On the rank of an elliptic surface
type: Item
instance of: scholarly article
title: On the rank of an elliptic surface
zbMATH Open document ID: 0905.14019
author: Joseph H. Silverman
author: Michael Rosen
published in: Inventiones Mathematicae
publication date: 1998-08-09T00:00:00Z
review text: \textit{K. Nagao} [Manuscr. Math. 92, No. 1, 13-32 (1997; Zbl 0870.11034)] has recently given a conjectural limit formula for the rank of an elliptic surface \({\mathcal E}\) in terms of a weighted average of fibral Frobenius trace values. We show that Tate's conjecture on the order of vanishing of \(L_2 ({\mathcal E}, s)\) essentially implies Nagao's formula; in particular, we prove Nagao's formula for rational elliptic surfaces. In the case that \({\mathcal E}\) is a twist, we reduce Nagao's and Tate's conjectures to the case of products of curves, and we verify the conjectures for many new classes of elliptic surfaces of Kodaira dimension 0 and 1.
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 1186162
zbMATH Keywords: vanishing of \(L_2\)
zbMATH Keywords: rank of an elliptic surface
zbMATH Keywords: Frobenius trace
zbMATH Keywords: Tate's conjecture
MaRDI profile type: Publication
full work available at URL: s002220050238
OpenAlex ID: W1972431694
DOI: 10.1007/S002220050238
description: scientific article; zbMATH DE number 1186162