Hodge theaters and label classes of cusps

Transcription

1 1 Hodge theaters and label classes of cusps Fucheng Tan for talks in Inter-universal Teichmüller Theory Summit 2016 July 21th, 2016

2 2 Initial Θ-data (F /F, X F, l, C K, V, V bad mod, ε) Recall the geometry: lz Y X µ l lz Y µ l X Z Z/lZ X {1} C deg l {1} C Aut K (C) Aut K (C/C) {1}. Aut K (X ) µ l {1}. Aut K (X ) Z/lZ {1}. Note: Π C K Π X K. We apply AAG to Π X K (called the Θ-approach ) so that Hom(H 2 ( X, Ẑ), Ẑ) = M X Θ = [ X, X ]/[[ X, X ], X ] M X l Θ. (cyclotomic rigidity) V mod V V(K), a chosen section of V(K) Vmod. ɛ is a cusp of C K arising from an element of Q. For v V bad, it is chosen that ɛ v corresponds to the canonical generator 1 of the quotient Π tp X Z.

3 3 κ-coric rational functions C K is a K-core a unique model over F mod : C Fmod. L := F mod or (F mod ) v with v V non mod. (similarly for archimedean places) L C : =function field of C L ( algebraic closures L C, L). Note C L A 1 L. For the proper smooth curve determined by some finite extension of L C, a closed point is called critical if it maps to (a closed point of C L cpt coming from) the 2-torsion points of E F. Among them, those not mapping to the cusp of C L is called strictly critical. A rational function f L C is κ-coric if (i) If f / L, then it has exactly 1 pole and 2 (distinct) zeros. (ii) The divisor of f is defined over a finite extension of F mod and avoids the critical points. (iii) f restricts to roots of unity at any strictly critical point of C L cpt. Every element of L can be realized as a value of a κ-coric function on C L at some non-critical L-valued point. f L C is κ-coric if f n is κ-coric for some n Z >0. f L C is κ -coric if c f is κ-coric for some c F mod (resp. O F mod,v ).

4 4 Frobenioid at a bad place v V bad Tempered Frobenoid F v, F birat. v Base category D v := B tp (X v ) B(K v ) =: Dv. F v D v left adjoint to the inclusion D v. The reciprocal of l-th root of the Frobenioid-theoretic theta function Ç å Θ v O (T birat Y ) = ker Aut(T birat Y ) Aut( Y v ) K v v Y v is determined category-theoretically by F v up to µ 2l (T birat Y ) lz-indeterminacy. v (lz Aut(T Y )) v F v p v -adic Frobenioid (base-field-theoretic hull) C v F v over D v. Θ v (» q v ) = q 1 2l v =: q v O (T X v )( O K v ). Constant section of the divisor monoid: N log Φ (q ) Φ Cv. v Φ C v := N log Φ (q ) v D v, p v -adic Frobenioid Cv Dv. q K v µ 2l ( )-orbit of characteristic splittings τ v v on Cv. (τv is a subfunctor of O ( ) : (Cv ) lin Mon.)

5 5 Ob(D v ) A A Θ := A Y v Ob(D v ) full subcategory D Θ v D v Y. v O (T birat ) A Θ O C Θ v : A Θ O (T A Θ) Θ v TA Θ. O C v O C Θ v, O C v O C Θ v compatible with q v TA Θ v TA, O (T A ) O (T A Θ). O C Θ v p v -adic Frobenioid Cv Θ over Dv Θ (a subcategory of F birat ). v Θ v µ 2l ( )-orbit of characteristic splittings τ Θ v on C Θ v. Split Frobenioids: (C v, τ v ) =: F v F Θ v := (C Θ v, τ Θ v ). Note: F v C v Fv since D v theta values, but F v (up to the lz-indeterminacy in Θ v ) Fv Θ.

6 6 Frobenioid at a good nonarchimedean place v V good B(X ) = D v D v := B(K v ). Monoid on D v : Φ Cv : Ob(D v ) Spec L ord(o L )pf Monoid on D v : Φ C v C v := F v. : Ob(D v ) Spec L ord(z p v ) pf C v. Formal symbol log p v log Θ p v characteristic splitting τ v. O C Θ v := O C v N log p v log Θ O C v := O C v N log p v. O C Θ v C Θ v, τ Θ v. Split Frobenioids: (C v, τ v ) =: F v Note: F v F v, F Θ v. F Θ v := (C Θ v, τ Θ v ).

7 7 Global realified Frobenioid associated to F mod C mod =realification of the Frobenioid associated to (F mod, trivial Galois extension). Prime(C mod ) V mod V. Φ C mod Φ C mod,v ord(o F mod,v ) pf R 0. p v log mod(p v ) Φ C mod,v. v V, have log Φ (p v ) Φ rlf C v The restriction functor C ρv given by p v. : C mod (C v ) rlf induces isomorphism of top. monoids ρ v : Φ C mod,v Φ rlf C v, log mod(p v ) log Φ(p v ) [K v : F mod,v ]. Formal symbol log Θ Φ C tht := Φ C mod log Θ, C tht, Prime(C tht) V mod. log mod(p v ) log mod(p v ) log Θ Φ C tht,v Φ C tht. The restriction functor C ρv : C tht (C v ) rlf induces isomorphism of top. monoids ρ v : Φ C tht,v Φ rlf C v, log mod(p v ) log Θ log Φ(p v ) log Θ, at v V good, [K v : F mod,v ] log mod(p v ) log Θ log Φ(p v ) log Θ v [K v : F mod,v ] log Φ (q v ), at v Vbad.

8 8 Θ-Hodge theaters and Θ-links between them Data: F v F v a category, for any v V. ( F v D v, D v, F v, F Θ v ) Cmod C mod a category (category-theoretically constructible Frobebioid strucuture) Prime( C mod ) V a bijection of sets. v V, an isomorphism of top. monoids ρ v : Φ C mod,v Φ rlf C v Require: F mod := Ä C mod, Prime( C mod) V, { F v }, { ρ v } ä F mod := Ä C mod, Prime(C mod) V, {F v }, {ρ v } ä. HT Θ := ( ) { F v } v V, F mod. F mod F tht ( F tht := (C tht, Prime(C tht) V, {F Θ v }, {ρ Θ v })). The Θ-link ( Θ ) is defined to be the full poly-isomorphism HT Θ Θ HT Θ ( Θ ) F tht F mod. n Θ v Θ n+1 q v is NOT a conventional evaluation map!

9 9 Cusp labels at local places D v D v a category (v V non ) or an Aut-holomorphic orbispace (v V arc ). Dv Dv a category (v V non ) or an object of TM (v V arc ). D-prime-strip D := { D v } v V. D -prime-strip D := { D v } v V. ( D D ) Morphisms between prime-strips := collection of morphisms between the constituent objects of prime-strips, indexed by V.. D v D v ( C v ) Recall that {cusps of D v }, {cusps of D v } are group-theoretic via π 1 ( D v ), π 1 ( D v ). A label class of cusps of D v := the set of cusps of D v over a nonzero cusp of D v (arising from a nonzero element of Q). (LabCusp( D v ) Aut Kv (X v /C v )-orbits of nonzero cusps of X v, for v bad.) LabCusp( D v ) F l as an F l -torsor. (F l Q) D v a canonical element η v LabCusp( D v ) determined by ɛ v.

10 10 Global cusp labels D := B(C K ). D D a category, V( D ) ( V(F )), V( D ) := V( D )/π 1 ( D ) ( V(K)). Recall that {cusps of D } is group-theoretic via π 1 ( D ). A label class of cusps of D = a nonzero cusp of D. LabCusp( D ) F l as an F l -torsor. For D = { D v } v V a D-prime-strip, a poly-morphism D D is a collection { D v D } v V of poly-morphisms D v D. v, w V,! isomorphism of F l -torsors LabCusp( D v ) LabCusp( D w ), s.t. η v η w. identify them, LabCusp( D) F l.

11 Model base-nf-bridges Aut( ab X F l) GL 2 (F l ) with chosen basis adapted to ab X F l E F [l] Q. (Note Π C K ab X.) Aut(C K )( Out(Π C K )) GL 2 (F l )/{1}. (Inn(Π C K ) acts by 1.) The model C Fmod Gal(K/F mod ) GL 2 (F l )/{1} Ç å (Aut(D ) ) Aut(C K ) /{1} Im(Gal(K/F 0 mod )) Ç å (Aut ɛ (D ) ) Aut ɛ (C K ) /{1} Im(Gal(K/F 0 1 mod )) Aut(D )/ Aut ɛ (D ) Aut(C K )/ Aut ɛ (C K ) F l. For v V non : (Look at the covers X v C v C K and X v C v C K.) α Aut(D v ) D v D v φ NF,v β φ NF,v α D D β Aut ɛ(d ) φ NF j φ NF v = {β φ NF,v α}, : D j = {D v j =(v,j)} {φnf v } D j D, j F l, φ NF (analogue for v V arc ) = {φ NF j } j F : D = {D j } l j F D, l F l -equivariant.

12 Model base-θ-bridges v V bad : F l = F l /{1} = F l {0} {cusps of C v }. µ X v (K v ) =image of 1 under Tate uniformization. Evaluation points of X v =µ -translations of the cusps with labels in F l. Θ v (a point over an evaluation point with label j F l ) µ 2l -orbit of ßq j2 v j j,. j Z By definition of X v X v, the points of X v over the evaluation points of X v are all defined over K v. We call them the evaluation points of X v. evaluation sections G v Π v := Π tp X v, (group-theoretic via Π v ). D > = {D >,v } a copy of the D-prime-strip D = {D v }. For any j F l, v V good : φ Θ arbitrary iso. v j : D v j B tp (Π v ) B tp (Π v ) arbitrary iso. D >,v natural sur. ev. sec. with label j B(G v ) φ Θ v j : D v j D >,v is defined to be the full poly-isomorphism. Finally, φ Θ j = {φ Θ v j } v V : D j = {D v j =(v,j)} D > = {D >,v } v V, φ Θ = {φ Θ j } j F l : D = {D j } j F l 12 D >.

13 13 Transport of label classes of cusps via model base-bridges For v V non, look at Π X v Π X v Π C K and Π X v Π X v Π C K. The arrow φ NF v j : D v j D π 1 (D v j ) π 1 (D ), isomorphism of F l -torsors on cusp labels LabCusp(D ) LabCusp(D v j ). (Consider the cuspidal inertias I π 1 (D ) whose unique index l subgroup Im(Π X v ) (resp. Im(Π X v )). Similarly for archimedean places. In summary, for any v V and j F l, have isomorphisms of F l -torsors: LabCusp(D ) φ NF v j LabCusp(D v j ) φ LC j = LabCusp(D j ) φ Θ j LabCusp(D > ) LabCusp(D ) φlc j LabCusp(D > ) F l, [ɛ] j.

14 14 Base-NF-bridges A D-NF-bridge is a poly-morphism φ NF : D J = { D j } j J D from a capsule of D-prime-strips to a category equivalent to D, which fits into the following commutative diagram: φ NF D J D φ NF D D A morphism of D-NF-bridges is a pair of poly-morphisms fitting into the following diagram: D J φ NF D capsule-full poly-iso. D J φ NF D Aut ɛ( D )-orbit of iso. Isom( φ NF, φ NF ) forms an F l -torsor.

15 Base-Θ-bridges A D-Θ-bridge is a poly-morphism φ Θ : D J = { D j } j J D > from a capsule of D-prime-strips to a D-prime-strip, which fits into the following commutative diagram: D J φ Θ D > D φ Θ D > A morphism of D-Θ-bridges is a pair of poly-morphisms fitting into the following diagram: D J φ Θ D > capsule-full poly-iso. D J φ Θ D > full poly-iso. Isom( φ Θ, φ Θ ) = { }.

16 16 Base-ΘNF-Hodge theaters A D-ΘNF-Hodge theater is a collection of data HT D-ΘNF = ( D φ NF φ Θ D J D > ) fitting into the following commutative diagram D φ NF φ Θ D J D > D φ NF D φ Θ A morphism of D-ΘNF-Hodge theaters is a pair of morphisms between the respective associated D-bridges fitting into the following diagram: D > D φ NF D J φ Θ D > φ NF φ Θ D D J D > χ : π 0 ( D J ) = J F l. j J = F l, φ LC j : LabCusp( D ) LabCusp( D > ).![ ɛ] LabCusp( D ) s.t. under LabCusp( D > ) F l, φ LC j ([ ɛ]) = φ LC 1 ( χ(j) [ ɛ]) χ(j). synchronized indeterminacy: LabCusp( D ) J.

17 Figure: D-ΘNF-Hodge theater (Fig. 4.4 of [IUT-I]) 17

18 18 Some symmetries surrounding base-θnf-hodge theaters For the forgetful functor {D-ΘNF-Hodge theaters} {D-NF-bridges} HT D-ΘNF = ( D φ NF φ Θ D J D > ) ( D φ NF D J ) the output data has F l -symmetry. (J F l ) For the forgetful functor {D-ΘNF-Hodge theaters} l -capsules of D-(resp. D -)prime-strips HT D-ΘNF D J (resp. D J ) the output data has S l -symmetry. (forgetting J F l ) Reduce (l ) l -indeterminacy to l!-indeterminacy: ( φ Θ D J ) ( φ Θ Proc( D J )), ( φ Θ Proc( D J )). (A procession in a category is a diagram P 1 P 2 P n with P j a j-capsule of objects and the collection of all capsule-full poly-morphisms. A morphism of processions is an order-preserving injection ι : {1,, n} {1,, m} plus the capsule-full poly-morphisms P j Q ι(j).)

19 19 The D-NF-link HT D-ΘNF D HT D-ΘNF between two D-ΘNF-Hodge theaters is the (induced) full poly-isomorphism D > D >. (mono-analytic core) Figure: Étale-picture of D-ΘNF-Hodge theaters (Fig. 4.7 of [IUT-I])

20 Global Frobenioids Let D D be a category. The Θ-approach to π 1 ( D ) M ( D ) F, M ( D ) F. π 1 ( D ) π 1 ( D ) ( C K C Fmod ), D D := B(π 1 ( D )). π 1 ( D )-invariants M mod ( D )( (F mod ), M mod( D )( F mod ). Belyi cuspidalization (G KCFmod )π1 rat( D ) π 1 ( D ) (well-defined up to inner action of kernel), pseudo-monoids π rat 1 ( D ) M κ( D ), M κ ( D )(= M κ( D ) πrat 1 ( D ) ), M κ ( D ). M ( D ) V( D )( V(F )) Φ ( D ) : Ob( D ) A monoid of arithmetic divisors on M ( D ) A model Frobenioid F ( D ) over D. Let F F ( D ) be a category ( Frobenioid structure on it). Suppose we are given D base( F ) isomorphic to D D. Then identify (by F -cority of C F ) base( F ) = D. Define F := F D, F mod := F terminal objects of D, ( = Frobenioid of arithmetic line bundles on [Spec O K / Gal(K/F mod )]).

21 21 A Ob( F ) O (A birat ) (= mult. group of the finite extension of F mod corresponding to A). Thus varying Frobenius-trivial objects A Ob( F ) over Galois objects of D, π 1 ( D ) M. p Prime(Φ F (A)) O p := Ä O (A birat ) Φ F (A)gpä 1 (p {0}). For A 0 lying over a terminal object of D and p 0 Prime(Φ F (A 0)): Consider the elements of Aut F (A) fixing O p for a system of p p 0, the closed subgroup (well-defined up to conjugation) Π p0 π 1 ( D ). Look at a pair π rat 1 ( D ) M κ isomorphic to π rat 1 ( D ) M κ ( D ) : (Q >0 Ẑ = {1} )! µ Θ Ẑ (π 1( D )) µẑ( M κ ) s.t. M κ ( D ) M κ Similarly have canonical isomorphism lim H π rat 1 ( D ) open H1 (H, µ Θ Ẑ (π 1( D ))) lim H π rat 1 ( D ) open H1 (H, µẑ( M κ )) M κ( D ) M κ, and canonical isomorphism compatible with the integral submonoids O p M ( D ) M.

22 22 So, F carries natural structures π rat 1 ( D ) M κ, M κ, M κ = ( M κ) πrat 1 ( D ). π rat 1 ( D ) M κ π1 rat( D ) M κ. (Consider the subset of elements for which the Kummer class restricted to some subgroup of π1 rat( D ) corresponding to an open subgroup of the decomposition group of some strictly critical point of C Fmod is a root of unity.) π rat 1 ( D ) M κ M plus the field structure on M {0}. (Consider Kummer classes arising from M κ restricted to subgroups of π1 rat( D ) corresponding to decomposition groups of non-critical F -valued points of C Fmod.) In particular, F Prime( F mod ) V mod = V( D )/π 1 ( D ). p p 0 (assumed nonarchimedean) determines a valuation on O (A birat ) {0}, O p (= mult. monoid of nonzero integral elements of the completion at p of the number field corresponding to A). Varying A (considered up to conjugation by Π p0 ) Π p0 O p 0 (ind-topological monoid) ( MLF-Galois TM-pair of strictly Belyi type )

23 23 Frobenioid-prime-strips F-prime-strip: F = { F v } v V, s.t. at v V non, F v = C v Cv ; at v V arc,... associated D-prime-strip D = { D v }. At v V non, π 1 ( D v ) D v D v ( X v, X v C v, v V mod ). π rat 1 ( D v ) π 1 ( D v ). π 1 ( D v ) M v ( D v )( O F v ), π rat 1 ( D v ) M κv ( D v ), M κv ( D v ), M κ v ( D v ). For an isomorph M = M v, M κv, M κ v of M v ( D v ), M κv ( D v ), M κ v ( D v ) respectively:! µ Θ Ẑ (π 1( D v )) µẑ( M), s.t. M( D v ) M. Thus, F v carries natural structures: π 1 ( D v ) M v, π rat 1 ( D v ) M κv, M κ v, M κv := M πrat 1 ( D v ) κv. plus the field structures on M gp v (Analogue at v V arc.) M κ v M κv M κv M gp v, {0}, ( M gp v ) π 1( D v ) {0}. ( M gp v ) π 1( D v ),

24 24 ( F ) F -prime-strip: F = { F v } v V, with F v F v (splitting Frobenioid), v V non ; F v =..., v V arc. A morphism of F-prime-strips is a collection of isomorphisms indexed by V. Similarly for F -prime-strips. Globally realified mono-analytic Frobenioid-prime-strip: F = Å C, Prime( C ) V, F, { ρ v : Φ C,v ã Φ rlf C } v v V := collection of data F mod. Isom( 1 F, 2 F ) Isom(base( 1 F ), base( 2 F )), Isom( 1 F, 2 F ) Isom(base( 1 F ), base( 2 F )), Isom( 1 F, 2 F) Isom( 1 D, 2 D), Isom( 1 F, 2 F ) Isom( 1 D, 2 D ).

25 25 Θ-bridges, NF-bridges Recall HT Θ = ({ F v } v V, F mod), HT D-ΘNF = ( D φ NF φ Θ D J D > ). Assume the D-prime-strip associated to HT Θ is equal to D >. Thus the F-prime-strip F > of HT Θ has D > as its associated D-prime-strip. j J, F-prime-strip F j = { F v j } v j V j with associated D-prime-strip D j, φ Θ j : D j D > (unique) ψ Θ j : F j F >, ψ Θ : F J F >. (The associated D-prime-strip of ( φ Θ j ) ( F > ) is equal to D j, F j ( φ Θ j ) ( F > ).) For any δ LabCusp( D ),! Aut ɛ ( D )-orbit of isomorphisms D δ to [ɛ]. D mapping A δ-valuation of V( D ) is an element mapping to an element of V un := Aut ɛ (C K ) V via this Aut ɛ ( D )-orbit of isomorphisms. At a δ-valuation v V( D ), Π p0 O p 0 open subgps. of Πp0 π (tp) 1 ( D ) corresponding to X, X, determined by δ (Analogue at v V arc.) F-prime-strip p v -adic Frobenioids. F δ ( π 1 ( D )). (only well-defined up to isomorphism, because V un V mod is not injective.)

26 26 For an F-prime-strip F, a poly-morphism F F :=a full poly-iso. F F δ for some δ LabCusp( D ). (Such F F is fixed by composition with elements in Aut( F) or Aut ɛ ( F ).) Over a given φ NF : D J D,! poly-morphism ψ NF : F J F. V J diag. V J = V j, F j = { F mod, V j F J = { F mod, V J Prime( F mod )}. Prime( F mod )}, Any poly-morphism F J F induces an isomorphism class of functors ( F ) F mod F res. J F v J, v J V J, (independent of choice of F J F among its F l -conjugates) hence isomorphism classes of restriction functors F mod F J F J. Similarly, have collections of isomorphism classes of restriction functors F J F J, F j F j.

27 An NF-bridge:= ( ψ NF F J F F ) : F J a capsule of F-prime-strips. ( D J := associated capsule of D-prime-strips.) F F, F F categories. ( D, D =bases.) ψ NF the poly-morphism lifting a D-NF-bridge φ NF : D J D. F F a morphism D D (abstractly) equivalent to D D plus an isomorphism F F D. A morphism of NF-bridges ( 1 F J1 1 F 1 F ) ( 2 F J2 2 F 2 F ) consists of (capsule-full poly-)isomorphisms 1 F J1 2 F J2, 1 F 2 F, 1 F 2 F compatible with the (D-)NF-bridges. A Θ-bridge:= ( ψ Θ F J F > HT Θ ) : F J as above. HT Θ a Θ-Hodge theater. F > the F-prime-strip associated to HT Θ. ( D > =associated D-prime-strip.) ψ Θ a poly-morphism lifting a D-Θ-bridge φ Θ : D J D >. A morphism of Θ-bridges is defined similarly.

28 ΘNF-Hodge theaters Constructions above HT ΘNF = ( F F ψ NF ψ Θ F J F > HT Θ ) v V bad, Aut(F v ) Aut(D v ). (Consider the rational function and divisor monoids of F v.) Isom(Θ-Hodge theaters) Isom(associated D-prime-strips) (The global data F mod admits no nontrivial automorphisms.) Isom(NF-bridges) Isom(associated D-NF-bridges), Isom(Θ-bridges) Isom(associated D-Θ-bridges), Isom(ΘNF-Hodge theaters) Isom(associated D-ΘNF-Hodge theaters). Given an NF-bridge ψ NF and a Θ-bridge ψ, Θ capsule-full poly-isomorphisms F J F J gluing them into HT ΘNF F l.

29 29 -label classes of cusps F l := F l {1}. ({1} F l ) F l -group:=set E plus {1}-orbits of bijections E F l. F l -torsor:=set T plus F l -orbits of bijections T F l. (F l F l, z z + λ) For D = { D v } v V, D v D v ( X v for v V non,...for v V arc ) A -label class of cusps of D v = D = { D v } v V. {cusps of D v lying over a single cusp of D v } ( elements of Q). {LabCusp ( D v )\ η 0 v }/{1} LabCusp( D v )( F l ) η v η v LabCusp ( D v ) F l as an F l -group, 1 Aut + ( D v ) Aut( D v ) {1} 1. For α {1} V, have Aut α ( D) Aut( D) of α-signed automorphisms. Given another D-prime-strip D = { D v }: A +-full poly-isomorphism D v D v := an Aut + ( D v )-orbit of an isomorphism D v D v. A +-full poly-isomorphism D D:= an Aut + ( D)-orbit of an isomorphism D D. (If D = D, these poly-isomorphism {1} V.)

30 D = B(X K ). Outer homomorphism (adapted to ab X Aut(D ) Aut(X K ) GL 2 (F l ) GL 2 (F l )/{1} F l Q) has image containing a Borel of SL 2 (F l )/{1} 1 Aut (D ) Aut(D ) F l 1. ( crucial F l -rigidity for the Hodge-Arakelov-theoretic evaluation of étale theta function.) (Aut K (X K ) )Aut (D ) transitively {cusps of X K }. Aut csp (D ) := automorphisms which fix the cusps of X K. Aut + (D ) Aut (D ) := unique index 2 subgroup Aut csp (D ). Choice of the cusp ɛ of C K thus (Aut K (X K /X K ) {1} ) Aut K (X K ) Aut (D )/ Aut csp (D ) ɛ F l, (Aut K (X K /X K ) ) Aut + (D )/ Aut csp (D ) F l, as an F l -group; {cusps of D } =: LabCusp (D ) F l, (Fix this from now on.) as an F l -torsor.

31 31 Base-Θ -bridges Copies of D-prime-strips: D = {D,v } v V, D t = {D v t } v V, t F l (as F l -group). Positive +-full poly-isomorphisms (tautological): φ Θ v t : D v t D,v, φ Θ t : D t D. φ Θ = {φ Θ t } t Fl : D = {D t } t Fl D (φ Θ 1 Fl : D t D t, D D +-full poly-iso. with 1-sign at all v) ( α {1} V, φ Θ α Θ : D t = D t, D D α-signed +-full poly-iso.) Let T be an F l -group. T := T /{1}, T := T \{0} D T, D T. A D-Θ -bridge φ Θ is a poly-morphism fitting into the following diagram: inducing F l φ Θ D T D T D φ Θ D A morphism of D-Θ -bridges is a pair of poly-morphisms fitting into: D T φ Θ D capsule-+-full poly-iso. inducing T T D T φ Θ D +-full poly-iso.

32 32 Base-Θ ell -bridges For v V non : (Look at the cover X v X v X K or X v X v X K.) (analogue for v V arc ) φ Θell t α Aut +(D v ) D v D v φ Θell v φ Θ ell,v β φ Θell D,v α D β Aut csp(d ) = {β φ Θell,v α} α,β, : D t = {D v t =(v,t)} {φθ ell v } D t D, φ Θell = {φ Θell t } t Fl : D = {D t } t Fl D. t F l as F l -torsor, (φ Θell γ F l : D t D γ(t) +-full poly-iso. with sign sign(γ) at all v) Let T be an F l -torsor. A D-Θell -bridge φ Θell is a poly-morphism s.t. inducing F l φ Θell D T D T φ Θell D D A morphism of D-Θ ell -bridges is a pair of poly-morphisms s.t. D T φ Θell D capsule-+-full poly-iso. inducing T T D T φ Θell D Aut csp( D )-orbit of iso.

33 33 Transport of -cusp labels via base-bridges Bijection compatible with F l -torsor structures: LabCusp ( D v t ) identification of F l -groups: φ Θell LabCusp ( D ). LabCusp ( D t ) := LabCusp ( D v t ) = LabCusp ( D w t ). Thus natural bijection compatible with F l -torsor structures: ζ Θell t : LabCusp ( D t ) LabCusp ( D ). synchronized indeterminacy: T LabCusp ( D ) t ζ Θell t (0) Bijection of F l -groups: LabCusp ( D v t ) identification of F l -groups: φ Θ LabCusp ( D,v ). LabCusp ( D ) := LabCusp ( D,v ) = LabCusp ( D,w ). Thus natural bijection of F l -groups: LabCusp ( D t ) LabCusp ( D ).

34 Base-Θ ell -Hodge theaters A D-Θ ell -Hodge theater is a collection of data HT D-Θell := ( D φ Θell φ Θ D T D ) fitting into the following commutative diagram D φ Θell φ Θ D T D φ Θell D D φ Θ (A morphism of D-ΘNF-Hodge theaters is defined in the obvious way as before.) D Isom(D-Θ -bridges) forms a ({1} {1} V )-torsor. Isom(D-Θ ell -bridges) forms an F l -torsor. Isom(D-Θ ell -Hodge theaters) forms a {1}-torsor. Given a D-Θ -bridge and a D-Θ ell -bridge, {capsule-+-full poly-isomorphisms D T D T gluing them into HT D-Θell} forms a ({1} V F l )-torsor.

35 Figure: D-Θ ell -Hodge theater (Fig. 6.1 of [IUT-I]) 35

36 Some symmetries surrounding base-θ ell -Hodge theaters For the forgetful functor D-Θ ell -Hodge theaters D-Θ ell -bridges ell HT D-Θell = ( φ Θ D φ Θ D T D ) ( φ Θ D T D ) the output data has F l -symmetry. For the forgetful functor D-Θ ell -Hodge theaters l -capsules of D-(resp. D -)prime-strips ell the output data has S l -symmetry. HT D-Θell D T (resp. D T ) Reduce (l ) l -indeterminacy to l!-indeterminacy: ( φ Θ D T ) ( φ Θ Proc( D T )), ( φ Θ Proc( D T )). Compatibility: For j {1,, l } and t = j + 1, the inclusion determines natural transformations {1,, j} {0, 1,, t 1} φ Θ Ä Proc( D T ) Proc( D T ) ä, φ Θ Ä Proc( D T ) Proc( D T ) ä.

37 37 D-Θ -bridge D-Θ-bridge: The D-Θ ell -link is the full poly-isomorphism ( φ Θ : D T D ) ( φ Θ : D T D > ) D T T \{0} D T, D D 0, D D >. HT D-Θell D HT D-Θell D. (mono-analytic core) Figure: Étale-picture of D-ΘNF-Hodge theaters (Fig. 6.3 of [IUT-I])

38 38 Θ -bridges, Θ ell -bridges and Θ ell -Hodge theaters A Θ -bridge is a poly-morphism ψ Θ F T F between a capsule of F-prime-strips indexed by an F l -group T and a F-prime-strip, which lifts a D-Θ -bridge φ Θ : D T D. A Θ ell -bridge is a poly-morphism ψ Θell F T D between a capsule of F-prime-strips indexed by an F l -torsor T and a category equivalent to D, which lifts a D-Θ ell -bridge φ Θell : D T D. A morphism of bridges is defined to be a pair of poly-isomorphisms on the domains and codomains, which lifts a morphism of the associated base-bridges. HT Θell := ( D ψ Θell ψ Θ F T F ). Isom(Θ -bridges) Isom(D-Θ -bridges), Isom(Θ ell -bridges) Isom(D-Θ ell -bridges), Isom(Θ ell -Hodge theaters) Isom(D-Θ ell -Hodge theaters). Given a Θ -bridge and a Θ ell -bridge, {capsule-+-full poly-isomorphisms gluing them into HT Θell} forms a ({1} V F l )-torsor.

39 39 Gluing ΘNF-Hodge theater and Θ ell -Hodge theater Recall HT Θell = ( D ψ Θell ψ Θ F T F ), HT D-Θell = ( D φ Θell φ Θ D T D ). HT ΘNF = ( F F ψ NF ψ Θ F J F > HT Θ ), D-Θ -bridge D-Θ-bridge: HT D-ΘNF = ( D φ NF φ Θ D J D > ). ( φ Θ : D T D ) ( φ Θ : D T D > ). Assuming D-Θ-bridge D T φ Θ D > is associated to the Θ-bridge ψ Θ F J F > HT Θ, we glue the Θ-bridge to the Θ -bridges ψ Θ F T F ). Such a gluing is unique because Isom( φ Θ, φ Θ ) = { }, Isom(Θ-bridges) Isom(associated D-Θ-bridges). Θ ell NF-Hodge-theater HT Θell NF.

40 40 Summary Figure: Θ ell NF-Hodge-theater (Fig. 6.5 of [IUT-I])