The Birch and Swinnerton-Dyer conjecture Proposal Text Summary: The Birch and Swinnerton-Dyer conjecture Proposal Text Amod Agashe 1 Project description This project lies in the area of number theory, and in particular in a subarea called arithmetic geometry. Our main objects of interest are elliptic curves, which we define in Section 1.1 below. The main question regarding elliptic curves that we plan to tackle is the Birch and Swinnerton- Dyer conjecture, which we discuss in Section 1.2. Finally we describe what we plan to accomplish regarding this conjecture in Section 1.3. 1.1 Elliptic curves For centuries, mathematicians have been interested in finding integer or rational solutions to polynomial equations. For example, consider the equation x2 + y2 = z2, whose solutions (x, y, z) correspond to the sides of right-angled triangles, and hence are called Pythagorean triples. An example of a solution to this equation is the familiar triple x = 3, y = 4, and z = 5. In fact, there are infinitely many rational solutions to this equation: take any integer t; then x = 2t, y = t2 - 1, and z = t2 - 1 gives a solution. Polynomial equations whose coefficients are integers or rational numbers are called Diophantine equations, and the study of their integer or rational solutions is called arithmetic geometry. In the example of the equation x2 + y2 = z2 above, the parametrization x = 2t, y = t2 - 1, and z = t2 - 1, where t ranges over all integers, describes all the solutions. Some equations may not have any rational solutions, e.g., x2 + y2 = -1 does not Collections: Mathematics