Arithmetic and Differential Swan conductors. The Rank one case via - PowerPoint PPT Presentation

Arithmetic and Differential Swan conductors. The Rank one case via π -exponentials. Andrea Pulita (Joint work with B.Chiarellotto ) Cetraro, 2007 October 8

SUMMARY • Arithmetic Swan conductor – Kato’s definition in the non perfect case • Differential Swan conductor – Kedlaya’s definition in the non perfect case • Co-monomials and explicit description of H 1 ( k ( ) , Q p / Z p ) ( t ) • π − exponentials as solutions of Differential equations – Explicit computation of the monodromy functor in the rank one case  • Decomposition in pure co-monomials    Proof : • Radius of the differential equation    attached to a pure co-monomial

� � � � � � � � � � NOTATIONS K :=finite extension of Q p , F q :=residue field of K , k :=field containing F q , E :=c.d.v.f. with residue field k , O L 0 :=a Cohen ring of k , O E L 0 :=a Cohen ring of E O L := O L 0 ⊗ W ( F q ) O K , O E L := O L ⊗ O L 0 O E L 0 � O L � O E L . O K = O Z p W ( F q ) O E L 0 O L 0 � F q � k � E ∼ F p = k ( ( t ) ) • E ∼ = k ( ( t ) ) (once we have chosen an uniformizer element t ) = { � a i T i | lim i →−∞ a i = 0 , ∀ i | a i | ≤ 1 , a i ∈ L (0) } • O E L (0) ∼ (Amice-Fontaine ring)

NOTATIONS • A L ( I ) = { � + ∞ −∞ a i T i , s.t. lim i →±∞ | a i | ρ i =0 , ∀ ρ ∈ I } (= Analytic Funct. on the annulus | T | ∈ I ) • R L := � ε> 0 A L (]1 − ε, 1[) (= Robba ring) • E † L,T = E † L := R L ∩ E L (= Bounded Robba ring). • ϕ :=Lifting of the Frobenius x �→ x q • G E := Gal(E sep / E), I E :=inertia, P E :=wild inertia • Rep fin O K (G E ) = { α : G E → GL ( V ) | V = finite free O K − module, such that α ( I E ) is finite }

FONTAINE’s EQUIVALENCE: THE PERFECT CASE • Assume k =perfect. • (Fontaine-Tsuzuki’s) classical case (1998): D † : Rep fin ∼ O K (G E ) − − − → ( ϕ, ∇ ) − Mod( O E † L / O L ) where: L / O L ) := { (D , ϕ D , ∇ ) | D := finite free / O E † • ( ϕ, ∇ ) − Mod( O E † L , ϕ D : D → D is ϕ -semilinear ∇ : D → D ⊗ Ω 1 / O L connection } O E† L Note: If t =uniformizer of E ∼ = k ( ( t ) ), T =lifting of t in O E † L , then ∼ ∼ Ω 1 D ⊗ Ω 1 O E † L · dT , D · dT = = O E† / O L O E† / O L L L the data of ∇ : D → D ⊗ Ω 1 / O L is equivalent to a connection O E† L ∇ T : D → D .

FONTAINE’s EQUIVALENCE REVISITED: THE NON PERFECT CASE • Assume k =arbitrary. • Kedlaya’s generalization (December 2006): D † : Rep fin ∼ O K (G E ) − − − → ( ϕ, ∇ ) − Mod( O E † L / O K ) where: L ) := { (D , ϕ D , ∇ ) | with D := finite free / O E † • ( ϕ, ∇ ) − Mod( O E † L , ϕ D : D → D is ϕ -semilinear ∇ : D → D ⊗ Ω 1 / O K integrable } O E† L • If t =uniformizer of E ∼ = k ( ( t ) ), if { ¯ u 1 , . . . , ¯ u r } = p -basis of k , T, u 1 , . . . , u r ∈ O E † L are lifting of t, ¯ u 1 , . . . , ¯ u r , then � � ∼ Ω 1 O E † L · dT ⊕ ⊕ i =1 ,...,r O E † L · du i . = O E† / O K L

The data of ∇ : D → D ⊗ Ω 1 / O K is equivalent to a family of O E† L connections ∇ T : D → D ∇ u 1 : D → D ∇ u 2 : D → D · · · · · · · · · ∇ u r : D → D commuting with ϕ , and commuting between them. • Note: If k is perfect, then Ω 1 / O L = Ω 1 / O K , hence this O E† O E† L L theory refines that of Fontaine-Tsuzuki. • We are interested only to these differential equations. We set L / O K ) = { (D , ∇ ) | ∇ : D → D ⊗ Ω 1 MCF( O E † / O K , + commutations , O E† L with an (unspecified) Frobenius ϕ D : ϕ ∗ (D) → D }

� � � � � Perfect case VS non perfect case Assume k non necessarily perfect. We notice that ∼ can � Gal( k ( ) sep /k ( Gal( k perf ( ) sep /k perf ( ( t ) ( t ) )) ( t ) ( t ) )) ∪ ∪ ∼ I k perf ( I k ( ( t ) ) ( t ) ) can hence: � � � � can Rep fin Rep fin Gal( k perf ( ) sep /k perf ( ) sep /k ( ( t ) ( t ) )) Gal( k ( ( t ) ( t ) )) O K O K ∼ ≀ ⊙ ≀ D † D † can ( ϕ, ∇ ) − Mod( O E † L perf / O L perf ) ( ϕ, ∇ ) − Mod( O E † L / O K ) ∼ where the last horizontal functor is given by L perf , ϕ D ⊗ ϕ, ∇ T ⊗ 1) (D , ϕ D , ∇ T , {∇ u i } i ) �− → (D ⊗ O E† L O E †

List of main results D † : Rep fin ∼ O K (G E ) − − − → ( ϕ, ∇ ) − Mod( O E † L / O K ) • On the left hand side: One has a complete description of H 1 (G E , Q / Z ) p − tor = H 1 (G E , Q p / Z p ) (Pulita 2006). • On the right hand side: We obtain a complete description of the group Pic Frob ( O E † L ) =group, under tensor product, of isomorphism classes of rank one objects in MCF( O E † L / O K ). ∼ • After choosing an identification ( Q p / Z p ) ⊃ ( Z /p n Z ) → µ p ∞ ( O K ) we make explicit the isomorphism ∼ Hom fin (G E , Q p / Z p ) ⊃ Hom fin (G E , ( O K ) × ) p -tor → Pic Frob ( O E † − L ) p -tor , induced by the functor D † (Pulita 2006). • Understanding of the Kato’s filtration on the L.H.S., and of the Kedlaya’s Irregularity on the R.H.S. proof of the equality (Kato) Swan Arithm = Swan Diff (Kedlaya) .

� � � � � � � Abbes-Saito’s filtration in rank one case: Kato’s filtration The Artin-Schreier-Witt theory describes H 1 (G E , Q p / Z p ): ¯ F − 1 δ 0 → Z /p m +1 Z Hom(G E , Z /p m +1 Z ) → 0 W m (E) W m (E)  ı ⊙ ⊙ ⊙ V V F − 1 � W m +1 (E) ¯ δ � Hom(G E , Z /p m +2 Z ) → 0 � W m +1 (E) 0 → Z /p m +2 Z ↓ ↓ ↓ ¯ F − 1 δ → Hom cont (G E , Q p / Z p ) → 0 0 → Q p / Z p − → CW (E) − − − → CW (E) − where � � V V CW (E) = lim W m (E) − → W m +1 (E) − → · · · , − → m and where Hom cont means that a character α : G E → Q p / Z p factorizes by a finite quotient of G E . Note : Elements in CW (E) are ( · · · , 0 , 0 , f 0 , . . . , f m ) , f i ∈ E.

Abbes-Saito’s filtration in rank one case: Kato’s filtration • K.Kato (1989) defined a filtration on H 1 (G E , Q / Z ) in 3 steps: 1) The setting v ( · · · , 0 , 0 , f 0 , . . . , f m ) := min( v t ( f 0 ) /p m , v t ( f 1 ) /p m − 1 , · · · , v t ( f m )) defines a valuation on CW (E). Define a filtration on CW (E) as: Fil d ( CW (E)) = { c := ( · · · , 0 , 0 , f 0 , . . . , f m ) | v ( c ) ≥ − d } . 2) Thank to the A-S-W sequence ¯ F − 1 δ → H 1 (G E , Q p / Z p ) → 0 . 0 → Q p / Z p → CW (E) − − − → CW (E) − Fil d (H 1 (G E , Q p / Z p )) := δ (Fil d ( CW (E))) . we define: Hom cont (G E , Q / Z ) ⊕ ℓ =prime Hom cont (G E , Q ℓ / Z ℓ ) 3) = ⇒ Fil d (H 1 (G E , Q / Z )) Inverse image of Fil d (H 1 (G E , Q p / Z p )) . = := � � d ≥ 0 | α ∈ Fil d (H 1 (G E , Q / Z )) Swan Arithm ( α ) = min 4)

Description of the Kato’s filtration of H 1 (G E , Q p / Z p ) • Once chosen a uniformizer element t ∈ E, one has E ∼ = k ( ( t ) ) and )) = CW ( t − 1 k [ t − 1 ]) ⊕ CW ( k ) ⊕ CW ( tk [ CW ( k ( ( t ) [ t ] ]) • One has CW (E) ∼ H 1 (E , Q p / Z p ) = (¯ F − 1)( CW (E)) H 1 (G k , Q p / Z p ) � �� � CW ( t − 1 k [ t − 1 ]) CW ( k ) = F − 1)( CW ( t − 1 k [ t − 1 ])) ⊕ F − 1)( CW ( k )) . (¯ (¯  CW ( t − 1 k [ t − 1 ]) = Pontriagyn dual of P G ab  (¯ F − 1)( CW ( t − 1 k [ t − 1 ]))  E     CW ( k ) (G ab = Pontriagyn dual of k ) p -tor   (¯ F − 1)( CW ( k ))    H 1 (G k , Q p / Z p )  =

Description of the Kato’s filtration of CW (E) Definition (Pulita 2006): A co-monomial of degree − d is a co-vector of the form λ t − d := ( · · · , 0 , 0 , λ 0 t − n , λ 1 t − np , . . . , λ m t − np m ) ∈ CW ( k ( ( t ) )) where d = np m , ( n, p ) = 1, λ := ( λ 0 , . . . , λ m ) ∈ W m ( k ). We call CW ( − d ) ( k ) the sub-group of CW (E) formed by such elements. Proposition: CW (E) together with the Kato’s filtration is graduated: CW (E) := ⊕ d ≥ 0 Gr d ( CW (E)) . Moreover:  CW ( k [ [ t ] ]) if d = 0 ,    Gr d ( CW (E)) =   CW ( − d ) ( k )  if d > 0 .

� � � � Note: For all char. p ring R (not necessarily with unit el.t) one has CW ( R ) � � ¯ ¯ F F F − 1)( CW ( R )) = lim CW ( R ) − → CW ( R ) − → · · · (¯ − → • Hence we are interested to the action of ¯ F . Gr 0 ( CW (E)) is a sub-¯ F-module, while ¯ F( CW ( − d ) ( k )) CW ( − pd ) ( k ) , ⊂ in fact ¯ F → ( · · · , 0 , 0 , λ p ( · · · , 0 , 0 , λ 0 t − n , . . . , λ m t − d ) 0 t − pn , . . . , λ p m t − pd ) �− • If d = np m , ( n, p ) = 1, one has a isomorphism: ∼ � ( λ 0 , . . . , λm ) CW ( − d )( k ) � W m ( k ) ( · · · , 0 , 0 , λ 0 t − n, . . . , λmt − d ) V¯ ¯ ¯ · p F= p F F ∼ � (0 , λp 0 , . . . , λp ( · · · , 0 , 0 , λp 0 t − pn, . . . , λp CW ( − pd )( k ) � W m +1( k ) mt − pd ) m )