A note on the Mordell-Weil rank modulo n

作者:Dokchitser Tim*; Dokchitser Vladimir
来源:Journal of Number Theory, 2011, 131(10): 1833-1839.
DOI:10.1016/j.jnt.2011.03.010

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.

  • 出版日期2011-10

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