caa a22 4500 378892347 CHVBK 20180305123458.0 cr unu---uuuuu 161128e20041130xx s 000 0 eng 10.1515/crll.2004.2004.577.153 doi (NATIONALLICENCE)gruyter-10.1515/crll.2004.2004.577.153 Silverman Joseph H. 1. Providence. The rank of elliptic surfaces in unramified abelian towers [Elektronische Daten] [Joseph H. Silverman] 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 × C C′ 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(n k /log log n ), which may be compared with the elementary bound of O(n 2). © Walter de Gruyter Grammar, syntax, linguistic structure nationallicence Phonetics, phonology, prosody nationallicence Historical & comparative linguistics nationallicence Journal für die reine und angewandte Mathematik (Crelles Journal) Walter de Gruyter 2004/577(2004-11-30), 153-169 0075-4102 2004:577<153 2004 2004 crll https://doi.org/10.1515/crll.2004.2004.577.153 text/html Onlinezugriff via DOI 1 research article jats NATIONALLICENCE 856 40 https://doi.org/10.1515/crll.2004.2004.577.153 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Silverman Joseph H. 1. Providence NATIONALLICENCE 773 0- Journal für die reine und angewandte Mathematik (Crelles Journal) Walter de Gruyter 2004/577(2004-11-30), 153-169 0075-4102 2004:577<153 2004 2004 crll CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-gruyter