October 31, 2003: Prof. Fedor Bogomolov, CIMS
Algebraic Curves and Algebraic Numbers
In algebraic geometry an algebraic curve, or
better, an affine algebraic curve, is roughly speaking a subset of
solutions of a polynomial equation P(x,y)
= 0 where x,y run
though elements of some field (rational numbers, complex numbers, real
numbers  for example, in the latter case this set is indeed in general
a curve in the plane  hence the name). I will touch on some
aspects of the theory which in its most elementary form investigates
the relation between geometry of the algebraic curve P(x,y) = 0 and the properties of
the set of rational solutions P(x,y)
= 0 if P is a
polynomial with rational coefficients.
