Konferenz: Symposium about Pure Mathematics

Vortrag von Prof. Dr. Anna Cadoret

Datum: 08.02.17 Zeit: 09.30 - 10.10 Raum: Y27H28

Arithmetic geometry aims at solving number-theoretic problems via geometric and representation-theoretic methods. It relies on the principle that `geometry should govern arithmetic'. This principle is embodied in fascinating conjectures, of diophantine nature like the Lang conjecture - a higher-dimensional generalization of the Mordell conjecture, or of motivic nature like the companion conjectures of Hodge and Grothendieck-Serre-Tate. The latter are related to Grothendieck's theory of motives, which predicts that the `arithmetico-geometrical complexity' of an algebraic variety is encoded in an algebraic group, its motivic Galois group. In this talk, starting from the concrete problem of determining rational solutions to cubic equations, I will develop an approach based on etale fundamental groups to control the variation of the motivic Galois group in a family of motives. In particular, I will explain how this problem is related to the Lang conjecture. I will also give applications in the case of 1-dimensional families of motives.