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On the rank of the fibers of elliptic K3 surfaces

On the rank of the fibers of elliptic K3 surfaces Let X be an elliptic K3 surface endowed with two distinct Jacobian elliptic fibrations π i , i = 1, 2, defined over a number field k. We prove that there is an elliptic curve C ⊂ X such that the generic rank over k of X after a base extension by C is strictly larger than the generic rank of X. Moreover, if the generic rank of π j is positive then there are infinitely many fibers of π i (j ≠ i) with rank at least the generic rank of π i plus one. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

On the rank of the fibers of elliptic K3 surfaces

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References (19)

Publisher
Springer Journals
Copyright
Copyright © 2012 by Sociedade Brasileira de Matemática
Subject
Mathematics; Theoretical, Mathematical and Computational Physics; Mathematics, general
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-012-0002-6
Publisher site
See Article on Publisher Site

Abstract

Let X be an elliptic K3 surface endowed with two distinct Jacobian elliptic fibrations π i , i = 1, 2, defined over a number field k. We prove that there is an elliptic curve C ⊂ X such that the generic rank over k of X after a base extension by C is strictly larger than the generic rank of X. Moreover, if the generic rank of π j is positive then there are infinitely many fibers of π i (j ≠ i) with rank at least the generic rank of π i plus one.

Journal

Bulletin of the Brazilian Mathematical Society, New SeriesSpringer Journals

Published: Apr 12, 2012

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