# Birch and Swinnerton-Dyer Conjecture

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## 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, 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.

## Sources

- 1965: B.J. Birch and H.P.F. Swinnerton-Dyer:
*Notes on Elliptic Curves (II)*(*J. reine angew. Math.***Vol. 218**: pp. 79 – 108)

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Millennium Prize problems** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Birch and Swinnerton-Dyer conjecture** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Millennium Prize problems** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous): Appendix $18$: Millennium Prize problems: $6$. - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next): Appendix $23$: Millennium Prize problems: $6$.