A note on the Mordell-Weil rank modulo n

TL;DR

This paper explores the limitations of extending the parity conjecture for Mordell-Weil ranks of elliptic curves to mod n for n>2, showing such analogues do not exist beyond parity.

Contribution

It proves that the parity phenomenon is unique and cannot be generalized to other moduli n>2 for the Mordell-Weil rank of elliptic curves.

Findings

No analogue for rank modulo 3, 4, 5, or any n>2 exists.

Standard conjectures imply the uniqueness of the parity phenomenon.

The rank modulo n for n>2 cannot be determined by local data.

Abstract

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.