**Seminario Teoría de Números UAM-ICMAT**

**Title: RELATIVE MOTIVES OVER SHIMURA VARIETIES **

**SPEAKER: Alex Torzewski (University of Warwick) **

**DATE: Thursday 10th May 2018 - 13:30 h **

**VENUE: Aula 420, Módulo 17, Departamento de Matemáticas, UAM (Campus de Cantoblanco, Madrid) **

**ORGANISER: UAM-ICMAT **

**ABSTRACT: **

**Given a Shimura variety S defined by a group G (e.g. a modular curve, in which case G=GL_2), there is a "canonical construction" which assigns to a representation of G variation of Hodge structures on the Shimura variety. Similarly, there is also a construction which gives an l-adic sheaf for each representation. It is conjectured that these should both be motivic in origin, i.e. for each representation of G, there should be a way to construct a motive over S whose cohomology coincides with that given by the previous constructions. We shall introduce all the concepts listed in the abstract and explain why such results are useful before giving a very brief outline of existing and current work in this direction.**