University of Sydney Algebra Seminar
Stefano Morra
Friday 30 August, 12-1pm, Place: Carslaw 275
Jantzen's generic decomposition pattern and the Breuil--Mezard conjecture
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.