The investigator and his students study capacity theory and its applications to arithmetic geometry. Capacity is a measure of size for sets, which arises in potential theory and has applications in probability, complex analysis, and number theory. The chief goal of this project is to prove a very strong version of the Fekete-Szego theorem on algebraic curves, which asserts that if an adelic set on a curve has large enough capacity, then there exist algebraic points with all their conjugates near it, in an adelic sense. The new theorem will be a basic existence theorem producing algebraic points which are subject to real and p-adic rationality conditions, as well as topological constraints. To fix ideas, for sets in the complex plane, the capacity of a circle turns out to equal its radius, and the capacity of a line segment is a quarter of its length. In analysis, the primary distinction is between sets of capacity 0 and sets of positive capacity: sets of capacity 0 are 'invisible' to holomorphic functions. In number theory, the main distinction is between sets of capacity greater than 1, and less than 1. The classical theorem of Fekete and Szego says that for a set in the complex plane stable under complex conjugation, if the capacity of the set is greater than 1, then every neighborhood of the set contains infinitely many Galois orbits of algebraic integers. David Cantor generalized the theorem to adelic sets on the projective line, and subsequently the investigator generalized it to adelic sets on algebraic curves. As an application, the investigator proved an existence theorem for algebraic integer points on affine algebraic varieties, which has been the object of considerable work by Moret-Bailly and Szpiro and their students. Earlier, Raphael Robinson had extended the Fekete-Szego theorem in another direction, showing that if a set of capacity greater than 1 were contained in the real line, then every real neighborhood contained infinitely many Galois orbits of totally real algebraic integers. Recently the investigator proved an adelic version of Robinson's theorem, proving the existence of algebraic numbers which were totally real and totally p-adic at a finite number of places. The goal of the project is to generalize this theorem to algebraic curves. The methods used will involve p-adic analysis, potential theory, and approximation theory for algebraic functions.
The study of diophantine equations (looking for integer solutions to polynomial equations in several variables) is a very old and very difficult subject, going back to the Greeks. It is only within the last half-century that much progress has been made, using methods of modern number theory. Two famous results were a negative solution to "Hilbert's Tenth problem" (by Matiyasevich in 1970), which asks if there is an algorithm for determining whether or not a given equation has integer solutions; and the negative resolution of Fermat's Last Theorem (by Wiles in 1995), which asks if sums of n-th powers of integers can be n-th powers. The investigator's work pursues a different direction, showing that for much larger arithmetic domains than the integers, under appropriate conditions there do exist solutions; and moreover there exist algorithms for telling whether or not they exist. The current work will considerably reduce the size of the domains where solutions are known to exist.
