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

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

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

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 for all g in G . � g ·� = �·� � unitary continuous actions � continuous actions of GL n ( Q p ) on of Gal ( Q p / Q p ) on � � ? ↔ p -adic Banach spaces p -adic vector spaces of (usually) infinite dimension of dimension n

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 � � étale ϕ , Γ -modules � ∼ ↔ p -adic vector spaces over E of dimension n of dimension n then � unitary continuous actions of GL n ( Q p ) on étale ϕ , Γ -modules � � � → p -adic Banach spaces over E of dimension n of (usually) infinite dimension

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

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

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

Corollary Let E : = ring of p -adic power series in X ± 1 lifting F p (( t )) � continuous actions semilinear injective actions � � � of Gal ( Q p / Q cyc p ) on ∼ − − → of ϕ on vector spaces over E p -adic vector spaces 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 . �

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 t n where Γ ∼ ◮ Γ � E by t γ : = ( 1 + t ) γ − 1 = � � γ = Z ∗ � p n � continuous actions � semilinear injective actions � � of Gal ( Q p / Q p ) on of commutative ϕ and Γ ∼ − − → p -adic vector spaces on vector spaces over E Proof. By the preceding theorem for H = Gal ( Q p / Q cyc p ) using H � Q cyc Q p − − − p − − − Q p . Γ �

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

Let K be a finite extension of Q p with valuation ring o K . Denote ◮ 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 } . Theorem Then D 0 ( Z p , K ) → K ⊗ o K o K [[ X ]] as normed K -algebras. ∼ − − Proof. ◮ By density of the locally constant functions, that is, o K [ Z / p Z ] ∪ o K [ Z / p 2 Z ] ∪ . . . inside C 0 ( Z p , o K ) D 0 ( Z p , o K ) − o K [ Z / p n Z ] = : o K [[ Z p ]] , and ∼ → lim − − ← − ◮ by the Iwasawa isomorphism that maps the generator 1 of Z p to 1 + X ∼ → o K [[ X ]] . o K [[ Z p ]] − −

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

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

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 ) .

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 , p = Γ and p N � N by p = ϕ p � N by Z ∗ ◮ Z ∗ Thus, p N Z ∗ � � Z p p M : = , 0 1 that is, M + = p N Z ∗ p ⋉ Z p acts on N .

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 ∗ � Q p 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 )

p -adic Langlands program 1 From Characteristic 0 to p 2 Fourier Transform 3 Notes available at http://imj-prg.fr/~enno.nagel .