Shimura varieties, Galois representation andAutomorphic forms

Elena Mantovan

California Institute of Technology

Modern Mathemathics WorkshopOctober 3, 2013

MSRI Program in Fall 2014

New Geometric Methods in Number Theoryand Authomorphic Forms

I Connection for Women on August 14-15, 2014

I Introductory Workshop on August 18–22, 2014

I Automorphic fomrs, Shimura varieites, Galois representationsand L-functions on December 01–05, 2014

Concurrent MSRI program in Geometric Representaion Theory

Appointment Categories:Research Professors, Research Members, Postdoctoral Fellows

Galois groups

I Let L be a field, L an algebraic closure of L,ΓL the absolute Galois group of L (topological group).

I Let l be a rational prime, Ql algebraic closure of Ql .

Defn. a n-dim’l l-adic rep of ΓL is ρ : ΓL → GLn(Ql)a continuous group homomorphism.

E.g. L a number field (i.e. finite extension of Q),L a p-adic local field (i.e. finite extension of Qp).

Local Galois groups

Let K be a p-adic local field.

I ΓK = IK o 〈frobv 〉 wherefrobv is a lift to char 0 of Frobenius in char p, i.e.

frobv (x) ≡ xq mod p

for q the size of the residue field of K ,〈frobv 〉 free top. group on one generator,IK contains a pro-p-group of finte index.

Defn. The Weil group WK := IK o frobZv ⊂ ΓK (dense subgroup).

Defn. A representation σ of ΓK is unramified if it is trivial on IK .

Global Galois groups

Let F be a number field.

I for every place v |p of F : let Fv the v-adic completion of F

Fact. For each place v |v of F : there is a natural embedding

ΓFv ↪→ ΓF

Thm. (Chebotarev) For any finite set of places S , the induced map∏v /∈S

ΓFv → ΓF

has dense image.

Galois representations

I F a number field :

Repn(ΓF ) =

irreducible n-dim’ll-adic reps of ΓF

a.e. unramifiedde Rham at l

/'

I K a p-adic local field, assume l 6= p:

Repn(ΓK )) =

{Frobenius-semisimple

n-dim’l l-adic reps of ΓK

}/'

Global-to-Local Restriction

I For every place v |p of F : ΓFv ⊂ ΓF (up to conjugation)

resv : Repn(ΓF )→ Repn(ΓFv )

Σ 7→ σ = (Σ|ΓFv)F−ss

I For Σ ∈ Repn(ΓF ):{resv (Σ)}v uniquely determines Σ (up to semisimplification).

I Let S a finite set containing all ramified primes of Σ:Σ is uniquely determined by the action of Frobv ∈ ΓFv ⊂ ΓF ,for all v /∈ S .

Automorphic forms

I F a number field: AF be the adeles of F , AF =∏′

v Fv .

Irr(GLn(AF )) =

cuspidal automorphicC-reps of GLn(AF )

algebraic at ∞

/'

I K a p-adic field, OK the ring of integers (e.g. Zp ⊂ Qp):

Irr(GLn(K )) =

{irreducible admissibleC-repr’s of GLn(K )

}/'

Defn. A representation π ∈ Irr(GLn(K )) is spherical if it is trivial onGLn(OK ).

Global-to-Local Restriction

From AF =∏′

v Fv : GLn(Fv ) ⊂ GLn(AF ), for every place v of F .

I If Π ∈ Irr(GLn(AF )):Π = ⊗′vπv

for πv ∈ Irr(GLn(Fv )).

resv : Irr(GLn(AF ))→ Irr(GLn(Fv ))

Π 7→ πv

Fact. If Π ∈ Irr(GLn(AF )): Π is a.e. spherical(i.e. πv is spherical for all but finitely many v).

Thm. ????

Local Langlands’ correspondence

Let K be a p-adic local field, l a prime number, l 6= p.Fix a field isomorphism Ql ' C.Theorem (Harris-Taylor, ’01; Henniart, ’00)There is a collection of bijections

recK : Irr(GLn(K ))→ Repn(ΓK )for every n ≥ 1 satisfying

1. for π ∈ Irr(GLn(K )): L(π, s) = L(recK (π), s)ε(π, s) = ε(recK (π), s);

2. compatible for all m ≤ n (e.g. with Artin reciprocity, m = 1).

3. compatible with duality, twist by a character, and takingcentral character.

The case n = 1: Class Field Theory

Theorem (Artin, 1924–30) Therey is a bijection

recK : Char(K×)→ Char(ΓK )

χ 7→ χ ◦ θ

where θ : K× → ΓabK satisfies

1. θ(K×) = W abK , θ(O∗K ) = IK

2. if $K ∈ K× uniformazer: θ($K ) ∈ frobK IK ,i.e. if χ unramified character:

χ($K ) = reck(χ)(frobK )

I The case n = 2: Deligne, 1980’s.

Unramified Galois reps ↔ Spherical Authomorphic reps

I if σ ∈ Repn(ΓK ) unramified: uniquely determined by the(semisimple) conjugacy class of σ(frobK ) ∈ GLn(Ql)(i.e. its e-values in Ql)

L(σ, s) = det(I − σ(frobK )q−s)−1

I Satake isomorphism gives a one-to-one correspondencebetween the set of spherical π ∈ Irr(GLn(K )) and semisimpleconjugacy classes of tπ ∈ GLn(C)(i.e. its e-values in C, the Satake parameters)

L(π, s) = det(I − tπq−s)−1)

Compatibility for all m ≤ n: Parabolic Induction

Let n =∑

mi : L =∏

i GLmi Levi subgroup of GLn.

I If σi ∈ Repmi(ΓK ): σ = ⊕iσi ∈ Repn(ΓK ).

I If πi ∈ Irr(GLmi (K )): ⊗iπi is an admissible rep of L(K ),

IndGLn(K)L(K) (⊗πi ) a admissible rep of GLn(K )

containing a canonical subquotient π ∈ Irr(GLn(K )).

recK (π) = ⊕i recK (πi )

Defn. π ∈ Irr(GLn(K )) is supercuspidal if it is not a subquotient ofa parabolically induced rep.

π supercuspidal ⇐⇒ recK (π) irreducible

Global Langlands’ Correspondence

Let F be a number field.For l a prime number, fix a field isomorphism Ql ' C.For n ≥ 1.

Conjecture (Langlands, Fointaine-Mazur)There is a bijection

RecF : Irr(GLn(AF ))→ Repn(ΓF )satisfying

L(Π, s) = L(RecK (Π), s)

ε(Π, s) = ε(RecK (Π), s)

for all Π ∈ Irr(GLn(AF )).

Equivalent statement

I if Π ∈ Irr(GLn(AF )), Π = ⊗′vπv :

πv spherical ⇐⇒ resv (RecF (Π)) unramified

and for all spherical/unramified v :

L(πv , s) = L(resv (RecK (Π)), s)

ε(πv , s) = ε(resv (recK (Π)), s)

Local-Global Compatibility

Let F be a number field, l a prime number, Ql ' C, n ≥ 1

Conjecture (Langlands)The bijection

RecK : Irr(GLn(AF ))→ Repn(ΓF )satisfies (

RecF (Π)|ΓFV

)F−ss= recFv (πV )

for all finite places v of F , Π = ⊗′vπv .

Construction of Galois reps and Modularity results

Irr(GLn(AF ))←→ Repn(ΓF )

I → Construction of Galois representations (Langlands)

E.g. ... Harris-Taylor ’01, Morel ’10, Shin ’10(F CM field, Π polarizable, regular algebraic),Harris-Lan-Taylor-Thorne ’13 (F CM field, Π algebraic).

I ← Modularity (automorphy) of Galois representations(Fointaine-Mazur)

E.g. Barnet-Lamb, Gee, Gerarthy, Harris, Taylor, Thorne et al.

Functoriality Principle

G a reductive algebraic group.

I Existence of reciprocity laws forG ,e.g. for G = GSP2g :

{reps of GSp2g} →{

orthogonal Galois repsρ : Γ→ O2g (Ql)

}I Compatibily under group homomorphism G → G ′,

e.g. for O2g ⊂ GL2g :

{reps of GSp2g} → {2g-dim’l orth. Galois reps }↓ ∩

{reps of GL2g} ↔ {2g-dim’l Galois reps }

Parabolic Induction

K a p-adic local field.

I n =∑

i mi , L =∏

i GLmi ⊂ GLn Levi subgroup:{irred. admissible

reps of L(K )

}↔

{n-dim’l Galois reps

s.t. σ = ⊕iσi

}↓ ∩{

irred. admissiblereps of GLn(K )

}↔ {n-dim’l Galois reps }

Jacquet Langlands Correspondence

K a p-adic local field.

I n ≥ 2: D a n2-dim’l central division algebra over K ,i.e. D× inner form of GLn:{

irred. admissiblereps of D×

}→ {n-dim’l Galois reps }

↓ ‖{irred. admissiblereps of GLn(K )

}↔ {n-dim’l Galois reps }

Shimura varieties

I G/Q connected reductive group,(e.g. GL2, GSp2g , or ResF/Q(G ′), F/Q finite field extension).

I Af the finite adeles, i.e. Af =∏′

p-∞Qp; and Z =∏′

p-∞ Zp.

I U ⊂ G (Af ) open compact subgroup,e.g. U0 = G (Z) ⊂ G (Af ), U ⊂ U0 finite index.

Defn. Shimura varieties associated with G are a compactible system:

I for U ⊂ G (Af ): XU smooth quasi-projective variety over E/Q.

I for each V ⊂ U: XV → XU finite etale morphism.

I for g ∈ G (Af ), U: g : Xg−1Ug → XU .

An example: GL2 and Modular Curves

I h complex upperhalf plane, h = {z ∈ C|Im(z) ≥ 0} SL2(Z)

I if A =

(a bc d

)∈ SL2(R) : z ∈ h 7→ az+b

cz+d ∈ h

I U(n) = {A ∈ SL2(Z)|A ≡ I mod n} ⊂ SL2(Z)

I Y (n) = U(n)\h is the modular curve of (full) level n,it is a smooth curve over C

I Y (n) has a moduli interpretation as classifying space forelliptic curves with marked n-torsion points:

if τ ∈ U(n)\h : (Eτ = C/〈1, τ〉Z,P = 1/n,Q = τ/n)

I Y (n) has a canonical model over Q

E.g. If G = GSp2g : moduli spaces of g -dim’l abelian varieties.

Constructing global Galois reps

Let G/Q connected reductive group.{XU}U G (Af ) Shimura varieties over E (n = dim XU).E algebraic closure of E , ΓE absolute Galois group of E .

Prop. 1. For ξ a representation of G over Ql :Lξ a lisse Ql -sheaf on XU (e.g. if ξ = 1: L1 = Ql)H ·ξ = lim−→U

H ·(XU ×E E ,Lξ) G (Af )× ΓE is admissible.

Prop. 2. If Π ∈ IrrG (A) algebraic (i.e. Π∞ algebraic):HomG(Af )(Πf ,H

·ξ) 6= ∅ for ξ appropriate.

Kottwitz-Langlands’ conjecture

Conjecture (Theorem in many cases) For i ≥ 0:

lim−→U

H i (XU ×E E ,Lξ) =⊕

Π

Πf ⊗ R iξ(Πf )

and [Rξ(Π)] = (−1)n−1∑

i (−1)i [R iξ(Πf )] virtual Galois rep.

The correspondence

Π 7→ Rξ(Πf ) for ξ appropriate

satisfies Langlands’ conjecture.

Rapoport-Zink Local Models

I G/Qp connected reductive group,(e.g. GLn, GSp2g , or ResK/Q(G ′), K/Qp finite field ext.).

I U ⊂ G (Qp) open compact subgroup,e.g. U0 = G (Zp) ⊂ G (Qp), U ⊂ U0 finite index.

Defn. (Kottwitz) B(G ) = G (Knr )/ ∼σ partially ordered finite set.

Defn. (Rapoport-Zink) For b ∈ B(G ): Local models associated with(G , b) are a compactible system:

I for U ⊂ G (Qp): MU smooth rigid analytic spaces over E/Qp.

I for each V ⊂ U: MV → MU finite etale morphism.

I for g ∈ G (Qp), U: g : Xg−1Ug → XU .

E.g. For G = GLn:B(GLn) = {isogeny classes BT groups over Fp of height n}MU are moduli spaces of BT groups H satisfying HFp

∈ b.

The algebraic group Jb/Qp

Let G/Qp connected reductive group, b ∈ B(G ).{MU}U G (Qp) RZ spaces over E

Defn. For b ∈ B(G ): Jb/Qp the alg. group of quasi-isogeny of b,

Jb(Qp) =(EndFp

(Xb)⊗Qp

)×for Xb/Fp a BT group in b.

Facts 1. Jb is an inner form of a Levi subgroup Lb of G .

2. If b ∈ B(G ) maximal (basic): Jb is inner form of G .

3. B(Lb) ⊂ B(G ), and b ∈ B(Lb) basic element.

4. For each U: MU Jb(Qp) (MU only locally of finite type).

The cohomology of RZ spaces

Fix E algebraic closure of E , ΓE absolute Galois group of E .

I Let l 6= p, i ≥ 0, n = dim MU :

H i = lim−→U

H ic(MU ×E E ,Ql(n)) Jb(Qp)× G (Qp)× ΓE

Prop. (M.) If ρ ∈ Irr(Jb(Qp)), i , j ≥ 0:

H i ,j(ρ) = ExtjJQp(ρ,H i ) G (Qp)× ΓE is admissible.

H i ,j(ρ) vanishes for almost all i , j ≥ 0.

Defn. For ρ ∈ Irr(Jb(Qp)):

[H(ρ)] = (−1)n−1∑i ,j

(−1)i+j [H i ,j(ρ)]

virtual admissible rep of G (Qp)× ΓE .

A conjecture of Kottiwitz

Conjecture (If G = ResK/F (GLn): Harris-Taylor, Fargues)Assume b ∈ B(G ) basic.For ρ ∈ Irr(Jb(Qp)): [H(ρ)] = JL(ρ)⊗ R(ρ)where:

I the correspondence Irr(Jb(Qp))→ Irr(G (Qp)),ρ 7→ JL(ρ), is the Jacquet-Langlands correspondence;

I the correspondence Irr(G (Qp))sc → Repn(ΓE ),π 7→ R

(JL−1(π)

), is the local Langlands correspondence.

A Conjecture of Harris

Assume b ∈ B(G ) is non-basic.

I L Levi subgroup of G : b ∈ B(L) ⊂ B(G ), b ∈ B(L) basic.

I {M(G , b)U}U G (Qp) RZ spaces associated with (G , b):

H ·(G , b) = lim−→U

H ·c(M(G , b)U×E E ,Ql(nG )) Jb(Qp)×G (Qp)×ΓE

I {M(L, b)V }V L(Qp) RZ spaces associated with (L, b):

H ·(L, b) = lim−→V

H ·c(M(L, b)V×E E ,Ql(nL)) Jb(Qp)×L(Qp)×ΓE

Conjecture (Theorem in few case, M.) For ρ ∈ Irr(Jb(Qp)):

[H(G , b)(ρ)] = IndG(Qp)L(Qp) [H(L, b)(ρ)]

Skect of Proof (I)Let P parabolic subgroup of G , L Levi of P.

Defn. {M(P, b)W }W P(Qp) rigid spaces over E ,H ·(P, b) = lim−→U

H ·c(M(P, b)U ×E E ,Ql(nP))H ·(P, b) Jb(Qp)× P(Qp)× ΓE

For U ⊂ G (Qp): let V = U ∩ L(Qp) ⊂W = U ∩ P(Qp):there are compatible Jb(Qp)× ΓE -equivariant morphisms

M(L, b)V ιL// M(P, b)W ιP

//

θuu

M(G , b)U

I ιLM(L, b)V ↪→ M(P, b)W is L(Qp)-equivariant,via L(Qp) ⊂ P(Qp).

I ιP : M(P, b)W → M(G , b)U is P(Qp)-equivariant,via P(Qp) ⊂ G (Qp).

I θ : M(P, b)W → M(L, b)U is P(Qp)-equivariant,via the projection P(Qp)→ L(Qp).

Skect of Proof (II)

M(L, b) ιL// M(P, b) ιP

//

θuu

M(G , b)

1. θ ◦ ιL = id and θ∗ : H ·(L, b) ' H ·(P, b).The action of P(Qp) on H ·(P, b) factors thru L(Qp).

2. Let XP be the orbit of Im(ιp) unde G (Qp):

XP ⊂ M(G , b) is closed, and H ·(XP) = IndG(Qp)P(Qp) (H ·(P, b)).

3. For certain b: there exists P s.t. XP = M(G , b).

IndG(Qp)L(Qp) (H ·(L, b)) = Ind

G(Qp)P(Qp) (H ·(P, b)) = H ·(XP) = H ·(G , b)

Recent Developments, New Directions

I Shimura varieties and local models of Hodge type (andabelian type) [Kisin, K-Chen-Viehmann]

I Boundary of compactifications of Shimura varieties [Pink,Faltings-Chai] [Lan, Harris-L-Taylor-Thorne, Pera]

I Torsion cohomology [Emerton-Calegari, Scholze]

I p-adic Local Langlands [Emerton, Brueil]