SPEAKER: Jordan Ellenberg
TITLE: Descent bounds for rational points on varieties
ABSTRACT:
(joint w/ A.Venkatesh)
We present some new upper bounds for numbers of rational points of bounded
height on curves and some higher-dimensional varieties. The main idea is
to combine the ideas of Bombieri-Pila and Heath-Brown with descent arguments
involving etale covers of the variety. For instance: Heath-Brown shows
that the number of rational points of height at most H on a degree-d plane
curve is bounded by C H^{2/d}, where C is a constant not depending on the
curve. This theorem is sharp in general, but we show that the additional
hypothesis that the curve has positive geometric genus allows an improvement
of the exponent.