Construction of Hodge Theaters

Construction of Hodge Theaters

Yu Yang

RIMS, Kyoto University

September 2, 2021

In this talk, I will explain the first paper of Prof. Mochizuki’s theory of IUT. The talk is divided into three parts as follows:

The motivation of Hodge theaters The goal of Hodge theaters

The construction of Hodge theaters (in particular, the ´etale picture!)

§1: Motivation of Hodge theaters

F/Q: a number field

OF: the ring of integers ofF

E/F: an elliptic curve s.t. E admits a semi-stable model E/OF

V(F): the set of places of F

V(F)^non,bad: the set of non-arch. placesv s.t. Ev def

= E|Fv has bad reduction, where F_v denotes the local field at v

ℓ: a prime number distinct fromp_v for all v∈V(F)^non,bad, wherep_v denotes the characteristic of the residue field of F_v

ht_E: Faltings height of E

Goal: We hope that ht_E can be bounded forallelliptic curves satisfying the above conditions.

Note thatE_v ∼=Gm/q^Z_v ↞Gm ⊇µ_ℓ for all v∈V(F)^non,bad, and that 0→µ_ℓ →Ev[ℓ]→Z/ℓZ→1.

Global multiplicative subspaces (=GMS)

We shall callH ⊆E[ℓ] a “GMS” ifH|Fv coincides withµ_ℓ for all v∈V(F)^non,bad. This means that there exists a Galois ´etale covering Y →E corresponding to H such that Yv →Ev is a topological covering of dual semi-graphs for all v∈V(F)^non,bad.

If “GMS” exists, then by some standard discussions of Diophantine geometry, we may obtain that htE can be bounded. However, we have

#{E/F s.t. GMS exists}<∞.

Goal of IUT: We want to do similar discussions forarbitrary elliptic curves over number fields.

The first step: We need an analogue of “GMS” forarbitrary elliptic curves.

IUT’s answer:It’s “Hodge theaters”

§2: Goal of Hodge theaters Reference

Section 1 of “S. Mochizuki, The ´etale theta function and its

Frobenioid-theoretic manifestations. Publ. Res. Inst. Math. Sci. 45 (2009), 227-349.”

Local theory (over non-archimedean bad places) ℓ >>0: a prime number

p≥3: a prime number s.t. p̸=ℓ k: ap-adic field

Ok: the ring of integers ofk

X∼=Gm,k/q^Z: an elliptic curve overkwith bad reduction (i.e., Tate curve)

X^log: the log stable curve overk determined by the zero point ofX Moreover, we assume that

√−1∈k

X[2ℓ](k) =X[2ℓ](k), where kdenotes an algebraic closure ofk

Then we have the following commutative diagram of cartesian squares of tempered coverings:

Y¨^log −−−−→^µℓ Y¨^log

µ2



y ^µ2y Y^log −−−−→^µℓ Y^log

ℓZ



y ^ℓZ

 y

X^log −−−−→^µℓ X^log −−−−→^Z/ℓZ X^log

The above coverings are defined in the next page via a picture of special fibers.

The special fibers of the above commutative digram is as follows:

On the other hand, let us fix a cusp OX (i.e., zero cusp) ofX. Then the image of O_X ofX →X is the zero cusp (or the zero point)O_X. The curve(X, O_X) can be regarded as an elliptic curve overk. Thus, we obtain

X^log −−−−→^Z/ℓZ X^log

±1



y ^±1y C^log −−−−−→^{degree ℓ} C^log, where C^{log def}= [

X^log/{±1}]

andC^{log def}= [

X^log/{±1}]

denote the quotient stacks.

Moreover, there exists a unique irreducible component 0X_s ∈Irr(X_s) such that the reduction of O_X is contained in0_Xs, where Irr(X_s) denotes the set of irreducible components ofX.

Let 0_Ys ∈Irr(Y_s) be a lifting of0_Xs. Then we obtain a labeling Z→^∼ Irr(Y_s)→^∼ Irr( ¨Y_s)→^∼ Irr( ¨Y

s) such that 07→0_Ys. Moreover, we put

µ₋∈X(k): 2-torsion point whose reduction is contained in0_Xs µ^Y₋∈Y(k): the unique lifting of µ₋ s.t. the reduction is contained in 0Ys

ξ_j^Y ∈Y(k): j·µ^Y₋ (with the action ofj ∈Z∼=Aut(Y^log/X^log))

We have the following definition.

Definition 1

We shall call a lifting of ξ_j^Y ∈Y(k) in Y¨(k)an evaluation point ofY¨^log labeled byj ∈Z(→^∼ Irr(Ys)). Moreover, we shall call a lifting of an evaluation point of Y¨(k) labeledj in Y¨(k)an evaluation point ofY¨^log labeled byj ∈Z(→^∼ Irr( ¨Y_s)→^∼ Irr(Y_s)).

Moreover, we have the following picture:

We have the following diagram of special fibers:

Non-archimedean Θ-functions We put

Y¨, Y, Y¨, Y

thep-adic formal schemes whose Raynaud generic fibers areY¨,Y,Y¨,Y, and whose special fibers are Y¨_s,Y_s,Y¨s,Ys, respectively.

Write 0_Y¨

s ∈Irr( ¨Y_s) for the irreducible component over0_Ys ∈Irr(Y_s) and U ⊆¨ Y¨,U ⊆ Y for the open formal subschemes such that

U¨s∼= 0Y¨s\Y¨_s^sing (∼=Gm), Us ∼= 0Ys \Y_s^sing (∼=Gm).

Then U is isomorphic to the p-adic formal completion of Gm,Ok with multiplicative coordinate U ∈Γ(U,O_U). Moreover, we put

U¨ ^def= √

U ∈Γ( ¨U,OU¨).

We have the following function on Y¨: Θ( ¨¨ U) =q⁻¹⁸ ·∑

n∈Z

(−1)ⁿ·q^12(n+12)2 ·U¨²ⁿ⁺¹. Moreover, we define a function

Θ^def= ¨Θ(an evaluation pt labeled by 0)·Θ¨⁻¹

onY¨ which can be regarded as an “ℓ-th root” ofΘ¨ (in the sense of cohomological classes). Note that there are exactly two evaluation points labeled by0 in Y¨s, and that we have

Θ(an ev. pt labeled by¨ 0) =−Θ(another ev. pt labeled by¨ 0).

Values of Θat evaluation points

We putq ^def= q^2ℓ1. Then Θ(an evaluation pt labeled byj)∈µ_2ℓ·q^j2.

Let V(F)^non,bad be the notation introduced in§1. Moreover, we denote by Θ_v the function defined above at the placev∈V(F)^non,bad such thatv is not over2. Then we have

±1 ±2 . . . ±j . . . Θv 7→ q

v q⁴

v . . . q^j2

v . . .

The Goal of Hodge theaters:Roughly speaking, Hodge theater (at least, the ´etale part) is a virtual “GMS” for an arbitrary elliptic curve over a number field which manages

Θv-values for all non-archimedean bad places (with their labels) via anabelian geometry.

§3: Initial Θ-data Reference

Section 3 of “S. Mochizuki, Inter-universal Teichm¨uller theory I:

Construction of Hodge theaters. Publ. Res. Inst. Math. Sci. 57(2021), 3–207.”

Firstly, we have the following notation:

F :a number field s.t. √

−1∈F

E :an elliptic curve overF s.t. E has stable reduction at all v∈V(F)^non

ℓ>5 :a prime number s.t. ℓ̸=pv for all v∈V(F)^non,bad K ^def= F(E[ℓ])

F_mod:the field of moduli ofE X^def= E\ {O_E}

C ^def= [X/{±1}]

Furthermore, we assume that

E[6](F) =E[6](F), whereF denotes an algebraic closure ofF CK def

= C×F K is a “K-core” (i.e., a terminal object in the category of ´etale coverings and quotients of X_K overK)

F/F_mod is Galois

SL2(Fℓ)⊆Im(GF →Aut(E[ℓ]) (∼=GL2(Fℓ))

Let X_K →XK def

= X×F K be an ´etale covering with Galois group Z/ℓZ. Note that #(X^cpt_K \X_K) =ℓ. Just like the local theory recalled above, we fix a cuspO_XK ∈X^cpt_K \X_K and call it zero cusp. Then(X^cpt_K , O_XK) is an elliptic curve over K. In particular, there exists a{±1}-action on X_K. Thus, we have

X_K −−−−→^Z/ℓZ XK

±1



y ^±1y C_K −−−−−→^{degree ℓ} C_K,

where C_K ^def= [X_K/{±1}]denotes the quotient stack. Moreover, we fix a non-zero cusp

ϵ

of C_K (i.e., the image of a cusp∈X^cpt_K \(X_K∪ {O_XK})).

We put the following

V⊆V(K) : a subset s.t. the natural mapV,→V(K)↠V(F_mod) is a bijection

V^bad ⊆V:a non-empty subset s.t. for everyv ∈V^bad, the following are satisfied: (i)E has bad reduction atv; (ii)X_v →X_v induces a topological covering of their dual semi-graphs; (iii) the reductionϵ_v of ϵis the cusp ofC^log_v labeled by 1∈Fℓ/{±1} (∼=Cusp(C^log_v )); (iv) the image of the natural map V^bad,→V(K)↠V(Fmod)↠V(Q)does not contain 2. Then for eachv ∈V^bad, we have the local theory explained in §2.

V^{good def}= V\V^bad. Note that Ev,v∈V^good, has bad reduction in general.

We shall call

(F /F, E, ℓ, C_K,V,V^bad, ϵ)

an initialΘ-data. From now on, we fix an initialΘ-data, and in the reminder of my talk, I will explain the following diagram which is

constructed from the given initial Θ-data (I only explain the constructions at non-archimedeanplaces which are the most important cases in the original form of IUT):

Many constructions appeared in the above picture are not difficult to understand from geometry of coverings of curves. On the other hand, in IUT, we need to share information via various links between different Hodge theaters (or different universes) by using fundamental groups (via anabelian geometry), then we need some group-theoretical descriptions.

§4: Construction of D-ΘNF-HT Reference

Section 4 of “S. Mochizuki, Inter-universal Teichm¨uller theory I:

Construction of Hodge theaters. Publ. Res. Inst. Math. Sci. 57(2021), 3–207.”

In the section, I explain the right-hand side:

D-prime-strips We put

Dv def=

{ π^tp₁ (X^log_v ) (or B^tp(X^log_v )⁰), ifv∈V^bad π^´et₁(X−→^v) (or B^´et(−→X^v)⁰), ifv∈V^good where B(−)⁰ denotes the subcategory of the Galois category B(−) consisting of connected objects, and −→X^vdef= X−→^K×KK_v is determined by (C_K, ϵ) via the picture in the next page:

Let D^def= {Dv}v∈V. Then we put D>def

= {D>,v}v∈V (∼=^†D),

where D_>,v∼=Dv for all v∈V.

Let F^⋇_ℓ ^def= F^×_ℓ/{±1},Vj,j∈F^⋇_ℓ, a copy ofV, andDj

def= {Dv_j}v_j∈Vj. Then we put

DJ

def= {Dj}j∈J, where J ^def= F^⋇_ℓ. Moreover, we put

D^⊚def= π^´et₁(C_K) (or B^´et(C_K)⁰).

⊠-symmetry (on cusps)

For every v∈V, recall the following commutative diagram:

X_v ^def= X_K×KKv −−−−→Z/ℓZ Xv

def= XK×KKv

±1



y ^±1y

C_v ^def= C_K×KKv

degreeℓ

−−−−−→ Cv

def= CK×KKv. We put

LabCusp_v ^def= the set of non-zerocusps of C_v. Then we have

LabCusp_v →^∼ F^⋇_ℓ (^def= F^×_ℓ /{±1}) ={ℓ^⋇, ℓ^⋇−1, . . . ,2,1,−2, . . . ,−ℓ^⋇}, where ϵ_v 7→the image of1 in F^⋇_ℓ and ℓ^{⋇ def}= (ℓ−1)/2.

Moreover, we put

LabCusp_K ^def= the set of non-zero cusps of C_K. Then there exists a natural bijection

LabCusp_v →^∼ LabCusp_K, v∈V,

via the natural homomorphismC_v →C_K. This means that the sets {LabCusp_v}v∈V

can be managed by LabCusp_K via the above bijections.

On the other hand, since C_K is a “K-core”, we have Aut(C_K),→Gal(K/F_mod).

Moreover, we define a subgroup

Aut_ϵ(C_K)^def= {σ ∈Aut(C_K) s.t.σ(ϵ) =ϵ}.

Let E[ℓ](F)↠Q(∼=Fℓ) be the quotient determined by the Galois ´etale covering X_K →X_K. Then we have the following

LabCusp_K →^∼ (

{Q↷O_XK} \ {O_XK}) /{±1}

→∼ (

Q\ {0})

/{±1}. Thus, we obtain an exact sequence

1→Aut_ϵ(C_K)→Aut(C_K)→Aut(Q)/{±1}(∼=F^⋇_ℓ)→1.

This means

Aut(C_K)/Autϵ(C_K)∼=F^⋇_ℓ.

By using anabelian geometry, we have a group-theoretical version of the above isomorphism:

Aut(D^⊚)/Autϵ(D^⊚)→^∼ F^⋇_ℓ

and Aut(D^⊚)/Aut_ϵ(D^⊚)is a sub-quotient of the Galois group of the extension of number fields Gal(K/Fmod). Moreover, we obtain the following action (=⊠-symmetry arising from arithmetic):

Aut(D^⊚)/Autϵ(D^⊚)↷LabCusp_K (=F^⋇_ℓ ↷ F^⋇_ℓ).

(Model) D-NF-bridge Let v∈V. We put

ϕ^NF_•,v :Dv → D^⊚

induced by X_v →C_v →C_K ifv∈V^bad and X−→^v →C_v →C_K if v∈V^good. We put (as a poly-morphism (i.e., a set of morphisms))

ϕ^NF_v ^def= Autϵ(D^⊚)◦ϕ^NF_•,v ◦Aut(Dv) :Dv→ D^⊚. Recall that Vj,j∈F^⋇_ℓ, a copy of V. We write

Dj def

= {Dv_j}v_j∈Vj, where Dv_j ∼=Dv. Let

ϕ^NF₁ ^def= {ϕ^NF_v

1 }v₁∈V1 :D1→ D^⊚ be the poly-morphism determined by ϕ^NF_v

1 ,v₁∈V1.

Since Aut_ϵ(D^⊚)·ϕ^NF₁ =ϕ^NF₁ (i.e., stable under the action of Aut_ϵ(D^⊚) by definition), we obtain an action of F^⋇_ℓ →^∼ Aut(D^⊚)/Aut_ϵ(D^⊚)on ϕ^NF₁ . Moreover, we put

ϕ^NF_j ^def= j·ϕ^NF₁ :Dj → D^⊚, j ∈F^⋇_ℓ. We shall call the poly-morphism

ϕ^NF_⋇ ^def= {ϕ^NF_j }_j∈F^⋇

ℓ :DJ (orD⋇)^def= {Dj}_j∈F^⋇

ℓ → D^⊚

the(model) D-NF-bridge (recallJ =F^⋇_ℓ).

(Model) D-Θ-bridge We put

D>

def= {D>,v}v∈V, where D>,v ∼=Dv.

Let v∈V^bad. Then we have the following morphism ϕe^Θ_v

j :Dv_j (∼=Dv)⃝1

↠Gal(K_v/K_v)⃝→ D2 v, j ∈F^⋇_ℓ,

where 1⃝is the natural surjection π₁^tp(X_v)↠Gal(Kv/Kv). Let us explain

⃝2.

Recall the tempered coverings whose special fibers are as following:

Note that{e^j₁, . . . , e^j_ℓ}_j∈F^⋇

ℓ areK_v -rational points ofX_v. Then we define

⃝2 to be “the Galois section determined by a point of {e^j₁, . . . , e^j_ℓ}_j∈F^⋇

ℓ” (roughly speaking,{e^j₁, . . . , e^j_ℓ}_j∈F^⋇

ℓ is a finite approximation of evaluation points explained in §2 and the Galois sections contains the informations of values of Θ_v explained in §2). Then we have information about values of theta functions.

We put (as a poly-morphism) ϕ^Θ_v

j

def= Aut(D>,v)◦ϕe^Θ_v

j◦Aut(Dv_j) :Dv_j → D>,v, j ∈F^⋇_ℓ. On the other hand, let v∈V^good. We put (as a full poly-isomorphism)

ϕ^Θ_v

j :Dv_j(∼=Dv)→ D^∼ >,v(∼=Dv), j ∈F^⋇_ℓ.

Moreover, for global case, we put ϕ^Θ_j ^def= {ϕ^Θ_v

j}v_j∈Vj :Dj

def= {Dv_j}v_j∈Vj → D>

def= {D>,v}v∈V. Then we have

ϕ^Θ_⋇ ^def= {ϕ^Θ_j}_j∈J^def_=F⋇ ℓ

:DJ (orD⋇)^def= {Dj}j∈J → D>

and shall call ϕ^Θ_⋇ the(model)D-Θ-bridge.

Summary

We have D-ΘNF-HT as following

D> def

= {D>,v}v∈V ϕ^Θ_⋇

← DJ def

= {Dj}_j∈J^def_=F⋇ ℓ

def= {{Dv_j}v_j∈Vj}j∈J ϕ^NF_⋇

→ D^⊚ and, for each j∈J, the maps of sets of cusps (asF^⋇_ℓ-torsors)

ϕ^LC_j :LabCusp(D^⊚)→^∼ LabCusp(Dj)→^∼ LabCusp(D>), ϵ7→j.

§5: Construction of D-Θ^ell-HT Reference

Section 5 and Section 6 of “S. Mochizuki, Inter-universal Teichm¨uller theory I: Construction of Hodge theaters. Publ. Res. Inst. Math. Sci. 57 (2021), 3–207.”

In this section, we explain the left-hand side:

⊞-symmetry (on cusps) For each v∈V, we put

LabCusp^±_v ^def= the set of cusps of X_v. Then we have the natural action of Gal(X_v/C_v)∼={±1} on

LabCusp^±_v →^∼ Fℓ={ℓ^⋇, . . . ,1,0,−1, . . . ,−ℓ^⋇}. On the other hand, we put

LabCusp^±_K^def= the set of cusps of X_K.

Then we may manage the sets of cusps {LabCusp^±_v}v∈V via the natural bijection induced by X_v →X_K:

LabCusp^±_v →^∼ LabCusp^±_K, v∈V.

We put

D_≻ ^def= {D_≻,v}v∈V,

where D≻,v ∼=Dv. Note that LabCusp^±_v can be mono-anabelian reconstructed from D_≻,v. On the other hand, we put

D^⊚±def= π^´et₁(X_K) (or B^´et(X_K)⁰).

Note that LabCusp^±_K can be mono-anabelian reconstructed fromD^⊚±. Then we obtain a group-theoretical version of the above bijection of cusps:

LabCusp^±(D_≻)→^∼ LabCusp^±(D^⊚±).

We may identify LabCusp^±(D_≻) with LabCusp^±_K(D^⊚±) via the above bijection. Moreover, there is a natural action

AutK(X_K)(∼=F^⋊±_ℓ ^def= Fℓ⋊{±1})↷LabCusp^±_K(∼=Fℓ).

In fact, the above action can be expressed group-theoretically. We put Aut_±(D^⊚±)^def= ker(Aut(D^⊚±)→F^⋇_ℓ),

where the homomorphism is determined by the quotient E[ℓ](F)→Q and Q∼=Fℓ is introduced in§4, and put

Aut_cusp(D^⊚±)⊆Aut(D^⊚±)

the subgroup of automorphisms which fix the cusps of X_K. Then we have the following action (=⊞-symmetry arising from geometry):

(Aut_K(X_K)∼=)Aut_±(D^⊚±)/Aut_cusp(D^⊚±)↷LabCusp^±_K(D^⊚±) which can be regarded as

F^⋊±_ℓ ↷ Fℓ

by using ϵ.

(Model) D-Θ^±-bridge

Let T ∼=LabCusp^±_K(D^⊚±)∼={ℓ^⋇, . . . ,1,0,−1, . . . ,−ℓ^⋇} with action of F^⋊±_ℓ andVt,t∈T, a copy ofV. We put a poly-isomorphism

ϕ^Θ_v^±

t :Dv_t(∼=Dv)

Aut+(D^⊚±)-orbit

→∼ D_≻,v,

where, roughly speaking, Aut+(D^⊚±)⊆Aut_±(D^⊚±) is the subgroup such that σ(“positive labels”) =“positive labels” for all σ∈Aut+(D^⊚±).

Moreover, we put

ϕ^Θ_t^{± def}= {ϕ^Θ_vt^±}v_t∈Vt :Dt

def= {Dv_t}v_t∈Vt → D_≻^def= {D_≻,v}v∈V. Then we shall put

ϕ^Θ_±^±def= {ϕ^Θ_t^±}t∈T :DT (or D±)^def= {Dt}t∈T → D≻

and callϕ^Θ_±^± the(model)D-Θ^±-bridge.

(Model) D-Θ^ell-bridge For v∈V, we put

ϕ^Θ_•,v^ell :Dv→ D^⊚±

the morphism determined by the natural morphism X_v →X_v →X_K if v∈V^bad and X−→^v →X_v →X_K ifv∈V^good. Write

ϕ^Θ_v^ell

0

def= Autcusp(D^⊚±)◦ϕ^Θ_•,v^ell ◦Aut+(Dv₀) :Dv₀ → D^⊚±, ϕ^Θ₀^{ell def}= {ϕ^Θ_v^ell

0 }v₀∈V0 :D0

def= {Dv₀}v₀∈V0 → D^⊚±. Note that since

t∈T (∼=Fℓ)⊆F^⋊±_ℓ ∼=Aut_±(D^⊚±)/Aut_cusp(D^⊚±)↷ϕ^Θ₀^ell, we put

ϕ^Θ_t^{ell def}= t·ϕ^Θ₀^ell :Dtdef

= {Dv_t}v_t∈Vt → D^⊚±.

We shall put

ϕ^Θ_±^{ell def}= {ϕ^Θ_t^ell}t∈T :DT def

= {Dt}t∈T → D^⊚±

and call ϕ^Θ_±^ell the(model)D-Θ^ell-bridge.

Summary

We have D-Θ^±ell-HT as following:

D_≻^def= {D_≻,v}v∈V ϕ^Θ_±^±

← DT

def= {Dt}t∈T(∼=Fℓ)

def= {{Dv_t}v_t∈Vt}t∈T ϕ^Θell_±

→ D^⊚±. Note that we do not have any information about theta functions by the definition of D-Θ^±ell-HT. To obtain that, we need to “glue”D-Θ^±ell-HT with D-ΘNF-HT.

§6: Θ^±ellNF-Hodge theaters Reference

Section 6 of “S. Mochizuki, Inter-universal Teichm¨uller theory I:

Construction of Hodge theaters. Publ. Res. Inst. Math. Sci. 57(2021), 3–207.”

D-ΘNF-Hodge theater We shall call

†D>

ϕ^Θ_⋇

←^†DJ ϕ^NF_⋇

→ ^†D^⊚

a D-ΘNF-Hodge theater if it is “isomorphic” to (i.e., poly-isomorphisms

†D> → D∼ >,^†DJ → D∼ J,^†D^⊚→ D^{∼ ⊚} satisfy certain compatible conditions) D>

ϕ^Θ_⋇

← DJ ϕ^NF_⋇

→ D^⊚.

D-Θ^±ell-Hodge theater We shall call

†D_≻^ϕΘ

±±

← ^†DT ϕ^Θell_±

→ ^†D^⊚±

a D-Θ^±ell-Hodge theater if it is “isomorphic” to (i.e., poly-isomorphisms

†D≻ → D∼ ≻,^†DT → D∼ T,^†D^{⊚± ∼}→ D^⊚± satisfy certain compatible conditions)

D_≻^ϕΘ

±±

← DT ϕ^Θell_±

→ D^⊚±.

D-Θ^±ellNF-Hodge theater

We put the following identifications (i.e., “a gluing”) (T\ {0})/{±1}=J.

Then we can construct a D-Θ-bridge

†ϕ^Θ_⋇[^†ϕ^Θ_±^±] :^†DJ

def= {Dt}t∈J →^†D>

from any D-Θ^±-bridge^†ϕ^Θ_±^± :^†DT →^†D_≻. Then we shall call a triple (D-Θ^±ell-HT,D-ΘNF-HT,^†ϕ^Θ_⋇[^†ϕ^Θ_±^±]∼=^†ϕ^Θ_⋇)

D-Θ^±ellNF-Hodge theater.

Then we obtain the following diagram mentioned above.

Frobenioids

Let Dbe aD-prime-strip (i.e., a fundamental group or (the subcategory of connected objects of) a Galois category). We put

F

the Frobenioid whose base isD. Roughly speaking, Fis a category over D whose objects are “rational functions” on objects (i.e., coverings) of D.

Θ^±ellNF-Hodge theater

We obtain the following which is called Θ^±ellNF-Hodge theater, and whose base is D-Θ^±ellNF-Hodge theater:

Thank you for the attention!