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Arithmetic on elliptic threefolds

Published online by Cambridge University Press:  04 December 2007

Rania Wazir
Rania Wazir
Affiliation:
Dipartimento di Matematica, Università degli Studi di Torino, Via Carlo Alberto, 10, 10129 Torino, [email protected]
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Abstract

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In a recent paper, Rosen and Silverman showed that Tate's conjecture on algebraic cycles implies a formula of Nagao, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. In this article, we extend their result to the case of elliptic threefolds. The main ingredients of our argument are a Shioda–Tate-like formula for elliptic threefolds, and a relation between the ‘average’ number of rational points on singular fibers and the Galois action on those fibers.

Keywords

elliptic n-foldMordell–Weil groupShioda–Tate formulaNagao formulaFrobenius trace values11G0514D06
Type
Research Article
Information
Compositio Mathematica , Volume 140 , Issue 3 , May 2004 , pp. 567 - 580
Copyright
Foundation Compositio Mathematica 2004