The Arithmetic Geometry group presently consists of the following people:
Owen Biesel Postdoc Martin Djukanovic Graduate student Bas Edixhoven Full Professor David Holmes Postdoc Jinbi Jin Graduate student Robin de Jong Scientific staff Martin Lübke Scientific Staff Maxim Mornev Graduate student Jacob Murre Emeritus Professor Wouter Zomervrucht Graduate student
We participate in the ALGANT programme and the International Research Training Group Moduli and Automorphic Forms
SeminarWe have a local algebra, geometry and number theory seminar.
DescriptionGeometers study geometric properties of sets of solutions of systems of equations. According to the possible kinds of equations (continuous, differentiable, analytic, polynomial), and of the structures that one studies, one distinguishes kinds of geometry (topology, differential topology and differential geometry, analytic geometry, algebraic geometry, arithmetic geometry).
In algebraic geometry the equations are given by polynomials. Classically, the coefficients and solutions were complex numbers. Number theorists consider integer or rational coefficients and solutions. The goal of arithmetic geometry is to understand the relations between algebraic geometry and number theory.
Three important notions in arithmetic geometry are ''algebraic variety'' (abstraction of system of polynomial equations), ''zeta function'' and ''cohomology''. Zeta functions associated to algebraic varieties are generating functions defined using the numbers of solutions in finite fields. Cohomology associates vector spaces equipped with certain structures to algebraic varieties. One important aim of arithmetic geometry is to understand the relations between the values of zeta functions at integers and properties of the set of rational solutions. Cohomology plays an important role here. Cohomology also provides representations of Galois groups, which is essential for Langlands's program (relations between such representations and ''automorphic'' representations of matrix groups). The most striking results obtained in this field are the proof of Weil's conjectures (Dwork, Grothendieck, Deligne), Faltings's proof of Mordell's conjecture, Fontaine's theory (comparison between certain cohomologies), Wiles's proof of Fermat's Last Theorem, Lafforgue's result on Langlands's conjectures, the proof of Serre's modularity conjecture (Khare, Wintenberger, Kisin....), and Taylor's proof of the Sato-Tate conjecture.
Apart from its numerous applications within mathematics, algebraic geometry over finite fields provides error correcting codes and crypto systems, both used in everyday life.