Joseph H. Silverman
The rank of elliptic surfaces in unramified abelian towers
Let ? ? C be an elliptic surface defined over a number field K. For a finite covering C? ? C defined over K, let ?? = ? ×C C? be the corresponding elliptic surface over C?. In this paper we give a strong upper bound for the rank of ?? (C?/K) in the case of geometrically abelian unramified coverings C? ? C and under the assumption that the Tate conjecture is true for ??/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(??(C?/K)) takes the form O(nk/log log n), which may be compared with the elementary bound of O(n2).
Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter
Print ISSN: 0075-4102
Volume: 2004, 11/2004
Pages: 153 - 169
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