Birch and Swinnerton-Dyer Conjecture

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When the solution to a Diophantine equation in polynomials are the points of an Abelian variety, the order of the group of rational points is related to the behavior of an associated $\zeta$ (zeta) function $\map \zeta s$ near $s = 1$.

In particular:

if $\map \zeta 1 = 0$ then there is an infinite set of rational points
if $\map \zeta 1 \ne 0$ then there is a finite set of rational points.


As of now, the Birch and Swinnerton-Dyer Conjecture has been confirmed only for special cases.

Also known as

The Birch and Swinnerton-Dyer Conjecture is also presented as the Birch–Swinnerton-Dyer Conjecture, where the first hyphen is longer than the second.

This style of presentation is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$, as the long dash is not simple to implement.

Source of Name

This entry was named for Bryan John Birch and Henry Peter Francis Swinnerton-Dyer.