University of Georgia
Mathematics Department Colloquium
Joseph L. Wetherell
University of Southern California
January 27 2000
Rational points on algebraic curves
Abstract:
Suppose we are given a polynomial equation with rational coefficients; it
is quite reasonable to ask whether there are any rational solutions to this
equation (solutions consisting entirely of rational numbers). If we are
very ambitious we might want to describe all possible rational solutions to
our equation.
For example, problem VI.17 of Diophantus's {\it Arithmetica} asks for a
positive rational solution to $x^8 + x^4 + x^2 = y^2$. As is typical in
the {\it Arithmetica}, Diophantus provides a solution to this problem,
namely $x = 1/2$ and $y= 9/16$.
The question of finding all rational solutions is more difficult. The
equation we are considering describes an algebraic curve of genus 2, so by
Faltings' theorem we know that there are at most finitely many rational
solutions. Unfortunately, Faltings doesn't tell us how to find the set of
rational solutions; indeed, we do not know any general method for
determining the set of rational solutions on a curve.
I will survey a variety of standard techniques which can sometimes
determine the set of rational solutions on a curve. In particular, I will
describe how to extend one of these techniques so that it works on our
curve from the {\it Arithmetica}. The answer? The only positive rational
solution to problem VI.17 is the one given by Diophantus.