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Introduction to Inter-universal Teichmuller theory
Fucheng Tan
RIMS, Kyoto University
SUSTech, January 26, 29, 2018
1
Studying IUT
To my limited experiences, the following seems to be an option for people who wish to get toknow IUT without spending too much time on all the details.
I Regard the anabelian results and the general theory of Frobenioids as blackbox.
I Proceed to read Sections 1, 2 of [EtTh], which is the basis of IUT.
I Read [IUT-I] and [IUT-II] (briefly), so as to know the basic definitions.
I Read [IUT-III] carefully. To make sense of the various definitions/constructions in thesecond half of [IUT-III], one needs all the previous definitions/results.
I The results in [IUT-IV] were in fact discovered first. Section 1 of [IUT-IV] allows one tosee the construction in [IUT-III] in a rather concrete way, hence can be read together with[IUT-III], or even before.
S. Mochizuki, The etale theta function and its Frobenioid-theoretic manifestations.
S. Mochizuki, Inter-universal Teichmuller Theory I, II, III, IV, Preprint.
2
p-adic numbers
Look at a prime number p.
I Any x ∈ Q can be written as x = pn · ab (p - ab).
I Define |x |p = p−n (|0|p := 0). This defines a metric on Q.
I ∏c∈{primes,∞}
|x |c = 1, ∀x ∈ Q×.
I Qp:=completion of (Q, | · |p).
Qp =
{x =
∞∑i=k
aipi , ai ∈ {0, 1, · · · , p − 1}
}
Zp =
{x =
∞∑i=0
aipi , ai ∈ {0, 1, · · · , p − 1}
}
I Alternatively,
Zp = lim←−Z/pn = {(an)n≥1 | an ∈ Z/pn, am ≡ an(pn),∀m ≥ n}Qp = (pN)−1Zp
Fp = Zp/p = Z/p
I Any x ∈ Qp has the form
x = pr · u, with r ∈ Q, u ∈ Z×p .
I (p) has a divided power structure:pn
n!∈ Zp.
3
Local and global Galois groups
I Number field F = finite extension of Q.Fix an algebraic closure F , and set GF = Gal(F/F ).
I Mixed char. local field K = finite extension of Qp.Fix algebraic closure K of K . Set GK = Gal(K/K ).
I GQp ⊂ GQ (up to conjugation): Choose an embedding Q ↪→ Qp, get
Q �� // Qp
Q �� //?�
GQ
OO
Qp?�
GQp
OO
I Inertia and Frobenius:
0→ IQp → GQp → GFp ' (Frobp)Z → 0.
(Neukirch-Uchida) For two number fields F1,F2,
Isomfield(F1,F2) ' Out(GF1 ,GF2) := Isomtop. gp.(GF1 ,GF2)/Inn(GF2).
(Mochizuki) Let K1,K2 be local fields over Qp.
IsomQp(K1,K2)∼−→ OutFil(GK1 ,GK2)
where Fil on GKiis given by the higher ramification groups.
4
What is a Galois representation?
I Consider G = GF ,GK . (K over Qp or Q`)
I A p-adic Galois representation is a finite dimensional Qp-vector space V equipped with aQp-linear continuous action of G , i.e. a continuous homomorphism
ρ : G −→ AutE (V ) ' GLd(Qp).
I By continuity condition, ρ factors through some Zp-lattice:
ρ : G −→ AutZp(T ) ' GLd(Zp).
I residual representationρ : G −→ GLd(Fp).
Galois representations coming from geometry:
I L=field of char. 0 (say Q,Qp,Q`), E = elliptic curve over L.
Tp(E ) = lim←−E (L)[pn], ,
ρE : GL → Aut(Tp(E )) ' GL2(Zp).
I X = “reasonable” scheme (say over Qp or Q),
H∗(Xet,Zp) := lim←−H∗(Xet,Z/pn) (' H∗(Xproet,Zp))
is a finite Zp-module. (SGA, Gabber.)
5
What is a modular form?
A modular form of weight k ∈ Z≥1 on SL2(Z) is a holomorphic function on the upper halfplane
f : H→ C
which is
(i) Modular invariant:
fÅaz + b
cz + d
ã= (cz + d)k f (z),
(ii) Holomorphic at ∞ (“an = 0,∀n < 0”):
The action ofÅ
1 10 1
ãshows f (z + 1) = f (z), so
f (q) =∑
an(f )qn, q := e2πiz .
(A modular form on a congruence subgroup Γ ⊂ SL2(Z) is what you think it is!)
A cuspidal (i.e. a0 = 0) modular form of weight 12 on Γ = SL2(Z):
∆(z) = q∏n≥1
(1− qn)24 :=∞∑n=1
τ(n)qn.
(Ramanujan Conjecture (Deligne 1973): |τ(p)| ≤ 2p11.)
6
Galois representations attached to modular forms
Theorem (Eichler-Shimura, Deligne). Let f be a normalized (a1(f ) = 1) cuspidal newformof weight k ≥ 2 on Γ1(N) of Nebentypus ε : (Z/N)× → C×.∃ Galois rep.
ρf : GQ → GL2(Qp)
s.t. for any ` - Np, ρf |GQ`is unramified (i.e. intertia IQ` has trivial image), and
det(1− Frob−1` x) = 1− a`(f )x + ε(`)`k−1x2
What about those `|Np? (Grothendieck, Tsuji, Faltings, Saito)
(1) ` 6= p but ` | N: ρf (IQ`) is potentially unipotent.
(2) ` = p but p - N: ρf |GQpis crystalline.
(3) ` = p and p | N: ρf |GQpis potentially semi-stable.
(In fact, semi-stable if ap(f ) 6= 0; potentially crystalline if ap(f ) = 0.)
Note: For E an elliptic curve over Q:
L(E , s) =∏p-N
(1− tpp−s + p1−2s)−1
∏p|N
(1− tpp−s)−1.
7
The modularity conjecture
Conjecture (Fontaine-Mazur, 1980’s). Any (irreducible) geometric (i.e. unramified atalmost all places and potentially semi-stable at p) Galois representation
ρ : GQ → GLn(Qp)
comes from a sub-quotient of H i (XQ,et,Qp)(j) for X/Q a proper smooth variety and some i , j .
Taniyama-Shimura-Weil conjecture (1950’s) (Wiles et al, 1990’s):
For any elliptic curve E over Q, ∃ cuspidal newform of weight 2 on Γ0(N)for some N, s.t. ρE ' ρf .
Theorem (Kisin, Hu-T).Assume p > 3. Let ρ : GQ → GL2(Zp) be such that
I ρ is odd, i.e. det ρ(c) = −1 for c the complex conjugation,
I ρ is unramified at almost all places,
I ρ|GQ(µp)is absolutely irreducible,
I ρ|GQpis potentially semi-stable of weight (0, k − 1) for some k ≥ 2.
Then ρ ' ρf for some modular form of weight k .
8
Taniyama-Shimura ⇒ Fermat’s Last Theorem.
Let (a, b, c) be a triple of nonzero relatively prime numbers and p ≥ 5 a prime number, whichsatisfy the Fermat equation (WMA: 2 | b and a ≡ 3( mod 4)).
ap + bp + cp = 0.
I (Frey) Set an elliptic curve E/Q to be
y2 = x(x − ap)(x + bp).
I (Frey-Serre) The residual rep.ρE : GQ → GL2(Fp)
is (abs. irr. and) unramified away from 2p and arises from finite flat group scheme overZp at p.
I (Wiles) There is a newform f of weight 2 on Γ0(NE ) (NE= product of the primes dividingabc) s.t.
ρE ' ρf .
I (Ribet) ρf has to come from a cuspidal newform of weight 2 on Γ0(2).
I We all know such a modular form does not exist!
9
The period rings in p-adic Hodge theory
I For X = SpecC[t, t−1],
H idR(X/C)
∼−→ Hom(Hi (Xan),C),
ω =dt
t7→
Äγ = (x 7→ e2πix , 0 ≤ x ≤ 1) 7→ 2πi
ä.
I X = smooth scheme of finite type over a field K ↪→ C.
H idR(X/K )⊗K C ∼−→ H i (X an
C ,Q)⊗Q C.
(In fact, the following construction works for any perfectoid field.)
I O[Cp= lim←−x 7→xp
OCp/p.
I W (O[Cp) = “O[Cp
⊗ Zp”.
I θ : W (O[Cp)→ OCp , (x0, x1, · · · ) 7→
∑∞n=0 p
nx(n)n , x
(n)n := limm→∞ xp
m
n,n+m,
xn = (xn,i )i∈N ∈ O[Cp, xn,n+m any lifting to OCp .
I Acris = p-adic completion of the PD-envelope w.r.t. ker(θ) of W (O[Cp).
I t := log([ε]) for [·] : O[Cp→W (O[Cp
). (p-adic “2πi”)
I Bcris = Acris[1/t].
I Bst = Bcris[u], u = log[(−p)[], (−p)[ := (−p, · · · ) ∈ O[Cp.
(equipped with a Bcris-linear operator N such that N(u) = −1, Nϕ = pϕN.)
A continuous Qp-rep. V of GK is semi-stable if
V ⊗Qp Bst∼←− (V ⊗Qp Bst)
GK ⊗K Bst.
The notion potentially semi-stable is defined by allowing replacing K with a finite extension.
10
Mysterious functor and comparison isomorphisms
Grothendieck (1970): For G a p-divisible group,
Tate module Tp(G) ←→ Dieudonne module D(G).
Theorem (Tusji, Faltings, Beilinson).For X a variety over a p-adic field K , ∃ isomorphism compatible with Galois action, filtration,Frobenius, and monodromy:
H i (XK ,et,Qp)⊗Qp Bst ' H ilog-cris(X0/OK0)⊗K0 Bst.
Theorem (Faltings-Andreatta-Iovita, T-Tong). Assume K is absolutely unramified overQp. Let X be a proper smooth formal scheme over OK , and X = XK . Let L be a lisseZp-sheaf on Xet and E a filtered convergent F -isocrystal on X0/OK . If they are associated,then ∃ natural isomorphism compatible with Galois action, filtration, and Frobenius action:
H i (XK ,et,L)⊗Zp Bcris ' H icris(X0/OK , E)⊗K Bcris.
Theorem (Faltings, T-Tong). The analogue for a proper smooth morphism of smoothformal schemes over OK holds.
11
The Vojta conjecture for curves (⇔ ABC conjecture ⇒ Szpiro conjecture)
X proper smooth geometrically connected curve over a number field; D reduced effectivedivisor on X s.t. U = X\D is hyperbolic.
I x ∈ X (Q), F ⊂ Q the minimal subfield where x is defined.
I deg=normalized degree map on arithmetic divisors on F .
I log-diffX (x) := deg(δx), for δx the different ideal of F/Q.
I log-condD(x) := deg((Dx)red), well-defined up to bounded discrepancy.(D the closure of D in X , a regular proper model of X . )
Theorem (S. Mochizuki). ∀d ∈ Z>0 and ∀ε > 0, ∃ constant C = C (X ,D, d , ε) such that onU(Q)≤d (points defined over an extension of Q with degree ≤ d):
htωX (D)(x) ≤ (1 + ε) (log-diffX (x) + log-condD(x)) + C .
Theorem (ABC/Masser-Oesterle conjecture). N(n):=product of distinct prime divisors ofn. ∀ε > 0, ∃ constant Kε such that for any triple (a, b, c) of coprime positive integers witha + b = c :
c < Kε · N(abc)1+ε.
FLT again (IUT will tell us more: for ε = 1, we can take Kε = 1.):
Suppose there is a positive integral solution to
X n + Y n = Zn, n ≥ 3.
(WMA X n,Y n,Zn coprime.) Take a = X n, b = Y n, c = Zn. Then
Zn ≤ 1 · N(X nY nZn)1+1 = N(XYZ )2 < Z 6, ⇒ n = 3, 4, 5.
12
Inter-universal Teichmuller theory
IUT studies the arithmetic structures surrounding::::::elliptic
::::::curves
:::::over
::::::::number
:::::fields
::::::::::(equipped
::::with
::::::::auxiliary
::::::data), and arrows among different copies of them. One of these arrows is called
Θ-link, which dismantle ring structures while preserving the data associated to local Galoisgroups.
The use of Θ-link eventually leads to two ways of computing the degree of certain arithmeticline bundle associated to theta function.The discrepancy between them can becontrolled/estimated, which leads to a proof of the Vojta/ABC conjecture.
IUT is in some sense a global simulation of the comparisonisomorphism in p-adic Hodge theory.
13
I I = {0, · · · , j} with j ≥ 1. For i ∈ I , ki/Qp a finite extension with ramification index ei .
I Ri = Oki , di the valuation of any generator of the different ideal.
RI = ⊗i∈IRi , logp(R×I ) = ⊗i∈I logp(R×i ). dI =∑i∈I
di .
I ai = dei/(p−2)eei
, bi = b log(pei/(p−1))log p c − 1
ei
paiRi ⊂ logp(R×i ) ⊂ p−biRi , (equalities if ei < p − 1)
I log : O× → O. Have a common container for its domain and codomain, despite the factthat the map itself dismantles the ring structures.
O× ⊂ O ⊂ I =1
p(log(O×)) ⊃ log(O×).
I For an automorphism (RI ⊂ ‹RI normalization)
φ : logp(R×I )Q∼−→ logp(R×I )Q with logp(R×I )
∼−→ logp(R×I ),
φ(pλ‹RI ) ⊂ pbλ−dI−aI c logp(R×I ) ⊂ pbλ−dI−aI c−bI ‹RI , λ ∈ 1
ejZ.
I K/Q fin. ext., Ri = OKvi.
IIQ = ⊗i∈I (⊕v |vQKv )i ' ⊕(⊗vi |vQ,i∈IKvi ).
I
pbλ−dI−aI c−bI ‹RI bound of averaged (over vi |vQ) volume µlog((UΘ)vQ) (1)
Theorem: For K = F (F [`]) with suitable EF and `, λ = vp((q1
2`vj )j
2), j = `−1
2 ,∑vQ
µlog((UΘ)vQ) ≥ −deg. of arith. l.b. determined by q-parameters of EF .
I The following are devoted to the construction of UΘ.
14
Overview
In IUT theory, one starts with a suitable elliptic curve E over a number field F and a primenumber `. A Hodge theater is designed to carry two kinds of symmetries associated to a fixedquotient
E [`]� F`.
The multiplicative symmetry will be applied to copies of the moduli field Fmod, while theadditive symmetry assures that the conjugacies of various values of theta function aresynchronized. These theta values and the number field Fmod will determine the so-called
Θ-pilot object. This set of theta values at each bad place v (modulo torsion) has the form
{qj2
v}j=1,··· , `−1
2,
with qv
a 2`-th root of the q-parameter of E at this place.
The main construction of IUT is the so-called multiradial representation of the Θ-pilotobject, which can only be achieved under the indeterminacies (Ind1, 2, 3), concerning theautomorphisms of local Galois groups, automorphisms of local unit groups, and change of ringstructures caused by log-operation, respectively.
The multiradial representation is in particular compatible with the crucial Θ-link between twocopies of Hodge theaters, which dismantle ring structures so as to extract information onarithmetic degrees. It follows from this compatibility that the so-called holomorphic hull ofthe multiradial representation of the Θ-pilot object, which is an arithmetic line bundle, hasdegree bigger than that determined by q-parameters. Such a result finally leads to the proof ofABC/Vojta conjecture.
15
Once-punctured elliptic curves
I k = finite extension of Qp.
I X = stable log curve of type (1, 1) over SpfOk , whose special fiber is singular and split,and whose generic fiber X is a smooth log curve.
I ΠtpX = tempered fundamental group of X ; ∆tp
X ⊂ ΠtpX the geometric tempered
fundamental group.1→ ∆tp
X → ΠtpX → Gk → 1.
Their profinite completion: ΠX ,∆X .
I Natural exact sequences
1→ Z(1)→ ∆ellX =
∆X
[∆X ,∆X ]→ Z→ 1,
1→ ∆Θ(' Z(1))→ ∆ΘX =
∆X
[∆X , [∆X ,∆X ]]→ ∆ell
X → 1,
where ∆Θ := Im(∧2∆ellX ↪→ ∆Θ
X ).
I
1→ Z(1)→ (∆tpX )ell → Z→ 1.
I The universal cover of the dual graph of X0 determines a cover
Y → X , Gal(Y /X ) ' Z.
The special fiber of Y (normalization) is a chain of copies of P1, labeled by Z, joined at0,∞.
16
Some Kummer theory
I qX ∈ Ok the q-parameter of the elliptic curve associated to X .
I Form kN = k(ζN , q1/NX ),N ≥ 1.
I The restriction to GKNof any cuspidal decomposition group of Πtp
Y Galois cover
YN → Y .
I Normalization YN → Y of Y in YN .
Pic(YN) ' ZGal(Y /X ) ' ZZ.
The (isomorphism class of) line bundle determined by 1Gal(Y /X ) is denoted by LN .
I Divisor of cusps of Y s1 ∈ Γ(Y,L1)
(well-defined up to ·O×k ).
I (With suitable base field extension)(i) There exists an N-th root sN of s1 on some Galois cover of YN .(ii) sN an action of Πtp
X on LN .
I
YN = Y2N .
I DN = effective divisor on YN supported on the special fiber and is equal to, at the irr.
component j , the zero locus of qj2/2NX .
I ∃τN ∈ Γ(YN , LN) (well defined up to O×-multiples), whose zero locus is equal to DN . Πtp
Y-action on LN . This action differs from the previous action by µN -multiples,
O×k2· ηΘ ⊂ H1(Πtp
Y,∆Θ). (2)
17
The etale theta function
I U ⊂ Y the open subscheme obtained by removing the node at the irr. component-0 of thespecial fiber, U = U×Y Y. The isomorphism U ' “Gm provides coordinate U ∈ Γ(U,O×
U).
I
Θ = Θ(U) =∑n∈Z
(−1)nq12
(n2+n)
X U2n+1. (3)
I (etale theta function)Θ extends uniquely to a meromorphic function on Y, whose zeros are of multiplicity oneand exactly the cusps, and whose divisor of poles = D1. Thus, O×k2
· ηΘ coincide with the
Kummer classes associated to O×k2· Θ.
I From now on, suppose k = k2 and that X is not k-arithmetic.
I (Constant Multiple Rigidity) The pullback via section determined by a 4-torsion point ofY gives
±ηΘ,Z ⊂ H1(Πtp
Y,∆Θ)
which corresponds to ΘΘ(√−1)
.
18
More covers
I Let k be a field of char. 0. Let X be a smooth log curve of type (1, 1) over k. Assumethat X is not k-arithmetic.
I The “theta quotient”
∆ΘX '
Ö1 Z(1) Z(1)
1 Z1
èI ` ≥ 3 a prime number. The `-th power map yields the quotient exact sequence
1→ ∆Θ,`(' µ`)→ ∆ΘX ,` → ∆ell
X ,` → 1.
I Let x0 denote the cusp of X . Consider
1 // ∆Θ,`// Dx0,` := Im(Dx0 ↪→ ΠΘ
X � ΠΘX ,`)
// Gk// 1
I Let∆ell
X ,` � Q (' Z/`Z)
be a quotient whose restriction to Dx0,` is trivial. Galois cover X → X withGal(X/X ) ' Q. Thus
∆ΘX '
Ö1 Z(1) Z(1)
1 `Z1
è.
I ιX the natural inversion of X (relative to the origin x0), ιX the inversion of X relative tosome cusp of X lying over x0.
XQ //
ιX
��
X
ιX��
Cdeg ` // C
I One checksAutk(C/C ) = Autk(C ) = {1}.
19
I Thus, any splitting of Dx0,` � Gk determines a cover X → X ,
∆ΘX '
Ö1 Z(1) `Z(1)
1 `Z1
è.
I ΠtpX Πtp
C ,ΠtpX ,Π
tpY ,Π
tp
Y,Πtp
X ,ΠtpC .
ΠtpX Πtp
C ,ΠtpX ,Π
tpY ,Π
tp
Y.
I Key assumptions:
I The quotient ∆ΘX ,` � Q in the definition of X is compatible with the natural quotient
ΠtpX � Gal(Y /X ) ' Z.
I Assume that the choice of splitting of Dx0,` → Gk is compatible with the ±1-structure on the”k×-torsor at x0 determined by ηΘ,Z.
I Always keep in mind:
Yµ` //
µ2��
Yµ2��
Yµ` //
`Z��
YZ
&&LLLLL
LLLLLL
LLL
`Z��
Xµ` //X
Z/`Z//
{±1}��
X{±1}��
Cdeg `
//C
20
`-th roots of etale theta function
I An “`-th root of the etale theta function”
ηΘ,`Z×µ2 ⊂ H1(Πtp
Y, `∆Θ). (4)
I (Labels of cusps)
|F`| := F`/{±1} ' Cusp(∗)/Autk(∗), ∗ = X ,C .
The rigidity property above is false for the curve X .
I
[∆ΘX ,∆
ΘY ] '
Ö1 0 `Z(1)
1 01
è' `∆Θ.
21
Initial Θ-data
1. F = number field containing√−1.
2. XF = EF\{oEF} once-punctured elliptic curve which admits stable reduction at all
nonarchimedean places V(F )non of F . Suppose EF [6] is rational over F .CF = stack-theoretic quotient of XF by its unique F -involution.Fmod = field of moduli of XF , with the set of valuations Vmod.Vbad
mod ⊂ Vmod a nonempty set of nonarchimedean places of Fmod away from 2, where XF
has bad multiplicative reductions at Vbad = Vbadmod ×Vmod
V(F ).Assume F/Fmod is Galois of degree prime to ` (see below).
3. ` ≥ 5 is a prime number away from Vbadmod and the orders of the q-parameters of EF .
Suppose furtherGF → Aut(EF [`]) ' GL2(F`)
has image containing SL2(F`). The kernel determines a number field K .
4. Hyperbolic orbicurve CK (K -core CK = CF ×F K ), XK , XF
.
5. A chosen sectionVmod
∼−→ V ⊂ V(K )
such that at v ∈ Vbad the cover X v → XFv is compatible with Tate uniformization.
(Such sections for all v ∈ Vbad exist thanks to (3)!) Write
Πv = ΠtpX
v
.
6. ε is a cusp of CK arising from an element of the quotient
EF [`]� Q (5)
in the definition of CK , such that at v ∈ Vbad the corresponding cusp εv of C v arisesfrom the canonical generator “±1” under Tate uniformization.
22
Frobenioids at bad places
I Xv
determines tempered Frobenioid Fv
over the base category
Dv := Btp(ΠXv),
D`v := B(GKv ) ⊂ Dv .
I Fv (up to µ2` × `Z-indeterminacy) the reciprocal of `-th root of theta function
Θv∈ K×
Yv
.
Also, Fv
determines category-theoretically the base-field-theoretic hull
Cv ⊂ Fv.
Θv
(»−qv ) = q
12`v =: q
v, up to µ2`-multiples.
Then qv
determines a pv -adic Frobenioid and a µ2`-orbit of characteristic splittings on it:
C`v , τ`v .
I Split Frobenioids:F`v := (C`v , τ`v ).
23
Global realified prime-strips
I (i) Holomorphic F-prime-strip
†F = {†Fv = †Cv}v∈V.
(ii) Mono-analytic F`-prime-strip
†F` = {†F`v }v∈V.
I Realification of the Frobenioid associated to (Fmod, the trivial Galois extension of Fmod):
C mod. (6)
At each v ∈ Vmod, we have the submonoid of ΦC mod
:
ΦC mod,v
' ord(OBFmod,v)pf ⊗ R≥0 (' R≥0).
The restriction functor Cρv : C mod → (C`v )rlf induces (for v |v)
ρv : ΦC mod,v
∼−→ ΦrlfC`v, log`mod(pv ) 7→
logΦ(pv )
[Kv : Fmod,v ].
I
F = F mod :=ÄC mod,Prime(C mod) ' V,F` = {F`v }v∈V, {ρv}v∈V
ä. (7)
a well-defined degree map on F , invariant under automorphisms.
I†HT Θ :=
({†F
v}v∈V, †F mod
).
24
κ-coric rational functions
I Look at the categoryD} := B(CK ).
M~(†D}) := M~(†D}) ∪ {0} ' F .
I The (unique) model CFmodover Fmod of CF
π1(†D~) ' ΠCFmod, †D~ := B(π1(†D~)).
M~mod(†D}) ' F×mod, M~mod(†D}) ' Fmod.
I (κ-coric functions) Let L = Fmod. Denote by KC the function field of CL, and KC analgebraic closure of KC , which then determines an algebraic closure L of L.(i) For the curve determined by some finite extension of KC , a closed point is calledcritical if it maps to the 2-torsion points of EF . A critical point not mapping to the cuspof CL is called strictly critical.(ii) A rational function f ∈ KC is κ-coric if
I if f /∈ L, then it has exactly 1 pole and ≥ 2 (distinct) zeros,I the divisor of zeros and poles of f is defined over a number field and avoids the critical
points, andI f restricts to roots of unity at strictly critical points of |CL|cpt.
(iii) ∞κ-coric if f n is κ-coric for some n ∈ Z>0.
(iv) ∞κ×-coric if c · f is ∞κ-coric for some c ∈ L×
.
I π1(†D}) a profinite group (absolute Galois group of KCFmod)
πrat1 (†D~)(� π1(†D~)),
pseudo-monoids of κ-, ∞κ- and ∞κ×-rational functions:
M~κ (†D}), M~∞κ(†D}), M~∞κ×(†D}).
25
Cyclotomic rigidity of global and local κ-coric structures
I Now the assignment
Φ~(†D}) : Ob(†D~) 3 A 7→ monoid of arithmetic divisors on M~(†D})A
Frobenioid over †D~:F~(†D}).
I Let†F~ ' F~(†D})
be a category with Base(†F~) = †D~. Define
†F} := †F~|†D} ,
†F~mod := †F~|terminal objects of †D~ .
Note the natural isomorphism of realified Frobenioids
†F~Rmod '†C mod.
I Look at an isomorphπrat
1 (†D~) y †M~∞κ×.
I
µet(π1(†D})) = MZ := Hom(H2(∆Z , Z), Z)
for ∆Z corresponding to the geometric fundamental group of the canonicalcompactification of some hyperbolic curve of genus ≥ 2 which is a finite etale cover ofXK appearing in the Belyi cuspidalization of XK .
µFr((−)) = Hom(Q/Z, (−)×).
26
I The consideration of divisors of zeros and poles of κ-coric functions and the fact
Q>0 ∩ Z× = {1},
∃! isomorphism of cyclotomes
µet(π1(†D}))∼−→ µFr(†M~∞κ×) (8)
commutative diagram:
M~∞κ×(†D})
'��
� � // lim−→HH1(H, µet(π1(†D})))
'��
†M~∞κ×� � // lim−→H
H1(H, µFr(†M~∞κ×))
I We conclude that †F~ carries natural structures of ∞κ×, ∞κ-, κ-rational functions
πrat1 (†D~) y †M~∞κ×,
†M~∞κ,†M~κ := (†M~∞κ)π
rat1 (†D~).
I In each local case, the p-adic Frobenioid †Fv carries the following natural structures:
π1(†Dv ) y †Mv (' OBF v
),
πrat1 (†Dv ) y †M∞κv , †M∞κ×v , †Mκv := †Mπrat
1 (†Dv )∞κv .
27
Local label classes of cusps
I For each v ∈ Vnon, †Dv ' Btp(Xv
). At each v ∈ Varc...
D-prime-strip:†D := {†Dv}v∈V.
D`-prime-strip:†D` := {†D`v }v∈V.
I A label class of cusps of †Dv is the set of cusps of †Dv over a (single) cusp of†Dv ' Btp(ΠC v
) arising from a nonzero element of the quotient Q ' F`.I At a bad place v , the action of AutKv (X
v) on the set of cusps of X
vfactors through its
quotient {±1}.I At each v ∈ V, the natural action of F×` on Q ' F` equips the set LabCusp(†Dv ) with
an F>` -torsor structure, where
F>` := (F`\{0})/{±1}, |F>
` | = `> :=`− 1
2.
I
LabCusp(†Dv )∼−→ LabCusp(†Dw ), †η
v7→ †η
w.
LabCusp(†D).
28
Global label classes of cusps of arithmetic nature
I The existence of the model CFmodover Fmod
Gal(K/Fmod) −→ GL2(F`)/{±1},
Aut(CK )∼−→
®Ç∗ ∗0 ∗
å´/{±1}
⋂Im(Gal(K/Fmod)),
Autε(CK )∼−→
®Ç∗ ∗0 ±1
å´/{±1}
⋂Im(Gal(K/Fmod)),
Aut(CK )/Autε(CK ) ' Aut(D})/Autε(D})∼−→ F>
` . (9)
I For v ∈ Vbad, the consideration of the covers Xv→ C v → CK yields morphisms
φNF•,v : Dv → D},
φNFv = {β ◦ φNF
•,v ◦ α}α∈Aut(Dv ),β∈Autε(D}), ∀v ∈ V,
φNFj : Dj = {Dv j
}v∈V,
φNF> = {φNF
j }j∈F>`
: D> = {Dj}j∈F>`→ D}.
By its very construction, φNF> is F>
` -equivariant.
29
Evaluation points at bad places
I Let µ− ∈ X v (Kv ) be the unique 2-torsion point whose closure intersects irr. component-0(of the special fiber of (any) given stable model) of X v .Define the evaluation points of X v (Kv ) as µ−-translations of the cusps with labels in |F`|.
I This way, the value of the function Θv
at an evaluation point of Yv
with label j ∈ |F`|lies in the µ2`-orbit of
qj2
v.
I For any j ∈ F>` and v ∈ Vbad, we have the poly-morphism
φΘv j
: Dv j
∼−→ Btp(Πv ) −→ Btp(Πv )∼−→ D>,v
with the middle arrow induced by the composition of the evaluation section labeled by jwith the natural surjection Πv � Gv .
I
†HT D-ΘNF = (†D}†φNF
>←− †DJ
†φΘ>−→ †D>).
I Synchronized indeterminacy
LabCusp(D})φNFj−→ LabCusp(Dj)
φΘj−→ LabCusp(D>)
can.∼−→ F>
`
I Recall the two kinds of Hodge theaters we have seen:
†HT Θ = ({†Fv}v∈V, †F mod),
†HT D-ΘNF = (†D}†φNF
>←− †DJ
†φΘ>−→ †D>).
Assume the F-prime-strip †F> associated to †HT Θ (via the †Fv
) has D-prime-strip †D>.
I †φΘ> : †DJ → †D> determines (uniquely)
†ψΘ> : †FJ −→ †F>.
I
†HT ΘNF = (†F~ L99 †F}†ψNF
>←− †FJ
†ψΘ>−→ †F> 99K
†HT Θ).
30
±-label classes of cusps of geometric nature
I I Fo±` := F` o {±1}.
I F±` -group, defined as a set E ' F` equipped with {±1}-action.I F±` -torsor, defined as a set T ' F`, where Fo±
` acts on F` by z 7→ ±z + λ, λ ∈ F`.
I A ±-label class of cusps of †Dv is the set of cusps of †Dv lying over a single cusp of †D±v ,a category corresponding to X v .
I There are two canonical elements in LabCusp±(†Dv ), corresponding to the zero cusp andthe canonical generator of Q. F±` -group
LabCusp±(†Dv )∼−→ F`.
I Now the global situation. Consider the category
D}± := B(XK )
and the homomorphism (adapted to the quotient Q)
Aut(D}±)∼−→ Aut(XK )→ GL2(F`)→ GL2(F`)/{±1}
Aut±(D}±) = kerÄ
Aut(D}±)� F>`
ä.
I
(∆CK/∆XK
') AutK (XK )∼−→ Aut±(D}±)/Autcsp(D}±)
ε∼−→ Fo±
` .
I Remark (Conjugate synchronization). We shall use
∆CK/∆XK
y LabCusp±(D}±) ' LabCusp±(†Dv t),∀t ∈ F`
to establish the crucial “conjugate synchronization” for the evaluations of etale thetafunction on the cusps.
31
The Hodge theater
I
†HT D-ΘNF = (†D}†φNF
>←− †DJ
†φΘ>−→ †D>),
†HT ΘNF = (†F~ L99 †F}†ψNF
>←− †FJ
†ψΘ>−→ †F> 99K
†HT Θ),
†HT D-Θ±ell= (†D}±
†φΘell
±←− †DT
†φΘ±±−→ †D�),
†HT Θ±ell= (†D}±
†ψΘell
±←− †FT
†ψΘ±±−→ †F�).
I T = F`. Write |T | = T/{±1}, T> := |T |\{0}.
(†φΘ±± : †DT → †D�) → (†φΘ
> : †DT> → †D>)†DT |T\{0} 7→ †DT> ,
†D0,†D� 7→ †D>.
the Θ±ellNF-Hodge-theater†HT Θ±ellNF
Note that such a gluing is unique because:
Isom(†φΘ>,‡φΘ
>) = {∗},
Isom(Θ-bridges)∼−→ Isom(associated D-Θ-bridges).
32
Mono-theta environments
I Let N be a positive integer.
I Take (a cocycle of) the reduction mod N of one class in ηΘ,`Z×µ2 ⊂ H1(Πtp
Y, `∆Θ) and
subtract it from the tautological section
stauY
: Πtp
Y→ Πtp
Y[µN ] ↪→ Πtp
Y [µN ] (10)
This yields a homomorphismsΘY
: Πtp
Y→ Πtp
Y [µN ]. (11)
I Modifying a cocycle above by coboundaries corresponds to replacing sΘY
by its
µN -conjugates.
I (Mono-theta environment) A mod N mono-theta environment
MΘ = {Π,DΠ, sΠ} :
(i) a topological group Π ' ΠtpY [µN ];
(ii) a subgroup DΠ ⊂ Out(Π) such that DΠ ' DY ⊂ Out(ΠtpY [µN ]), generated by the images of
K×,Gal(Y /X ) via the natural outer actions;
(iii) a collection of subgroups sΘΠ ⊂ Π, isomorphic to the µN -conjugacy class of the image
sΘY
(Πtp
Y) ⊂ Πtp
Y [µN ] of sΘY
.
I
ΠtpX = Aut(Πtp
Y [µN ])×Out(ΠtpY [µN ]) DY .
33
Rigidities of mono-theta environments
I (Cyclotomic rigidity) †MΘ ∼−→MΘ(Y ) isomorphs of splittings sΘY, stau
Y. Their
difference yields (g 7→ sΘ(g)/stau(g))
(`∆Θ)(†MΘ)/N =: µetN (†MΘ)
∼−→ µFrN (†MΘ) := ker
Ä(`∆Θ[µN ])(†MΘ)� (`∆Θ)(†MΘ)
ä(12)
I (Discrete rigidity) Any projective system
· · · → †MΘN′
†γN′,N−→ †MΘN → · · ·
is isomorphic to the projective system · · · →MΘN′(Y )
γN′,N−→ MΘN(Y )→ · · · .
I (Constant multiple rigidity) Given †MΘ = {†MΘN}N one may construct (an isomorph
†ΠtpX ' Πtp
X , hence) some µ`-multiple of an isomorph of ηΘ,`Z×µ2 .
I The commutator map [−,−] which determines the tautological section, isΠC (†MΘ)-equivariant. In particular, the inducedΠC (†MΘ)/ΠX (†MΘ) ' ∆C (†MΘ)/∆X (†MΘ) ' Fo±
` -action is compatible with (12).
34
Splittings of integer rings via theta values
I Let ιX be the unique inversion of X over k , which lies over an inversion ιX of X .
I Note that ιX fixes µ− and (hence) ιX fixes (µ−)X . Thus, ιX lifts to an inversion ιY of Y .
Thus the restriction to the decomposition group Dµ− ⊂ Πtp
Yof (µ−)Y of etale theta
function takes values in µ2`.
I Let Π ' ΠtpX be a topological group. The set of µ`-multiples of the reciprocal of the
isomorph of ηΘ,`Z×µ2(Π):
θ(Π) ⊂ H1(Πtp
Y(Π), µet(Π)).
Elements whose certain positive integral power (up to torsion) lies in θ(Π):
∞θ(Π) ⊂ lim−→H
H1(Πtp
Y(Π)|H , µet(Π)).
I Let (ι,D) ' (ιY ,Dµ−) be isomorph via Π. Restriction to D yields splittings
(O× ·∞θι(Π))/Oµ //
��
O×µ(Π)× (∞θι(Π)/Oµ)
��(O× ·∞θιenv
(MΘ(Π)))/Oµ // O×µ(MΘ(Π))× (∞θιenv
(MΘ(Π))/Oµ).
I Let v ∈ Vnon. For an isomorph (G y O×µ) of (Gv y O×µK v
), define
Ism(G ) (⊂ AutG (O×µ))
as the subset of G -equivaraint automorphisms of O×µ preserving the imageIm((O×)H → O×µ) for any open subgroup H ⊂ G . A ×µ-Kummer structure on(G y O×µ) is an Ism-orbit of O×µ ' O×µ
K v.
35
Conjugate synchronization of theta values
I (i) ΓIX ⊂ ΓX the unique connected subgraph which is a tree, stabilized by ιX , andcontains all vertices of ΓX .(ii) Γ•X ⊂ ΓX the unique connected subgraph which is stabilized by ιX and consists ofexactly one vertex of ΓX .
(iii) finite connected subgraphs associated to X , Y , Y .
I (Decomposition groups associated to subgraphs)
Π• ⊂ ΠI ⊂ ΠtpX =: Π.
I Let t ∈ LabCusp±(Π) and the decomposition group Π•t ⊂ ΠI. For � = •t,I, writeΠ� = Π� ∩ Πtp
Y, ∆� = ∆ ∩ Π�.
I Given a projective system MΘ, have subgroups ΠI(MΘ) ⊂ ΠI(MΘ) ⊂ Π(MΘ)corresponding to ΠI ⊂ ΠI ⊂ Π.
I Let It ⊂ Π be a cuspidal ineria of ∆•t . Suppose γ′ ∈ ∆X ,•t , γ ∈ “∆X and set
δ = γγ′ ∈ “∆X . Decomposition group of µ−-translation of the cusp giving rise to I δt :
Dδt,µ− ⊂ Πδ
I = ΠγI,
compatible with the conjugacy of ∆C/∆X ' Fo±` .
I Finally, get rid of the dependence on ιY = ι. The restriction to Dδt,µ− yields µ2`, µ-orbits
of elements
(θenv
)|t|((MΘI)γ) ⊂ ∞(θ
env)|t|((MΘ
I)γ) ↪→ lim−→H
H1(G ((MΘI)γ)|H , µFr((MΘ
I)γ)).
I (Canonical splittings) The restriction to Dδ0,µ− induces splittings
(O× ·∞θιenv((MΘ
I)γ))/µ // O×µ((MΘI)γ)× (∞θ
ιenv
((MΘI)γ)/µ).
36
Constant and Theta monoids
I (Constant Monoids) MΘ := MΘ(Fv
).
Ψcns(MΘ)v = O.(Gv (MΘ)) ⊂ lim−→H
H1(ΠY (MΘ)|H , µFr(MΘ)),
Ψcns(Dv ) = O.(Gv (D`v )) (' O.F v
),
Ψcns(Fv ) = O.Cv (AΘ∞),
with AΘ∞ a universal covering pro-object of Dv .
I Kummer mapκ : Ψcns(Fv )
∼−→ Ψcns(MΘ) ' Ψcns(Dv )
I
Ψenv(MΘ)v =¶
Ψιenv(MΘ)v := Ψcns(MΘ)×v · θιenv
(MΘ)N©ι,
ΨFenv,v =ß
ΨFΘv ,α
:= O×CΘv
(AΘ∞) · (Θα
v)N|AΘ
∞
™α∈Πv
,
I Matching the image of α under Πv � `Z and ι ∈ `Z:
κ : (∞)ΨFenv,v∼−→ (∞)Ψenv(MΘ).
Consider the full-poly-isomorphism MΘ(Πv ) 'MΘ(Fv
). The construction in Πv of
(∞)Ψenv(MΘ(Πv )) ' (∞)ΨFenv,v
is compatible with arbitrary automorphisms of the pair G (MΘ) y Ψ×µFenv,v.
37
Gaussian monoids and Gaussian evaluation isomorphisms
I
The action of (∆Cv /∆X v)(MΘ) ' Fo±
` on ΠX v(MΘ) induces isomorphisms among
Gv (MΘI)t y Ψcns(MΘ)t , t ∈ F`.
The diagonal embedding (recall F>` = |F`|\{0})
Gv (MΘI)〈|F`|〉 y Ψcns(MΘ)〈|F`|〉, Gv (MΘ
I)〈F>`〉 y Ψcns(MΘ)〈F>
`〉
is compatible with the pair
(Gv (MΘI)�) ΠvI(MΘ) y Ψcns(MΘ)
ThusΨcns(MΘ)0 ' Ψcns(MΘ)〈F>
`〉 (13)
compatible with the respective actions of Galois groups.
I From now on, we use the subscript 4 to indicate the identification of labels 0 = 〈F>` 〉.
F`4 := {Ψcns(Fv )}v∈V.
I (Mono-theta-theoretic Gaussian monoids)
Ψgau(MΘ)v ={
Ψξ(MΘ)v := Ψ×cns(MΘ)〈F>`〉 · ξ
N}ξ∈∏|t|∈F>
`(θ
env)|t|(MΘ
I).
I The restriction θι(ΠvI)∼−→ θ|t|(ΠvI) induces evaluation isomorphism
(∞)Ψenv(MΘ)v∼−→ (∞)Ψgau(MΘ)v .
Concretely, (modulo µ2`)
Θv7→ {qj2
v}j∈F>
`.
38
I Define
ΨFgau(Fv
) =
ΨFξ(Fv
) = Im
ÖΨξ(MΘ)v ↪→
∏|t|∈F>
`
Ψcns(MΘ)|t|∼−→∏|t|
Ψcns(Fv )
èξ
.
I The evaluation (plus splittings) isomorphism at Frobenius level
(∞)ΨFenv
∼−→ (∞)ΨFgau (14)
I At Frobenius level, construct Frobenioids C env = C env(HT Θ), C gau = C gau(HT Θ) and
C env∼−→ C gau. (15)
I Define global realified prime-strips
F env =ÄC env,Prime(C env)
∼−→ V,F`env := F`(ΨFenv), {ρv}ä, F gau
I F` = {F`v }v∈V F`I×µ = {F`I×µv }v∈V
of split Frobenioids equipped with ×µ-Kummer structures. Thus F (−) F I×µ(−) and
F I×µenv
ev∼−→ F I×µgau .
I Recall the localization functors
F~mod,j → Fj , j ∈ F>`
and thus (via C (Fj)∼−→ F~Rmod,j) the inclusion
C gau(HT Θ) ↪→∏j∈F>
`
F~Rmod,j . (16)
Here we have used the multiplicative symmetry on the set of labels F>` .
39
Definition of log-links
At each bad place, this section concerns the action of theta values on the log-shell
{qj2
v}j=1,··· , `−1
2y
1
pvlog(O×Fv
).
I In the following, we always look at
F = {Fv := Cv}v∈V
D = {Dv ' Btp(ΠXv)}v∈V
D` = {D`v ' B(GKv )}v∈V.
I Consider the log-operation:
Ψcns(Fv )×log−→ (Ψcns(Fv )×)pf =: log(Fv ) (' F v ).
Monoid/Frobenioid:
Ψlog(Fv ) (' O.F v
), log(Fv ) (' Fv ).
I (i) A log-link is defined to be a poly-isomorphism log(‡F)∼−→ †F:
log : ‡F→ †F.
(ii) A log-link between Hodge theaters is the collection of log-links on F-prime-strips†F>,
†F�,†FJ ,
†FT which lifts some isomorphisms on the bases:
log : ‡HT → †HT .
I The Kummer isomorphismsκ : Ψcns(F)
∼−→ Ψcns(D)
are incompatible with log-links.
40
Additive and multiplicative global Frobenioids
I An object of F~mod is of the form J = {Jv}v∈V with Jv ⊂ Kv a fractional ideal at v in V,almost all of which are the integer rings. Obviously,
F~mod ' F~mod.
I The Frobenioid of pairs (T , {tv}v∈V) with T an F×mod-torsor and a trivialization at each v
F~MOD (' F~mod).
I
log(FvQ) = ⊕v |vQ log(Fv ), log(FVQ) =∏
vQ∈VQ
log(FvQ).
I Localization F~mod → F induces an injective ring homomorphism
M~mod ↪→ log(FVQ). (17)
M~mod := Im(M~mod ↪→ log(FVQ)).
I To construct the Frobenioid F~mod, we need the number field M~mod AND the additivemodules Ψlog(†Fv )
⋃{0} ⊂ log(Fv ).
I A nonzero element in the number field which is integral at all local places has to be aroot of unity (and that log kills torsion) For any m ∈ Z, the set mM~mod,j is invariant under the log-links.
The global Frobenioid F~MOD associated to the number field Fmod only uses the latter asan entire set (and local data), hence is invariant under the log-links.
41
Log-shells
I
I(Fv ) =1
2pvlogÄ
(Ψcns(Fv )×)Gvä⊂ log(Fv ),
I(Dv ) = I(Fv (D)),
I(F`×µv ) = · · · ,
I(D`v ) = I(Gv ) =1
2pvlog(O×Kv
(Gv )).
I ∼ // I
O ∼ //?�
OO
O?�
OO
log(O×)∼ //
?�
OO
log(O×)?�
OO
O× ∼ //
log
OO
log����
O×log
OO
log����
(O×)pf ∼ // (O×)pf
42
Log-volumes and global degrees
I On an arbitrary compact open subset A of a finite extension of Qp, there is a well-definedvolume µ(A). The log-volume of A:
µlog(A) := log(µ(A)).
I
I(iFvQ) = ⊕v |vQI(iFv ), ∀i ∈ I ,
I(IFvQ) = ⊗i∈II(iFvQ),
I(I ,jFv ) = I(jFv )⊗ I(I\jFvQ),
IQ((−)FVQ) =∏
vQ∈VQ
IQ((−)FvQ).
I (i) For an object J = {Jv}v∈V of F~mod, regarded as an element in IQ(IFVQ), we take thesum of the log-volumes of the Jv as v varies:
µlogI ,VQ
(J ).
(ii) µlogI ,VQ
(J ) is equal to the degree of the arithmetic line bundle corresponding to J .
(iii) The log-volumes on the log-shells are invariant under the log-links betweenHodge-theaters.
43
Logarithmic Gaussian Procession monoids
I A procession in a category is a diagram
P1 ↪→ P2 ↪→ · · · ↪→ Pn
with Pj a j-capsule of objects and ↪→ the collection of all capsule-full poly-morphisms.
I Given a full log-link log : m−1HT → mHT ,
mΨFLGP,v = Im
ÖΨFgau(mFv ) ↪→
∏j∈F>
`
(ΨmFv )j →∏j∈F>
`
log(m−1Fv )j
è,
mΨDFLGP,v= Im
ÖΨenv(mDv ) ↪→
∏j∈F>
`
Ψcns(mDv )j →
∏j∈F>
`
log(m−1Fv (Dv ))j
è,
so that they are compatible with the processions on the m−1Fj and the mFj .
I In more concrete terms,I(S1Dv ) ↪→Å
q1
vy I(S2,1Dv )
ã↪→Åq22
vy I(S3,2Dv )
ã↪→ · · · ↪→
Åq(`>)2
vy I(S`>+1,`
>Dv )ã.
The first component I(S1Dv ) = I(Dv ) is considered as coric data.
I Consider the splitting monoid Ψ⊥FLGP,vof the monoid ΨFLGP,v
given by the split
Frobenioid F`v , i.e. the value group portion corresponding to {qj2
v}j∈F>
`.
I Observe thatΨ⊥FLGP,v
∩Ψ×FLGP,v= µ2`.
Ψ⊥FLGP,vis invariant under log-links.
44
Rigidifying the theta values
I The coric dataGv y O×µ
F v
is compatible with the theta functions/values for the very reason that the former aretorsion-free while the latter are constructed via torsions.
I
C LGP = Im(C gau(HT Θ) ↪→∏j
F~Rmod,j∼−→∏j
F~RMOD,j).
F LGP =ÄC LGP,Prime(†C LGP)
∼−→ V,F`LGP := F`(ΨFLGP), {ρv}
ä.
I A Θ-pilot object PΘ is an element in the global realified Frobenioid
PΘ ∈ C LGP
Ö↪→
∏j∈F>
`
(F~RMOD)j
èdetermined by any collection of generators (up to torsion) of the monoids Ψ⊥FLGP,v
for
v ∈ Vbad.
I Concretely, this is, at each v ∈ Vbad, the dataÅ{qj2
v}j∈F>
`
ã· O×µ
F v.
45
Etale-like multiradial representation
I •HT D = {mHT D}m∈Z linked by the full log-link. It carries procession. The log-volumeon a procession is normalized by taking elementary average.
I •HT D the following data •RLGP:
(a) The topological modules carrying the procession-normalized log-volume map
I(Sj+1,j(•D`v )) ⊂ IQ(Sj+1,j(•D`v )).
(b) The number field embedded into the product of local fields
•M~Dmod,j ⊂ IQ(Sj+1 (•D`VQ))
(c) For each v ∈ Vbad, the splitting monoid + action on the log-shells by multiplication:
•Ψ⊥DFLGP,v ⊂∏j∈F>
`
I(Sj+1,j(•D`v )).
I (Ind1) is the automorphisms Aut(Prc(•D`T )).(Ind2) is the collection of actions of Ism(Gv ) on each component of IQ(Sj+1,j(•D`v )), forevery v .(Ind3) refers to the following inclusion occurring in upper semi-compatibility:
I(Sj+1,j (•D`v )) ' I(Sj+1,j (•Dv )) ⊃⋃m∈Z
κ ◦ logn(mΨ×cns(Fv )Gv ), n = 0, 1, ∀m ∈ Z.
I The natural poly-isomorphisms induced by switching
0,•RLGP sw−→ 1,•RLGP, Prc(0,•D`T )sw−→ Prc(1,•D`T )
are both compatible, up to (Ind1) and (Ind2), with the full poly-isomorphism
0,•D`0 −→ 1,•D`0 .
46
Frobenius-like multiradial representation
I (Multiradial representation of theta values) Let PΘ ∈ C LGP be a Θ-pilot object.
UΘ :=⋃
Ind1,2,3
κ(PΘ) ⊂∏j∈F>
`
IQ(Sj+1(•D4,VQ)),
i.e. the union of the translations by the actions of Aut(Prc(•D`T )) and the Ism-action(for each v ∈ V) on each direct summand of the j + 1 factors of
IQ(Sj+1,j(•Dv )) ' IQ(Sj+1,j(•D`v ))
of the set κ(PΘ), equipped with their actions on the tensor-packets of log-shells.
I Theorem (Multiradiality).The construction of multiradial representation UΘ of Θ-pilot objects is invariant underarbitrary automorphism of the abstract prime-strip F I×µ.
47
The Θ-link
I Test object: A q-pilot object is an element in the global realified Frobenioid
Pq ∈ C 4
determined by any collection of generators (up to torsion) of the splitting monoids ofF`4,v for v ∈ Vbad.
I Define the ΘLGP-link to be the full poly-isomorphism
ΘLGP : ‡F I×µLGP∼−→ †F I×µ4 . (18)
At each v ∈ Vbad, the identificationÅ{qj2
v}j∈F>
`
ã· O×µ
F v= q
v· O×µ
F v.
I Pq is not equipped with local trivializations at each v ∈ V, while the PΘ object is.
48
Statement of the main theorem
I At each vQ, define the holomorphic hull of (UΘ)vQ as the smallest subset of IQ((−)FvQ)containing (UΘ)vQ , which is of the form
(UΘ)vQ = λ · O(−)FvQ
with λ ∈ IQ((−)FvQ), which is non-zero in each direct summand of the decomposition of
IQ((−)FvQ) into direct sum of local fields. (Recall (1).)
I Let Pq ∈ C 4 be a q-pilot object and denote by −| log(q)| its log-volume in the log-shells.
LetUΘ ⊂
∏j∈F>
`
IQ(Sj+1(•D4,VQ))
be the holomorphic hull of the multiradial representation UΘ of a Θ-pilot object PΘ.Write −| log(Θ)| for the log-volume of UΘ. (It can > 0!)Then we have the following inequality in R:
− | log(Θ)| ≥ −| log(q)|. (19)
I Remark.Actually we have a strict inequality −| log(Θ)| > −| log(q)|.
I Remark. There are obvious variants of (19). For instance, let’s consider q1N ,N ∈ Z>0.
Increasing N, modulo the effect of enlargement of the field F , we see that the RHS of(19) gets larger and larger.
49
Proof of the Theorem
(1) First consider the 0-column of Hodge theaters 0,•HT .
0µ(0UΘ) ≥ 0µ(κ(0,0PΘ)) = 0µ(0,0PΘ). (20)
(2) Now regard the domain and codomain of the ΘLGP-link
ΘLGP : 0,0F I×µLGP∼−→ 1,0F I×µ4
as associated to the Hodge theaters 0,0HT and 1,0HT , respectively. (Keep in mind that theuse of the ΘLGP-link implicitly requires that we regard both sides as abstract prime-strips.)We then have the association of log-volumes
α : 0R≥0∼−→ 1R≥0,
0µ(0,0PΘ) 7→ 1,0 deg(1,0Pq) (= −| log(q)|). (21)
(3) Let 1,0PΘ ∈ 1,0C LGP be a Θ-pilot object.
1UΘ =⋃
Ind1, 2, 3
κ(1,0PΘ).
(4) 1UΘ is by definition the holomorphic hull of 1UΘ, which can be regarded as an object of1,0F I×µ4 . Then
(1µ(1UΘ) <) 1µ(1UΘ) = 1,0 deg(1UΘ) (= −| log(Θ)|).
(5) By Multiradiality Theorem, the construction of 0UΘ out of 0,0F I×µLGP is compatible withthe ΘLGP-link, hence is compatible (i.e. has no interference) with the holomorphic structureof 1,0HT . Then, the assignment
µ0(0UΘ) 7→ µ1(1UΘ). (22)
is compatible the map (21). Therefore (20, 21) gives rise to
1,0 deg(1UΘ) ≥ 1,0 deg(1,0Pq), (23)
i.e.−| log(Θ)| ≥ −| log(q)|.
50
Writei ,logHT = {· · · i ,−1HT log→ i ,0HT · · · }.
1,0HT �� //
��
1,logHT
1µ
��
0,logHT
0µ
��
0,0HT
��
? _oo 1,0HT
��1,0F I׵4
1,0deg��
'
""0,0F I×µLGP
0µ��
ΘLGP // 1,0F I×µ4
1,0deg��
1R≥0
**UUUUUUU
UUUUUUUU
UUUUUUUU
1R≥0hoo
%%JJJJJ
JJJJJ
0R≥0
��
γoo 0R≥0lKoo
yyssssss
sssss
α // 1R≥0
ttiiiiiiii
iiiiiiii
iiiiiiii
R`≥0
Here the arrow lK represents the operation 0,0PΘ 7→ 0UΘ, the arrow γ denotes the operation0UΘ 7→ 1UΘ, R`≥0 denotes the codomain of the log-volume map µ` on the mono-analytic
log-shells, and the arrow h : 1UΘ 7→ 1UΘ denotes the operation of taking holomorphic hull.The final inequality (23) is a consequence of the compatibility between α and γ.
51
Computing the log-volume of multiradial representation
Theorem. Fix the initial Θ-data
(F/F ,XF , `,CK ,V,Vbadmod, ε)
and assume further that
(i) EF has good reduction at any place in V(F )good ∩ V(F )non which does not divide 2`;
(ii) Write emod the maximal ramification index of Fmod/Q. Denote
d∗mod = 212335 · dmod, e∗mod = 212335 · emod;
(iii) Write Ftpd = Fmod(EFmod[2]). Assume the 3 · 5-torsion points of EF are defined over F ,
and thatF = Fmod(
√−1,EFmod
[2 · 3 · 5]) = Ftpd(√−1,EFtpd
[3 · 5]).
(iv) Let L be any of the intermediate fields between K/Fmod mentioned above; they are allGalois over Fmod. Consider the (effective) arithmetic divisors on L
dLADiv, fLADiv, qLADiv,
determined by the different ideal of L/Q, the conductor, and the q-parameters of EF atV(L)bad = Vbad
mod ×VmodV(L).
(v) For Vmod 3 v |vQ ∈ V(Q), write V(L)v = V(L)×Vmod{v}, V(L)vQ = V(L)×V(Q) {vQ}.
Denote by degL the degree map on arithmetic divisors on L and by degS for the portionsupported in a set of places S . Consider the following normalized degree:
log(dL) =degL(dLADiv)
[L : Q], log(dLv ) =
degV(L)v (dLADiv)∑w∈V(L)v [Lw : QvQ ]
, log(dLvQ) =degV(L)vQ
(dLADiv)∑w∈V(L)vQ
[Lw : QvQ ].
In the same way, we define log(fL), log(fLv ), log(fLvQ), and
log(q), log(qv ), log(qvQ). Note that the global degrees | log(q)| = log(q)2` .
(vi) Denote by ηprm ∈ R>0 an element with the property (via the Prime Number Theorem)
# {prime numbers ≤ x} ≤ 4x
3 log(x), ∀x ≥ ηprm.
We have
log(q)
6≤ (1 +
20dmod
`)Ä
log(dFtpd) + log(fFtpd)ä
+ 20(`e∗mod + ηprm). (24)
52
(1) Note that F/Ftpd is tamely ramified at places away from 2 · 3 · 5 and K/F is tamelyramified at places away from `, and
Gal(F/Ftpd) ↪→ GL2(F3)× GL2(F5)× Z/2Z, Gal(K/F ) ↪→ GL2(F`).
Then(log(dK ) ≤) log(dFtpd) + log(fFtpd) + log(211 · 33 · 52) + 2 log(`). (25)
(2) Let L be an intermediate field between K/Ftpd which is Galois over Fmod. Consider the set
V(L)dst ⊂ V(L)non
of places that extends to a place in V(K ) which is ramified over Q. Set
sLADiv =∑
v∈V(L)dst
ev · v
and the respective images V(Fmod)dst, V(Q)dst of V(Ftpd)dst in V(Fmod), V(Q). Applying
1 ≤∑
w∈V(Fmod)dst,w |vQ
[Fmod,w : QvQ ] ≤ dmod,
one seeslog(sQ) ≤ dmod log(sFmod).
Furtherlog(sQ) ≤ 2dmod(log(dFtpd) + log(fFtpd)) + log(2 · 3 · 5 · `). (26)
(3) Recall that we want to estimate, for each vQ ∈ V(Q), the log-volume of
(UΘ)vQ ⊂∏j∈F>
`
IQ(Sj+1(•D`vQ))
and then average it over F>` . We thus reduce to computing the weighted average over all
possible collections {vi |vQ, i ∈ Sj+1}.
53
(3.1) The case vQ ∈ V(Q)dst. Take I = Sj+1, ki = Kv i, and λ the order of qj
2
v j
if v j ∈ Vbad
and 0 if v j ∈ Vgood. Then the upper bound for the log-volume of the component of (UΘ)vQ at{vi |vQ, i ∈ Sj+1} is given by
(−λ+ dI + 1) log pvQ +∑i∈I∗
(log(ev i) + 3).
Write ιwQ = 1 if pwQ ≤ e∗mod` and ιwQ = 0 otherwise. Define s≤ADiv =∑
wQ∈V(Q)dstιwQ
log pwQ·wQ.
The weighted average upper bound with respect to the asymmetry of the λ:
−j2 log(qvQ)
2`+ (j + 1) log(dKvQ) + log(sQvQ) + 4(j + 1) log(e∗mod`) log(s≤vQ).
Now, as j runs over F>` , the upper bound for the log-volume of (UΘ)vQ
−(`+ 1) log(qvQ)
24+`+ 5
4log(dKvQ) + log(sQvQ) + (`+ 5)
4
3(e∗mod`+ ηprm).
(Here one uses elementary estimates for log(e∗mod`) log(s≤vQ).)
(3.2) The case vQ ∈ V(Q)non\V(Q)dst: the upper bound is equal to zero.(3.3) The case vQ ∈ V(Q)arc: upper bound is given by `+5
4 log(π).(4) Finally, taking the sum of the procession-normalized upper bounds over vQ ∈ V(Q) andapplying (25, 26) an upper bound of the −| log(Θ)| in (19) (hence an upper bound of
−| log(q)| = − log(q)2` ):
`+ 1
4
®(1 +
12dmod
`)(log(dFtpd) + log(fFtpd))− (1− 12
`2)
log(q)
6+ 10(e∗mod`+ ηprm)
´− log(q)
2`.
The Theorem follows from this and some elementary computations.
54
Application to the ABC conjecture
I STP Vojta conjecture holds on a compactly bounded subset K ⊂ UX (Q) whose supportcontains the prime 2, in the case
X = P1Q, D = {0, 1,∞}.
I Denote by log(q∀(−)) (resp. log(q-2(−))) the function on UX (Q) given by the normalized
degree of the arithmetic line bundle determined by the q-parameters at allnonarchimedean places (resp. away from 2).
I Let ≈ denote equality up to bounded discrepancy. On a suitable compactly boundedsubset K ⊂ UX (Q),
log(q-2(−))
6≈
log(q∀(−))
6≈ htωX (D)
Theorem. Let d ≥ 1 be an integer. Let EF be an elliptic curve over a number field Fcorresponding to a point xE ∈ UX (F ) ∩ UX (Q)≤d . Define Ftpd = Fmod(EFmod
[2]), andassume F = Fmod(
√−1,EFmod
[2 · 3 · 5]).For any real number ε ≥ 0, there is a finite subset Excd ,ε ⊂ UX (Q)≤d depending only on
K, d , ε such that if xE ∈ (K ∩ UX (Q)≤d)\Excd ,ε, then there is a constant CK dependingonly on K such that
log(q∀xE )
6≤ (1 + ε) (log-diffX (xE ) + log-condD(xE )) + CK.
55
(1) First let Excd be the set consisting of the 4 possibilities of EF which do not admit anF -core. That is, by taking xE ∈ (K ∩ UX (Q)≤d)\Excd , WMA EF admits an F -core.(2) For A any finite set of prime numbers, there exists a prime number p (via the PrimeNumber Theorem) such that
p /∈ A, p ≤ 2(θA + ξprm), θA :=∑l∈A
log(l). (27)
(3) Write
h = log(q∀xE ) =1
[F : Q]
∑v∈V(F )non
hv fv log(pv )
where hv ≥ 1 is the local height of EF for v where EF has multiplicative bad reduction, andhv = 0 for v where EF has good reduction.(4) Let α ∈ [ 1
2 , 1) be a real number and β be real number such that 1− α ≤ β ≤ α. Considerthe finite set A of primes numbers p which satisfy one of the following (not mutuallyexclusive) conditions:
I p ≤ hβ;
I p = pv for some v ∈ V(F )non with hv ≥ h1−α;
I p|hv 6= 0 for some nonzero hv with v ∈ V(F )non. (In this case, hv < h1−α implies thatp ≤ hv < hβ.)
Then by some computations and (27) there exists a prime number ` such that (δ := 212335d)
hβ < ` ≤ 10δhα log(2δh) ≤ 20δ2h1+α (< 20δ2h2); (28)
hv < h1−α, if ` = pv for some v ∈ V(F )non; (29)
` - hv 6= 0, ∀v ∈ V(F )non. (30)
(5) We may enlarge Excd by adding finitely many possibilities of j-invariants of EF , so as toassume that EF admits no `-cyclic subgroup schemes.(6) Denote by Vbad
mod the set of nonarchimedean place of V(Fmod) which do not divide 2` andwhere EF has bad multiplicative reduction. If it is empty, then by (29) and α ≥ 1
2 h isbounded. Then we may assume
Vbadmod 6= ∅.
(7) Step (5) and (30) the image of the natural homomorphism Gal(Q/F )→ GL2(F`)determined by EF [`] contains SL2(F`).
56
(8) Now we can apply Theorem 0.13 to obtain (noting δ ≥ e∗mod and (28))
log(q)
6≤ (1 + δh−β)
Älog(dFtpd) + log(fFtpd)
ä+ 200δ2hα log(2δh) + 20ηprm.
Observe (recalling (28, 29))
log(q-2xE )
6− log(q)
6≤ h1−α log(`)
6≤ h1−α log(2δh).
Then
h
6≤ (1 + δh−β)
Älog(dFtpd) + log(fFtpd)
ä+ 200δ2hα log(2δh) + h1−α log(2δh) +
CK2
(31)
for some constant CK. As β ≥ 1− α and α ≥ 12 ,
h
6≤ (1 + δhα−1)
Älog(dFtpd) + log(fFtpd)
ä+
h
6· λ ·
Çhα−1 2400δ2 log(2δh)
λ
å+
CK2
(32)
for any positive number λ.
(9) If εE = hα−1 2400δ2 log(2δh)λ > ε, then (since α < 1) h is bounded above by a quantity
depending only on ε and d . We may assume εE ≤ ε. Note hα−1 δλ ≤ εE . So
δhα−1 ≤ λεE ≤ λε.
Taking any λ ∈ (0, 11+2ε ], one has
h
6≤ (1 + ε)
Älog(dFtpd) + log(fFtpd)
ä+ (1− λε)−1CK
2,
≤ (1 + ε)Ä
log(dFtpd) + log(fFtpd)ä
+ CK.
(10) Note log(dFtpd) = log-diffX (xE ), log(fFtpd) ≤ log-condX (xE ). Done!
57
