Lectures on the Birch-Swinnerton-Dyer Conjecture

Lectures on the Conjecture of Birch and Swinnerton - Dyer Benedict

The Birch-swinnerton-dyer Conjecture

We give a brief description of the Birch-Swinnerton-Dyer conjecture which is one of the seven Clay problems.

The Birch and Swinnerton-Dyer Conjecture

A polynomial relation f(x, y) = 0 in two variables defines a curve C. If the coefficients of the polynomial are rational numbers then one can ask for solutions of the equation f(x, y) = 0 with x, y ∈ Q, in other words for rational points on the curve. If we consider a non-singular projective model C of the curve then over C it is classified by its genus. Mordell conjectured, and in 1983 Falting...

The conjecture of Birch and Swinnerton-Dyer

This essay starts by first explaining, for elliptic curves defined over Q, the statement of the conjecture of Birch and Swinnerton-Dyer. Alongside, it contains a discussion of some results that have been proved in the direction of the conjecture, such as the theorem of Kolyvagin-Gross-Zagier and the weak parity theorem of Tim and Vladimir Dokchitser. The second, third and fourth part of the ess...

The Birch and Swinnerton-Dyer Conjecture

In this talk I shall attempt to introduce some of the main features of the Birch and Swinnerton-Dyer conjecture, (BSD). The congruent number problem, deciding whether an integer D is the area of a right angle triangle with rational sides, is not easy. It turns out that the problem is equivalent to finding out if a certain elliptic curve has an infinite number of rational points. In 1983 Tunnell...