Members' Colloquium

Essentially optimal estimates have been obtained for mean values of Vinogradov’s exponential sum as a consequence of the decoupling method (by Bourgain, Demeter and Guth), and the efficient congruencing method (by the speaker). Such work makes essential use of the fact that the system of Diophantine equations associated with these mean values is translation-dilation invariant. We sketch out such ideas, and report on progress for systems which are not translation-dilation invariant obtained by exploiting the arithmetic of function fields over the field of p-adic numbers. Counting problems for Diophantine equations feature prominently.