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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


Source: Agashe, Amod - Department of Mathematics, Florida State University


Collections: Mathematics