Contents The analogy between knots and primes etale 2 Ypi.math.cornell.edu/~dkmiller/bin/primes-knots.pdfp) which has a presentation ˇtame 1 (Q p) = h˙;˝: ˙˝˙ 1 = ˝pi: We can

AUTOMORPHIC REPRESENTATIONS AND DEFORMATION

THEORY IN ARITHMETIC GEOMETRY

DANIEL MILLER

Abstract. This is a brief expository note, motivated by [Mor12], on the anal-

ogy between the character variety of the fundamental group of a hyperbolicknot, and the p-ordinary deformation space of a two-dimensional modular

Galois representation. Hopefully this analogy should generalize to arbitraryreductive groups, or at least GL(n). In light of this, much of this note is

an introduction to the relevant objects from a general perspective. So we

introduce notion of a p-adic family of automorphic representations, and the(conjecturally) corresponding family of p-adic Galois representations in full

generality.

Contents

1. The analogy between knots and primes 12. Automorphic representations 23. Shimura varieties 54. Modular representations 65. Deformation theory 96. Deformations of hyperbolic structures and Hida theory 12References 13

1. The analogy between knots and primes

Recall that a morphism f : X → Y of schemes is etale if it is flat and unramified.Equivalently, f is flat and for each y ∈ Y , the fiber Xy is the spectrum of a finiteproduct of finite separable extensions of k(y). If X is a connected noetherian schemewith chosen base point x ∈ X, the category of finite etale covers of X is canonicallyequivalent to the category of finite sets with continuous action of π1(X,x). Hereπ1(X,x) is the etale fundamental group of X at x, which we will denote π1(X)when x is clear. To save space, we will write π1(A) instead of π1(SpecA). A goodreference for all of this is [Mil13].

We start by recalling the analogy described in [Mor12, ch.3-4]. We should thinkof the circle S1 as a K(Z, 1)-space. The arithmetic analogue is Spec(Fq) for any

prime power q. Indeed, as is the case for any field, π1(Fq) = Gal(Fq/Fq) ' Z,generated by the Frobenius frq(x) = xq. The arithmetic analogue of a 3-manifold

Date: July 2014.

2000 Mathematics Subject Classification. 11F41,11F70,11F80,11G18,13D10,14G35.Key words and phrases. Automorphic representation, Shimura variety, deformation theory,

modular form, Hecke algebra, Hida theory.

1

REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 2

is X = Spec(OF ) r S for any number field F and finite set S of primes in OF .Indeed, for constructible sheaves F on X, there is a “3-dimensional duality”Ext•(F ,Gm) = H3−•

c (X,F )∨ [Maz73, 2.4] similar to the classical Artin-Verdierduality Ext•(L ,Q) = H3−•

c (M,L )∨ for local systems on a 3-manifold M .In topology, a knot is an embedding K : S1 ↪→ S3. More generally, we could

consider any K(Z, 1) ↪→ M , where M is a 3-manifold. The arithmetic analogueis a map Spec(Fq) → Spec(OF ) coming from a prime ideal p ⊂ OF with residuefield Fq. In topology, the standard way of studying a knot K : S1 ↪→ S3 is toconsider a small tubular neighborhood VK of S1 in S3. For simplicity, we assumeF = Z; then the arithmetic analogue is an “infinitesimal etale neighborhood” ofSpec(Fp) ↪→ Spec(Z), namely Spec(Zp) → Spec(Z). The peripheral group of K isπ1(VKrK) ' Z2, corresponding to π1(Spec(Zp)rSpec(Fp)) = π1(Qp). Finally, theknot group is ΓK = π1(S3 rK), corresponding to ΓQ,p = π1(Z[ 1

p ]). The peripheral

map π1(VK rK)→ ΓK corresponds to the map π1(Qp)→ ΓQ,p.Unfortunately, this analogy is not perfect. While the topological periphery group

π1(VK rK) is abelian, free on generators l,m (l for latitude, m for meridian) andπ1(VK rK)→ π1(VK) has kernel 〈m〉, the arithmetic case is a more complicated.The group Dp = π1(Qp) is a very complicated pro-solvable group. It can be writtendown in terms of generators and relations as in [NSW08, VII §5], but this is notvery helpful. The group Dp does have a canonical quotient classifying tame coversof Spec(Qp) which has a presentation

πtame1 (Qp) = 〈σ, τ : στσ−1 = τp〉.

We can think of the degenerate case “p = 1” as being in exact analogy with topology.The kernel of Dp → π1(Zp) is written Ip, and (in analogy with topology) called theinertia group at p.

2. Automorphic representations

2.1. Adeles. Let F be a number field. If v is a place of F , write Fv for thecompletion of F at v, and Ov for the ring of integers of Fv. Write A = AF for thering of adeles of F ; the most important fact about A is that it is a locally compactF -algebra. It can be defined in many ways:

• The topological direct limit lim−→SA(S), where S ranges over all finite sets

of valuations of F , and

A(S) =∏v∈S

Fv ×∏v/∈S

Ov.

• The topological tensor product (R× Z)⊗ F .• A restricted direct product

∏′v(Fv, Ov), consisting of those tuples (av) ∈∏

v Fv for which av ∈ Ov for almost all v.• Via [GS66], an initial object in the category of locally compact F -algebras

with no proper open ideals, and with the intersection of all closed maximalideals being 0.

Write Af = Z⊗ F for the ring of finite adeles. It is totally disconnected.In [Con12], it is shown that there is a unique fiber-product-preserving functor

(−)(A) from affine schemes of finite type over F to topological spaces, compatiblewith closed embeddings, for which (SpecF [t])(A) = A with its usual topology. Inparticular, if G is an algebraic group over F , the abstract group G(A) carries the


structure of a locally compact topological group. As such, it has a unique (up toscalar) Haar measure dg.

2.2. Hecke algebras. A good reference for this section is [Fla79]. Let G be aconnected reductive group over F , and let g = Lie(G). There exists an open subsetof U ⊂ Spec(OF ) and a “spreading out” of G to a reductive group scheme on U .Up to finitely many places, this spreading out is well-defined. In particular, foralmost all finite places v, the group G(Ov) is well-defined for almost all v. It isa maximal (open) compact subgroup of G(Fv). We normalize Haar measures onG(Fv) so that G(Ov) has volume 1, and choose the Haar measure on G(A) to bethe product of measure on each G(Fv).

Let v be a finite place. Write Hv for the Hecke algebra consisting of continuous,locally constant, compactly supported functions G(Fv) → Q. Multiplication isconvolution:

(f ? g)(x) =

∫G(Fv)

f(g)g(y−1x) dy.

Even though this is written as an integral, it is a finite sum over double cosets ofopen compact subgroups of G(Fv), so no analysis is involved. For almost all v,the algebra Hv comes with a canonical idempotent, ev = χG(Ov). Write Hf for therestricted tensor product (in the sense of [Fla79, §2]) of the Hv with respect to theev. It is the direct limit lim−→S

H(S), where H(S) =⊗

v∈S Hv, and for T ⊃ S, the

injection H(S) → H(T ) is induced by the ev for v ∈ T r S. We will also think ofHf as the algebra of locally constant, compactly supported functions f on G(Af).

The ring F∞ = F ⊗ R is a finite product of copies of R and C, so G(F∞)is naturally a Lie group. Fix a maximal compact subgroup K∞ ⊂ G(F∞). TheHecke algebra H∞ = H∞(G) is the convolution algebra of K∞-finite distributionson G(F∞) with support in K∞. There is an isomorphism

U(gC)⊗U(kC) M(K∞),∼−→ H∞ D ⊗ µ 7→ D ? µ,

where k = Lie(K∞) and M(K∞) is the algebra of measures on K∞.The global Hecke algebra of G is H = H∞ ⊗Hf . We will be interested in special

classes of representations of H.For a sufficiently large set S of finite places, it makes sense to define eS ∈ Hf to

be the characteristic function of∏v/∈S G(Ov).

An admissible representation of Hf is an Hf -module V such that for each v ∈ V ,there is a finite set S of places for which eS · v = v.

2.3. Automorphic representations. A good reference for this section is [BJ79].Let F , G, . . . be as above. Let Z be the center of G, and choose a character

ω : Z(F )\Z(A) → C×. Write L2(G,ω) for the space of measurable functionsf : G(F )\G(A)→ C such that

f(zx) = ω(z)f(x) z ∈ Z(A)

‖f‖2 =

∫G(F )Z(A)\G(A)

|f(x)|2 dx <∞.

The space L2(G,ω) is a representation ofG(A) in the obvious way. Write L2disc(G,ω)

for the closed subspace generated by all irreducible closed subrepresentations. LetA(G,ω) ⊂ L2

disc(G,ω) be the space of K-finite vectors which are also Z(g∞)-finite.(It is a non-trivial fact that K-finite vectors are smooth.)


Then A(G,ω) is naturally a H-module, and as such, decomposes as a countabledirect sum of irreducible representations with finite multiplicities:

(1) A(G,ω) =⊕

π∈Gc.a(ω)

m(π)π.

We call the irreducible admissible representations of H appearing in (1) automor-phic representations of G. By [Fla79, th.4], each automorphic representation πdecomposes as a restricted tensor product

⊗πv of irreducible admissible represen-

tations of the Hv.In the remainder, we will often pass without comment between admissible repre-

sentations ofG(Af) and admissible representations ofHf . This is not hard. Supposeπ : G(Af)→ GL(V ) is an admissible representation. For f ∈ Hf and v ∈ V , put

f ? v =

∫G(Af )

f(x)π(x) · v dx.

This integral is actually a finite sum. Indeed, we can write f as a finite sum of scalarsmultiples of characteristic functions χgK , where K ⊂ G(Af) is open, compact, andfixes v. For such a function, we see that

χgK ? v =

∫G(Af )

χgK(x)π(x)v dx =

∫K

gv dx = vol(K)gv.

So the action of Hf on V makes sense. Going the other way is also easy. If V is anadmissible Hf -module and g ∈ G(Af), choose open compact normal K such thatχK ? v = v. Inspired by the above, put gv = vol(K)−1χgK ? v.

2.4. Hecke eigensystems and L-functions. Let π be an automorphic repre-sentation of G and choose a nonzero vector u in π. For almost all places v, theidempotent ev = χG(Ov) in Hv fixes u (in this case, we say that π is unramified atv). In particular, the action of Hv on π factors through that of

Hv(Ov) = evHvev = C∞c (G(Ov)\G(Fv)/G(Ov)).

Let S be a set of places outside which ev fixes u. Let H(S) =⊗

v/∈S Hv(Ov).Then π is an irreducible admissible module over H(S) ⊗

⊗v∈S Hv. Since H(S) is

central in this algebra, it must act via a character χ : H(S) → C. The system ofhomomorphisms {χv : Hv(Ov)→ C : v /∈ S} is called a Hecke eigensystem.

In the case G = GL(n), Hecke eigensystems have a particularly easy description.A character χ : Hv(Ov) → C is uniquely determined by a semisimple conjugacyclass σv(χ) ∈ GL(n,C). If π =

⊗v πv is an automorphic representation of GL(n),

put σv(π) = σ(χπv ) and (for finite v):

Lv(s, π) = det(1−N(v) · σv(π)−s

)−1.

For S sufficiently large, we can define the partial L-function of π as

LS(s, π) =∏v/∈S

Lv(s, π).

This has the expected properties including analytic continuation, a functional equa-tion. . . . In the case G = GL(n), an automorphic representation π is determined byL(s, π).


3. Shimura varieties

For the rest of this note, the reader should keep in mind the example F = Q,G = GL(2). Many of the definitions work in greater generality, but technicalities(which we wish to avoid) multiply endlessly.

3.1. Locally symmetric spaces and their cohomology. Classically, one stud-ies representations of a real semisimple group G by fixing a maximal compact K,setting X = G/K, and studying the regular representation of G on C∞(Γ\X) forΓ ⊂ G a discrete group. Big examples are the (affine) modular curves Y0(n), comingfrom Γ0(n) ⊂ SL(2,R). We will carry out this construction adelically.

Let G be a connected reductive group over Q. Put X = Z∞\G(F∞)/K∞. LetK ⊂ G(Af) be an open compact subgroup. We define

ShK(G) = G(Q)\(X ×G(Af))/K.

A priori, this is only a topological space, but the quotient map X × G(Af)/K →ShK(G) gives ShK(G) the structure of a Riemannian manifold. Let (V, ρ) be arepresentation of G. There is an induced local system Vρ of F -vector spaces onShK(G), whose (global) sections are locally constant sections s : X×G(Af)/K → Vsuch that s(γx) = ρ(γ)s(x) for γ ∈ G(Q).

The cohomology H•(ShK(G),Vρ) is naturally an admissible Hf -module. Foropen compact C ⊂ K, the function χgC acts via the correspondence

ShK(G)� ShK∩C(G)·g−→ ShK∩g−1Cg(G)� ShK(G).

There is a standard compactification of ShK(G), namely its Borel-Serre compacti-fication ShK(G). Define the cuspidal cohomology to be

H•cusp(ShK(G),Vρ) = ker(H•(ShK(G),Vρ)→ H•(∂ ShK(G),Vρ)

).

The cuspidal cohomology is also an admissible Hf -module. In fact, we have ageneralized Eichler-Shimura isomorphism [Sch09, 4.1].

H•cusp(ShK(G),Vρ) =⊕

π∈Acusp(G,χρ)K-spherical

H•(H∞, π∞ ⊗ ρ)⊗ πKf .

The notation H•(H∞,−) needs explanation. There is a good category of admissibleH∞-modules, and hom(C,−) is left-exact. Its derived functor is the (g∞,K∞)-cohomology H•(H∞−).

Better:

H•cusp(Sh(G),Vρ) =⊕

π∈Gc(ω)

H•(H∞, π∞ ⊗ ρ)⊗ πf .

3.2. Canonical models. A good reference for this is [Moo98]. In subsection 3.1we constructed ShK(G) as a Riemannian manifold. It turns out that there is agood definition of “canonical model” for Shimura varieties over number fields, andin that sense, all Shimura varieties ShK(G) have a canonical model over a numberfield called the reflex field. (We have been intentionally avoiding use of the Shimuradatum necessary to define ShK(G) in full generality – the reflex field depends onthis.)

Let E be the reflex field. Not only does the projective system Sh(G) = lim←− ShK(G)

descend to the reflex field, but the action of G(Af) via correspondences descends in


a canonical way. Moreover, in [Har85] it is shown that our local systems Vρ descendin a functorial way to G(Af)-equivariant local systems on Sh(G).

The main reason we care about this is that if ShK(G) is smooth and E = Q,then general theorems about etale cohomology tell us that

H•sing,c(ShK(G),Vρ) = H•et,c(ShK(G)Q,Vρ(Ql))

after choice of an isomorphism C ' Ql. The choice of an arithmetic compactifica-tion of ShK(G) lets us extend the action of Hf to the etale cohomology of ShK(G).

4. Modular representations

4.1. Modular curves. In this section, algebraic groups, adeles, etc. will be takenover Q. Let n > 1 be an integer. We define the following congruence subgroup ofSL2(Z):

Γ0(n) =

{(a bc d

)∈ SL(2,Z) : c ≡ 0 (mod n)

}.

Let K0(n) be the induced subgroup of GL2(Af). Write Y0(n) for the induced locallysymmetric space:

Y0(n) = ShK0(n)(GL2) = GL2(Q)\GL2(A)/Z∞K∞K0(n).

Here Z∞ = Z(GLd(R)) and K∞ = SO(2) ⊂ GL2(R). Put X = GL2(R)+/Z∞K∞.Note that GL2(R)+/Z = SL2(R), so

GL2(R)+/Z∞K∞∼−→ H = {z ∈ C : =z > 0}

via γ 7→ γ · i. The strong approximation theorem tells us that

GL2(A) = GL2(Q) GL2(R)K0(n),

so the quotient ShK0(n)(GL2) is just

(GL2(Q) ∩K0(n))\H = Γ0(n)\H = Y0(n).

This will be a singular complex-analytic orbifold. There are two ways of realizingY0(n) and its compactication X0(n) as curves over Q:

(1) Interpret Y0(n) as a moduli space for elliptic curves with level structure.This moduli problem makes sense over Q, so Y0(n) descends in a canonicalway to Q.

(2) Use the general theory of canonical models of Shimura varieties.

The former approach generalizes to a special class of Shimura varieties consisting ofthose of PEL type (standing for Polarization, Endomorphism, and Level structure).The theory of PEL-type Shimura varieties is interesting and useful, but we won’tgo into it here.

Instead, note that the space

H± = Z∞\GL2(R)/ SO2(R)

can be interpreted as the set of GL2(R)-conjugacy classes of homomorphisms S =RC/R Gm → GL(2)R containing

h : (x, y) 7→(x y−y x

).


This is all defined over Q, so the theory of canonical models discussed in subsec-tion 3.2 tells us that if K ⊂ GL2(Af) is any open compact subgroup, the quotient

ShK(GL2) = GL2(Q)\(H± ×GL2(Af))/K

descends to a uniquely determined curve over Q. Moreover, this curve has a well-defined smooth compactification also defined over Q, so we don’t need to worryabout the difference between minimal and toroidal compactifications.

4.2. The Eichler-Shimura construction. Let n > 3. As above, write Y0(n) forthe Shimura variety ShK0(n))(GL2), and write X0(n) for its arithmetic compactifica-

tion. We are interested in the cohomology H1cusp(X0(n),Vsymk−2). At the moment,

this is just a C-vector space with an action of Hf . However, in subsection 3.1, thereis an automorphic decomposition

H1cusp(X0(n),Vsymk−2) =

⊕π∈A(GL2)

Γ0(n)-spherical

H1(gl2,SO(2), π∞ ⊗ symk−2)⊗ πΓ0(n)f .

The computation in [Har87, §3.4-3.6] tells us that H1(gl2,SO(2), π∞⊗ symk−2) = 0unless π is the automorphic representation coming from a weight-k cuspidal eigen-form of level n, in which case H1(gl2,SO(2), π∞⊗ symk−2) = C⊕C. In particular,

H1cusp(X0(n),Vsymk−2) =

⊕f eigen-cusp

C⊕C = Sk(Γ0(n))⊕ Sk(Γ0(n)).

Recall that our modular curves are defined over Q. So we can consider thecohomology spaces

H1et,cusp(X0(n)Q,Ql) ' S2(Γ0(n),C).

These have commuting actions of ΓQ,ln = π1(Z[ 1ln ]) and Hf . So ΓQ,ln acts on each

Hf -irreducible piece. These pieces are 2-dimensional, so we get, for each cuspidaleigenform f , a Galois representation ρf,l : ΓQ,nl → GL2(Ql). The Eichler-Shimurarelation basically tells us that the Hecke and Frobenius parameters for πf andρf,l match up. That is, for all p - ln, we have ρf,l(frp) = σp(πf ), or equivalentlyLp(ρf,l, s) = Lp(πf , s).

It is known that ρf,l : ΓQ,ln → GL2(Ql) factors through GL2(Kf,λ), whereKf = Q(ap(f) : p prime) is a number field and λ is a place of Kf dividing l.An elementary argument shows that we can conjugate the image of ρf,l to lie inGL2(Of,λ), where Of = OKf . In particular, we can reduce ρf,l modulo λ to get acontinuous representation

ρf,l : ΓQ,ln → GL2(Of,λ/λ) = GL2(Fλ).

We say that a mod-l representation of ΓQ is modular (better, automorphic) if it isof the form ρf,l for some f . Similarly, we say that an l-adic representation of ΓQ ismodular (better, automorphic) if it is of the form ρf,l for some l.

In what follows, we will ignore the modular form f and just write π for thecorresponding cuspidal automorphic representation of GL(2), keeping in mind thatfor some automorphic representations (those corresponding to Maass forms) westill don’t know how to construct the associated Galois representations. For anautomorphic representation π, write ρπ,l for the corresponding l-adic representation(assuming it exists).


4.3. Interpolating modular representations. The Hida families we will seelater on are essentially p-adic families of cuspidal automorphic representations ofGL(2). Even better, Hida constructed the corresponding p-adic family of Galoisrepresentations. Here we will give the (conjectural) bigger picture.

Let G be a connected reductive group over Q. Recall we have a Hecke alge-bra H = Hf ⊗ H∞, and automorphic representations of G are, by definition, aspecial class of irreducible representations of H. Remember that Hf is the con-volution algebra of locally constant, compactly supported functions on G(Af). IfK ⊂ G(Af) is open compact, then eK = 1

µ(K)χK is an idempotent in Hf , and we

write H(K) = eKHfeK for algebra of locally constant, compactly supported, K-bi-invariant functions on G(Af). If eK acts trivially on an automorphic representationπ, we say π is K-spherical.

The basic idea of a p-adic family of automorphic representations is that weshould fix Kp ⊂ G(Ap

f ) (the tame level) and consider families of automorphicrepresentations that are KpKp-spherical, where Kp ranges over a special class ofsubgroups of G(Qp). One makes this rigorous via a modified Hecke algebra. Herewe mainly follow [Urb11, 4.1]. Fix a prime p and assume G is split at p. Let (T,B)be a Borel pair, and let N be the unipotent radical of B. Define

Ir = {g ∈ G(Zp) : g ∈ B(Z/pr)}T− = {t ∈ T (Qp) : tN(Zp)t

−1 ⊂ N(Zp)}∆−r = IrT

−Ir.

There is an isomorphism u : Zp[T−/T (Zp)] → C∞c (Ir\∆−r /Ir) by t 7→ ut = χIrtIr .

Here it is crucial that we normalize the Haar measure so that Ir has volume 1.So we call Up = Zp[T

−/T (Zp)] and think of Up as a single avatar for all of theC∞c (Ir\∆−r /Ir,Zp). Our big Hecke algebra is

h = C∞c (Kp\G(Apf )/Kp,Qp)⊗ Up.

Note that h ↪→ Hf ⊗Qp. So, if σ is an irreducible representation of h, we will call

σ automorphic if σ = πKpIr

f |h for some honest automorphic representation π. Notethat if G 6= GL(n), the representation π may not be unique.

A p-adic family of automorphic representations of G will contain KpIr-sphericalrepresentations for varying r. We will also require the weight to vary p-adically.So, let W be the p-adic weight space determined by

W(A) = homcts(T (Zp), A×).

Let Wcl = X∗(T ) be the set of classical weights. We say an automorphic represen-tation π is cohomological of weight λ if π appears in some H•cusp(ShKpIr (G),Vλ(C)).

Given such a representation, we have the trace map trπ : h → Qp, which charac-terizes π as an h-module.

Definition. A p-adic family σ of automorphic representations of G of tame levelKp consists of:

• An rigid subset U ⊂W.• A finite flat surjection w : T→ U.• A Qp-linear map J : h→ O(T).• A dense set σcl ⊂ Ucl.


We require that for each σ ∈ σcl, the weight λ = w(σ) is dominant and the com-posite

Jσ : h→ O(T)evσ−−→ Qp

is m(π, λ) trπ for a cohomological representation π of weight λ.

Here m(π, λ) is the Euler-Poincare characteristic

m(π, λ) =∑

(−1)i dim homh(πKpIr ,Hi

cusp(ShKpIr (G),Vλ(C)).

In [Urb11], Urban constructed p-adic families containing the “finite slope” represen-tations, for groups satisfying the Harish-Chandra condition (having discrete seriesat infinity).

Recall that for “nice” automorphic representations π, there should be a Ga-lois representation ρπ,p : ΓQ → LG(Qp) unramified almost everywhere, such thatfor unramified v, ρπ,p(frv) is conjugate to the Satake parameter of π at v. For

G = GL(n), this will just be a representation ΓQ → GLn(Qp). Note that evenif multiplicity-one theorems fail for G, the trace trπ determines ρπ,p. So we willspeak of “the Galois representation associated to trπ,” bearing in mind that wedon’t currently know how to construct ρπ,p for general G.

Conjecture. Let σ be a p-adic family of automorphic representations of G withL-algebraic classical points. Then there is a continuous representation ρσ : ΓQ →LG(O(T)) such for all σ ∈ σcl, the composite

ρσ : ΓQ → LG(O(T))evσ−−→ LG(Qp)

is the Galois representation associated to Jσ.

This conjecture is wide open – we don’t even know how to construct individ-ual ρπ,p for most G. In [SU14], Skinner and Urban construct p-adic families ofpseudorepresentations for “unitary similitude groups” associated to an imaginaryquadratic field. For G = GL(2) and π corresponding to a p-ordinary form, Hidahas constructed a big p-adic family containing π, and the associated p-adic familyof Galois representations.

5. Deformation theory

5.1. Motivation. First let’s consider the motivation for studying deformations ofGalois representations. If X is a nice (that is smooth, projective and geometricallyintegral) variety over Q, its etale cohomology H•et(XQ,Ql) carries a continuousaction of ΓQ. The “right” way to see this is as follows. Spread out X to a smoothproper scheme X over an open U = Spec(Z)rS. Write π : X → U for the structuremap. Then R•π∗Ql is a local system on Uet. The etale version of covering spacetheory tells us that local systems correspond to representations of π1(U) = ΓQ,S .

The motivating example was the representation ρE,l coming from an elliptic curve

Eπ−→ U via R1π∗Ql. Langlands’ conjectural framework tells us that there should

exist an automorphic cuspidal representation π of GL(2) for which ρE,l ' ρπ,l. Wedon’t know how to construct π this directly. However, we do know (via Serre’sconjecture) that ρE,l is automorphic. One of the main applications of deformationtheory is to prove that the automorphy of ρE,l implies that of ρE,l.

More generally, given an l-adic Galois representation ρ : ΓQ,S → GLn(Ql) thatis suitably nice (geometric, in the sense of Fontaine-Mazur [FM95]), Langlands’


program tells us that we should expect there to be a cuspidal automorphic repre-sentation π of GL(n) such that ρ ' ρπ,l (assuming we knew how to construct ρπin general). Proving that ρ is automorphic is very hard! However, we have a muchbetter chance (in theory and in practice) of showing that ρ : ΓQ,S → GLn(Fl)is automorphic. A theorem to the effect that “ρ automorphic ⇒ ρ automorphic”is known as a automorphy lifting theorem. In practice, one has to impose manytechnical conditions on ρ and the automorphic representation with ρπ ' ρ, and oneuses groups like GSp(n) instead of GL(n).

5.2. Representations of knot groups. Our exposition here follows that of [Mor12,ch.13-14]. Let K ⊂ S3 be a hyperbolic knot, M = S3 rK the knot complement,π = π1(M) the knot group. The uniformization H3 → M induces a represen-tation π → Aut(H3) = PSL2(C) which lifts to ρ : π → SL2(C). Introduce therepresentation variety Rep(π,SL2) of homomorphisms π → SL2(C). There are twoways of describing Rep(π,SL2). One elementary but useful approach is to writeπ = 〈g1, . . . , gm|r1, . . . , rn〉. The variety Rep(π,SL2) is just the subset of SL2(C)m

cut out by the relations r1, . . . , rn. A more functorial definition is to require thatfor all C-algebras A, a natural isomorphism

homgrp(π,SL2(A)) ' homsch/C(SpecA,Rep(π,SL2)).

The character variety of K is the geometric quotient

XK = Rep(π,SL2)//SL2 = Spec(C[Rep(π,SL2)]SL2(C)

),

via the obvious action of SL(2) on Rep(π,SL2) via conjugation. The representationρ is a point in XK , and one is interested in the connected component XK(ρ).

5.3. Deformation functors. The analogous situation in number theory is muchmore complicated, partly because ΓS = π1(Spec(Z) r S) is not a finitely pre-sented group – it’s a compact topological group which is (conjecturally) topologi-cally finitely presented. So instead of looking for representations ΓS → GL2(C), weshould look for continuous representations ΓS → GL2(A), where A is a topologicalZp-algebra.

Briefly, a scheme X over k can be thought of in terms of its functor of pointsX(−) : k-Alg → Set. In the arithmetic context, our deformation spaces will beformal schemes over Zp. For us, this just means that the test category consists ofcomplete local pro-artinian Zp-algebras with residue field Fp. If R is such an ring,we write Spf(R) to denote the functor A 7→ hom(R,A). There is a way of makingSpf(R) into a topological space with structure sheaf, but we will not need this.

The functorial approach to defining representation schemes works well. If π is

an arbitrary profinite group, there is a formal scheme Rep(π,GLn), satisfying

Rep(π,GLn)(A) = homcts(π,GLnA),

for any local pro-artinian Zp-algebra A with residue field Fp. The problem is,

Rep(π,GLn) is really horrible as a space – it generally has infinitely many differentconnected components. So before we do anything else, let’s restrict to the connectedcomponent of ρ, where ρ : π → GLn(Fp) has been fixed beforehand. The component

X�(ρ) represents continuous homomorphisms π → GLn(A) that reduce to ρ modulop.


As before, we will quotient out by the natural action of GL(n), but here weshould be careful because GL(n) does not preserve the component X�(ρ). Thecorrect thing to do is to first define

GLn(A) = {g ∈ GLn(A) : g ≡ 1 (mod p))} = ker (GLn(A)→ GLn(Fp)) .

The action of GL(n) on Rep(π,GLn) preserves X�(ρ). Now a miracle happens.Define

X(ρ)(A) = X�(A)/GLn(A).

Then in [Maz99, pr.1], it is proved that if ρ is absolutely irreducible and π satisfiesa certain technical hypothesis (which will hold for all our examples), the functorX(ρ) is represented by a complete local noetherian Zp-algebra Rρ with residue fieldFp. That is, there is a representation ρ : π → GLn(Rρ) lifting ρ such that any

GLn(A)-equivalence class of lifts π → GLn(A) is induced by a unique continuoushomomorphism Rρ → A.

Suppose we have fixed a subgroup I ⊂ π. We call a representation ρ : π →GL2(A) I-ordinary if ρI is a free, rank-one, direct summand of ρ. Suppose ρ isabsolutely irreducible and I-ordinary. Then we can define a subfunctor X◦(ρ) ofX(ρ) by

X◦(ρ)(A) = {ρ ∈ X(ρ)(A) : ρ is I-ordinary}.By [Maz99, pr.3], X◦(ρ) is represented by a complete local noetherian Zp-algebraR◦ρ with residue field Fp.

5.4. The case n = 1. The easiest example is when our representations take valuesin GL(1). Let π = π1(Z[ 1

p ]) and ρ = κ : π → GL1(Fp) be the mod-p cyclotomic

character. This is defined, for σ ∈ ΓQ, by

σ(ζp) = ζ κ(σ)p .

Let’s start by computing Rep(π,GL1). Deformations ρ : π → A× factor throughπab. Class field theory tells us that πab ' Z×p . So

Rep(π,GL1) = Spf(Zp

qZ×p

y)'

∐ε:π→F×

p

Spf (ZpJZpK) ,

where each Spf(ZpJZpK) is the connected component of some ε. We see that Rκ 'ZpJZpK ' ZpJXK, via [p]↔ X + 1.

5.5. The main example. Let U ⊂ Spec(Z) be open, and put π = π1(U). Let

Ee−→ U be an elliptic curve. Choose a prime p /∈ U . The p-torsion subscheme E[p]

is an etale cover of U , so we get an action of π on the underlying set E[p] ' (Z/p)2.This action preserves the group structure, so we have a representation

ρ = ρE,p : π1(U)→ GL2(Fp).

Another way of constructing ρ is to use the equivalence between etale-local Fp-systems on U and Fp-representations of π to realize Re∗Fp as a mod-p representa-tion of π.

We could consider the universal deformation ring Rρ and its associated formalspectrum X(ρ). Recall that if K ↪→ S3 is a hyperbolic knot, the “knot decomposi-tion group” DK = π1(torus) is abelian, so

ρK |DK ∼(ε ∗

ε−1

),


for some character ε : ΓK → C×. In particular, ε(IK) = 1. So to make the analogybetween knots and primes more precise, we should require that our residual Galoisrepresentation ρ : π → GL2(Fp) be p-ordinary in the sense that

ρ|Dp ∼(ϕ ∗

ψ

),

where ψ(Ip) = 1. This is exactly “Ip-ordinary” as defined above. So there isa p-ordinary representation ρ◦ : π → GL2(R◦ρ) that is universal for p-ordinarydeformations, i.e. Spf(R◦ρ) represents the functor

X◦(ρ)(A) = {ρ ∈ X(ρ)(A) : ρ is p-ordinary}.

6. Deformations of hyperbolic structures and Hida theory

We put together all the machinery we’ve developed to see an analogy between thespace of hyperbolic structures on a 3-manifold and the formal spectrum of Hida’sbig Hecke algebras.

6.1. Deformation of hyperbolic structures. Let K be a hyperbolic knot, M =S3 rK the knot complement. Put Γ = π1(M). Let TK = Teich(Γ) be the space ofinjective homomorphisms Γ→ Iso(H3) with discrete image, taken up to conjugacy;this is the space of hyperbolic structures on M . Since Iso(H3) = PSL2(C), we geta map

φ : TK → XK .

It’s not quite straightforward – you have to fudge for this to be well defined. On asmall neighborhood, T ◦K → X◦K is an isomorphism.

Also, X◦K → C, ρ 7→ tr ρ(m) is locally biholomorphic. Similarly, T◦(ρ)→ X◦(ρ),choose a generator τ of the Zp-quotient of π1(Z[ 1

p ]). Then ρ 7→ tr ρ is p-adically

bianalytic near ρ.

6.2. Hida theory. We showed explicitly how to construct the Galois representa-tions associated to cuspidal eigenforms of weight 2. In fact, in [Del73], Deligneshowed how to construct ρf,l for any cusp-eigenform f of weight k > 2. Recall thatsuch forms have a Fourier expansion

f(z) =∑n>1

an(f)e2πnz.

Say that f is p-ordinary if ap(f) is a p-adic unit. This is equivalent to ρf,p beingan extension of an unramified character by a character. In [Hid86a, Hid86b], Hidap-adically interpolated the Galois representations ρf,p coming from p-ordinary f ofvarying weight and level.

Fix a prime p > 5 and an integer n prime to p. Let f be a p-ordinary modularform of level npr. Then there is a p-adic family f of automorphic representationscontaining f . In fact, Hida explicitly constructs a completion h◦ of h such that forT◦ = Spf(h◦), all p-ordinary forms of tame level n which are congruent to f lie inT◦. Moreover, there is the associated Galois representation ρf : ΓQ → GL2(h◦),such that all p-ordinary modular Galois representations congruent to ρf,p comefrom h◦.


6.3. The analogy. Let f be a p-ordinary cuspidal eigenform. Put ρ = ρf,p. LetT◦ = Spf(h◦) be the corresponding p-adic family of modular forms, and ρf : ΓQ →GL2(h◦) the Galois representation. Since ρf ≡ ρf,p (mod p) and ρf is p-ordinary,we get a map R◦ρf → h◦, equivalently

φ : T◦ → X◦(ρ).

The very hard theorem of Wiles, etc. is that φ is an isomorphism if f satisfiessome technical hypotheses. This is the analogy of the map T ◦K → X◦K being anisomorphism in a small neighborhood.

References

[BJ79] A. Borel and H. Jacquet. Automorphic forms and automorphic representations. In Au-

tomorphic forms, representations and L-functions, volume 33, part 1 of Proc. Sympos.

Pure. Math., pages 189–207. Amer. Math. Soc., 1979.[Con12] Brian Conrad. Weil and Grothendieck approaches to adelic points. Enseign. Math. (2),

58(1-2):61–97, 2012.

[Del73] Pierre Deligne. Formes modulaires et representations de GL(2). In Modular functions ofone variable, II, volume 349 of Lecture Notes in Mathematics, pages 55–105. Springer,

1973.

[Fla79] D. Flath. Decomposition of representations into tensor products. In Automorphic forms,representations and L-functions, volume 33, part 1 of Proc. Sympos. Pure. Math., pages

179–183. Amer. Math. Soc., 1979.[FM95] Jean-Marc Fontaine and Barry Mazur. Geometric Galois representations. In Elliptic

curves, modular forms & Fermat’s last theorem (Hong Kong, 1993), Ser. Number The-

ory, I, pages 41–78. Int. Press, Cambridge, MA, 1995.[GS66] Oscar Goldman and Chih-han Sah. On a special class of locally compact rings. J. Alge-

bra, 4:71–95, 1966.

[Har85] Michael Harris. Arithmetic vector bundles and automorphic forms on Shimura varieties.I. Invent. Math., 82(1):151–189, 1985.

[Har87] Gunter Harder. Eisenstein cohomology of arithmetic groups. The case GL2. Invent.

Math., 89(1):37–118, 1987.[Hid86a] Haruzo Hida. Galois representations into GL2(ZpJXK) attached to ordinary cusp forms.

Invent. Math., 85(3):545–613, 1986.

[Hid86b] Haruzo Hida. Iwasawa modules attached to congruences of cusp forms. Ann. Sci. Ecole

Norm. Sup. (4), 19(2):231–273, 1986.

[Maz73] Barry Mazur. Notes on etale cohomology of number fields. Ann. Sci. Ecole Norm. Sup.(4), 6:521–552, 1973.

[Maz99] Barry Mazur. Deforming Galois representations. In Galois groups over Q, volume 16 of

Math. Sci. Res. Inst. Publ., pages 385–437. Springer, New York, 199.

[Mil13] James S. Milne. Lectures on etale cohomology (v2.21), 2013. Available atwww.jmilne.org/math/.

[Moo98] Ben Moonen. Models of Shimura varieties in mixed characteristics. In Galois represen-tations in arithmetic algebraic geometry (Durham, 1996), volume 254 of London Math.Soc. Lecture Note Ser., pages 267–350. Cambridge Univ. Press, 1998.

[Mor12] Masanori Morishita. Knots and primes. Universitext. Springer, 2012.[NSW08] Jurgen Neukirch, Alexander Schmidt, and Kay Wingberg. Cohomology of number fields,

volume 323 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, sec-

ond edition, 2008.[Sch09] Joachim Schwermer. The cohomological approach to cuspidal automorphic representa-

tions. In Automorphic forms and L-functions I. Global aspects, volume 488 of Contemp.

Math., pages 257–285. Amer. Math. Soc., 2009.[SU14] Christopher Skinner and Eric Urban. The Iwasawa main conjectures for GL2. Invent.

Math., 195(1):1–277, 2014.

[Urb11] Eric Urban. Eigenvarieties for reductive groups. Ann. of Math. (2), 174(3):1685–1784,2011.


Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY 14853,

USA

E-mail address: [email protected]

URL: http://www.math.cornell.edu/~dkmiller

Documents

Contents The analogy between knots and primes etale 2 Ypi.math.cornell.edu/~dkmiller/bin/primes-knots.pdfp) which has a presentation ˇtame 1 (Q p) = h˙;˝: ˙˝˙ 1 = ˝pi: We can