Seminars

Abstract: We describe a new method to compute the reductions mod p of irreducible two-dimensional semi-stable representations of the absolute Galois group \GQp of \Qp. This method uses the compatibility with respect to reduction mod p between the p-adic Local Langlands Correspondence and the Iwahori theoretic version of the mod p Local Langlands Correspondence. By estimating certain logarithmic functions on \Qp by polynomials on open subsets of \Zp, we compute the reductions mod p completely for weights at most p+1. We also state how this method can be used, in theory, to compute the reductions mod p of semi-stable representations of arbitrarily large weights. 

This talk is based on a joint work with Eknath Ghate. For details, please refer to arxiv.org/abs/2311.03740.

Overview : The main subjects of this course are the moduli stack of p-adic Galois representations (so-called the Emerton-Gee stack) and the theory of local models (developed by Le-Le Hung-Levin-Morra). The Emerton-Gee stack has its significance in the Langlands program. First, its local geometry is equivalent to the geometry of p-adic Galois deformation rings. Second, it provides a framework for the categorical p-adic local Langlands program recently proposed by Emerton-Gee-Hellmann. Therefore, it is desirable to understand the geometry of the Emerton-Gee stack. Although it is a highly complicated geometric object, the theory of local models provides concrete projective algebraic varieties whose singularities model those of the Emerton-Gee stacks. We will introduce these two subjects as well as some related topics. We will focus on conceptual understanding with a brief discussion of technical details. We hope to provide references to the subjects that can guide enthusiastic audiences to further studies.

For details, please refer to Heejong Lee.