\(\mathbb{Q} (T)\)-rank elliptic curves and certain limit coming from the local points
type: Item
instance of: scholarly article
title: \(\mathbb{Q} (T)\)-rank elliptic curves and certain limit coming from the local points
zbMATH Open document ID: 0870.11034
DOI: 10.1007/BF02678178
author: Koh-ichi Nagao
published in: Manuscripta Mathematica
publication date: 1997-04-13T00:00:00Z
full work available at URL: https://eudml.org/doc/156249
review text: Let \({\mathcal E}\) denote an elliptic curve defined by a Weierstrass equation with \(\mathbb{Z}[T]\)-coefficients over \(\mathbb{Q}(T)\), the field of rational functions in one variable over the rationals. For \(t\in \mathbb{Z}\), let \(E_t\) denote the curve defined over \(\mathbb{Z}\) by specializing \(T\) to \(t\). For \(p\) a prime, let \(E_{t,p}\) denote the curve over \(\mathbb{F}_p\) defined by reducing the coefficients of \(E_t\) modulo \(p\). Put \[ A_p({\mathcal E})={1\over p}\sum_{t=0}^{p-1}p+1-\# E_t(\mathbb{F}_p). \] In a previous article [Kobe J. Math. 11, No. 2, 211-219 (1994; Zbl 0848.11026)] \textit{K. Nagao} considered the function \[ S(N,{\mathcal E})=-{1\over N}\sum A_p({\mathcal E})\cdot \log p, \] where the sum is over all primes less than or equal to \(N\), and conjectured that \(S(N,{\mathcal E})\) converges to the \(\mathbb{Q}(T)\)-rank of \({\mathcal E}\) as \(N\rightarrow\infty\). In the present article, the author shows that this conjecture holds for several families of elliptic curves over \(\mathbb{Q}(T)\), including curves of the form \(Y^2=f(X)+T^i\), where \(i=1\) or 2 and \(f(X)\in \mathbb{Z}[X]\) is a monic cubic polynomial without double roots.
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 980714
zbMATH Keywords: elliptic curve
zbMATH Keywords: \({\mathbb{Q}}(T)\)-rank
zbMATH Keywords: local points
zbMATH Keywords: rational points
reviewed by: Robert F. Lax
MaRDI profile type: Publication
cites work: Notes on elliptic curves. I.
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cites work: Construction of elliptic curves with high rank via the invariants of the Weyl groups
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description: scientific article; zbMATH DE number 980714