Compatible systems along the boundary Weizhe Zheng Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences June 15, 2018 Weizhe Zheng Compatible systems along the boundary June 15, 2018 1 / 30
References K. Fujiwara. Independence of ℓ for intersection cohomology (after Gabber). In Algebraic geometry 2000, Azumino (Hotaka) , 2002. I. Vidal. Crit` eres valuatifs (d’apr` es O. Gabber). Appendice ` a “Courbes nodales et ramification sauvage virtuelle”, Manuscripta Math. , 2005. Weizhe Zheng Compatible systems along the boundary June 15, 2018 2 / 30
Serre’s conjectures on ℓ -independence Plan of the talk Serre’s conjectures on ℓ -independence 1 Compatible systems along the boundary 2 Relation with wild ramification 3 Weizhe Zheng Compatible systems along the boundary June 15, 2018 3 / 30
Serre’s conjectures on ℓ -independence References J.-P. Serre, J. Tate. Good reduction of abelian varieties. Ann. Math. (1968). J.-P. Serre. Facteurs locaux des fonctions zˆ eta des vari´ et´ es alg´ ebriques (d´ efinitions et conjectures). S´ eminaire Delange-Pisot-Poitou (1970). Serre proposed conjectures C1–C8 related to the definition of the Hasse-Weil zeta functions of projective smooth varieties over global fields. Weizhe Zheng Compatible systems along the boundary June 15, 2018 4 / 30
Serre’s conjectures on ℓ -independence Arithmetic zeta function Riemann zeta function: 1 1 � � ζ ( s ) = n s = 1 − p − s . n ≥ 1 p Weizhe Zheng Compatible systems along the boundary June 15, 2018 5 / 30
Serre’s conjectures on ℓ -independence Arithmetic zeta function Riemann zeta function: 1 1 � � ζ ( s ) = n s = 1 − p − s . n ≥ 1 p Let X be a scheme of finite type over Spec( Z ). Arithmetic zeta function: 1 1 � � ζ X ( s ) = ( NC ) s = 1 − ( Nx ) − s C ∈ Z eff x ∈|X| ( X ) 0 � Z X v (( Nv ) − s ) = v ∈| V | for X over V of finite type over Spec( Z ). Weizhe Zheng Compatible systems along the boundary June 15, 2018 5 / 30
Serre’s conjectures on ℓ -independence Cohomological interpretation Let X be a variety (= scheme separated of finite type) over a field k . For each ℓ � = char( k ), Grothendieck defined a finite-dimensional Q ℓ -vector k , Q ℓ ), equipped with a continuous action of Gal(¯ space H i ℓ, c = H i c ( X ¯ k / k ). Weizhe Zheng Compatible systems along the boundary June 15, 2018 6 / 30
Serre’s conjectures on ℓ -independence Cohomological interpretation Let X be a variety (= scheme separated of finite type) over a field k . For each ℓ � = char( k ), Grothendieck defined a finite-dimensional Q ℓ -vector k , Q ℓ ), equipped with a continuous action of Gal(¯ space H i ℓ, c = H i c ( X ¯ k / k ). Theorem (Grothendieck) Let X be a variety over k = F q . For each ℓ ∤ q, P i ,ℓ ( t ) ( − 1) i +1 , � Z X ( t ) = i where P i ,ℓ ( t ) = det(1 − Fr t , H i ℓ, c ) . Weizhe Zheng Compatible systems along the boundary June 15, 2018 6 / 30
Serre’s conjectures on ℓ -independence Weil conjectures (continued) Let X be a proper smooth variety over k = F q . Theorem (Deligne, C2) The reciprocal roots of P i ,ℓ are of weight i (algebraic numbers with all complex conjugates of absolute value q i / 2 ). Weizhe Zheng Compatible systems along the boundary June 15, 2018 7 / 30
Serre’s conjectures on ℓ -independence Weil conjectures (continued) Let X be a proper smooth variety over k = F q . Theorem (Deligne, C2) The reciprocal roots of P i ,ℓ are of weight i (algebraic numbers with all complex conjugates of absolute value q i / 2 ). Corollary (C1) P i ,ℓ ∈ Z [ t ] and is independent of ℓ . Weizhe Zheng Compatible systems along the boundary June 15, 2018 7 / 30
Serre’s conjectures on ℓ -independence Weil conjectures (continued) Let X be a proper smooth variety over k = F q . Theorem (Deligne, C2) The reciprocal roots of P i ,ℓ are of weight i (algebraic numbers with all complex conjugates of absolute value q i / 2 ). Corollary (C1) P i ,ℓ ∈ Z [ t ] and is independent of ℓ . Corollary Let X be a proper smooth variety over an arbitrary field k. Then the Betti number dim H i ( X ¯ k , Q ℓ ) is independent of ℓ � = char( k ) . Weizhe Zheng Compatible systems along the boundary June 15, 2018 7 / 30
Serre’s conjectures on ℓ -independence Hasse-Weil zeta function Let X be a proper smooth variety over a global field F . L i ( s ) ( − 1) i +1 , � ζ X ( s ) = i ℓ ) I v ) , � det(1 − Fr q − s v , ( H i L i ( s ) = v where v runs over finite places of F , and I v denotes the inertia group at v . Weizhe Zheng Compatible systems along the boundary June 15, 2018 8 / 30
Serre’s conjectures on ℓ -independence ℓ -independence Let K be a local field: a complete discrete valuation field of finite residue field F q . Let X be a proper smooth variety over K . Conjecture ℓ ) I K ) ∈ Z [ t ] and is independent of ℓ ∤ q. (Serre, C5) det(1 − Fr t , ( H i Weizhe Zheng Compatible systems along the boundary June 15, 2018 9 / 30
Serre’s conjectures on ℓ -independence ℓ -independence Let K be a local field: a complete discrete valuation field of finite residue field F q . Let X be a proper smooth variety over K . Conjecture ℓ ) I K ) ∈ Z [ t ] and is independent of ℓ ∤ q. (Serre, C5) det(1 − Fr t , ( H i (Serre-Tate, C8) For each lifting F ∈ Gal( ¯ K / K ) of Fr , det(1 − Ft , H i ℓ ) ∈ Z [ t ] and is independent of ℓ ∤ q. Weizhe Zheng Compatible systems along the boundary June 15, 2018 9 / 30
Serre’s conjectures on ℓ -independence Monodromy Weight Conjecture Let M denote the monodromy filtration. Conjecture Eigenvalues of F lifting Fr on gr M n H i ℓ are of weight i + n. Weizhe Zheng Compatible systems along the boundary June 15, 2018 10 / 30
Serre’s conjectures on ℓ -independence Monodromy Weight Conjecture Let M denote the monodromy filtration. Conjecture Eigenvalues of F lifting Fr on gr M n H i ℓ are of weight i + n. C8 + Monodromy Weight Conjecture ⇒ det(1 − Ft , gr M n H i ℓ ) ∈ Z [ t ] and is independent of ℓ ⇒ C5 Weizhe Zheng Compatible systems along the boundary June 15, 2018 10 / 30
Serre’s conjectures on ℓ -independence Monodromy Weight Conjecture Let M denote the monodromy filtration. Conjecture Eigenvalues of F lifting Fr on gr M n H i ℓ are of weight i + n. C8 + Monodromy Weight Conjecture ⇒ det(1 − Ft , gr M n H i ℓ ) ∈ Z [ t ] and is independent of ℓ ⇒ C5 (Monodromy Weight Conjecture ⇒ C6 + C7) Weizhe Zheng Compatible systems along the boundary June 15, 2018 10 / 30
Serre’s conjectures on ℓ -independence General residue field Let K be a complete discrete valuation field of residue field k . Let X be a proper smooth variety over K . Conjecture (Serre-Tate, C4) For each F ∈ I K , det(1 − Ft , H i ℓ ) ∈ Z [ t ] and is independent of ℓ � = char( k ) . Weizhe Zheng Compatible systems along the boundary June 15, 2018 11 / 30
Serre’s conjectures on ℓ -independence Local monodromy theorem Let X be a variety over K . Theorem (Grothendieck) An open subgroup of I K acts on H i ℓ, c unipotently. Weizhe Zheng Compatible systems along the boundary June 15, 2018 12 / 30
Serre’s conjectures on ℓ -independence Local monodromy theorem Let X be a variety over K . Theorem (Grothendieck) An open subgroup of I K acts on H i ℓ, c unipotently. (Deligne, Gabber, Illusie) There exists an open subgroup I ′ of I K , independent of ℓ , such that for every g ∈ I ′ , ( g − 1) i +1 acts by 0 on H i ℓ and H i ℓ, c . Weizhe Zheng Compatible systems along the boundary June 15, 2018 12 / 30
Serre’s conjectures on ℓ -independence Equal characteristic case Theorem (Deligne, Terasoma, Ito) Monodromy Weight Conjecture holds in equal characteristic. Weizhe Zheng Compatible systems along the boundary June 15, 2018 13 / 30
Serre’s conjectures on ℓ -independence Equal characteristic case (continued) Let K be a complete discrete valuation field of residue field k , both of characteristic p > 0. Let X be a proper smooth variety over K . Theorem (Lu-Z., C4) For each F ∈ I K , det(1 − Ft , H i ℓ ) ∈ Z [ t ] and is independent of ℓ � = p. (Deligne, Terasoma, Lu-Z., C8) Assume k = F q . For each lifting F ∈ Gal( ¯ K / K ) of Fr , det(1 − Ft , H i ℓ ) ∈ Z [ t ] and is independent of ℓ � = p. Weizhe Zheng Compatible systems along the boundary June 15, 2018 14 / 30
Serre’s conjectures on ℓ -independence Equal characteristic case (continued) Let K be a complete discrete valuation field of residue field k , both of characteristic p > 0. Let X be a proper smooth variety over K . Theorem (Lu-Z., C4) For each F ∈ I K , det(1 − Ft , H i ℓ ) ∈ Z [ t ] and is independent of ℓ � = p. (Deligne, Terasoma, Lu-Z., C8) Assume k = F q . For each lifting F ∈ Gal( ¯ K / K ) of Fr , det(1 − Ft , H i ℓ ) ∈ Z [ t ] and is independent of ℓ � = p. Corollary (C5) Assume k = F q . For each lifting F ∈ Gal( ¯ K / K ) of Fr , ℓ ) I K ) ∈ Z [ t ] and is independent of ℓ ∤ q. det(1 − Ft , ( H i Weizhe Zheng Compatible systems along the boundary June 15, 2018 14 / 30
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