Re: The Wearing O' the Orange

In article <20060321.1154.106239snz@xxxxxxxxxxxxxxx>,
David G. Bell <dbell@xxxxxxxxxxxxxxx> wrote:
On 21 Mar, in article <87r74wdxkq.fsf@xxxxxxxxxxx>
dd-b@xxxxxxxx "David Dyer-Bennet" wrote:

"Keith F. Lynch" <kfl@xxxxxxxxxxxxxx> writes:

David Dyer-Bennet <dd-b@xxxxxxxx> wrote:
I'm not really the expert on this; and the papers from Burke's
aren't near the top of the pile at the moment, so I can't cite
exactly what the connection actually *is*. The current baronet
(16th) is Prof. Sir Peter Swinnerton-Dyer, FRS.

Interesting. The Swinnerton-Dyer conjecture is one of the seven Clay
Math millinnium math problems, problems whose solution they will
award one million dollars for, as they're considered to be the most
important unsolved problems in mathematics. I'm pretty sure that
that professor is the person who came up with the problem.

The "Birch and Swinnerton-Dyer Conjecture" (or I've seen it
hyphenated, but then it looks like three people if you don't know).
But yes, that's him. A fairly major figure in modern math. My father
knew him somewhat -- distant family connection and all, and my father
was a math prof (but not a researcher, after his thesis).

Can anyone give a useful summary of what that's about?

Elliptic curves have both a group structure and an analytic structure.

The B--S-D conjecture states that the order of vanishing at s=1 of the
L-function of the elliptic curve is equal to the number of generators
of the group, and that the residue can be written as a product of
various terms including one which depends on the value of the
generators of the group.

It's almost precisely analogous to a connection between class groups
and the Riemann zeta-functions of number fields which *is* known to
hold; it's clearly making a deep connection between two classes of
objects one of which is elliptic curves, but it's very unclear (to the
point that answering the question would likely bring you close to
resolving the conjecture) what the other class is.

Tom
.

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