Functorial Construction E := a ring (in fact field) of p -adic power - - PowerPoint PPT Presentation

Fourier Transformation in the p-adic Langlands program — p-ADICS 2015

Enno Nagel http://www.math.jussieu.fr/~nagel

Belgrade, 7 September 2015

1

p-adic Langlands program

2

From Characteristic 0 to p

3

Fourier Transform

Number Theory ... global Langlands local Langlands p-adic Langlands p-adic linear group actions on a p-adic Banach space

• f usually infinite dimension

p-adic Galois group actions on a p-adic vector space

• f finite dimension

...

p-adic Langlands correspondence

p-adic vector space := vector space over (an extension of) Qp p-adic Banach space := complete normed p-adic vector space

Definition

An action of a group G on a normed space with norm · is unitary if

g · = ·

for all g in G.

continuous actions

• f Gal(Qp/Qp) on

p-adic vector spaces

• f dimension n
• ?

↔ unitary continuous actions

• f GLn(Qp) on

p-adic Banach spaces

• f (usually) infinite dimension

Functorial Construction

◮ E:= a ring (in fact field) of p-adic power series in X±1 ◮ étale ϕ, Γ-module over E:= a module over Ewith a

semilinear action of two commuting matrices ϕ and Γ First

continuous actions

• f Gal(Qp/Qp) on

p-adic vector spaces

• f dimension n
• ∼

↔

• étale ϕ, Γ-modules
• ver Eof dimension n
• then
• étale ϕ, Γ-modules
• ver Eof dimension n
• →

unitary continuous actions

• f GLn(Qp) on

p-adic Banach spaces

• f (usually) infinite dimension

1

p-adic Langlands program

2

From Characteristic 0 to p

3

Fourier Transform

Cyclotomic Extension

Put

◮ 1, ζp, ζp2, . . . := roots of unity of p-power order ◮ Qcyc

p

:= Qp(1, ζp, ζp2, . . .)

Then

Qp − − −

H Qcyc

p − − −

Γ

Qp

where

Γ := Gal(Qcyc

p /Qp)

∼

− − → Z∗

p

σ → x given by ζσ = ζx for all ζ = 1, ζp, ζp2 . . .

From characteristic 0 to p

Theorem (Field of Norms)

The absolute Galois groups of Fp((t)) and Qcyc

p

are isomorphic. Put ϕ := Frobenius of Fp((t))

Theorem

Let E be a field of characteristic p.

continuous actions

• f Gal(E/E) on

vector spaces over Fp

• ∼

− − →

• semilinear injective actions
• f ϕ on vector spaces over E

Corollary

Let E:= ring of p-adic power series in X±1 lifting Fp((t))

continuous actions

• f Gal(Qp/Qcyc

p ) on

p-adic vector spaces

• ∼

− − →

• semilinear injective actions
• f ϕ on vector spaces over E
• Proof.

By the preceding theorem using

◮ Gal(Qp/Qcyc

p ) ∼

= Gal(Fp((t))/Fp((t))), and

◮ lifting the vector space coefficients from Fp to Qp by

applying the functor of Witt vectors and inverting p.

Theorem (Fontaine)

Let E:= ring of p-adic power series in X±1 lifting Fp((t)) with

◮ ϕ Eby t ϕ := (1 + t)p − 1, and ◮ Γ Eby t γ := (1 + t)γ − 1 = γ

n

• t n where Γ ∼

= Z∗

p

continuous actions

• f Gal(Qp/Qp) on

p-adic vector spaces

• ∼

− − → semilinear injective actions

• f commutative ϕ and Γ
• n vector spaces over E
• Proof.

By the preceding theorem for H = Gal(Qp/Qcyc

p ) using

Qp − − −

H Qcyc

p − − −

Γ

Qp.

1

p-adic Langlands program

2

From Characteristic 0 to p

3

Fourier Transform

Let K be a finite extension of Qp with valuation ring oK. Denote

◮ C0(Zp, K) := { all continuous f : Zp → K}, and ◮ D0(Zp, K) := { all continuous linear ν: C0(Zp, K) → K}.

Theorem

Then D0(Zp, K)

∼

− − → K ⊗oK oK[[X]] as normed K-algebras. Proof.

◮ By density of the locally constant functions, that is,

• K[Z/pZ] ∪ oK[Z/p2Z] ∪ . . . inside C0(Zp, oK)

D0(Zp, oK)

∼

− − → lim ← − − oK[Z/pnZ] =: oK[[Zp]],

and

◮ by the Iwasawa isomorphism that maps the generator 1 of

Zp to 1 + X

• K[[Zp]]

∼

− − → oK[[X]].

Mahler Basis

Theorem (Schikhof Duality)

Let ·

n

: Zp → K with x

n

:= x(x − 1) · · · (x − n + 1)/n!. Then { all zero sequences over K }

∼

− − → C0(Zp, K) (an) →

• an

·

n

• Proof.

Because D0(Zp, K)

∼

− − → { all bounded sequences over K }.

Let r ≥ 0. Denote

◮ C

r(Zp, K) := { all r -times differentiable f : Zp → K},

◮ Dr(Zp, K) := { all continuous linear ν: C

r(Zp, K) → K},

and dr(N, K) := { all

• anXn in K[[X]] with {|an|/nr } bounded }.

Theorem

We have Dr(Zp, K)

∼

− − → dr(N, K) as normed K-vector spaces.

Back to ϕ, Γ-modules

Let n = 2, that is,

V = K ⊕ K.

If Gal(Qp/Qp) V is “effective crystalline” then its

ϕ, Γ-module D over Eis

◮ base extended from

a ϕ, Γ-module N over d0(N, K), and

◮ and, for some r, s ≥ 0,

N is a ϕ, Γ-submodule of dr(N, K) ⊕ ds(N, K).

Matrix Action

Fourier transform the ϕ, Γ-module N is

◮ a module over D0(Zp, K), ◮ a submodule of Dr(Zp, K) ⊕ Ds(Zp, K).

In particular

◮ Zp N by δx in D0(Zp, K) for all x in Zp, ◮ Z∗

p N by Z∗ p = Γ and pN N by p = ϕ

Thus,

M :=

• pNZ∗

p

Zp 1

• ,

that is, M+ = pNZ∗

p ⋉ Zp acts on N.

N ¯ N = submodule of Dr(Qp, K) ⊕ Ds(Qp, K) over D0(Qp, K)

Then there is an action of

Q∗

p

Qp Q∗

p

• n ¯

N which extends uniquely to one of GL2(Qp) on ¯ N.

Dualizing gives the sought-for Banach space

¯ N GL2(Qp) B := subquotient of C

r(Qp, K) ⊕ Cs(Qp, K)

1

p-adic Langlands program

2

From Characteristic 0 to p

3

Fourier Transform

Notes available at

http://imj-prg.fr/~enno.nagel.