**The work of Bhargava and the Birch and Swinnerton-Dyer conjecture**+Manjul Bhargava [1] is, in my mind (and many other peoples'), a top contender for a Fields medal later this year at the International Congress of Mathematicians in Seoul. A report on some recent work of his is here:

http://mattbakerblog.wordpress.com/2014/03/10/the-bsd-conjecture-is-true-for-most-elliptic-curves/He has announced that the Birch and Swinnerton-Dyer conjecture [2] is true for at least (just under) 2/3 of all elliptic curves. This is an improvement of past results of his (and coauthors) in this direction (showing a 'positive proportion' satisfy BS-D), and a comment to the post indicates that this is expected to improve to 4/5 given some ongoing work by Chris Skinner and collaborators.

That this work is considered by many to be Fields medal-worthy is analogous to how Stephen Smale [3] and Michael Freedman [4] won the Fields medal (essentially) for their proofs of the Poincaré conjecture in dimensions 5 and above, and in dimension 4, respectively [5]; it wasn't the original big conjecture, but it shed light on the general problem and why proving it in dimension 3 (the 'real' Poincaré conjecture [6], eventually proved by Grisha Perelman [7]) was going to need completely different tools.

So what is the Birch and Swinnerton-Dyer conjecture? Essentially, it tells us something about the number of rational-number solutions to certain two-variable cubic equations in two variables, with rational coefficients, defining a so-called

*elliptic curve*. Such curves are

*not* ellipses, but there is an indirect historical link, via functions which calculate lengths of arcs on ellipses. If we look at solutions over the complex numbers, then one finds they form a surface looking like the skin of a donut (the one with a hole), possibly pinched or 'degenerate'.

However, if we look for solutions among the rational numbers a/b, with a and b whole numbers (and possibly b=0, corresponding to a 'point at infinity'!), then this is a lot harder, and then the problem falls into two cases: a finite number of solutions and infinitely many solutions. In the infinitely many solutions case, we can further distinguish the 'number' of solutions, in a manner of speaking, by the dimension, or

*rank*, of the infinite part. (And the finite number of solutions in fact corresponds to rank 0.) The fact that this dimension is finite is a hard theorem all on its own, due to Mordell [8]. There is an algebraic structure on the rational solutions to the equation underlying an elliptic curve, namely that of an

*abelian group*, and the animated gif below shows one way of thinking about this (source:

http://en.wikipedia.org/wiki/File:EllipticGroup.gif). This algebraic structure means that the possible solutions are very constrained and there is a rich theory behind elliptic curves for this very reason (and this algebraic structure is why elliptic curves can be used for cryptography, including the infamous set with a built-in back door [9])

Now there is another way to calculate a 'rank' for an elliptic curve, which uses complex analysis, or more roughly, infinite products built from the elliptic curve. One defines a so-called

*L-function*, similar in character to the famous Riemann zeta function [10] (but unrelated), and then the Modularity Theorem [11], proven in part by Andrew Wiles [12] on his way to proving Fermat's Last Theorem [13], tells us that this L-function makes sense for inputs other than when it was first defined, namely we can plug in any complex number and get a sensible result. If one analyses the behaviour of this function at the input 1, roughly how fast it grows heading away from that point, then this defines a non-negative integer called the

*analytic rank* of the original elliptic curve.

That these two numbers, the rank and the analytic rank, have anything to do with each other, especially since the second one wasn't even

*defined* for all elliptic curves until the late 1990s, is hugely surprising. Birch and Swinnerton-Dyer made their conjecture based on computer work in the early 1960s on elliptic curves for which the various quantities were known.

So at last, what is the conjecture? Namely that the rank and analytic rank are equal, and in fact there is a conjectured formula for the rate at which the values of the L-function associated to the elliptic curve increase as one heads away from the point 1 in the complex plane. This is terrifically scary, involving the size of the Tate-Shafarevich group Ш (one of the few, if only, instances of Cyrillic used internationally by mathematicians), which is mentioned in the linked blog post, the number of solutions in the 'finite part' of the elliptic curve (for those who know group theory: the order of the torsion subgroup) and other quantities which are associated to each elliptic curve.

Now the really interesting thing is that we expect

*most* elliptic curves to have rank (either sort!) equal to either 0 or 1, and by 'most' I mean 100% of them. Thus

*0%* of elliptic curves should have rank bigger than 1. This may seem weird, but there are infinitely many elliptic curves, and so 0% really means something like 'given any tiny positive number t, then taking larger and larger samples of elliptic curves, I expect less than t% of them have rank bigger than 1'. And for the rank 0 and 1 curves, these are expected to be split 50-50 (the 'parity conjecture') between the two cases. The large-rank curves are indeed rare: the largest explicitly-known rank of an elliptic curve is 18, with cubic equation

y^2 + xy = x^3 − 26175960092705884096311701787701203903556438969515x + 51069381476131486489742177100373772089779103253890567848326,

due to

+Noam Elkies, though there are examples with a rank at least 28, but with actual rank unknown. Interestingly, the 'finite part' of the elliptic curve (the

*torsion subgroup*), can only be one of 15 things: the trivial group, the cyclic group C_n of order n=2,3,..,10 or 12, or the product of C_2 with C_2m for m=1,2,3,4.

To summarise: what Bhargava has done (with his collaborators) is show that in at least 2/3 of all elliptic curves, the rank and the analytic rank coincide, showing the BSD conjecture for this many curves, with a view to extend this to 4/5 of all elliptic curves.

[1]

http://en.wikipedia.org/wiki/Manjul_Bhargava [2]

http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture[3]

http://en.wikipedia.org/wiki/Stephen_Smale[4]

http://en.wikipedia.org/wiki/Michael_Freedman[5]

http://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture[6]

http://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture[7]

http://en.wikipedia.org/wiki/Grigori_Perelman[8]

http://en.wikipedia.org/wiki/Mordell%27s_theorem[9]

http://jiggerwit.wordpress.com/2013/09/25/the-nsa-back-door-to-nist/[10]

http://en.wikipedia.org/wiki/Riemann_zeta_function[11]

http://en.wikipedia.org/wiki/Modularity_theorem[12]

http://en.wikipedia.org/wiki/Andrew_Wiles[13]

http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem(Thanks to

+Peter Woit for the link to Matt Baker's blog posting)