delta (a,b)= -16(4a^ 3+27b^ 2) \neq 0\). In easier terms his result shows that the average rank of an elliptic curve in this family is less than 2,3. In his paper Brumer also obtains unconditional results for elliptic curves of this family but defined over \(\mathbb{F}_ q (t)\). In the paper under review the authors consider elliptic curves \…
reviewed by: Amílcar Pacheco
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 490682
zbMATH Keywords: elliptic curves with high rank
zbMATH Keywords: Birch and Swinnerton-Dyer conjectures
zbMATH Keywords: Weil-Taniyama conjectures
MaRDI profile type: Publication
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OpenAlex ID: W1519630552
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description: scientific article; zbMATH DE number 490682