p-adic representation theory and the p-adic Langlands program
Final Report Abstract
Let p be a prime number. The p-adic local Langlands program is a comparatively new branch of number theory and arithmetic geometry. It is an internationally highly active field and may be seen as a significant refinement of Robert Langlands’ general philosophy to study arithmetic problems concerning local fields. In the last ten years, the p-adic approach has led to considerable new insights. On the other hand, it has also raised deep foundational problems in representation theory. In my time as a Heisenberg scholar, I was able to successfully address one of those problems. I developed a good duality theory for smooth representations of p-adic reductive groups on vector spaces over fields of characteristic p. It is a simple observation that the classical smooth duality functor is not of much use in characteristic p. However, following ideas of Björk, Schneider, Teitelbaum and Venjakob, I showed that its derived functors lead to a duality theory enjoying many good formal properties. Moreover, I was able to explicitly compute these derived functors in a large class of examples. My results actually show that this enhanced smooth duality theory appears quite naturally in a particular case of the mod-p local Langlands correspondence constructed by Breuil and Colmez.
