Hodge Tate Weights Research Articles

Irreducibility of some crystalline loci with irregular Hodge–Tate weights

We show that loci of crystalline representations of G K G_K for K / Q p K/\mathbb {Q}_p an unramified extension are irreducible when the Hodge–Tate weights are fixed and sufficiently small. This was previously known for weights in the interval [ − p , 0 ] [-p,0] and in this paper we show how that this bound can be relaxed provided the Hodge–Tate weights are sufficiently irregular at certain embeddings. This is motivated by the desire to extend the conjectures of Breuil–Mézard [Duke Math. J. 115 (2002), pp. 205–310] on loci of potentially crystalline representations to irregular weights.

Irregular loci in the Emerton–Gee stack for GL2

Abstract Let K / Q p K/\mathbf{Q}_{p} be unramified. Inside the Emerton–Gee stack X 2 \mathcal{X}_{2} , one can consider the locus of two-dimensional mod 𝑝 representations of Gal ⁢ ( K ̄ / K ) \mathrm{Gal}(\overline{K}/K) having a crystalline lift with specified Hodge–Tate weights. We study the case where the Hodge–Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms. We prove that if the gap between each pair of weights is bounded by 𝑝 (the irregular analogue of a Serre weight), then this locus is irreducible. We also establish various inclusion relations between these loci.

Colength one deformation rings

Let K / Q p K/\mathbb {Q}_p be a finite unramified extension, ρ ¯ : G a l ( Q ¯ p / K ) → G L n ( F ¯ p ) \overline {\rho }:\mathrm {Gal}(\overline {\mathbb {Q}}_p/K)\rightarrow \mathrm {GL}_n(\overline {\mathbb {F}}_p) a continuous representation, and τ \tau a tame inertial type of dimension n n . We explicitly determine, under mild regularity conditions on τ \tau , the potentially crystalline deformation ring R ρ ¯ η , τ R^{\eta ,\tau }_{\overline {\rho }} in parallel Hodge–Tate weights η = ( n − 1 , ⋯ , 1 , 0 ) \eta =(n-1,\cdots ,1,0) and inertial type τ \tau when the shape of ρ ¯ \overline {\rho } with respect to τ \tau has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre’s conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [Mém. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150].

Local models for Galois deformation rings and applications

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.

Potential automorphy for $$GL_n$$

We prove potential automorphy results for a single Galois representation $G_F \rightarrow GL_n(\overline{\mathbb{Q}}_l)$ where $F$ is a CM number field. The strategy is to use the $p,q$ switch trick and modify the Dwork motives employed in \cite{HSBT} to break self-duality of the motives, but not the Hodge-Tate weights. Another key result to prove is the ordinarity of certain $p$-adic representations, which follows from log geometry techniques. One input is the automorphy lifting theorem in \cite{tap}.

Semi-stable deformation rings in even Hodge–Tate weights: The residually reducible case

Let [Formula: see text] be a prime number and [Formula: see text] a positive even integer less than [Formula: see text]. In this paper, we find the strongly divisible modules corresponding to the Galois stable lattices in each 2-dimensional semi-stable non-crystalline representation of [Formula: see text] with Hodge–Tate weights [Formula: see text] whose mod-[Formula: see text] reductions are corresponding to nontrivial extensions of two distinct characters. We use these results to construct the irreducible components of the semi-stable deformation rings in Hodge–Tate weights [Formula: see text] of the non-split reducible residual representations of [Formula: see text].

REDUCTIONS OF -DIMENSIONAL SEMISTABLE REPRESENTATIONS WITH LARGE -INVARIANT

AbstractWe determine reductions of $2$ -dimensional, irreducible, semistable, and non-crystalline representations of $\mathrm {Gal}\left (\overline {\mathbb {Q}}_p/\mathbb {Q}_p\right )$ with Hodge–Tate weights $0 < k-1$ and with $\mathcal L$ -invariant whose p-adic norm is sufficiently large, depending on k. Our main result provides the first systematic examples of the reductions for $k \geq p$ .

Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

AbstractWe consider the Kisin variety associated to ann-dimensional absolutely irreducible modpGalois representationρ¯{\bar{\rho}}of ap-adic fieldKtogether with a cocharacter μ. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin’s conjecture holds ifKis totally ramified withn=3{n=3}or μ is of a very particular form. As an application, we get a connectedness result for the deformation ring associated toρ¯{\bar{\rho}}of given Hodge–Tate weights. We also give counterexamples to show Kisin’s conjecture does not hold in general.

A classification of Breuil modules

Let p be a prime number and f a positive integer with f < p. In this paper, we determine the structure of simple Breuil modules of type $$ \oplus _{i = 1}^n\omega _f^{{k_i}}$$ corresponding to n-dimensional irreducible representations of $${G_{{{\bf{Q}}_p}}}$$ . We also describe the extensions of those simple Breuil modules if they correspond to mod p reductions of strongly divisible modules that correspond to Galois stable lattices in potentially semistable representations of $${G_{{{\bf{Q}}_p}}}$$ with Hodge-Tate weights {0, 1, …, n − 1} and Galois type $$ \oplus _{i = 1}^n\widetilde\omega _f^{{k_i}}$$ .

Potentially diagonalisable lifts with controlled Hodge-Tate weights

Motivated by the weight part of Serre's conjecture we consider the following question. Let K/\mathbb{Q}_p be a finite extension and suppose \overline{\rho} : G_K \rightarrow \operatorname{GL}_n(\overline{\mathbb{F}}_p) admits a crystalline lift with Hodge-Tate weights contained in the range [0,p] . Does \overline{\rho} admit a potentially diagonalisable crystalline lift of the same Hodge-Tate weights? We answer this question in the affirmative when K = \mathbb{Q}_p and n \leq 5 , and \overline{\rho} satisfies a mild 'cyclotomic-free' condition. We also prove partial results when K/\mathbb{Q}_p is unramified and n is arbitrary.

Finiteness properties of the category of mod p representations of

Abstract We establish the Bernstein-centre type of results for the category of mod p representations of $\operatorname {\mathrm {GL}}_2 (\mathbb {Q}_p)$ . We treat all the remaining open cases, which occur when p is $2$ or $3$ . Our arguments carry over for all primes p. This allows us to remove the restrictions on the residual representation at p in Lue Pan’s recent proof of the Fontaine–Mazur conjecture for Hodge–Tate representations of $\operatorname {\mathrm {Gal}}(\overline {\mathbb Q}/\mathbb {Q})$ with equal Hodge–Tate weights.

A generalization of Colmez-Greenberg-Stevens formula

In this paper we study the derivatives of Frobenius and the derivatives of Hodge-Tate weights for families of Galois representations with triangulations. We give a generalization of the Fontaine-Mazur L-invariant and use it to build a formula which is a generalization of the Colmez-Greenberg-Stevens formula.

Relative crystalline representations and p-divisible groups in the small ramification case

Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension over $W(k)[\frac{1}{p}]$ of ramification degree $e$. Let $R_0$ be a relative base ring over $W(k)\langle t_1^{\pm 1}, \ldots, t_m^{\pm 1}\rangle$ satisfying some mild conditions, and let $R = R_0\otimes_{W(k)}\mathcal{O}_K$. We show that if $e < p-1$, then every crystalline representation of $\pi_1^{\text{\'et}}(\mathrm{Spec}R[\frac{1}{p}])$ with Hodge-Tate weights in $[0, 1]$ arises from a $p$-divisible group over $R$.

Reductions of Some Two-Dimensional Crystalline Representations via Kisin Modules

Abstract We determine rational Kisin modules associated with 2-dimensional, irreducible, crystalline representations of $\textrm{Gal}(\overline{{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$ of Hodge–Tate weights $0, k-1$. If the slope is larger than $\lfloor \frac{k-1}{p} \rfloor $, we further identify an integral Kisin module, which we use to calculate the semisimple reduction of the Galois representation. In that range, we find that the reduction is constant, thereby improving on a theorem of Berger, Li, and Zhu.

ON THE IRREDUCIBLE COMPONENTS OF SOME CRYSTALLINE DEFORMATION RINGS

We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_{K}$ for a finite extension $K/\mathbb{Q}_{p}$ . This is done by considering a moduli space of Breuil–Kisin modules, satisfying an additional Galois condition, over the unrestricted deformation ring. For $K$ unramified over $\mathbb{Q}_{p}$ and Hodge–Tate weights in $[0,p]$ , we study the geometry of this space. As a consequence, we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of $\mathbb{Q}_{p}$ , with Hodge–Tate weights in $[0,p]$ , are potentially diagonalizable.

SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE

We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $p$ . This is a generalization to $\text{GL}_{3}$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $n$ -dimensional Galois representations’, Duke Math. J. 149(1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math. 212(1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $\text{GL}_{3}(\mathbb{F}_{q})$ .

Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3 (\mathbb{Q}_p)$ and local-global compatibility

Let $\rho_p$ be a $3$-dimensional $p$-adic semi-stable representation of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ with Hodge-Tate weights $(0,1,2)$ (up to shift) and such that $N^2\ne 0$ on $D_{\mathrm{st}}(\rho_p)$. When $\rho_p$ comes from an automorphic representation $\pi$ of $G(\mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real field $F^+$ which is compact at infinite places and $\mathrm{GL}_3$ at $p$-adic places), we show under mild genericity assumptions that the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(\mathbb{A}_{F^+}^\infty)$ of arbitrary fixed tame level contain (copies of) a unique admissible finite length locally analytic representation of $\mathrm{GL}_3(\mathbb{Q}_p)$ which only depends on and completely determines $\rho_p$.

Inertial and Hodge\u2013Tate weights of crystalline representations

Let K be an unramified extension of {mathbb {Q}}_p and rho :G_K rightarrow {text {GL}}_n(overline{{mathbb {Z}}}_p) a crystalline representation. If the Hodge–Tate weights of rho differ by at most p then we show that these weights are contained in a natural collection of weights depending only on the restriction to inertia of {overline{rho }} = rho otimes _{overline{{mathbb {Z}}}_p} overline{{mathbb {F}}}_p. Our methods involve the study of a full subcategory of p-torsion Breuil–Kisin modules which we view as extending Fontaine–Laffaille theory to filtrations of length p.

Derivatives of Frobenius and Derivatives of Hodge—Tate Weights

In this paper we study the derivatives of Frobenius and the derivatives of Hodge—Tate weights for families of Galois representations with triangulations. We generalize the Fontaine—Mazur $$\mathcal{L}$$ -invariant and use it to build a formula which is a generalization of the Colmez—Greenberg—Stevens formula. For the purpose of proving this formula we show two auxiliary results called projection vanishing property and “projection vanishing implying $$\mathcal{L}$$ -invariants” property.

Semistable deformation rings in even Hodge–Tate weights

Semistable deformation rings in even Hodge–Tate weights