July 17 - August 11, 2006
Mathematisches Institute, Georg-August-Universität, Göttingen, Germany
Organizers: Jim Carlson, Henri Darmon, David Ellwood, Brendan Hassett, Yuri Tschinkel
Designed for graduate students and mathematicians within five years of their PhD, the program will introduce the participants to modern techniques and outstanding conjectures at the interface of number theory and algebraic geometry.
The main focus is rational points on algebraic varieties over non-algebraically closed fields. Do they exist? If not, can this be proven efficiently and algorithmically? When rational points do exist, are they finite in number and can they be found effectively? When there are infinitely many rational points, how are they distributed?
For curves, a cohesive theory addressing these questions has emerged in the last few decades. Highlights include Faltings' finiteness theorem and Wiles' proof of Fermat's Last Theorem. Key techniques are drawn from the theory of elliptic curves, including modular curves and parametrizations, Heegner points, and heights.
The arithmetic of higher-dimensional varieties is equally rich, offering a complex interplay of techniques including Shimura varieties, the minimal model program, moduli spaces of curves and maps, deformation theory, Galois cohomology, harmonic analysis, and automorphic functions. However, many foundational questions about the structure of rational points remain open, and research tends to focus on properties of specific classes of varieties.
This school will offer three core courses (on curves, surfaces, and higher-dimensional varieties), supplemented by seminars on computational and algorithmic aspects of arithmetic geometry, and by mini-courses on more advanced topics.