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Main Theorem

Chapter XII: Cohomological correspondences between Shimura varieties

12.2 Main Theorem

Let(𝐺1, 𝑋1) and(𝐺2, 𝑋2)be two Hodge type Shimura data (cf. [Mil05]) equipped with an isomorphism πœƒ : 𝐺1,

A𝑓 ' 𝐺2,

A𝑓. Let {πœ‡π‘–} denote the conjugacy class of Hodge cocharacters determined by 𝑋𝑖 and consider them as dominant characters of ˆ𝑇. In particular, πœ‡1 and πœ‡2 are both minuscule. Then [XZ17, Corollary 2.1.5]

implies that there is a canonical inner twistΞ¨R:𝐺1β†’ 𝐺2overC. Recall notations in Β§1.3. We defineπœ‡π‘–,ad to be the composition ofπœ‡π‘–with the quotient𝐺 β†’ 𝐺adand consider it as a character of ˆ𝑇sc. We assume that

πœ‡1,ad |

𝑍(𝐺ˆ

Ξ“Q

sc )= πœ‡2,ad |

𝑍(𝐺ˆ

Ξ“Q sc ) .

It follows from [XZ17, Corollary 2.1.6] thatΨRcomes from a unique global inner twistΨ : 𝐺

1 Β―Q β†’ 𝐺

2 Β―Q such thatΞ¨ = Int(β„Ž) β—¦πœƒ, for some πœƒ : 𝐺1,

A𝑓 ' 𝐺2,

A𝑓 and β„ŽβˆˆπΊ2,ad(A¯𝑓).

We assume that𝐾𝑖 βŠ‚ 𝐺(A𝑓) to be sufficiently small such thatπœƒ 𝐾1=𝐾2. Choose a prime 𝑝such that𝐾1, 𝑝 (and therefore 𝐾2, 𝑝) is hyperspecial. Let𝐺𝑖 be the integral model of𝐺𝑖,

Q𝑝overZ𝑝determined by𝐾𝑖, 𝑝. Then𝐺1 '𝐺2, and we can thus identify their Langlands dual groups (𝐺 ,Λ† 𝐡,Λ† 𝑇ˆ). Choose an isomorphism πœ„ : C ' Q¯𝑝. Let 𝜈 | 𝑝 be a place of the compositum of reflex fields of (𝐺𝑖, 𝑋𝑖) determined by our choice of isomorphismπœ„. We write Shπœ‡π‘–for the mod 𝑝fibre of the canonical integral model of Sh𝐾𝑖(𝐺𝑖, 𝑋𝑖) base change toπ‘˜πœˆ. We make the following assumption

πœ‡1 |

𝑍(𝐺ˆ

Ξ“Q𝑝)= πœ‡2 |

𝑍(𝐺ˆ

Ξ“Q𝑝) . (12.2)

The assumption guarantees the existence of the ind-scheme Shπœ‡1|πœ‡2 which fits into the following commutative diagram

Shπœ‡1,𝐾1 Shπœ‡1|πœ‡2 Shπœ‡2,𝐾2

Shtlocπœ‡1 Shtlocπœ‡

1|πœ‡2 Shtlocπœ‡2

β†βˆ’ β„Žπœ‡

1

loc𝑝

βˆ’

β†’ β„Žπœ‡

2

loc𝑝

β†βˆ’ β„Žloc

πœ‡1

βˆ’

β†’ β„Žloc

πœ‡2

, (12.3)

and makes both squares to be Cartesian.

Remark 12.2.1. In the case that(𝐺1, 𝑋1)= (𝐺2, 𝑋2),Shπœ‡1|πœ‡2is the perfection of the mod p fibre of a natural integral model of some Hecke correspondence. If(𝐺1, 𝑋1) β‰  (𝐺2, 𝑋2), thenShπœ‡1|πœ‡2can be regarded as β€œexotic Hecke correspondences” between mod p fibres of different Shimura varieties. We refer to [XZ17, Β§7.3.3, Β§7.3.4] for a detailed discussion.

Let (𝐺𝑖, 𝑋𝑖) 𝑖 = 1,2,3 be three Hodge type Shimura data, together with the iso- morphismsπœƒπ‘–, 𝑗 :𝐺𝑖,

A𝑓 '𝐺𝑗 ,

A𝑓 satisfying the natural cocycle condition. Choose a common level 𝐾 using the isomorphismπœƒπ‘–, 𝑗. Let 𝑝 be an unramified prime, such that the assumption(12.2)holds for each pair of( (𝐺𝑖, 𝑋𝑖),(𝐺𝑗, 𝑋𝑗)). Choose a half Tate twistQβ„“(1/2).

Let 𝑉𝑖 := π‘‰πœ‡

𝑖 be the highest weight representation of ˆ𝐺

Qβ„“ of highest weight πœ‡π‘–. Write 𝑉e𝑖 ∈ Coh𝐺ˆQβ„“(𝐺ˆ

Qβ„“πœŽ) for the vector bundle associated to 𝑉𝑖 analogous to

Β§11.4. Recall fromΒ§12.1 that, to each representationπ‘Š of𝐺

Qβ„“, we can attach the Γ©tale local systemLπ‘Š ,Qβ„“ on Shπœ‡π‘–. Let𝑑𝑖 =h2𝜌, πœ‡π‘–i=dim Sh𝐾(𝐺𝑖, 𝑋𝑖). Denote the global section of the structure sheaf on the quotient stack [𝐺 πœŽΛ† /𝐺ˆ] by J, and the prime-to-𝑝 Hecke algebra byH𝑝.

Theorem 12.2.2. There exists a map Spc : Hom

Coh𝐺ˆQβ„“(𝐺ˆ

Qβ„“πœŽ)(𝑉e1,𝑉e2) β†’HomHπ‘βŠ—J(Hβˆ—π‘(Shπœ‡1,Lπ‘Š ,Qβ„“h𝑑1i),Hβˆ—π‘(Shπœ‡2,Lπ‘Š ,Qβ„“h𝑑2i), (12.4)

which is compatible with compositions on the source and target.

Proof. Choose a latticeΛ𝑖 ∈RepZ

β„“(𝐺ˆ

Zβ„“)in𝑉𝑖. We denote byΞ›e𝑖 ∈Coh𝐺ˆZβ„“(𝐺ˆ

Zβ„“πœŽ) the coherent sheaf which corresponds toΛ𝑖 as in Β§11.1. Then

HomCoh𝐺ˆQβ„“(𝐺ˆ

Qβ„“πœŽ)(𝑉e1,𝑉e2) 'Hom𝐺ˆ

Qβ„“

(𝑉1, 𝑉2βŠ—Qβ„“[𝐺ˆ]) (12.5) 'Hom𝐺ˆ

Qβ„“

(Ξ›1βŠ—Z

β„“Qβ„“,(Ξ›2βŠ—Z

β„“ Zβ„“[𝐺ˆ]) βŠ—Z

β„“Qβ„“) 'Hom𝐺ˆ

Zβ„“

(Ξ›1,Ξ›2βŠ—Zβ„“ Zβ„“[𝐺ˆ]) βŠ—Zβ„“ Qβ„“ 'Hom

Coh𝐺ˆZβ„“(𝐺ˆ

Zβ„“πœŽ)(fΞ›1,Ξ›f2) βŠ—Zβ„“Qβ„“. By Theorem 11.2.1, we get a map

SΞ›1,Ξ›2 : Hom

Coh𝐺ˆZβ„“(𝐺ˆ

Zβ„“πœŽ)(Ξ›f1,Ξ›f2) β†’CorrShtloc(𝑆(Ξ›f1), 𝑆(Ξ›f2)). (12.6) Combining(12.5)with(12.6), we get the following map

HomCoh𝐺ˆQβ„“(𝐺ˆ

Qβ„“πœŽ)(𝑉e1,𝑉e2) β†’CorrShtloc(𝑆(Ξ›f1), 𝑆(Ξ›f2)) βŠ—Zβ„“ Qβ„“. (12.7) Choose a dominant coweight𝜈and a quadruple (π‘š1, 𝑛1, π‘š2, 𝑛2)that is (πœ‡1+𝜈, 𝜈)- acceptable and(πœ‡2+𝜈, 𝜈)-acceptable. We have the following diagram

Shπœ‡1 Shπœˆπœ‡

1|πœ‡2 Shπœ‡2

Shtlocπœ‡1 Sht𝜈,loc

πœ‡1|πœ‡2 Shtlocπœ‡2

Shtloc(πœ‡1 π‘š1,𝑛1) Sht𝜈,πœ‡loc(π‘š1,𝑛1)

1|πœ‡2 Shtloc(πœ‡2 π‘š2,𝑛2)

β†βˆ’ β„Žπœ‡

1

loc𝑝 locπœˆπ‘

βˆ’

β†’ β„Žπœ‡

2

loc𝑝

β†βˆ’ β„Žloc

πœ‡1

resπ‘š1, 𝑛1

βˆ’

β†’ β„Žloc

πœ‡2

resπœˆπ‘š1, 𝑛1 resπ‘š2, 𝑛2

β†βˆ’ β„Žloc(π‘š1

, 𝑛1) πœ‡1

βˆ’

β†’ β„Žloc(π‘š2

, 𝑛2) πœ‡2

, (12.8)

where

β€’ all squares are commutative (discussions on diagram (10.5) and diagram (12.3),

β€’ except for the square at the down right corner, and the other three squares are Cartesian (discussions on diagram(12.3)and diagram (12.5),

β€’ the morphismβ†βˆ’ β„Žπœ‡

1 is perfectly proper ([XZ17, Lemma 5.2.12]),

β€’ the morphisms loc𝑝(π‘šπ‘–, 𝑛𝑖) are perfectly smooth (Proposition 12.1.1).

Then the morphism locπœˆπ‘(π‘š1, 𝑛1) := resπœˆπ‘š1,𝑛1 β—¦locπœˆπ‘ is also perfectly proper. Thus we can pullback the cohomological correspondences (cf. [XZ17, A.2.11)]) on the right hand side of (12.6) along locπœˆπ‘(π‘š1, 𝑛1)to obtain a map

locπœˆπ‘(π‘š1, 𝑛1)β˜…: CorrShtloc(𝑆(Ξ›f1), 𝑆(Ξ›f2)) β†’ CorrSh𝜈

πœ‡|πœ‡(loc𝑝(π‘š1, 𝑛1)β˜…π‘†(fΞ›1),loc𝑝(π‘š2, 𝑛2)β˜…(𝑆(Ξ›f2)). Note that πœ‡π‘– are minuscule, then theβ˜…-pullback of𝑆(Ξ›e𝑖)along loc𝑝(π‘šπ‘–, 𝑛𝑖)equals

Zβ„“h𝑑𝑖i. Next, we construct a natural map β„­π‘Š : CorrSh𝜈

πœ‡1|πœ‡ 2

(Shπœ‡1,Zβ„“h𝑑1i),(Shπœ‡2,Zβ„“h𝑑2i)

β†’CorrSh𝜈

πœ‡1|πœ‡ 2

(Shπœ‡1,Lπ‘Š ,Zβ„“h𝑑1i),(Shπœ‡2,Lπ‘Š ,Zβ„“h𝑑2i) . (12.9)

For each𝑛 ∈Z+, we note that there exists an ind-scheme Sh(𝑛)

πœ‡1|πœ‡2 which fits into the following commutative diagram such that both squares are Cartesian

Shπœ‡1,𝐾(𝑛)

β„“ 𝐾ℓ Sh𝜈,(𝑛)

πœ‡1|πœ‡2 Sh

πœ‡2,𝐾(𝑛)

β„“ 𝐾ℓ

Shπœ‡1 Shπœˆπœ‡

1|πœ‡2 Shπœ‡2.

β†βˆ’ β„Žπœ‡(𝑛)

1

𝑝𝑛

1

βˆ’

β†’ β„Ž(𝑛)πœ‡

2

𝑝𝑛 𝑝𝑛

2

β†βˆ’ β„Žπœ‡

1

βˆ’

β†’ β„Žπœ‡

2

Here the three vertical maps are the natural quotients by the finite group𝐾ℓ/𝐾𝑛

β„“ and are thus Γ©tale.

Let (𝑓𝑛)𝑛 : (β†βˆ’ β„Žπœ‡

1)βˆ—(Z/ℓ𝑛Zh𝑑1i)𝑛 β†’ (β†’βˆ’ β„Žπœ‡

2)!(Z/ℓ𝑛Zh𝑑2i)𝑛 be a cohomological cor- respondence in CorrSh𝜈

πœ‡1|πœ‡2

(Shπœ‡1,Zβ„“h𝑑1i),(Shπœ‡2,Zβ„“h𝑑2i)

. For each 𝑛 ∈ Z+, the shifted pullback (cf. [XZ17, A.2.12]) of 𝑓𝑛gives rise to a cohomological correspon- dence

˜ 𝑓𝑛: (β†βˆ’

β„Ž(

𝑛)

πœ‡1 )βˆ—(Z/ℓ𝑛Zh𝑑1i) β†’ (β†’βˆ’ β„Ž(

𝑛)

πœ‡2)!(Z/ℓ𝑛Zh𝑑2i)

in CorrSh𝜈 ,(𝑛)

πœ‡1|πœ‡2

(Shπœ‡

1,𝐾(

𝑛) β„“

𝐾ℓ

,Z/ℓ𝑛Zh𝑑1i),(Shπœ‡

2,𝐾(

𝑛) β„“

𝐾ℓ

,Z/ℓ𝑛Zh𝑑2i)

. For any repre- sentationπ‘Šof𝐺

Qβ„“, recall theZ/ℓ𝑛ZmoduleΞ›π‘Š ,β„“/β„“π‘›Ξ›π‘Š ,β„“constructed in Β§12.2. The cohomological correspondence Λœπ‘“π‘›gives rise to a cohomological correspondence

˜

π‘”π‘›βˆˆCorr

Sh𝜈 ,(𝑛)πœ‡

1|πœ‡2

(Shπœ‡1,𝐾(𝑛)

β„“

𝐾ℓ Γ—Ξ›π‘Š ,β„“/β„“π‘›Ξ›π‘Š ,β„“h𝑑1i,Sh

πœ‡2,𝐾(𝑛)

β„“

𝐾ℓ Γ—Ξ›π‘Š ,β„“/β„“π‘›Ξ›π‘Š ,β„“h𝑑2i). In addition, the cohomological correspondence Λœπ‘“π‘› is𝐾ℓ/𝐾(𝑛)

β„“ -equivariant. Then it follows that the cohomological correspondence Λœπ‘”π‘› is also𝐾ℓ/𝐾(

𝑛)

β„“ -equivariant and descends to a cohomological correspondence

𝑔𝑛 ∈CorrSh𝜈

πœ‡1|πœ‡2

( (Shπœ‡1,Lπ‘Š ,β„“,𝑛h𝑑1i),(Shπœ‡2,Lπ‘Š ,β„“,𝑛h𝑑2i)). Definingβ„­π‘Š( (𝑓𝑛)𝑛) :=(𝑔𝑛)𝑛completes the construction ofβ„­π‘Š. Compose the maps we previously construct,

CorrSht𝜈 ,πœ‡loc(π‘š1, 𝑛1)

1|πœ‡2

(Shtlocπœ‡1(π‘š1,𝑛1), 𝑆(fΞ›1)loc(π‘š1,𝑛1)),(Shtlocπœ‡1(π‘š2,𝑛2), 𝑆(Ξ›f2)loc(π‘š2,𝑛2)) (12.10)

locπœˆπ‘(π‘š1,𝑛1)β˜…

βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’β†’CorrSh𝜈

πœ‡1|πœ‡2

( (Shπœ‡1,Zβ„“h𝑑1i),(Shπœ‡2,Zβ„“h𝑑2i))

β„­π‘Š

βˆ’βˆ’βˆ’β†’CorrSh𝜈

πœ‡1|πœ‡ 2

( (Shπœ‡1,Lπ‘Š ,Zβ„“h𝑑1i),(Shπœ‡2,Lπ‘Š ,Zβ„“h𝑑2i))

Hβˆ—π‘

βˆ’βˆ’β†’HomH𝑝(Hβˆ—π‘(Shπœ‡1,Lπ‘Š ,Zβ„“h𝑑1i),Hβˆ—π‘(Shπœ‡2,Lπ‘Š ,Zβ„“h𝑑2i)).

We justify that the composition of maps in(12.11)factors through CorrShtloc(𝑆(𝑉e1), 𝑆(𝑉e2)).

Note that the proof of Lemma 11.2.3.(3) and the definition of locπœˆπ‘(π‘š1, 𝑛1)β˜…imply that for a quadruple (π‘š0

1, 𝑛0

1, π‘š0

2, 𝑛0

2) of (πœ‡1 + 𝜈, 𝜈)-acceptable and (πœ‡2 + 𝜈, 𝜈)- acceptable integers, the functor locπœˆπ‘(π‘š1, 𝑛1)β˜…commutes with the connecting mor- phism in(10.12)(withπœ‡1,πœ‡2,πœ†fixed). Let𝜈 ≀ 𝜈0and(π‘š0

1, 𝑛0

1, π‘š0

2, 𝑛0

2)be a quadruple of non-negative integers satisfying appropriate acceptance conditions. The proper smooth base change shows that locπœˆπ‘(π‘š0

1, 𝑛0

1)β˜…commutes with enlarging𝜈 to𝜈0. In addition, the proper smooth base change together with the construction ofβ„­π‘Š show that the following diagram commutes:

CorrSh𝜈

πœ‡1|πœ‡ 2

( (Shπœ‡1,Zβ„“h𝑑1i),(Shπœ‡2,Zβ„“h𝑑2i)) CorrSh𝜈

πœ‡1|πœ‡ 2

( (Shπœ‡1,Lπ‘Š ,Zβ„“h𝑑1i),(Shπœ‡2,Lπ‘Š ,Zβ„“h𝑑2i))

CorrSh𝜈0

πœ‡1|πœ‡2

( (Shπœ‡1,Zβ„“h𝑑1i),(Shπœ‡2,Zβ„“h𝑑2i)) CorrSh𝜈0

πœ‡1|πœ‡2

( (Shπœ‡1,Lπ‘Š ,Zβ„“h𝑑1i),(Shπœ‡2,Lπ‘Š ,Zβ„“h𝑑2i)).

β„­π‘Š

π‘–βˆ— π‘–βˆ—

β„­π‘Š

Thus the map β„­π‘Š is compatible with the enlargement of 𝜈. Finally, by [XZ17, Lemma A.2.8], the composition of maps Hβˆ—π‘β—¦β„­π‘Š commutes with enlarging𝜈to𝜈0. We complete the proof of the statement at the beginning of this paragraph.

Composing(12.7)with(12.11), we get a canonical map HomCoh𝐺ˆQβ„“(𝐺ˆ

Qβ„“πœŽ)(𝑉e1,𝑉e2) β†’HomH𝑝(Hβˆ—π‘(Shπœ‡1,Lπ‘Š ,Qβ„“h𝑑1i),Hβˆ—π‘(Shπœ‡2,Lπ‘Š ,Qβ„“h𝑑2i)). (12.11) The fact that (12.9) is compatible with the compositions of the source and target can be proved in an analogous way as [XZ17, Lemma 7.3.12], and we omit the details. Then the action of J naturally translates to the right hand side of (12.9) and upgrades it to our desired map

Spc : Hom

Coh𝐺ˆ(𝐺 πœŽΛ† )(𝑉e1,𝑉e2) β†’HomHπ‘βŠ—J(Hβˆ—π‘(Shπœ‡1,Lπ‘Š ,Qβ„“h𝑑1i),Hβˆ—π‘(Shπœ‡2,Lπ‘Š ,Qβ„“h𝑑2i)). As discussed in loc.cit, the action of J on Hβˆ—π‘(Shπœ‡π‘–,Lπ‘Š ,Qβ„“h𝑑𝑖i) is expected to coincide with the usual Hecke algebra action, which may be understood as the Shimura variety analogue of V. Lafforgue’s "𝑆 = 𝑇" theorem (cf. [Laf18]). We prove this in the case of Shimura sets.

Proposition 12.2.3. LetSh𝐾(𝐺 , 𝑋) be a zero-dimensional Shimura variety. Then the action ofJonHβˆ—π‘(Shπœ‡π‘–,Lπ‘Š ,Qβ„“h𝑑𝑖i)is given by the classical Satake isomorphism.

Proof. Let 𝑓 ∈ J. Since the Shimura variety we consider is zero-dimensional, it follows from [XZ17, A.2.3(5)] that the cohomological correspondence locβ˜…π‘(SO(𝑓)) can be identified with a Zβ„“-valued function on Shπœ‡|πœ‡. By our construction of the map Spc, this function is given by the pullback of a function 𝑓0 on Shtlocπœ‡|πœ‡ = 𝐺(Z𝑝)\𝐺(Q𝑝)/𝐺(Z𝑝). Corollary 11.2.5(2) thus implies that the function 𝑓0 is exactly the function SO(𝑓) ∈ 𝐻

𝐺 ,𝐸[π‘βˆ’1/2, 𝑝1/2] which is the image of 𝑓 under the classical Satake isomorphism.

For any 𝑛 ∈ Z+, takeπ‘Š = Z𝑛ℓ. Recall our construction ofβ„­π‘Š, the cohomological correspondence Λœπ‘“π‘›is given by a finite direct sum of the function locβ˜…π‘(SO(𝑓))since the Shimura variety we consider is a set of discrete points. Then the action of Spc(𝑓) on Hβˆ—π‘(Shπœ‡π‘–,Lπ‘Š ,Qβ„“h𝑑𝑖i) is given by the classical Satake isomorphism. For generalπ‘Š, we take resolutions of it as in (12.10), and the statement follows from

the caseπ‘Š =Z𝑛ℓ.

12.3 Non-Vanishing of the Geometric Jacquet-Langlands Transfer

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