- Rational points on definable sets (Alex Wilkie)
Abstract - Lecture Notes
- Functional transcendence via o-minimality (Jonathan Pila)
- Diophantine Properties of Torsion Points and Special Points (Philipp Habegger)
Relative Manin-Mumford for abelian varieties
With an eye or two towards applications to Pell's equation and Davenport's work on integration of algebraic functions, Umberto Zannier and I have recently characterised torsion points on a fixed algebraic curve in a fixed abelian scheme of dimension bigger than one (when all is defined over the algebraic numbers): there are at most finitely many provided the natural obstacles are absent. I sketch the proof as well as the applications.
Hyperbolic Ax-Lindemann-Weierstrass theorem
This is a joint work with B. Klingler and E. Ullmo. We generalise the definability result of Peterzil and Starchenko to general Shimura varieties and prove the general hyperbolic Ax-Lindemann-Weierstrass theorem.
Integer valued definable functions
Wilkie has conjectured that for sets definable in the real field with exponentiation, the bound in the Pila-Wilkie theorem can be improved to a power of log H. At present this is only known for curves and certain surfaces. An easy consequence of the result for curves is that a definable analytic function on the real line that takes integer values at all natural numbers either grows faster than exp(x^epsilon) for some positive epsilon or is a polynomial. This can be thought of as a (rather weak) analogue of Polya's theorem on integer valued entire functions. I'll discuss the above and then show how some simple o-minimality arguments can be used to extend this result to functions of several variables.