Rational points on elliptic curves

Fridays, 9–11 (tentative) Seminar Room, CMAT

The goal of this seminar is to introduce the study of the arithmetic and geometry of a particular family of Diophantine equations: elliptic curves. They are a central object in modern number theory, and that is the perspective we will take. While the study of elliptic curves naturally calls for some basic tools from algebraic geometry, no prior knowledge of the subject will be assumed — we will develop the necessary background as we go.

Coordinator:

Photo of Gonzalo Tornaría Gonzalo Tornaría

Outline

  1. Geometry and arithmetic: geometry of conics and cubics, Weierstrass normal form, the group law.

  2. Torsion points: points of order 2 and 3, real and complex points, points of finite order, the Nagell–Lutz theorem.

  3. The group of rational points: heights and descent, the Mordell theorem.

References

J. Silverman, J. Tate, Rational points on elliptic curves (1992).