Generic local deformation rings when

Shotton, Jack

doi:10.1112/s0010437x22007461

Cited by 3 publications

(4 citation statements)

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“…[q] e . We therefore recover the geometric results of [Sho22] in this case (and there is no real need to assume that Σ is unipotent here). 5.2.…”

Section: (Non-)examplessupporting

confidence: 66%

“…However, one can still hope for a nice description at an open dense set of points. For G = GL n , in the tame case, we did this in [Sho22], finding a local description of X G around a dense subset of its fibre at each prime l. This description turns out to be related to the endomorphism algebra of the (integral) Gelfand-Graev representation, and we applied this to the l = p "Breuil-Mézard" conjecture.…”

Section: Introductionmentioning

confidence: 99%

“…where M runs over conjugacy classes of standard Levi subgroups, is fibrewise dense in X G (see [Sho22]). In general this is not true; there are two issues coming from the disconnectedness of component groups and the existence of non-regular distinguished unipotent conjugacy classes.…”

mentioning

confidence: 99%

“…In [Sho22] we were able to use this and the desciption of the coordinate ring of (T / /W ) [q] as the endomorphism ring of a Gelfand-Graev representation 4 to give a local proof of the author's "l = p Breuil-Mézard conjecture" in the tame case. It should be possible to do something similar for unramified groups, but only for modular L-parameters which lie only on components in the image of (1).…”

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confidence: 99%

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Irreducible components of the moduli space of Langlands parameters

Shotton¹

2023

Preprint

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Let F/Qp be finite and let X G be the moduli space of Langlands parameters valued in G, in characteristic distinct from p. First, we determine the irreducible components of X G . Then, we determine the local structure around tamely ramified points for which the image of 'tame inertia' is regular. This local structure is related to the endomorphism rings of Gelfand-Graev representations, by work of Li. Lastly, we determine an open dense set in X M , when M is a Levi subgroup of G, such that the natural map of moduli stacks [X M /M ] → [X G /G] is smooth on this set.

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“…[q] e . We therefore recover the geometric results of [Sho22] in this case (and there is no real need to assume that Σ is unipotent here). 5.2.…”

Section: (Non-)examplessupporting

confidence: 66%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Irreducible components of the moduli space of Langlands parameters

Shotton¹

2023

Preprint

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Lifting G-Valued Galois Representations when $\ell \neq p$

Booher,

Cotner,

Tang

2024

Forum of Mathematics, Sigma

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In this paper, we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$ . In particular, under certain technical conditions applicable to any root datum, we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\operatorname {\mathrm {GL}}_n$ -valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm {G}_2$ -valued representations.

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Irreducible Components of the Moduli Space of Langlands Parameters

Shotton

2023

International Mathematics Research Notices

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Let $F/\mathbb{Q}_{p}$ be finite, $G$ be an $L$-group, and let $\mathfrak{X}_{G}$ be the moduli space of Langlands parameters $W_{F} \to G$, in characteristic distinct from $p$. First, we determine the irreducible components of ${\mathfrak{X}}_{G}$. Then, we determine the local structure around tamely ramified points for which the image of inertia is regular. This local structure is related to the endomorphism rings of Gelfand–Graev representations, by work of Li. Lastly, we determine an open dense set in ${\mathfrak{X}}_{M}$, when $M$ is a Levi subgroup of $G$, such that the natural map of moduli stacks $[{\mathfrak{X}}_{M}/M^{\circ }] \to [{\mathfrak{X}}_{G}/G^{\circ }]$ is smooth on this set.

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Generic local deformation rings when

Cited by 3 publications

References 18 publications

Irreducible components of the moduli space of Langlands parameters

Irreducible components of the moduli space of Langlands parameters

Lifting G-Valued Galois Representations when $\ell \neq p$

Irreducible Components of the Moduli Space of Langlands Parameters

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