A note on the Mordell-Weil rank modulo n

JOURNAL OF NUMBER THEORY

卷:131

Article

Dokchitser, Tim¹  Dokchitser, Vladimir² 

[1] Univ Cambridge Robinson Coll, Cambridge CB3 9AN, England

[2] Univ Cambridge Emmanuel Coll, Cambridge CB2 3AP, England

关键词: Mordell-Weil rank;    Elliptic curves;   

DOI  :  10.1016/j.jnt.2011.03.010

来源: Elsevier

【 摘 要 】

Conjecturally, the parity of the Mordell-Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers, so the rank modulo 2 is (conjecturally) the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any n > 2, so this is purely a parity phenomenon. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

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