p-adic Hodge Theory and Applications

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The PI will continue his work in p-adic Hodge theory, with a particular focus on applications to the arithmetic of Shimura varieties. In particular, their integral models, arithmetic compactifications and the structure of their mod p points. This should eventually allow one to understand the cohomology of these varieties and the Galois representations which occur there.p-adic Hodge theory is branch of number theory which seeks to study the local properties of Galois representations which arise from geometry. The subject has been at the heart of many of the most important developments in number theory in the last 30 years (e.g the proof of Fermat's Last theorem). The project aims to use these techniques to study the arithmetic of a class of geometric objects - Shimura varieties - which have proved particularly important in arithmetic applications.