Related papers: Serre weights and Breuil's lattice conjecture in d…

We prove a conjecture of Conrad, Diamond, and Taylor on the size of certain deformation rings parametrizing potentially Barsotti-Tate Galois representations. To achieve this, we extend results of Breuil and Mezard (classifying Galois…

We give a new local proof of Breuil-M\'ezard conjecture for two dimensional representations of the absolute Galois group of $\mathbb{Q}_p$, when $p\ge 5$ and the representation has scalar endomorphisms.

Using $p$-adic local Langlands correspondence for $\operatorname{GL}_2(\mathbb{Q}_2)$ and an ordinary $R = \mathbb{T}$ theorem, we prove that the support of patched modules for quaternionic forms meet every irreducible component of the…

The conjecture of Serre referred in the title is the one about modularity of odd Galois representations into GL(2,F) where F is a finite field of characteristic p. We present an analogous conjecture where GL(2) is replaced by GL(n). We…

Let p > 2 be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call "pseudo-Barsotti-Tate representations", over arbitrary finite extensions of the…

In a previous paper [CG], we showed how one could generalize Taylor-Wiles modularity lifting theorems [Wil95, TW95] to contexts beyond those in which the automorphic forms in question arose from the middle degree cohomology of Shimura…

We prove, under mild hypotheses, that there are no irreducible two-dimensional_even_ Galois representations of $\Gal(\Qbar/\Q)$ which are de Rham with distinct Hodge--Tate weights. This removes the "ordinary" hypothesis required in previous…

Under some technical assumptions of a global nature, we establish the weight part of Serre's conjecture for mod $p$ Galois representations for CM fields that are tamely ramified and sufficiently generic at $p$.

We compute the deformation rings of two dimensional mod l representations of Gal(Fbar/F) with fixed inertial type, for l an odd prime, p a prime distinct from p and F/Q_p a finite extension. We show that in this setting (when p is also odd)…

We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of $U(n)$ which are compact at infinity and split at places dividing $p$ in generic situations. That is, we show that all modular…

The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using…

We formulate an analogue of the Breuil-M\'ezard conjecture for the group of units of a central division algebra over a $p$-adic local field, and we prove that it follows from the conjecture for $\mathrm{GL}_n$. To do so we construct a…

Let A be a basic connected finite dimensional algebra over an algebraically closed field, let G be a group, let T be a basic tilting A-module and let B the endomorphism algebra of T. Under a hypothesis on T, we establish a correspondence…

We consider closed subschemes in the affine grassmannian obtained by degenerating $e$-fold products of flag varieties, embedded via a tuple of dominant cocharacters. For $G= \operatorname{GL}_2$, and cocharacters small relative to the…

Gouv\^ea-Mazur [GM] made a conjecture on the local constancy of slopes of modular forms when the weight varies $p$-adically. Since one may decompose the space of modular forms according to associated residual Galois representations, the…

Let $Y$ be an abelian variety over a subfield $k \subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford-Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre…

We develop a local model theory for moduli stacks of $2$-dimensional non-scalar tame potentially Barsotti--Tate Galois representations of the Galois group of an unramified extension of $\mathbb{Q}_p$. We derive from this explicit…

Let $F$ be a finite unramified extension of $\mathbb Q\_p$ and $\bar\rho$ be an absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We conjecture that…

We prove some new cases of weight part of Serre's conjectures for mod $p$ Galois representations associated to automorphic representations on unitary groups $U(d)$. The approach is a generalization of the work of Gee-Liu-Savitt, namely, we…

A conjecture of Breuil, Buzzard, and Emerton says that the slopes of certain reducible $p$-adic Galois representations must be integers. In previous work we showed this conjecture for representations that lie over certain non-subtle…