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A note on the Mordell–Weil rank modulo n

By Tim Dokchitser and Vladimir Dokchitser

Abstract

AbstractConjecturally, 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

Publisher: Elsevier Inc.
Year: 2011
DOI identifier: 10.1016/j.jnt.2011.03.010
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