Franz Lemmermeyer

Algebraic Geometry


Part I is very elementary and should be a lot of fun. I will discuss the arithmetic of the rational function field in more detail than usually: resultants and discriminants, Fermat's Last Theorem for polynomials, Mason's ABC theorem, quadratic extensions, the Pell equation, and eventually valuations, divisors, and the Theorem of Riemann-Roch. Part II is an experiment and should also be a lot of fun even though the subject may seem very abstract at first. Prerequisites include the elementary notions of algebra and topology.



Homework is always due one week after hand-out except when stated otherwise.


Here are a few books covering what we did in the first part (and much more): Also, check out the lecture notes by David Marker on this page.

Extra Credit Problems

Solutions to the following problems will be accepted until the end of the semester.
  1. In Lecture 3 I proved, using the ABC theorem, that the elliptic curve y2 = x3 + x cannot be parametrized. Give a similar proof of the more general result that y2 = x3 + ax2 + bx cannot be parametrized if the polynomial on the right hand side does not have multiple roots.
  2. Assume that you are given a nondegenerate conic defined over Q with a rational point. Can it be rationally parametrized using trigonometric functions? I suspect the answer is yes. Here are a few examples: the hyperbola X2 - Y2 = 1 is parametrized by X = 1/cos t and Y = tan t; the parabola Y = X2 is parametrized by X = cos t and Y = (cos 2t +1)/2; and the circle X2 + Y2 = 2 is parametrized by X = cos t + sin t, Y = cos t - sin t.