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Several forms of Drinfelds lemma Kiran S. Kedlaya Department of - - PowerPoint PPT Presentation

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Several forms of Drinfelds lemma Kiran S. Kedlaya Department of Mathematics, University of California, San Diego kedlaya@ucsd.edu http://kskedlaya.org/slides/ Recent Advances in Modern -Adic Geometry virtual seminar November 12, 2020

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Several forms of Drinfeld’s lemma

Kiran S. Kedlaya

Department of Mathematics, University of California, San Diego kedlaya@ucsd.edu http://kskedlaya.org/slides/

Recent Advances in Modern π‘ž-Adic Geometry virtual seminar November 12, 2020

Supported by NSF (grant DMS-1802161) and UC San Diego (Warschawski Professorship). Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 1 / 30

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Drinfeld’s lemma for schemes

Contents

1

Drinfeld’s lemma for schemes

2

Drinfeld’s lemma for perfectoid spaces (and diamonds)

3

Drinfeld’s lemma for 𝐺-isocrystals

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 2 / 30

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Drinfeld’s lemma for schemes

References for this section

Eike Lau, On generalised 𝒠-shtukas, PhD thesis (Bonn, 2004), pdf. KSK, Sheaves, stacks, and shtukas, Arizona Winter School 2017 (pdf).

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 3 / 30

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Drinfeld’s lemma for schemes

Setup: a formal quotient by Frobenius

π‘Œ = a scheme over π”Ύπ‘ž 𝑙 = an algebraically closed fjeld of characteristic π‘ž π‘Œπ‘™ = π‘Œ Γ—π”Ύπ‘ž 𝑙 πœ’π‘™ = the pullback to π‘Œπ‘™ of the absolute Frobenius on Spec 𝑙 We will consider β€œπ‘Œπ‘™/πœ’π‘™β€ is a formal quotient: an object of some type

  • ver π‘Œπ‘™/πœ’π‘™ is an object of the same type over π‘Œπ‘™ equipped with an

isomorphism with its πœ’π‘™-pullback.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 4 / 30

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Drinfeld’s lemma for schemes

Coherent sheaves

Theorem (Drinfeld, Lau) For π‘Œ/π”Ύπ‘ž of fjnite type, the base extension functor (coherent sheaves on π‘Œ) β†’ (coherent sheaves on π‘Œπ‘™/πœ’π‘™) is an equivalence of categories and preserves cohomology. Idea of proof: reduce to π‘Œ projective, trivialize πœ’π‘™-action on 𝐼0(π‘Œ, β„°(π‘œ)).

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 5 / 30

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Drinfeld’s lemma for schemes

Finite Γ©tale covers and profjnite fundamental groups

Corollary For any π‘Œ, FEt(π‘Œ) β†’ FEt(π‘Œπ‘™/πœ’π‘™) is an equivalence. Corollary For π‘Œ connected, π‘Œπ‘™/πœ’π‘™ is connected and for any geometric point 𝑦 β†’ π‘Œπ‘™, 𝜌prof

1

(π‘Œπ‘™/πœ’π‘™, 𝑦) β‰… 𝜌prof

1

(π‘Œ, 𝑦). Warning: in general 𝜌0(π‘Œπ‘™) β‰  𝜌0(π‘Œ). For example, if π‘Œ = Spec β„“ is a geometric point, 𝜌0(π‘Œπ‘™) β‰… Μ‚ β„€ indexed by identifjcations of the copies of π”Ύπ‘ž in 𝑙 and β„“; but πœ’π‘™ acts on 𝜌0(π‘Œπ‘™) by translation by β„€.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 6 / 30

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Drinfeld’s lemma for schemes

Finite Γ©tale covers and profjnite fundamental groups

Corollary For any π‘Œ, FEt(π‘Œ) β†’ FEt(π‘Œπ‘™/πœ’π‘™) is an equivalence. Corollary For π‘Œ connected, π‘Œπ‘™/πœ’π‘™ is connected and for any geometric point 𝑦 β†’ π‘Œπ‘™, 𝜌prof

1

(π‘Œπ‘™/πœ’π‘™, 𝑦) β‰… 𝜌prof

1

(π‘Œ, 𝑦). Warning: in general 𝜌0(π‘Œπ‘™) β‰  𝜌0(π‘Œ). For example, if π‘Œ = Spec β„“ is a geometric point, 𝜌0(π‘Œπ‘™) β‰… Μ‚ β„€ indexed by identifjcations of the copies of π”Ύπ‘ž in 𝑙 and β„“; but πœ’π‘™ acts on 𝜌0(π‘Œπ‘™) by translation by β„€.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 6 / 30

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Drinfeld’s lemma for schemes

Products of two (or more) fundamental groups

Corollary For π‘Œ1, π‘Œ2 two connected qcqs π”Ύπ‘ž-schemes, put π‘Œ = π‘Œ1 Γ—π”Ύπ‘ž π‘Œ2 and let πœ’1, πœ’2 ∢ π‘Œ β†’ π‘Œ be the partial Frobenius maps. Then π‘Œ/πœ’2 is connected qcqs, and for any geometric point 𝑦 β†’ π‘Œ, 𝜌prof

1

(π‘Œ/πœ’2, 𝑦) β‰… 𝜌prof

1

(π‘Œ1, 𝑦) Γ— 𝜌prof

2

(π‘Œ2, 𝑦).

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 7 / 30

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Drinfeld’s lemma for schemes

Open subschemes and Γ©tale sheaves

Corollary For any π‘Œ, quasicompact open subschemes of π‘Œ and π‘Œπ‘™/πœ’π‘™ are the same. Corollary For π‘Œ any π”Ύπ‘ž-scheme and β„“ β‰  π‘ž prime, (lisse β„šβ„“-sheaves on π‘Œ) β†’ (lisse β„šβ„“-sheaves on π‘Œπ‘™/πœ’π‘™) (constructible β„šβ„“-sheaves on π‘Œ) β†’ (constructible β„šβ„“-sheaves on π‘Œπ‘™/πœ’π‘™) are equivalences of categories and preserve cohomology. (And so on.)

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 8 / 30

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Drinfeld’s lemma for schemes

Context: shtukas and excursion operators

These constructions are used to describe excursion operators on moduli stacks of shtukas, in order to describe the Langlands correspondence per

  • V. Lafgorgue. (See last week’s seminar!)

Similarly, other forms of Drinfeld’s lemma are needed to do likewise for local Langlands in mixed characteristic, or for π‘ž-adic coeffjcients in positive characteristic.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 9 / 30

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Drinfeld’s lemma for perfectoid spaces (and diamonds)

Contents

1

Drinfeld’s lemma for schemes

2

Drinfeld’s lemma for perfectoid spaces (and diamonds)

3

Drinfeld’s lemma for 𝐺-isocrystals

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 10 / 30

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Drinfeld’s lemma for perfectoid spaces (and diamonds)

References for this section

Carter–KSK–ZΓ‘brΓ‘di, Drinfeld’s lemma for perfectoid spaces and

  • verconvergence of multivariate (πœ’, Ξ“)-modules, arXiv:1808.03964v2

(2020). KSK, Sheaves, stacks, and shtukas, Arizona Winter School 2017 (pdf). KSK, Simple connectivity of Fargues-Fontaine curves, arXiv:1806.11528v3 (2018). Scholze–Weinstein, Berkeley Lectures on π‘ž-adic Geometry (pdf).

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 11 / 30

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Drinfeld’s lemma for perfectoid spaces (and diamonds)

Absolute products of perfectoid spaces

Let Pfd be the category of perfectoid spaces in characteristic π‘ž. This category admits absolute products. For example, if π‘Œ1 = Spa π”Ύπ‘ž((π‘’π‘žβˆ’βˆž)), π‘Œ2 = Spa π”Ύπ‘ž((π‘£π‘žβˆ’βˆž)), then π‘Œ1 Γ— π‘Œ2 = {𝑀 ∈ Spa π”Ύπ‘žπ‘’, 𝑣[π‘’βˆ’π‘žβˆž, π‘£π‘žβˆ’βˆž]∨

(𝑒,𝑣)[π‘’βˆ’1π‘£βˆ’1] ∢ 𝑀(𝑒), 𝑀(𝑣) < 1},

which is not quasicompact!

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 12 / 30

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Drinfeld’s lemma for perfectoid spaces (and diamonds)

Quotients by partial Frobenius

For π‘Œ1, π‘Œ2 ∈ Pfd, put π‘Œ = π‘Œ1 Γ— π‘Œ2. This space admits partial Frobenius operators πœ’1, πœ’2. Unlike for schemes, however, π‘Œ/πœ’2 is an

  • bject of Pfd! Moreover, if π‘Œ1, π‘Œ2 are quasicompact, then so is π‘Œ/πœ’2.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 13 / 30

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Drinfeld’s lemma for perfectoid spaces (and diamonds)

Product with a geometric point

Theorem For π‘Œ2 a geometric point, FEt(π‘Œ1) β†’ FEt(π‘Œ/πœ’2) is an equivalence. This reduces to the case where π‘Œ1 is itself a geometric point. When π‘Œ2 = Spa β„‚β™­

π‘ž, this can be proved by interpreting π‘Œ/πœ’2 in terms of the

Fargues-Fontaine curve for π‘Œ1.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 14 / 30

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Drinfeld’s lemma for perfectoid spaces (and diamonds)

Product with a geometric point

Theorem For π‘Œ2 a geometric point, FEt(π‘Œ1) β†’ FEt(π‘Œ/πœ’2) is an equivalence. For general π‘Œ2 = Spa 𝐿, we reduce from 𝐿 to 𝐿′ where 𝐿 is a completion of 𝐿′(𝑒). A direct calculation rules out abelian covers; one then uses π‘ž-adic difgerential equations to construct a β€œramifjcation fjltration” to reduce to the abelian case.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 15 / 30

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Drinfeld’s lemma for perfectoid spaces (and diamonds)

Products of two (or more) fundamental groups

Corollary For π‘Œ1, π‘Œ2 ∈ Pfd connected qcqs, π‘Œ/πœ’2 is connected. For 𝑦 β†’ π‘Œ a geometric point, 𝜌prof

1

(π‘Œ/πœ’2, 𝑦) β‰… 𝜌prof

1

(π‘Œ1, 𝑦) Γ— 𝜌prof

2

(π‘Œ2, 𝑦). A similar statement holds for diamonds. This can be used to describe π‘ž-adic representations of 𝜌prof

1

(π‘Œ1, 𝑦) Γ— 𝜌prof

2

(π‘Œ2, 𝑦) in terms of multivariate (πœ’, Ξ“)-modules (see Carter–KSK–ZΓ‘brΓ‘di).

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 16 / 30

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Drinfeld’s lemma for perfectoid spaces (and diamonds)

Products of two (or more) fundamental groups

Corollary For π‘Œ1, π‘Œ2 ∈ Pfd connected qcqs, π‘Œ/πœ’2 is connected qcqs. For 𝑦 β†’ π‘Œ a geometric point, 𝜌prof

1

(π‘Œ/πœ’2, 𝑦) β‰… 𝜌prof

1

(π‘Œ1, 𝑦) Γ— 𝜌prof

2

(π‘Œ2, 𝑦). When π‘Œ1 = π‘Œ2, are π‘ž-adic representations of 𝜌prof

1

(π‘Œ1, 𝑦) Γ— 𝜌prof

2

(π‘Œ2, 𝑦) related to vector bundles on the (relative) square of the relative Fargues–Fontaine curve? And how to classify the latter?

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 17 / 30

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Drinfeld’s lemma for perfectoid spaces (and diamonds)

More questions

Is there a version for constructible sheaves? (See Fargues–Scholze?) Does this build towards an β€œβ„“ = π‘žβ€ Langlands correspondence for β„šπ‘ž?

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 18 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

Contents

1

Drinfeld’s lemma for schemes

2

Drinfeld’s lemma for perfectoid spaces (and diamonds)

3

Drinfeld’s lemma for 𝐺-isocrystals

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 19 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

References for this section

Work in progress, but see... Abe, Langlands correspondence for isocrystals and the existence of crystalline companions for curves, J. Amer. Math. Soc. 31 (2019). KSK, Notes on isocrystals, arXiv:1606.01321v5 (2018). KSK, Γ‰tale and crystalline companions, I, arXiv:1811.00204v3 (2020). KSK, Γ‰tale and crystalline companions, II, arXiv:2008.13053v1 (2020).

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 20 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

Context: the Langlands correspondence again

Let π‘Œ be a curve over a fjnite fjeld of characteristic π‘ž. For a given reductive group 𝐻, the Langlands correspondence for 𝐻 is supposed to involve not just β„šβ„“-sheaves for primes β„“ β‰  π‘ž, but also some β€œcrystalline” replacement for β„“ = π‘ž. This happens in a β€œde Rham-style” Weil cohomology. The analogue of lisse sheaves are overconvergent 𝐺-isocrystals. (Today we’ll talk about a simpler construction: convergent 𝐺-isocrystals.) The analogue of constructive sheaves are arithmetic 𝒠-modules. Using these, Abe handles the case 𝐻 = GL(π‘œ) after L. Lafgorgue. (I won’t defjne these today.) We need a form of Drinfeld’s lemma to follow the approach of V. Lafgorgue for more general 𝐻. What we get today won’t be enough (because it won’t include arithmetic 𝒠-modules), but it’s progress...

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 21 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

Convergent 𝐺-isocrystals

Let π‘Œ be a smooth affjne scheme over a perfect fjeld 𝑙 of characteristic π‘ž. Fix a formal scheme 𝑄 smooth over 𝑋(𝑙) with 𝑄𝑙 β‰… π‘Œ and a lift 𝜏 of πœ’π‘Œ to 𝑄. A convergent 𝐺-isocrystal on π‘Œ is a fjnite projective module over Ξ“(𝑄, 𝒫)[π‘žβˆ’1] equipped with an integrable 𝑋(𝑙)[π‘žβˆ’1]-linear connection and a horizontal isomorphism with its 𝜏-pullback. The resulting β„šπ‘ž-linear tensor category F-Isoc(π‘Œ) does not depend on 𝑄

  • r 𝜏, and extends by glueing to general smooth π‘Œ. We refer to the

𝜏-action also as the πœ’π‘Œ-action.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 22 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

Newton polygons

For β„° ∈ F-Isoc(π‘Œ) and 𝑦 β†’ π‘Œ a geometric point, we may pull back β„° to F-Isoc(𝑦) and apply the Dieudonné–Manin classifjcation: that pullback decomposes as β¨π‘’βˆˆβ„š ℰ𝑒 where for 𝑒 = 𝑠

𝑑 ∈ β„š in lowest terms, ℰ𝑒 admits a

basis killed by πœ’π‘’

π‘Œ βˆ’ π‘žπ‘ .

Theorem (Grothendieck–Katz) The Newton polygon function on |π‘Œ| is upper semicontinuous.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 23 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

Slope fjltrations

Theorem (Katz) If the Newton polygon is constant, then β„° admits a fjltration 0 = β„°0 βŠ‚ β‹― βŠ‚ β„°π‘š = β„° in which ℰ𝑗/β„°π‘—βˆ’1 has all Newton slopes equal to πœˆπ‘—, and 𝜈1 < β‹― < πœˆπ‘š. Theorem If β„° has all Newton slopes equal to 0, then β„°πœ’π‘Œ is a lisse β„šπ‘ž-sheaf on π‘Œ.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 24 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

Convergent Ξ¦-isocrystals

For 𝑗 = 1, 2, let π‘Œπ‘— be a smooth affjne scheme over a perfect fjeld 𝑙𝑗 of characteristic π‘ž. Fix a formal scheme 𝑄𝑗 smooth over 𝑋(𝑙𝑗) with (𝑄𝑗)𝑙𝑗 β‰… π‘Œπ‘— and a lift πœπ‘— of πœ’π‘Œπ‘— to 𝑄𝑗. A convergent Ξ¦-isocrystal on π‘Œ = π‘Œ1 Γ—π”Ύπ‘ž π‘Œ2 is a fjnite projective module over Ξ“(𝑄1 Γ—β„€π‘ž 𝑄2, 𝒫)[π‘žβˆ’1] equipped with an integrable 𝑋(𝑙1 βŠ—π”Ύπ‘ž 𝑙2)[π‘žβˆ’1]-linear connection and commuting horizontal isomorphisms with its πœπ‘—-pullbacks. Let Ξ¦ Isoc(π‘Œ) be the resulting category.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 25 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

Total Newton polygons

We map Ξ¦ Isoc(π‘Œ) to F-Isoc(π‘Œ) by keeping the action of πœ’ = πœ’1 ∘ πœ’2. Theorem Suppose that π‘Œ2 is a geometric point. For β„° ∈ Ξ¦ Isoc(π‘Œ), the total Newton polygon of β„° (i.e., the Newton polygon of the image object in F-Isoc(π‘Œ)) factors through |π‘Œ1|. Idea of proof: by Grothendieck–Katz, we may apply Drinfeld’s lemma to the total Newton polygon stratifjcation.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 26 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

Relative Dieudonné–Manin

Theorem Suppose that π‘Œ2 is a geometric point. Then any β„° ∈ Ξ¦ Isoc(π‘Œ) decomposes as β¨π‘’βˆˆβ„š ℰ𝑒 where for 𝑒 = 𝑠

𝑑 ∈ β„š in lowest terms,

β„°πœ’π‘’

2βˆ’π‘žπ‘ 

𝑒

∈ F-Isoc(π‘Œ1). Idea of proof: fjrst do the case where the total Newton polygon is

  • constant. Then use:

Theorem For 𝑉𝑗 βŠ† π‘Œπ‘— open dense and 𝑉 = 𝑉1 Γ— 𝑉2, Ξ¦ Isoc(π‘Œ) β†’ Ξ¦ Isoc(𝑉) is fully faithful.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 27 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

Products of two (or more) schemes

Theorem (not just a corollary!) Any irreducible β„° ∈ Ξ¦ Isoc(π‘Œ) is a subobject of β„°1 ⊠ β„°2 for some ℰ𝑗 ∈ F-Isoc(π‘Œπ‘—). In general, we cannot write β„° = β„°1 ⊠ β„°2; think of irreducible representations of product groups. Again, we fjrst do the case where the total Newton polygon is constant, then use the full faithfulness of restriction.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 28 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

Footnotes

Similar statements (defjnitely!) apply to overconvergent 𝐺-isocrystals, and to logarithmic convergent 𝐺-isocrystals. One can (probably!) relax the smoothness hypothesis on π‘Œπ‘— by some descent arguments. This should even allow π‘Œπ‘— to be an algebraic stack (crucial for moduli of shtukas). One can (hopefully?) also consider constructible objects. One can (maybe?) give an analogue of the isomorphism 𝜌prof

1

(π‘Œ/πœ’2, 𝑦) β‰… 𝜌prof

1

(π‘Œ1, 𝑦) Γ— 𝜌prof

2

(π‘Œ2, 𝑦) in terms of Tannakian fundamental groups. One can (???) consider isocrystals without Frobenius structure.

Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 29 / 30

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Drinfeld’s lemma for 𝐺-isocrystals

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Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 30 / 30