We construct projective varieties in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of $\mathbb{Q}_p$ with small regular Hodge-Tate weights. We establish several significant facts about their geometry including a unibranch property at special points and a representation theoretic description of the irreducible components of their special fibers. We derive from these geometric results a number of local and global consequences: the Breuil-Mézard conjecture in arbitrary dimension for tamely potentially crystalline deformation rings with small Hodge-Tate weights (with appropriate genericity conditions), the weight part of Serre's conjecture for $U(n)$ as formulated by Herzig (for global Galois representations which satisfy the Taylor-Wiles hypotheses and are sufficiently generic at $p$), and an unconditional formulation of the weight part of Serre's conjecture for wildly ramified representations.
Presentations of non-commutative deformation rings via $A_\infty$-algebras and applications to deformations of Galois representations and pseudorepresentations
We survey some recent progress on generalizations of conjectures of Serre concerning the cohomology of arithmetic groups, focusing primarily on the "weight" aspect. This is intimately related to (generalizations of) a conjecture of Breuil and Mézard relating the geometry of potentially semistable deformation rings to modular representation theory. Recently, B. Levin, S. Morra, and the authors established these conjectures in tame generic contexts by constructing projective varieties (local models) in mixed characteristic whose singularities model, in generic cases, those of tamely potentially …
The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that $\barρ:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_2(\bar{\mathbb{F}}_p)$ lifts to a Hida line for which the weights range over a congruence class modulo-$p^2$. The advantage of the purely Galois theoretic approach is that it allows us to construct $p$-adic families of Galois representations lifting the actual representation $\barρ$, and not just the semisimplification.
In a recent work of Galatius and Venkatesh, the authors showed the importance of studying simplicial generalizations of Galois deformation functors. They established a precise link between the simplicial universal deformation ring $R$ prorepresenting such a deformation problem (with local conditions) and a derived Hecke algebra. Here we focus on the algebraic part of their study which we complete in two directions. First, we introduce the notion of simplicial pseudo-characters and prove relations between the (derived) deformation functors of simplicial pseudo-characters and that of simplicial …
In this article, we use deformation theory of Galois representations valued in the symplectic group of degree four to prove a freeness result for the cohomology of certain quaternionic unitary Shimura variety over the universal deformation ring for certain type of residual representation satisfying a property called rigidity. This result plays an important role in the proof of the arithmetic level raising theorem for the symplectic similitude group of degree four over the field of rational numbers by the author.
In this article, we study deformations of conjugate self-dual Galois representations. The study has two folds. First, we prove an R=T type theorem for a conjugate self-dual Galois representation with coefficients in a finite field, satisfying a certain property called rigid. Second, we study the rigidity property for the family of residue Galois representations attached to a symmetric power of an elliptic curve, as well as to a regular algebraic conjugate self-dual cuspidal representation.
In this paper, we study the Jacobi sums over Galois rings of arbitrary characteristics and completely determine their absolute values, which extends the work in \cite{feng1}, where the Jacobi sums over Galois rings with characteristics a square of a prime number were discussed.
We study the rigid generic fiber $\mathcal{X}^\square_{\overlineρ}$ of the framed deformation space of the trivial representation $\overlineρ: G_K \to \text{GL}_n(k)$ where $k$ is a finite field of characteristic $p>0$ and $G_K$ is the absolute Galois group of a finite extension $K/\mathbf{Q}_p$. Under some mild conditions on $K$ we prove that $\mathcal{X}^\square_{\overlineρ}$ is normal. When $p > n$ we describe its irreducible components, and show Zariski density of its crystalline points.
Let $p$ be an odd prime and $q$ a power of $p$. We examine the deformation theory of reducible and indecomposable Galois representations $\barρ:G_{\mathbb{Q}}\rightarrow \text{GSp}_{2n}(\mathbb{F}_q)$ that are unramified outside a finite set of primes $S$ and whose image lies in a Borel subgroup. We show that under some additional hypotheses, such representations have geometric lifts to the Witt vectors $\text{W}(\mathbb{F}_q)$. The main theorem extends that of Hamblen and Ramakrishna in which the $n=1$ case was treated.
We study irreducible odd mod $p$ Galois representations $\barρ \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show that any $\barρ$ that lifts locally, and at places above $p$ to de Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We also prove non-geometric lifting results without any oddness assumption.
For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete characterization of the freeness of the ring of integers $\mathcal{O}_L$ over its associated order $\mathfrak{A}_{L/K}$ in the unique Hopf-Galois structure on $L/K$, which is analogous to the one already known for cyclic degree $p$ extensions of $p$-adic fields. We shall derive positive and negative results on criteria for the freeness of $\mathcal{O}_L$ as $\mat…
The classification of the local Galois representations using $(φ,Γ)$-modules by Fontaine has been generalized by Kisin and Ren over the Lubin-Tate extensions of local fields using the theory of $(φ_q,Γ_{LT})$-modules. In this paper, we extend the work of (Fontaine) Herr by introducing a complex which allows us to compute cohomology over the Lubin-Tate extensions and compare it with the Galois cohomology groups. We further extend that complex to include certain non-abelian extensions. We then deduce some relations of this cohomology with those arising from $(ψ_q,Γ_{LT})$-modules. We also comput…
On the dimension conjecture for global Galois deformation rings in characteristic two
Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究院）
In this paper, we study the graph of homothety classes of stable free lattices in a two-dimensional representation over a local UFD. This generalizes a classical result of the case where the base ring is a discrete valuation ring due to Serre. As applications, we consider the case when the representation comes from a residually reducible Hida family and we study the control theorem of Selmer groups. These results enable us to know the precise statement of the main conjecture in residually reducible case as we will remark in section 4 and section 5.
