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Interview with Professor Henri Darmon
The Birch and Swinnerton-Dyer Conjecture: An Interview with Professor Henri Darmon Agn` es F. Beaudry
If you made a poll of number theorists and asked them, “What’s your favorite problem in number theory?”, you would probably have it equally divided between the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture, which are two of the Millennium Prize problems in number theory. My favorite problem is the Birch and Swinnerton-Dyer conjecture. -Prof. Henri Darmon The Birch and Swinnerton-Dyer conjecture (BSD) was stated in the sixties by Peter Swinnerton-Dyer and Bryan Birch who gathered considerable evidence, based on numerical data from the EDSAC computer at Cambridge, suggesting a relation between the rational solutions of special Diophantine equations called elliptic curves and their solutions in Zp for different prime values. In the early eighties, the work of Victor Kolyvagin, Benedict H. Gross and Don Zagier, combined with that of Andrew Wiles finally created tools to approach the previously obscure problem, eventually bringing it to the forefront, the Millennium prize stamping it as one of the most important problems of the century. prof. Henri Darmon has been working on this problem. We met with him to try to understand what the BSD really means and perhaps get a glimpse of his work on the problem.
Elliptic curves and projective modgroup law. To obtain this group, one needs to els In the official description by Andrew Wiles, the BSD is described as a relation between the Lfunction of an elliptic curve, terms I will clarify below, and its rank, a number that, to some extent, measures the size of the set of rational solutions (solutions in Q) of that elliptic curve. Pr. Darmon explains: “It started without involving L-functions at all, these are just part of the baggage that you need to make this conjecture very precise. But the idea is simple. You start with an an elliptic curve
look at the projective model of the equation.” The projective model of an equation is obtained by adding an extra variable, z, in order to transform it into a homogenous equation of degree three: y 2 z = x3 + axz 2 + bz 3 .
This equation has a trivial solution, (x, y, z) = (0, 0, 0), which we ignore. Also, if (x, y, z) is a solution to this equation, then so is (λx, λy, λz). We let two solutions be equivalent if they differ by a non-zero scalar. There are two possibilities, either z 6= 0 or z = 0. If z 6= 0 for a solution P = (x, y, z), then E = y 2 = x3 + ax + b, a, b ∈ Q. P is equivalent to a solution (x′ , y ′ , 1), because The set of rational solutions of these equations, we can multiply P by z −1 . This solves the origiE(Q), has a very nice structure, an abelian nal equation, thus we get a bijection between the The δelta-ǫpsilon
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