Moduli of finite flat group schemes, and modularity

By M. Kisin. 2009

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Local models of Shimura varieties, I. Geometry and combinatorics (2010, background): “In [51] Kisin constructs a kind of birational modification scheme of the universal flat deformation of a finite flat group scheme over a discrete valuation ring of unequal characteristic (0,p), and shows that the singularities in characteristic p of these schemes are modeled by certain local models ”
Quasi-semi-stable representations (2007, neutral): “Proof. The case r = 1 is done in proposition 1.1.11 of [ 11 ].”
Iwasawa Main Conjecture for Supersingular Elliptic Curves (2014, background): “The R3,x is modular by [35]: it must be modular unless the residual representation were induced from a Galois character for Q( √”
Lattices in the cohomology of Shimura curves (2013, methodology): “6) could be proved without relying on the main results of [33], and without using the potential diagonalizability of potentially Barsotti–Tate representations proved in [37] and [31], but rather just the explicit computations of potentially Barsotti–Tate deformation rings in Sect.”
Finding meaning in error terms (2008, background): “31See [28], where the [LC(v)] condition for v|p > 2 is that the local representation is Barsotti-Tate; [29] deals with p = 2; and [30] covers the case where the local representations for v|p become semi-stable over an abelian extension of Qp.”