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.

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: $6$.