Algebra Seminar: Stefano Morra -- Jantzen’s generic decomposition pattern and the Breuil--Mezard conjecture

Stefano Morra (l’Institut Universitaire de France) will speak in the algebra seminar
this week.  We will go out for lunch after the talk.  

Where: Carslaw 275 

When: 12-1pm, Friday 30 August 

Title: Jantzen’s generic decomposition pattern and the Breuil--Mezard conjecture 

Abstract: The Breuil--Mezard conjecture, generated from the proof of the
Shimura--Taniyama--Weil conjecture, predicts that discrete invariants coming from the
deformation theory of a continuous homomorphism Gal(\bar{Q_p}/Q_p)-->GL_2(\bar{F_p}) are
dictated by character formulas coming the mod-p reduction of GL_2(F_p)-representations
with \bar{Z_p}-coefficients.  This conjecture has now been geometrized into a statement
relating cycles on moduli spaces of (n\geq 2-dimensional) continuous Galois
representations with p-adic coefficients and decomposition numbers of finite dimensional
locally algebraic representations of GL_n(Z_p) with p-adic coefficients.  This involves
in particular decomposition patterns for the mod-p reduction of Deligne--Lusztig
representations, established in generic cases by Jantzen in the early eighties.  

Motivated by the desire to clarify the behavior of the Breuil--Mezard conjecture in
non-generic situation, we present an improvement of the generic decomposition pattern of
Jantzen, with arithmetic applications as the weight part of Serre conjecture (which is a
manifestation of the Breuil--Mezard conjecture).  

This is joint work with D.~Le, B.~Le Hung and B.~Levin.