IV Topics in Number Theory - ℓ-adic representations

3ℓ-adic representations

IV Topics in Number Theory

3 `-adic representations

In this section, we shall discuss

-adic representations of the Galois group, which

often naturally arise from geometric situations. At the end of the section, we

will relate these to complex representations of the Weil–Langlands group, which

will be what enters the Langlands correspondence.

Definition

(

-adic representation)

Let

be a topological group. An

-adic

representation consists of the following data:

– A finite extension E/Q

;

– An E-vector space V ; and

– A continuous homomorphism ρ : G → GL

(V )

∼

(E).

In this section,we will always take

= Γ

, where

F/Q

is a finite

extension with p 6= `.

Example.

The cyclotomic character

cycl

: Γ

→ Z

⊆ Q

is defined by the

relation

cycl

(γ)

= γ(ζ)

for all

ζ ∈

with

= 1 and

γ ∈

. This is a one-dimensional

-adic

representation.

Example. Let E/K be an elliptic curve. We define the Tate module by

E = lim

←−

E[`

](

K), V

E = T

E ⊗

Then V

E is is a 2-dimensional `-adic representation of Γ

over Q

Example. More generally, if X/K is any algebraic variety, then

V = H

´et

(X ⊗

K, Q

)

is an `-adic representation of Γ

We will actually focus on the representations of the Weil group

instead of

the full Galois group

. The reason is that every representation of the Galois

group restricts to one of the Weil group, and since the Weil group is dense, no

information is lost when doing so. On the other hand, the Weil group can have

more representations, and we seek to be slightly more general.

Another reason to talk about the Weil group is that local class field theory

says there is an isomorphism

Art

: W

∼

So one-dimensional representations of

are the same as one-dimensional

representations of F

For example, there is an absolute value map

→ Q

, inducing a represen-

tation

→ Q

. Under the Artin map, this sends the geometric Frobenius

. In fact, ω is the restriction of the cyclotomic character to W

Recall that we previously defined the tame character. Pick a sequence

∈

by π

= π and π

n+1

= π

. We defined, for any γ ∈ Γ

(γ) =



γ(π

)



∈ lim

←−

(

F ) = Z

(1).

When we restrict to the inertia group, this is a homomorphism, independent

of the choice of (

), which we call the tame character . In fact, this map is

-equivariant, where Γ

acts on

by conjugation. In general, this still defines

a function Γ

→ Z

(1), which depends on the choice of π

Example.

Continuing the previous notation, where

is a uniformizer of

, we

let T

be the `

-torsion subgroup of

/hπi. Then

= hζ

, π

i/hπ

∼

(Z/`

The

th power map

→ T

n−1

is surjective, and we can form the inverse limit

, which is then isomorphic to

. This then gives a 2-dimensional

-adic

representation of Γ

In terms of the basis (ζ

), (π

), the representation is given by

γ 7→



cycl

(γ) t

(γ)

0 1



Notice that the image of



1 Z

0 1



. In particular, it is infinite. This cannot

happen for one-dimensional representations.

Perhaps surprisingly, the category of

-adic representations of

over

does not depend on the topology of

, but only the

as an abstract field. In

particular, if we take

, then after taking care of the slight annoyance that

it is infinite over

, the category of representations over

does not depend

on `!

To prove this, we make use of the following understanding of

-adic represen-

tations.

Theorem

(Grothendieck’s monodromy theorem)

Fix an isomorphism

(1)

∼

. In other words, fix a system (

) such that

n−1

. We then view

a homomorphism I

→ Z

via this identification.

Let

→ GL

(

) be an

-adic representation over

. Then there exists

an open subgroup

⊆ I

and a nilpotent

N ∈ End

such that for all

γ ∈ I

ρ(γ) = exp(t

(γ)N) =

∞

j=0

(γ)N)

In particular, ρ(I

) unipotent and abelian.

In our previous example, N =



0 1

0 0



Proof. If ρ(I

) is finite, let I

= ker ρ ∩ I

and N = 0, and we are done.

Otherwise, first observe that

is any compact group and

G → GL

(

)

is an

-adic representation, then

contains a

-invariant lattice, i.e. a finitely-

generated

-submodule of maximal rank. To see this, pick any lattice

⊆ V

Then ρ(G)L

is compact, so generates a lattice which is G-invariant.

Thus, pick a basis of an

-invariant lattice. Then

→ GL

(

)

restricts to a map I

→ GL

We seek to understand this group

(

) better. We define a filtration on

) by

= {g ∈ GL

) : g ≡ I mod `

which is an open subgroup of

(

). Note that for

k ≥

1, there is an

isomorphism

k+1

→ M

/`O

sending 1 +

. Since the latter is an

-group, we know

is a pro-

group.

Also, by definition, (G

)

⊆ G

k+1

Since

−1

(

) is open, we can pick an open subgroup

⊆ I

such that

(

)

⊆ G

. Recall that

(

) is the maximal pro-

quotient of

, because the

tame characters give an isomorphism

∼

`-p

(1).

So ρ|

: I

→ G

factors as

) = `

` ν

using the assumption that ρ(I

) is infinite.

Now for r ≥ s, let T

= ν(`

) = T

r−s

∈ G

r+2−s

. For r sufficiently large,

= log(T

) =

m≥1

(−1)

m−1

− 1)

converges `-locally, and then T

= exp N

We claim that

is nilpotent. To see this, if we enlarge

, we may assume

that all the eigenvalues of N

are in E. For δ ∈ W

and γ ∈ I

, we know

(δγδ

−1

) = ω(δ)t

(γ).

ρ(δγδ

−1

) = ρ(γ)

w(σ)

for all γ ∈ I

. So

ρ(σ)N

ρ(δ

−1

) = ω(δ)N

Choose

lifting

(

) =

. Then if

is an eigenvector for

with eigenvalue

, then

(

)

is an eigenvector of eigenvalue

−1

. Since

has finitely many

eigenvalues, but we can do this as many times as we like, it must be the case

that λ = 0.

Then take

N =

for r sufficiently large, and this works.

There is a slight unpleasantness in this theorem that we fixed a choice of

roots of unity. To avoid this, we can say there exists an

(1) =

V ⊗

(1)

→

V nilpotent such that for all γ ∈ I

, we have

ρ(γ) = exp(t

(γ)N).

Grothendieck’s monodromy theorem motivates the definition of the Weil–

Deligne groups, whose category of representations are isomorphic to the category

of `-adic representations. It is actually easier to state what the representations

of the Weil–Deligne group are. One can then write down a definition of the

Weil–Deligne group as a semi-direct product if they wish.

Definition

(Weil–Deligne representation)

A Weil–Deligne representation of

over a field E of characteristic 0 is a pair (ρ, N), where

– ρ

→ GL

(

) is a finite-dimensional representation of

over

with open kernel; and

– N ∈ End

(V ) is nilpotent such that for all γ ∈ W

, we have

ρ(γ)Nρ(γ)

−1

= ω(γ)N,

Note that giving

is the same as giving a unipotent

exp N

, which

is the same as giving an algebraic representation of

. So a Weil–Deligne

representation is a representation of a suitable semi-direct product W

n G

Weil–Deligne representations form a symmetric monoidal category in the

obvious way, with

(ρ, N) ⊗(ρ

, N

) = (ρ ⊗ ρ

, N ⊗ 1 + 1 ⊗ N).

There are similarly duals.

Theorem.

Let

E/Q

be finite (and

` 6

). Then there exists an equivalence of

(symmetric monoidal) categories



`-adic representations

of W

over E



←→



Weil–Deligne representations

of W

over E



Note that the left-hand side is pretty topological, while the right-hand side

is almost purely algebraic, apart from the requirement that

has open kernel.

In particular, the topology of E is not used.

Proof.

We have already fixed an isomorphism

(1)

∼

. We also pick a lift

Φ ∈ W

of the geometric Frobenius. In other words, we are picking a splitting

= hΦi n I

The equivalence will take an

-adic representation

to the Weil–Deligne repre-

sentation (ρ, N ) on the same vector space such that

(Φ

γ) = ρ(Φ

γ) exp t

(γ)N (∗)

for all m ∈ Z and γ ∈ I

To check that this “works”, we first look at the right-to-left direction. Suppose

we have a Weil–Deligne representation (

ρ, N

) on

. We then define

→

Aut

(

) by (

∗

). Since

has open kernel, it is continuous. Since

is also

continuous, we know

is continuous. To see that

is a homomorphism,

suppose

γ · Φ

δ = Φ

m+n

where γ, δ ∈ I

and

= Φ

−n

γΦ

Then

exp t

(γ)N ·ρ(Φ

δ) =

j≥0

(γ)

ρ(Φ

δ)

j≥0

(γ)q

ρ(Φ

δ)N

= ρ(Φ

δ) exp(q

(γ)).

But

(γ

) = t

(Φ

−n

γΦ

) = ω(Φ

−n

(γ) = q

(γ).

So we know that

(Φ

γ)ρ

(Φ

δ) = ρ

(Φ

m+n

δ).

Notice that if

γ ∈ I

∩ ker ρ

, then

(

) =

exp t

(

)

. So

is the nilpotent

endomorphism occurring in the Grothendieck theorem.

Conversely, given an

-adic representation

, let

N ∈ End

be given by

the monodromy theorem. We then define

by (

∗

). Then the same calculation

shows that (

ρ, N

) is a Weil–Deligne representation, and if

⊆ I

is the open

subgroup occurring in the theorem, then

(

) =

exp t

(

)

for all

γ ∈ I

. So

by (∗), we know ρ(I

) = {1}, and so ρ has open kernel.

This equivalence depends on two choices — the isomorphism

(1)

∼

and

also on the choice of Φ. It is not hard to check that up to natural isomorphisms,

the equivalence does not depend on the choices.

We can similarly talk about representations over

, instead of some finite

extension

. Note that if we have a continuous homomorphism

→

(

), then there exists a finite E/Q

such that ρ factors through GL

(E).

Indeed,

(

)

⊆ GL

(

) is compact, since it is a continuous image of

compact group. So it is a complete metric space. Moreover, the set of finite

extensions of

E/Q

is countable (Krasner’s lemma). So by the Baire category

theorem,

(

) is contained in some

(

), and of course,

(Φ) is contained

in some GL

(E).

Recalling that a Weil–Deligne representation over

only depends on

as a

field, and

∼

for any `, `

, we know that

Theorem.

Let

`, `

. Then the category of

representations of

equivalent to the category of

representations of W

Conjecturally,

-adic representations coming from algebraic geometry have

semi-simple Frobenius. This notion is captured by the following proposi-

tion/definition.

Proposition.

Suppose

is an

-adic representation corresponding to a Weil–

Deligne representation (ρ, N). Then the following are equivalent:

(i) ρ

(Φ) is semi-simple (where Φ is a lift of Frob

(ii) ρ

(γ) is semi-simple for all γ ∈ W

\ I

(iii) ρ is semi-simple.

(iv) ρ(Φ) is semi-simple.

In this case, we say

and (

ρ, N

) are

-semisimple (where

refers to Frobenius).

Proof.

Recall that

∼

Z o I

, and

(

) is finite. So that part is always

semisimple, and thus (iii) and (iv) are equivalent.

Moreover, since

(Φ) =

(Φ), we know (i) and (iii) are equivalent. Finally,

(Φ) is semi-simple iff

(Φ

) is semi-simple for all Φ. Then this is equivalent

to (ii) since the equivalence before does not depend on the choice of Φ.

Example.

The Tate module of an elliptic curve over

is not semi-simple, since

it has matrix

(γ) =



ω(γ) t

(γ)

0 1



However, it is F -semisimple, since

ρ(γ) =



ω(γ) 0

0 1



, N =



0 1

0 0



It turns out we can classify all the indecomposable and

-semisimple Weil–

Deligne representations. In the case of vector spaces, if a matrix

acts nilpo-

tently on a vector space

, then the Jordan normal form theorem says there is a

basis of

in which

takes a particularly nice form, namely the only non-zero

entries are the entries of the form (

i, i

+ 1), and the entries are all either 0 or 1.

In general, we have the following result:

Theorem

(Jordan normal form)

is semi-simple,

N ∈ End

(

) is nilpotent

with

m+1

= 0, then there exists subobjects

, . . . , P

⊆ V

(not unique as

subobjects, but unique up to isomorphism), such that

→ N

is an

isomorphism, and N

r+1

= 0, and

V =

r=0

⊕ N P

⊕ ··· ⊕ N

r=0

⊗

Z[N]

r+1

)

For vector spaces, this is just the Jordan normal form for nilpotent matrices.

Proof.

If we had the desired decomposition, then heuristically, we want to set

to be the things killed by

but not in the image of

. Thus, using semisimplicity,

we pick P

to be a splitting

ker N = (ker N ∩ im N) ⊕ P

Similarly, we can pick P

ker N

= (ker N + (im N ∩ ker N

)) ⊕ P

One then checks that this works.

We will apply this when

is a representation of

→ GL

(

) and

is the

nilpotent endomorphism of a Weil–Deligne representation. Recall that we had

ρ(γ)Nρ(γ)

−1

= ω(γ)N,

is a map

V → V ⊗ ω

−1

, rather than an endomorphism of

. Thankfully,

the above result still holds (note that

V ⊗ω

−1

is still the same vector space, but

with a different action of the Weil–Deligne group).

Proposition. Let (ρ, N) be a Weil–Deligne representation.

(i) (ρ, N) is irreducible iff ρ is irreducible and N = 0.

(ii) (ρ, N) is indecomposable and F -semisimple iff

(ρ, N) = (σ, 0) ⊗ sp(n),

where

is an irreducible representation of

and

(

)

∼

is the

representation

ρ = diag(ω

n−1

, . . . , ω, 1), N =







0 1







Example. If

ρ =





ω ω





, N =





0 0 0

0 0 1

0 0 0





then this is an indecomposable Weil–Deligne representation not of the above

form.

Proof. (i) is obvious.

For (ii), we first prove (

⇐

). If (

ρ, N

) = (

σ,

⊗ sp

(

), then

-semisimplicity

is clear, and we have to check that it is indecomposable. Observe that the kernel

is still a representation of

. Writing

N=0

for the kernel of

, we

note that V

N=0

= σ ⊗ω

n−1

, which is irreducible. Suppose that

(ρ, N) = U

⊕ U

Then for each

, we must have

N=0

= 0 or

N=0

. We may wlog assume

N=0

= 0. Then this forces U

= 0. So we are done.

Conversely, if (

ρ, N, V

) is

-semisimple and indecomposable, then

is a

representation of

which is semi-simple and

V → V ⊗ ω

−1

. By Jordan

normal form, we must have

V = U ⊕ NU ⊕ ··· ⊕ N

with N

r+1

= 0, and U is irreducible. So V = (σ, 0) ⊗ sp(r + 1).

Given this classification result, when working over complex representations,

the representation theory of

(2) lets us capture the

bit of

-semisimple

Weil–Deligne representation via the following group:

Definition

(Weil–Langlands group)

We define the (Weil–)Langlands group to

= W

× SU(2).

A representation of

is a continuous action on a finite-dimensional vector

space (thus, the restriction to W

has open kernel).

Theorem.

There exists a bijection between

-semisimple Weil–Deligne repre-

sentations over

and semi-simple representations of

, compatible with tensor

products, duals, dimension, etc. In this correspondence:

–

The representations

that factor through

correspond to the

Weil–Deligne representations (ρ, 0).

–

More generally, simple

representations

σ ⊗

(

Sym

n−1

) correspond to

the Weil–Deligne representation (σ ⊗ ω

(−1+n)/2

, 0) ⊗ sp(n).

If one sits down and checks the theorem, then one sees that the twist in the

second part is required to ensure compatibility with tensor products.

Of course, the (

-semisimple) Weil–Deligne representations over

are in

bijection those over

, using an isomorphism

∼