A Quotient of the Lubin–tate Tower

Ludwig, Judith

doi:10.1017/fms.2017.15

Cited by 12 publications

(29 citation statements)

References 28 publications

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“…This theorem generalises [27,Theorem 4.6], which is the special case when n = 2, K = Q p and σ is a character.…”

Section: Introductionsupporting

confidence: 59%

“…5, which proves Theorem B. The calculations follow the same path as Section 4 of [27], the idea being that pushforward along the map π G H : M P(K ) → P n−1 is a geometric realisation of the parabolic induction functor, so étale cohomology of F π on P n−1 is equal to étale cohomology of an analogously defined sheaf F σ on M P(K ) . For the reader familiar with [27], we mention that our argument deviates somewhat from that of [27].…”

Section: Introductionmentioning

confidence: 62%

“…Here we take the quotient in Huber's category V of locally v-ringed spaces, as in [27]. The construction of the perfectoid structure on M P(K ) follows the strategy via globalisation from [27], where the quotient was constructed in the case when n = 2 and K = Q p . In that case, modular curves were used to globalise and one could rely on the perfectoidness results of [32].…”

Section: Introductionmentioning

confidence: 99%

“…Let us now describe the strategy of [27] and this paper in slightly more detail; the reader may also consult the introduction to [27]. The space M 1 has a GL n (O K )-…”

Section: Introductionmentioning

confidence: 99%

“…into pairwise isomorphic spaces (coming from the decomposition of the Lubin-Tate space at level 0 into connected components). As in [27] we reduce the construction of M 1 /P(K ) to the construction of M (0) 1 /P(O K ) using the geometry of the Gross-Hopkins period map. We can realize M , which is a subspace of X 1 defined in terms of the Hodge-Tate period map.…”

Section: Introductionmentioning

confidence: 99%

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A quotient of the Lubin–Tate tower II

Ludwig

2020

Math. Ann.

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In this article we construct the quotient $$\mathcal {M}_\mathbf {1}/P(K)$$ M 1 / P ( K ) of the infinite-level Lubin–Tate space $$\mathcal {M}_\mathbf {1}$$ M 1 by the parabolic subgroup $$P(K) \subset \mathrm {GL} _n(K)$$ P ( K ) ⊂ GL n ( K ) of block form $$(n-1,1)$$ ( n - 1 , 1 ) as a perfectoid space, generalizing the results of Ludwig (Forum Math Sigma 5:e17, 41, 2017) to arbitrary n and $$K/{\mathbb {Q}} _p$$ K / Q p finite. For this we prove some perfectoidness results for certain Harris–Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze’s candidate for the mod p Jacquet–Langlands and mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of perfectoidness of $$\mathcal {M}_\mathbf {1}/P(K)$$ M 1 / P ( K ) when $$n=2$$ n = 2 , and shows that $$\mathcal {M}_\mathbf {1}/Q(K)$$ M 1 / Q ( K ) is not perfectoid for maximal parabolics Q not conjugate to P.

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“…This theorem generalises [27,Theorem 4.6], which is the special case when n = 2, K = Q p and σ is a character.…”

Section: Introductionsupporting

confidence: 59%

Section: Introductionmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

“…Let us now describe the strategy of [27] and this paper in slightly more detail; the reader may also consult the introduction to [27]. The space M 1 has a GL n (O K )-…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A quotient of the Lubin–Tate tower II

Ludwig

2020

Math. Ann.

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Factorisation de la cohomologie étale p-adique de la tour de Drinfeld

Colmez

Dospinescu

Nizioł

2023

Forum of Mathematics, Pi

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Résumé For a finite extension F of ${\mathbf Q}_p$ , Drinfeld defined a tower of coverings of (the Drinfeld half-plane). For $F = {\mathbf Q}_p$ , we describe a decomposition of the p-adic geometric étale cohomology of this tower analogous to Emerton’s decomposition of completed cohomology of the tower of modular curves. A crucial ingredient is a finiteness theorem for the arithmetic étale cohomology modulo p whose proof uses Scholze’s functor, global ingredients, and a computation of nearby cycles which makes it possible to prove that this cohomology has finite presentation. This last result holds for all F; for $F\neq {\mathbf Q}_p$ , it implies that the representations of $\mathrm{GL}_2(F)$ obtained from the cohomology of the Drinfeld tower are not admissible contrary to the case $F = {\mathbf Q}_p$ .

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On Some Consequences of a Theorem of J. Ludwig

Paškūnas

2021

J. Inst. Math. Jussieu

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We prove some qualitative results about the p-adic Jacquet–Langlands correspondence defined by Scholze, in the $\operatorname {\mathrm {GL}}_2(\mathbb{Q}_p )$ residually reducible case, using a vanishing theorem proved by Judith Ludwig. In particular, we show that in the cases under consideration, the global p-adic Jacquet–Langlands correspondence can also deal with automorphic forms with principal series representations at p in a nontrivial way, unlike its classical counterpart.

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A Quotient of the Lubin–tate Tower

Cited by 12 publications

References 28 publications

A quotient of the Lubin–Tate tower II

A quotient of the Lubin–Tate tower II

Factorisation de la cohomologie étale p-adique de la tour de Drinfeld

On Some Consequences of a Theorem of J. Ludwig

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