On mod  local-global compatibility for in the ordinary case

Herzig, Florian; Le, Daniel; Morra, Stefano

doi:10.1112/s0010437x17007357

Cited by 18 publications

(67 citation statements)

References 46 publications

(214 reference statements)

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“…From [, Lemma A.3.6] (which builds on the classical result of Fontaine) we have an anti‐equivalence

\begin{matrix} true \\ right (() φ, F \otimes_{F_{p}} k false((\underset{̲}{π}) false)) - {Mod}_{dd} & \tilde{⟶} & left {Rep}_{F} (G_{{false(K_{0} false)}_{\infty}}) \\ right D & \mapsto & left Hom (() D, k {(false(\underset{̲}{π} false))}^{sep}) . \end{matrix}

…”

Section: Integral P‐adic Hodge Theory I: Preliminariesmentioning

confidence: 93%

“…None of the results in this section is new: they build on classical work of Breuil and Caruso (cf. for instance ) and we refer the reader to [, Section 2] for a concise reference.…”

Section: Integral P‐adic Hodge Theory I: Preliminariesmentioning

confidence: 99%

“…[, Definition 3.1.1]), where

{Rep}_{E}^{K -st} false(G_{{boldQ}_{p}} false)

denotes the category of

p

‐adic Galois representations of

G_{{boldQ}_{p}}

with

E

‐coefficients becoming semi‐stable over

K

. These functors depend on the choice of uniformizers and establish equivalence of categories (we choose the uniformizer

p

as in for consistency).…”

Section: Integral P‐adic Hodge Theory I: Preliminariesmentioning

confidence: 99%

“…The aim of this paper is to obtain evidence towards Serre‐type conjectures for ordinary Galois representations in semisimple rank two, developing the techniques introduced in where the maximally non‐split case is considered. The submodule structure of

\bar{r} |_{G_{F_{w}}}

(for

w | p

) plays now a crucial role (as in ) and we define, for all possible configurations of the submodule structure in

\bar{r} |_{G_{F_{w}}}

, a set of weights

W^{?} false(\bar{r} false)

in which

\bar{r}

will be modular.…”

Section: Introductionmentioning

confidence: 99%

“…The case that

\bar{r} |_{G_{F_{w}}}

is maximally non‐split (that is, the Loewy length of

\bar{r} |_{G_{F_{w}}}

is 3) is treated in .…”

Section: Introductionmentioning

confidence: 99%

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Serre weights for three-dimensional ordinary Galois representations

Morra

Park

2017

J. London Math. Soc.

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Let F/Q be a CM field where p splits completely and let r¯:Galfalse(Q¯/Ffalse)→ GL 3false(boldF¯pfalse) be a Galois representation whose restriction to Gal(Q¯p/Fw) is ordinary and strongly generic for all places w above p. In this paper, we specify the set of Serre weights in which r¯ can be modular. To this aim, we develop a technique in integral p‐adic Hodge theory to describe extensions of rank‐one Breuil modules.

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“…From [, Lemma A.3.6] (which builds on the classical result of Fontaine) we have an anti‐equivalence

\begin{matrix} true \\ right (() φ, F \otimes_{F_{p}} k false((\underset{̲}{π}) false)) - {Mod}_{dd} & \tilde{⟶} & left {Rep}_{F} (G_{{false(K_{0} false)}_{\infty}}) \\ right D & \mapsto & left Hom (() D, k {(false(\underset{̲}{π} false))}^{sep}) . \end{matrix}

…”

Section: Integral P‐adic Hodge Theory I: Preliminariesmentioning

confidence: 93%

“…None of the results in this section is new: they build on classical work of Breuil and Caruso (cf. for instance ) and we refer the reader to [, Section 2] for a concise reference.…”

Section: Integral P‐adic Hodge Theory I: Preliminariesmentioning

confidence: 99%

“…[, Definition 3.1.1]), where

{Rep}_{E}^{K -st} false(G_{{boldQ}_{p}} false)

denotes the category of

p

‐adic Galois representations of

G_{{boldQ}_{p}}

with

E

‐coefficients becoming semi‐stable over

K

. These functors depend on the choice of uniformizers and establish equivalence of categories (we choose the uniformizer

p

as in for consistency).…”

Section: Integral P‐adic Hodge Theory I: Preliminariesmentioning

confidence: 99%

\bar{r} |_{G_{F_{w}}}

(for

w | p

) plays now a crucial role (as in ) and we define, for all possible configurations of the submodule structure in

\bar{r} |_{G_{F_{w}}}

, a set of weights

W^{?} false(\bar{r} false)

in which

\bar{r}

will be modular.…”

Section: Introductionmentioning

confidence: 99%

“…The case that

\bar{r} |_{G_{F_{w}}}

is maximally non‐split (that is, the Loewy length of

\bar{r} |_{G_{F_{w}}}

is 3) is treated in .…”

Section: Introductionmentioning

confidence: 99%

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Serre weights for three-dimensional ordinary Galois representations

Morra

Park

2017

J. London Math. Soc.

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Can We Dream of a 1-Adic Langlands Correspondence?

Caruso

Dávid

Mézard

2022

Lecture Notes in Mathematics

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Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows

Hung

Левин

et al. 2017

Invent. math.

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We prove the weight part of Serre's conjecture in generic situations for forms of U (3) which are compact at infinity and split at places dividing p as conjectured by [Her09]. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights (2, 1, 0) for K/Qp unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mézard conjectures hold for these deformation rings. M denote the O-flat and reduced quotient of R τ,β M such that Spec R τ,β,∇ M

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On mod local-global compatibility for in the ordinary case

Cited by 18 publications

References 46 publications

Serre weights for three-dimensional ordinary Galois representations

Serre weights for three-dimensional ordinary Galois representations

Can We Dream of a 1-Adic Langlands Correspondence?

Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows

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