Fourier Transformation in the p-adic Langlands program

Transcription

1 Fourier Transformation in the p-adic Langlands program p-adics 2015 Enno Nagel Belgrade, 7 September 2015

2 1 p-adic Langlands program 2 From Characteristic 0 to p 3 Fourier Transform

3 ... Number Theory global Langlands... p-adic Langlands local Langlands p-adic Galois group actions on a p-adic vector space of finite dimension p-adic linear group actions on a p-adic Banach space of usually infinite dimension

4 p-adic Langlands correspondence p-adic vector space := vector space over (an extension of) Q p 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 } of Gal(Q p /Q p ) on p-adic vector spaces of dimension n? { unitary continuous actions } of GL n (Q p ) on p-adic Banach spaces of (usually) infinite dimension

5 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 E with a semilinear action of two commuting matrices ϕ and Γ First { continuous actions } of Gal(Q p /Q p ) on p-adic vector spaces of dimension n { } étale ϕ, Γ-modules over E of dimension n then unitary continuous actions { } { } étale ϕ, Γ-modules of GL n (Q p ) on over E of dimension n p-adic Banach spaces of (usually) infinite dimension

6 1 p-adic Langlands program 2 From Characteristic 0 to p 3 Fourier Transform

7 Cyclotomic Extension Put 1, ζ p, ζ p 2,... := roots of unity of p-power order Q cyc p := Q p (1, ζ p, ζ p 2,...) Then where Q p H Qcyc p Γ Γ := Gal(Q cyc p /Q p ) Z p σ x given by ζ σ = ζ x for all ζ = 1, ζ p, ζ p 2... Q p

8 From characteristic 0 to p Theorem (Field of Norms) The absolute Galois groups of F p ((t )) and Q cyc p Put ϕ := Frobenius of F p ((t )) Theorem Let E be a field of characteristic p. { continuous actions } of Gal(E/E) on vector spaces over F p are isomorphic. { } semilinear injective actions of ϕ on vector spaces over E

9 Corollary Let E := ring of p-adic power series in X ±1 lifting F p ((t )) { continuous actions } of Gal(Q p /Q cyc p ) on p-adic vector spaces { } semilinear injective actions of ϕ on vector spaces over E Proof. By the preceding theorem using Gal(Q p /Q cyc p ) = Gal(F p ((t ))/F p ((t ))), and lifting the vector space coefficients from F p to Q p by applying the functor of Witt vectors and inverting p.

10 Theorem (Fontaine) Let E := ring of p-adic power series in X ±1 lifting F p ((t )) with ϕ E by t ϕ := (1 + t ) p 1, and Γ E by t γ := (1 + t ) γ 1 = γ n t n where Γ = Z p { continuous actions } of Gal(Q p /Q p ) on p-adic vector spaces { semilinear injective actions of commutative ϕ and Γ on vector spaces over E } Proof. By the preceding theorem for H = Gal(Q p /Q cyc p ) using Q p H Qcyc p Γ Q p.

11 1 p-adic Langlands program 2 From Characteristic 0 to p 3 Fourier Transform

12 Let K be a finite extension of Q p with valuation ring o K. Denote Theorem C 0 (Z p, K) := { all continuous f : Z p K}, and D 0 (Z p, K) := { all continuous linear ν: C 0 (Z p, K) K}. Then D 0 (Z p, K) K ok o K [[X]] as normed K-algebras. Proof. By density of the locally constant functions, that is, o K [Z/pZ] o K [Z/p 2 Z]... inside C 0 (Z p, o K ) D 0 (Z p, o K ) lim o K[Z/p n Z] =: o K [[Z p ]], and by the Iwasawa isomorphism that maps the generator 1 of Z p to 1 + X o K [[Z p ]] o K [[X]].

13 Mahler Basis Theorem (Schikhof Duality) Let n : Zp K x with n := x(x 1) (x n + 1)/n!. Then { all zero sequences over K } C 0 (Z p, K) (a n ) a n n Proof. Because D 0 (Z p, K) { all bounded sequences over K }.

14 Let r 0. Denote and C r (Z p, K) := { all r -times differentiable f : Z p K}, D r (Z p, K) := { all continuous linear ν: C r (Z p, K) K}, d r (N, K) := { all a n X n in K[[X]] with { a n /n r } bounded }. Theorem We have D r (Z p, K) d r (N, K) as normed K-vector spaces.

15 Back to ϕ, Γ-modules Let n = 2, that is, V = K K. If Gal(Q p /Q p ) V is effective crystalline then its ϕ, Γ-module D over E is base extended from a ϕ, Γ-module N over d 0 (N, K), and and, for some r, s 0, N is a ϕ, Γ-submodule of d r (N, K) d s (N, K).

16 Matrix Action Fourier transform the ϕ, Γ-module N is a module over D 0 (Z p, K), a submodule of D r (Z p, K) D s (Z p, K). In particular Z p N by δ x in D 0 (Z p, K) for all x in Z p, Z p N by Z p = Γ and p N N by p = ϕ Thus, M := ( p N Z ) p Z p, 0 1 that is, M + = p N Z p Z p acts on N.

17 N N = submodule of D r (Q p, K) D s (Q p, K) over D 0 (Q p, K) Then there is an action of ( ) Q p Q p Q p on N which extends uniquely to one of GL 2 (Q p ) on N. Dualizing gives the sought-for Banach space N GL 2 (Q p ) B := subquotient of C r (Q p, K) C s (Q p, K)

18 1 p-adic Langlands program 2 From Characteristic 0 to p 3 Fourier Transform Notes available at