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The rank of elliptic surfaces in unramified abelian towers
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Joseph H. Silverman
Published/Copyright:
July 27, 2005
Abstract
Let ℰ → C be an elliptic surface defined over a number field K. For a finite covering C′ → C defined over K, let ℰ′ = ℰ ×CC′ be the corresponding elliptic surface over C′. In this paper we give a strong upper bound for the rank of ℰ′ (C′/K) in the case of geometrically abelian unramified coverings C′ → C and under the assumption that the Tate conjecture is true for ℰ′/K. In the case that C is an elliptic curve and the map C′ = C → C is the multiplication-by-n map, the bound for rank(ℰ′(C′/K)) takes the form O(nk/log logn), which may be compared with the elementary bound of O(n2).
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Published Online: 2005-07-27
Published in Print: 2004-11-30
© Walter de Gruyter
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Articles in the same Issue
- The resolution property for schemes and stacks
- Beyond Endoscopy and special forms on GL(2)
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- Wild Euler systems of elliptic units and the Equivariant Tamagawa Number Conjecture
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- The rank of elliptic surfaces in unramified abelian towers
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- On rigidity of Grauert tubes over homogeneous Riemannian manifolds
