Extensions of Vector Bundles on the Fargues-Fontaine Curve

Birkbeck, Christopher; Feng, Tony; Hansen, David J.; Hong, Serin; Li, Qirui; Wang, Anthony; Ye, Lynnelle

doi:10.1017/s1474748020000183

Cited by 13 publications

(12 citation statements)

References 8 publications

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“…is cartesian; then the right vertical morphism has the same degree as the left vertical morphism, which we already know has degree q n−1 . We now turn to part (2). The existence of canonical subgroups C m of level m again follows from Proposition 2.3.2.…”

Section: Recall That We Have a Decompositionmentioning

confidence: 89%

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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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Section: Recall That We Have a Decompositionmentioning

confidence: 89%

“…To see this, note that pr −1 (U ) is an inverse limit of qcqs spaces, hence qcqs and therefore a spectral space. It then follows that the quotient π −1 G H (V ) ∼ = pr −1 (U )/P(O K ) is a spectral space by [2,Lemma 3.2.3], so in particular qcqs. The intersection of two such subsets of…”

Section: Preliminariesmentioning

confidence: 98%

A quotient of the Lubin–Tate tower II

Ludwig

2020

Math. Ann.

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“…This is [BFH + 17, Lemma 3.2.3] (note that openness of the quotient map is automatic for group quotients).…”

Section: Diamonds and Cohomologymentioning

confidence: 99%

“…For our next result, we use the notion of a rank-1 point , following [BFH + 17]. Let be a locally spatial diamond (not necessarily over ).…”

Section: Diamonds and Cohomologymentioning

confidence: 99%

“…Let be a locally spatial diamond (not necessarily over ). By definition [BFH + 17, Definition 3.2.1], is a rank-1 point if it satisfies any of the following equivalent conditions: has no proper generalizations in ;there are a perfectoid field and a map with topological image ;, where ranges over the quasi-compact opens of containing .…”

Section: Diamonds and Cohomologymentioning

confidence: 99%

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Shimura varieties at level and Galois representations

Caraiani¹,

Gulotta

Hsu

et al. 2020

Compositio Math.

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We show that the compactly supported cohomology of certain $\text{U}(n,n)$- or $\text{Sp}(2n)$-Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\text{GL}_{n}/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945–1066; MR 3418533] and Newton–Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].

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Classification of quotient bundles on the Fargues–Fontaine curve

Hong¹

2023

Sel. Math. New Ser.

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Extensions of Vector Bundles on the Fargues-Fontaine Curve

Cited by 13 publications

References 8 publications

A quotient of the Lubin–Tate tower II

A quotient of the Lubin–Tate tower II

Shimura varieties at level and Galois representations

Classification of quotient bundles on the Fargues–Fontaine curve

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