# Birch and Swinnerton-Dyer Conjecture

## Unsolved Problem

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.

## Progress

As of now, this conjecture has been confirmed only for special cases.

## Also known as

This 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.