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I was wondering whether anyone knows of any good non-technical or even popular expositions of the Birch—Swinnerton-Dyer conjecture, for someone with minimal background in elliptic curves. I was thinking of something along the lines of du Sautoy's excellent book 'The Music of the Primes' on the Riemann hypothesis, though perhaps catering slightly less to the layman and more geared to a mathematical audience.

(Note when I say minimal background, I just mean a senior undergraduate level 'awareness' of the group law on elliptic curves, working with curves over finite fields, and the Mordell-Weil theorem.)

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Try Zagier's MPI preprint MPI/89-48 with the title The Birch and Swinnerton-Dyer conjecture from a naive point of view.

Addendum (2012/02/23). An even more elementary introduction is this one by Harder and Zagier.

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    $\begingroup$ appeared in: Arithmetic algebraic geometry (Texel, 1989), 377–389, Progr. Math., 89, Birkhäuser Boston, Boston, MA, 1991. $\endgroup$
    – user9072
    Commented Mar 12, 2011 at 10:00
  • $\begingroup$ This looks great -- I may order the whole volume. Do you know of anywhere I can find the article online in the meantime? $\endgroup$
    – Sputnik
    Commented Mar 12, 2011 at 13:08
  • $\begingroup$ Unfortunately MPIM (mpim-bonn.mpg.de) does not seem to have retrodigitised this article. You can find the book at books.google.com but the article is not shown online. $\endgroup$
    – Chandan Singh Dalawat
    Commented Mar 13, 2011 at 4:33
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    $\begingroup$ See mpim-bonn.mpg.de/… $\endgroup$
    – David E Speyer
    Commented Aug 1, 2011 at 22:33
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    $\begingroup$ Zagier's webpage contains pdf's of many of his papers including this one: see paper number 52 on people.mpim-bonn.mpg.de/zagier $\endgroup$
    – SGP
    Commented Apr 4, 2012 at 13:42
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In my opinion, the best non-technical overview has to be in the Clay Millenium Problems description of Wiles:

http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/birchswin.pdf

The conjecture, far from being resolved (or perhaps even formulated correctly), is very much an active area of research. Moreover, there is not yet a good conceptual framework in which to view the problem, which perhaps explains the lack of non-technical or popular literature on the subject.

If you are looking to learn about B+S-D in a more serious way, or would like a nice (historical) overview, you might enjoy the paper of John Tate, "The Arithmetic of Elliptic Curves", Inventiones 23 (1974).

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There is a short non-technical description of the Birch and Swinnerton-Dyer Conjecture in Keith Devlin's book The Millennium Problems. See Chapter 6, pages 189-211. Devlin's exposition is meant for a broad audience and may be at the level you are looking for. He tries hard to illustrate the problem and starts by comparing the Conjecture to the old Greek problem of finding sides that are rational numbers for a triangle with an area that is a positive whole number. He then provides elementary descriptions of the group of rational points of an elliptic curve, the rank of the group, reduction mod p and the Hasse-Weil L-function L(E,s).

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A popular book on the topic is planned to appear in march: Elliptic Tales: Curves, Counting, and Number Theory by Ash and Gross. Judging from their last book, it will probably contain some actual mathematical meat.

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While this is not a non-technical account:

William Stein: The Birch and Swinnerton-Dyer Conjecture, a Computational Approach. http://modular.math.washington.edu/books/bsd/bsd.pdf (It is not finalized; presently ca 70 pages)

I could well imagine that at least parts could be of interest to you. For example, the introduction contains various historical quotes. Also, as there is a computational aspect there is a need to be explicit. (There is even some sample code, in Sage, included.)

Other than that, I second the suggestion for Wiles's description. (And, from the little I could see on Google books, also the Zagier paper seems really good.)

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Here a are some videos: http://www.claymath.org/video/

See http://claymath.msri.org/birchandswinnertondyer.mov

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