| Abstract: | Shimura varieties are algebraic varieties defined over number fields which are importantobjects of study in arithmetic geometry. Although the theory was originally rooted in complex analysis and geometry, nowadays much of the interest stems from the close relationships between Shimura varieties and automorphic forms and Galois representations,which has been used to establish many new cases of the Langlands correspondences. Often, the key point is to study models of these objects over arithmetic rings such as $\mathbb{Z}_{(p)}$. In this talk, I will explain some recent progress on the study of thesemodels and their mod $p$ reductions, and discuss some applications to the Langlands program and other areas of arithmetic geometry. |
