RIMS-1933
Explicit Estimates in Inter-universal Teichmüller Theory

By

Shinichi MOCHIZUKI, Ivan FESENKO, Yuichiro HOSHI,
Arata MINAMIDE, and Wojciech POROWSKI

November 2020

R ESEARCH I NSTITUTE FOR MATHEMATICAL SCIENCES
KYOTO UNIVERSITY, Kyoto, Japan

EXPLICIT ESTIMATES IN INTER-UNIVERSALTEICHMÜLLER THEORYSHINICHI MOCHIZUKI, IVAN FESENKO, YUICHIRO HOSHI,
ARATA MINAMIDE, AND WOJCIECH POROWSKI

Abstract. In the final paper of a series of papers concerning interuniversal Teichmüller theory, Mochizuki verified various numerically
non-eﬀective versions of the Vojta, ABC, and Szpiro Conjectures overnumber fields. In the present paper, we obtain various numerically effective versions of Mochizuki’s results. In order to obtain these results,
we first establish a version of the theory of étale theta functions thatfunctions properly at arbitrary bad places, i.e., even bad places that
divide the prime “2”. We then proceed to discuss how such a modified version of the theory of étale theta functions aﬀects inter-universal
Teichmüller theory. Finally, by applying our slightly modified versionof inter-universal Teichmüller theory, together with various explicit estimates concerning heights, the j-invariants of “arithmetic” elliptic curves,and the prime number theorem, we verify the numerically eﬀective versions of Mochizuki’s results referred to above. These numerically effective versions imply eﬀective diophantine results such as an eﬀective
version of the ABC inequality over mono-complex number fields [i.e., therational number field or an imaginary quadratic field] and an eﬀective
version of a conjecture of Szpiro. We also obtain an explicit estimate
concerning “Fermat’s Last Theorem” (FLT) — i.e., to the eﬀect that
FLT holds for prime exponents > 1.615 · 1014 — which is suﬃcient togive an alternative proof of the first case of Fermat’s Last Theorem.

Contents
Introduction
Acknowledgements
0. Notations and Conventions
1. Heights
2. Auxiliary numerical results
3. µ6 -Theory for [EtTh]
4. µ6 -Theory for [IUTchI-III]
5. µ6 -Theory for [IUTchIV]
References

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54

2010 Mathematics Subject Classification. Primary 14H25; Secondary 14H30.
1

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MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

Introduction
In [IUTchIV], Mochizuki applied the theory of [IUTchI-IV] [cf. also
[Alien] for a detailed survey of this theory] to prove the following result[cf. [IUTchIV], Corollary 2.2, (ii), (iii)]:
Theorem. Write X for the projective line over Q; D ⊆ X for the divisor
consisting of the three points “0”, “1”, and “∞”; (Mell )Q for the moduli
stack of elliptic curves over Q. We shall regard X as the “λ-line” — i.e.,
we shall regard the standard coordinate on X as the “λ” in the Legendre
form “y 2 = x(x − 1)(x − λ)” of the Weierstrass equation defining an elliptic
curve — and hence as being equipped with a natural classifying morphism
def
UX = X \ D → (Mell )Q . Writelog(q∀(−) )for the R-valued function on (Mell )Q (Q), hence also on UX (Q), obtained byforming the normalized degree “deg(−)” of the eﬀective arithmetic divisor
determined by the q-parameters of an elliptic curve over a number field atarbitrary nonarchimedean places. Let
KV ⊆ UX (Q)be a compactly bounded subset that satisfies the following conditions:(CBS1) The support of KV contains the nonarchimedean place “2”.
(CBS2) The image of the subset “K2 ⊆ UX (Q2 )” associated to KV via the jinvariant UX → (Mell )Q → A1Q is a bounded subset of A1Q (Q2 ) =Q2 , i.e., is contained in a subset of the form 2Nj-inv · OQ2 ⊆ Q2 ,where Nj-inv ∈ Z, and OQ2 ⊆ Q2 denotes the ring of integers [cf. thecondition (∗j-inv ) of [IUTchIV], Corollary 2.2, (ii)].
Then there exist· a positive real number Hunif which is independent of KV and· positive real numbers CK and HK which depend only on the choiceof the compactly bounded subset KV
such that the following property is satisfied: Let d be a positive integer, ϵdand ϵ positive real numbers ≤ 1. Then there exists a finite subset
Excϵ,d ⊆ UX (Q)≤d— where we denote by UX (Q)≤d ⊆ UX (Q) the subset of Q-rational points
defined over a finite extension field of Q of degree ≤ d — which depends
only on KV , ϵ, d, and ϵd , and satisfies the following properties:• The function log(q∀(−) ) is
4+ϵd
+ HK
≤ Hunif · ϵ−3 · ϵ−3d ·don Excϵ,d .

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

3• Let EF be an elliptic curve over a number field F ⊆ Q that determinesa Q-valued point of (Mell )Q which lifts [not necessarily uniquely!] to a pointxE ∈ UX (F ) ∩ UX (Q)≤d such thatxE ∈ KV , xE ∈/ Excϵ,d .
Write Fmod for the minimal field of definition of the corresponding point∈ (Mell )Q (Q) anddef

Fmod ⊆ Ftpd = Fmod (EFmod [2]) ⊆ F
for the “tripodal” intermediate field obtained from Fmod by adjoining thefields of definition of the 2-torsion points of any model of EF ×F Q over
Fmod [cf. [IUTchIV], Proposition 1.8, (ii), (iii)]. Moreover, we assume thatthe (3 · 5)-torsion points of EF are defined over F , and that√√def
F = Fmod ( −1, EFmod [2 · 3 · 5]) = Ftpd ( −1, EFtpd [3 · 5])√— i.e., that F is obtained from Ftpd by adjoining −1, together with thefields of definition of the (3 · 5)-torsion points of a model EFtpd of the elliptic
curve EF ×F Q over Ftpd determined by the Legendre form of the Weierstrass equation discussed above. Then EF and Fmod arise as the “EF ” and“Fmod ” for a collection of initial Θ-data as in [IUTchIV], Theorem 1.10,
that satisfies the following conditions:(C1) (log(q∀xE ))1/2 ≤ l ≤ 10δ · (log(q∀xE ))1/2 · log(2δ · log(q∀xE ));(C2) we have an inequality∀
1
6 · log(qxE )≤ (1 + ϵ) · (log-diﬀX (xE ) + log-condD (xE )) + CK
def

— where we write δ = 212 ·33 ·5·d; log-diﬀX for the [normalized] log-diﬀerent
function on UX (Q) [cf. [GenEll], Definition 1.5, (iii)]; log-condD for the[normalized] log-conductor function on UX (Q) [cf. [GenEll], Definition 1.5,(iv)].
In the present paper, we prove a numerically eﬀective version of thistheorem without assuming the conditions (CBS1), (CBS2) [cf. the portionof Corollary 5.2 that concerns κ/κlog /K]. Moreover, we prove that if one
restricts one’s attention to the case where the point “xE ” is defined overa mono-complex number field [i.e., Q or an imaginary quadratic field — cf.
Definition 1.2], then one may eliminate the compactly bounded subset “KV ”
from the statement of this theorem [cf. the portion of Corollary 5.2 that
does not concern κ/κlog /K].
In order to obtain Corollary 5.2, we establish a version of the theoryof étale theta functions that functions properly at arbitrary bad places, i.e.,even bad places that divide the prime “2”. Roughly speaking, this is achievedby modifying the notion of evaluation points at which the theta function isevaluated [cf. the explanation of §3 below for more details].
We then proceed to apply Corollary 5.2 to verify the following eﬀective
diophantine results [cf. Theorems 5.3, 5.4; Corollary 5.8; the notations andconventions of §0]:

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MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

Theorem A. (Eﬀective version of the ABC inequality over monocomplex number fields) Let L be a mono-complex number field [i.e., Q
or an imaginary quadratic field — cf. Definition 1.2]; a, b, c ∈ L× nonzero
elements of L such thata + b + c = 0;ϵ a positive real number ≤ 1.Write ∆L for the absolute value of the discrimdef
inant of L; d = [L : Q];∏
def
HL (a, b, c) =max{|a|v , |b|v , |c|v };v∈V(L)def

IL (a, b, c) = {v ∈ V(L)non | ♯{|a|v , |b|v , |c|v } ≥ 2} ⊆ V(L)non ;defradL (a, b, c) =∏♯(OL /pv );v∈IL (a,b,c){def

hd (ϵ) =3.4 · 1030 · ϵ−166/81 (d = 1)
6 · 1031 · ϵ−174/85(d = 2).

Then we have
HL (a, b, c) ≤ 25d/2 · max{exp( d4 · hd (ϵ)), (∆L · radL (a, b, c))3(1+ϵ)/2 }≤ 25d/2 · exp( d4 · hd (ϵ)) · (∆L · radL (a, b, c))3(1+ϵ)/2 .
Theorem B. (Eﬀective version of a conjecture of Szpiro) Let a, b, c
be nonzero coprime integers such thata + b + c = 0;ϵ a positive real number ≤ 1. Then we have|abc| ≤ 24 · max{exp(1.7 · 1030 · ϵ−166/81 ), (rad(abc))3(1+ϵ) }≤ 24 · exp(1.7 · 1030 · ϵ−166/81 ) · (rad(abc))3(1+ϵ)— which may be regarded as an explicit version of the inequality( ∏ )3+ϵ“|abc| ≤ C(ϵ)
p
”
p|abc

conjectured in [Szp], §2 [i.e., the “forme
forte” of loc. cit., where we note∏that the “p” to the right of the “ ” in the above display was apparently
unintentionally omitted in loc. cit.].

Corollary C. (Application to “Fermat’s Last Theorem”) Let
p > 1.615 · 1014
be a prime number. Then there does not exist any triple (x, y, z) of positive
integers that satisfies the Fermat equation
xp + y p = z p .

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

5

The estimate in Corollary C is suﬃcient to give an alternative proof [i.e.,to the proof of [Wls]] of the first case of Fermat’s Last Theorem [cf. Remark
5.8.1]. The proof of Corollary C is obtained by combining• the slightly modified version of [IUTchI-IV] developed in the present
paper with• various estimates [cf. Lemmas 5.5, 5.6, 5.7] of an entirely elementary
nature.
In fact, the estimate in the first display of Corollary C may be strengthenedroughly by a factor of 2 by applying the [slightly less elementary] results of[Ink1], [Ink2] [cf. Remarks 5.7.1, 5.8.2].
[The authors have received informal reports to the eﬀect that one mathematician has obtained some sort of numerical estimate that is formallysimilar to Corollary C, but with a substantially weaker [by many orders ofmagnitude!] lower bound for p, by combining the techniques of [IUTchIV],§1, §2, with eﬀective computations concerning Belyi maps. On the other
hand, the authors have not been able to find any detailed written expositionof this informally advertized numerical estimate and are not in a position tocomment on it.]
We also obtain an application of the ABC inequality of Theorem B toa generalized version of Fermat’s Last Theorem [cf. Corollary 5.9], which
does not appear to be accessible via the techniques involving modularity ofelliptic curves over Q and deformations of Galois representations that playa central role in [Wls].
In the following, we explain the content of each section of the present
paper in greater detail.
In §1, we examine various [elementary and essentially well-known] properties of heights of elliptic curves over number fields. Let F ⊆ Q be a number
field; E an elliptic curve over F that has semi-stable reduction over the ringof integers OF of F . Suppose that E is isomorphic over Q to the elliptic
curve defined by an equationy 2 = x(x − 1)(x − λ)— where λ ∈ Q \ {0, 1}. For simplicity, assume further that
Q(λ) is mono-complex[i.e., Q or an imaginary quadratic field — cf. Definition 1.2]. Write j(E) ∈ Q
for the j-invariant of E. In Corollary 1.14, (iii), we verify that the [logarithmic] Weil heighth(j(E))[cf. Definition 1.1, (i)] of j(E) satisfies the following property:(H1) Let l be a prime number. Suppose that E admits an l-cyclic subgroup scheme, and that l is prime to the local heights of E at eachof its places of [bad] multiplicative reduction [i.e., the orders of theq-parameter at such places — cf. [GenEll], Definition 3.3]. Then the
nonarchimedean portion of h(j(E)) is bounded by an explicit absolute
constant ∈ R.

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MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

To verify (H1), we make use of the following two types of heights:• the Faltings height hFal (E) [cf. the discussion entitled “Curves” in§0],• the symmetrized toric height hS-tor (E) [cf. Definition 1.7].
These heights hFal (E) and hS-tor (E) may be related to h(j(E)) by means of
numerically explicit inequalities [cf. Propositions 1.8, 1.10, 1.12] and satisfy
the following important properties:(H2) Let E ′ be an elliptic curve over F ; ϕ : E → E ′ an isogeny of degree d.
Then it holds that hFal (E ′ ) − hFal (E) ≤ 12 log(d) [cf. [Falt], Lemma
5].
(H3) The archimedean portion of hS-tor (E) is bounded above by the nonarchimedean portion of hS-tor (E) [cf. Proposition 1.9, (i)].
[Here, we note that (H3) is an immediate consequence of the product formula,
together with the assumption that the cardinality of the set of archimedean
places of the mono-complex number field Q(λ) is one.] The property (H1)then follows, essentially formally, by applying (H2) and (H3), together with
the numerically explicit inequalities [mentioned above], which allow one tocompare the diﬀerent types of heights.
In §2, we review• a result concerning the j-invariants of “arithmetic” elliptic curves[cf. Proposition 2.1];• certain eﬀective versions of the prime number theorem [cf. Proposition 2.2].
In §3, we establish a version of the theory of étale theta functions [cf.
[EtTh], [IUTchII]] that functions properly at arbitrary bad places, i.e., evenbad places that divide the prime “2”. Here, we note that the original definition of the notion of an evaluation point — i.e., a point at which thetheta function is evaluated that is obtained by translating a cusp by a 2torsion point [cf. [EtTh], Definition 1.9; [IUTchI], Example 4.4, (i)] — does
not function properly at places over 2 [cf. [IUTchIV], Remark 1.10.6, (ii)].
Thus, it is natural to pose the following question:
Is it possible to obtain a new definition of evaluation points
that functions properly at arbitrary bad places by replacing
the “2-torsion point” appearing in the [original] definition ofan evaluation point by an “n-torsion point”, for some integern > 2?
Here, we recall that the definition of an evaluation point obtained by translating a cusp by an n-torsion point functions properly at arbitrary bad places
if the following two conditions are satisfied:EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

7(1) The various ratios of theta values at the Galois conjugates of [thepoint of the Tate uniformization of a Tate curve corresponding to aprimitive 2n-th root of unity] ζ2n are roots of unity [cf. [IUTchII],
Remark 2.5.1, (ii)].
(2) The theta value at ζ2n is a unit at arbitrary bad places [cf. [IUTchIV],
Remark 1.10.6, (ii)].
One fundamental observation — due to Porowski — that underlies the theoryof the present paper is the following:n satisfies the conditions (1), (2) if and only if n = 6[cf. Lemma 3.1; Proposition 3.2; the well-known fact that 1 − ζ4 , 1 − ζ8are non-units at places over 2]. Following this observation, in Definition
3.3, we introduce a new version of the notion of an “étale theta functionof standard type” [cf. [EtTh], Definition 1.9] obtained by normalizing étaletheta functions at points arising from 6-torsion points of the given elliptic
curve. In the remainder of §3, we then proceed to discuss how the adoption ofsuch “étale theta functions of µ6 -standard type” aﬀects the theory developedin [EtTh].
Next, in §4, we discuss how the modifications of §3 aﬀect [IUTchI-III].
Roughly speaking, we observe that, once one makes suitable minor technicalmodifications,(∗) the theory developed in [IUTchI-III] remains essentially unaﬀectedeven if, in the notation of [IUTchI], Definition 3.1, (b), one eliminates
the assumption “of odd residue characteristic” that appears in thediscussion of “Vbadmod ”.
In §5, we begin by proving a “µ6 -version” [cf. Theorem 5.1] of [IUTchIV],
Theorem 1.10, i.e., that applies the theory developed in §2, §3, §4. This allows us to obtain a “µ6 -version” [cf. Corollary 5.2] of [IUTchIV], Corollary
2.2, (ii), (iii) [i.e., the “Theorem” reviewed at the beginning of the present
Introduction] without assuming the conditions (CBS1), (CBS2) that appear
in the statement of this Theorem concerning the nonarchimedean place “2”.
The proof of Corollary 5.2 makes essential use of the theory of §1, §2 [cf., especially, Corollary 1.14; Propositions 2.1, 2.2]. In the case of mono-complex
number fields, we then derive• Theorem 5.3 from Corollary 5.2 by applying the product formula,
together with the essential assumption that the number field under consideration is mono-complex [cf. the property (H3) discussed
above];• Theorem 5.4 from [the proof of] Theorem 5.3, together with various
elementary computations.
Finally, we apply• Theorem 5.3, together with various elementary considerations, to“Fermat’s Last Theorem” [cf. Corollary 5.8] and8

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI• Theorem 5.4, again together with various elementary computations,to a generalized version of “Fermat’s Last Theorem” [cf. Corollary
5.9].
In this context, we note [cf. Remark 5.3.2] that it is quite possible that, inthe future, other interesting applications of Theorems 5.3, 5.4 to the studyof numerical aspects of diophantine equations can be found.
Acknowledgements
Each of the co-authors of the present paper would like to thank the otherco-authors for their valuable contributions to the theory exposed in thepresent paper. In particular, the co-authors [other than the first author]of the present paper wish to express their deep gratitude to the first author, i.e., the originator of inter-universal Teichmüller theory, for countlesshours of valuable discussions related to his work. Moreover, the authors are
grateful to J. Sijsling for responding to our request to provide us with thecomputations that underlie Proposition 2.1. The second and fifth authors
were partially supported by the ESPRC Programme Grant “Symmetries
and Correspondences”. The third author was supported by JSPS KAKENHI Grant Number 18K03239; the fourth author was supported by JSPS
KAKENHI Grant Number 20K14285. This research was supported by the
Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University, as well as by the Center
for Next Generation Geometry [a research center aﬃliated with the ResearchInstitute for Mathematical Sciences].
0. Notations and Conventions
Numbers:
Let S be a set. Then we shall write ♯S for the cardinality of S.
Let E ⊆ R be a subset of the set of real numbers R. Then for λ ∈ R, if□ denotes “< λ”, “≤ λ”, “> λ”, or “≥ λ”, then we shall write E□ ⊆ E for
the subset of elements that satisfy the inequality “□”. If E is finite, thenwe shall write max E for the smallest real number λ such that E≤λ = E andmin E for the largest real number λ such that E≥λ = E.
For any nonzero integer n ∈/ {1, −1}, we shall write rad(n) for the productof the distinct prime numbers p which divide n. We shall define rad(1) andrad(−1) to be 1.
def

Let F be a field. Then we shall write F ⋔ = F \ {0, 1}.
Let Q be an algebraic closure of the field of rational numbers Q, F ⊆ Q adef
subfield. Then we shall write OF ⊆ F for the ring of integers of F ; Z = OQ ;non
Primes ⊆ Z for the set of all prime numbers; V(F )(respectively, V(F )arc )for the set of nonarchimedean (respectively, archimedean) places of F ;∪defV(F ) = V(F )arc
V(F )non .
For v ∈ V(F ), we shall write Fv for the completion of F at v.

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

9

Now suppose that F is a number field, i.e., that [F : Q] < ∞.
Let v ∈ V(F )non . Write pv ⊆ OF for the prime ideal corresponding to v;
pv for the residue characteristic of Fv ; fv for the residue field degree of Fv
over Qpv ; ordv for the normalized valuation on Fv determined by v, where wetake the normalization to be such that ordv restricts to the standard pv -adic
valuation on Qpv . Then for any x ∈ Fv , we shall writev (x)||x||v = p−ord
,
v

def

def

[F :Qpv ]|x|v = ||x||v v.

Let v ∈ V(F )arc . Write σv : F ,→ C for the embedding determined, up tocomplex conjugation, by v. Then for any x ∈ Fv , we shall write
def||x||v = ||σv (x)||C ,def|x|v = ||x||v[Fv :R]— where we denote by || · ||C the standard [complex] absolute value on C.
Note that for any w ∈ V(Q) that lies over v ∈ V(F ), the absolute value|| · ||v : Fv → R≥0 extends uniquely to an absolute value || · ||w : Qw → R≥0 .
We shall refer to this absolute value on Qw as the standard absolute value
on Qw .
Curves:Let E be an elliptic curve over a field. Then we shall write j(E) for the
j-invariant of E.
Let E be an elliptic curve over a number field F that has semi-stablereduction over OF . Write hFal (E) for the Faltings height of E [cf. [Falt], §3,
the first Definition]. Then we shall write
def

hFal (E) = hFal (E) + 12 log π[cf. [Lbr], Definition 2.3; [Lbr], Remark 2.1]. Here, we note that the quantity
hFal (E) is unaﬀected by passage to a finite extension of the base field F of
E [cf., e.g., [Lbr], Proposition 2.1, (i)].
1. Heights
Let E be an elliptic curve over a number field. In the present section,
we introduce the notion of the symmetrized toric height hS-tor (E) of E [cf.
Definition 1.7]. We then compare hS-tor (E) with the [logarithmic] Weilheight h(j(E)) of j(E) [cf. Proposition 1.8]. Finally, we prove that if E
satisfies certain conditions, then the nonarchimedean portion of h(j(E)) isbounded by an absolute constant [cf. Corollary 1.14, (iii)].
Definition 1.1. Let F be a number field.
(i) Let α ∈ F . Then for □ ∈ {non, arc}, we shall write∑defh□ (α) = [F1:Q]
log max{|α|v , 1} (≥ 0),v∈V(F )□
defh(α) = hnon (α) + harc (α)
and refer to h(α) as the [logarithmic] Weil height of α. We shall also
write h⊚ (α) for h(α).

10MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI(ii) Let α ∈ F × . Then for □ ∈ {non, arc}, we shall write∑def1
log max{|α|v , |α|−1
htor
(α)
=
v } (≥ 0),□
2[F :Q]v∈V(F )□
def

tor
htor (α) = htor
non (α) + harc (α)

and refer to htor (α) as the [logarithmic] toric height of α. We shalltor
also write htor
⊚ (α) for h (α).
Remark 1.1.1. One verifies easily that for □ ∈ {non, arc, ⊚}, the quantitiesh□ (α) and htor
□ (α) are unaﬀected by passage to a finite extension of F .
Definition 1.2. Let F be a number field. Then we shall say that F ismono-complex if F is either
the field of rational numbers Q or an imaginary quadratic field.
One verifies easily that F is mono-complex if and only if the cardinality ofV(F )arc is one.

Lemma 1.3. (Properties of toric heights) Let F be a number field,
α ∈ F × . Then the following hold:(i) It holds thattor −1htor□ (α) = h□ (α );htor□ (α) =

−1
1
2 · {h□ (α) + h□ (α )}for □ ∈ {non, arc, ⊚}.
(ii) It holds thath(α) = htor (α).
In particular, we have h(α) = h(α−1 ) [cf. (i)].
(iii) Suppose that F is mono-complex. Then we havetorhtor
arc (α) ≤ hnon (α).

(iv) Let x, y ∈ F ; xtor , y tor ∈ F × . Then we haveh□ (x) + h□ (y) ≥ h□ (x · y);tor
tor tor
tor torhtor· y tor )□ (x ) + h□ (y ) ≥ h□ (xfor □ ∈ {non, arc, ⊚}.
Proof. First, we consider assertion (i). The first equality follows immediately from the various definitions involved. The second equality follows immediately from the various definitions involved, together with the following[easily verified] fact: For any s ∈ R>0 , it holds thatmax{s, s−1 } = max{s, 1} · max{s−1 , 1}.

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

11

def

Next, we consider assertion (ii). Write d = [F : Q]. Then we compute:∑∑2
2d · htor (α) =log max{|α|v , |α|−1}=log(|α|−1vv · max{|α|v , 1})v∈V(F )∑

= 2d · h(α) +v∈V(F )

log |α|−1= 2d · h(α)
vv∈V(F )

— where the final equality follows from the product formula. This completes
the proof of assertion (ii).
Next, we consider assertion (iii). Let w be the unique element of V(F )arc .
In light of the first equality of assertion (i), to verify assertion (iii), we may
assume without loss of generality that |α|w ≥ 1. Then we compute:∑
2d · htor
log |α|−1arc (α) = log |α|w =v≤∑

v∈V(F )nontor
log max{|α|v , |α|−1v } = 2d · hnon (α)

v∈V(F )non

— where the second equality follows from the product formula. This completes the proof of assertion (iii). Finally, we consider assertion (iv). It
follows immediately from the second equality of assertion (i) that to verify
the second inequality of assertion (iv), it suﬃces to verify the first inequality of assertion (iv). But the first inequality follows immediately from thefollowing [easily verified] fact: For any s, t ∈ R≥0 , it holds thatmax{st, 1} ≤ max{s, 1} · max{t, 1}.
□

This completes the proof of assertion (iv).

Remark 1.3.1. It may appear to the reader, at first glance, that the notionof the toric height of an element of a number field F is unnecessary [cf.
Lemma 1.3, (ii)]. In fact, however, the toric height of an element α ∈ F ×
satisfies the following important property [cf. Lemma 1.3, (iii)]:
If F is mono-complex, then the archimedean portion of thetoric height of α is bounded by the nonarchimedean portionof the toric height of α.
This property is an immediate consequence of the product formula [cf. theproof of Lemma 1.3, (iii)]. We note that, in general, the notion of the Weilheight does not satisfy this property. For instance, for any n ∈ Z>0 , we havehnon (n) = 0;
htor
non (n) =1
2 log(n);harc (n) = log(n);htorarc (n) =1
2 log(n).

Definition 1.4. Let F be a field; | · | : F → R≥0 a map satisfying thefollowing conditions:(i) The restriction of | · | to F × determines a group homomorphism
F × → R>0 [relative to the multiplicative group structures on F × ,R>0 ].
(ii) It holds that |0| = 0.

12

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI(iii) For any x ∈ F , it holds that |x + 1| ≤ |x| + 1.
Then for α ∈ F ⋔ , we shall write
J(α)
J0∞ (α)J1∞ (α)J01 (α)=|α2 − α + 1|3 · |α|−2 · |α − 1|−2=|α(1 − α) − 1|3 · |α|−2 · |1 − α|−2 ;{}max |α|, |α|−1 ;{}max |α − 1|, |α − 1|−1 ;{}max |α − 1| · |α|−1 , |α| · |α − 1|−1 .

def

def

=

def

=

def

=

Lemma 1.5. (Comparison between J(α) and |α|2 ) In the notation ofDefinition 1.4, suppose that |α| ≥ 2. Then we have|α|2 ≤ 28 · J(α).
Proof. First, we note that since |α − 1| ≤ |α| + 1, we have
|α2 − α + 1| = |α2 − (α − 1)| ≥ |α|2 − |α − 1| ≥ |α|2 − (|α| + 1).
Thus, we conclude that28 · |α2 − α + 1|3 · |α|−2 · |α − 1|−2 ≥ 28 · (|α|2 − |α| − 1)3 · |α|−2 · (|α| + 1)−2
≥ |α|2— where we observe that since x2 − x − 1 ≥ x(x − 23 ), −(x + 1)2 ≥ −(2x)2 ,3and the function 2x−3
is monotonically decreasing for x ∈ R≥2 , the final
inequality follows from the elementary fact that{}28 · (x2 − x − 1)3 − x4 · (x + 1)2 ≥ x3 · 28 · (x − 23 )3 − x · (x + 1)2{}≥ 22 · x3 · 26 · (x − 32 )3 − x3}{
3
≥ 28 · x3 · (x − 23 )3 · 1 − 2−6 · (1 + 2x−3
)3
≥ 0
for x ∈ R≥2 .

□Lemma 1.6. (Comparison between J(α) and J0∞ (α) · J1∞ (α) · J01 (α))
In the notation of Definition 1.4, the following hold:(i) Write z for the rational function given by the standard coordinate
on P1Z and
A = {z, z −1 , 1 − z, (1 − z)−1 , z · (z − 1)−1 , (z − 1) · z −1 },
def

def

B = {δ ⊆ A | ♯δ = 2; if we write δ = {a, b}, thenit holds that A = {a, a−1 , b, b−1 , −ab, −(ab)−1 }}.

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

13

Then the set B coincides with the set
B ′ = {{z, (1 − z)−1 }, {z, (z − 1) · z −1 }, {z −1 , 1 − z},def{z −1 , z · (z − 1)−1 }, {1 − z, z · (z − 1)−1 },{(1 − z)−1 , (z − 1) · z −1 }}.
Moreover, the map
ϕ: B

→ A

{a, b} 7→ −ab
is bijective. Here, we recall that the symmetric group on 3 letters
S3 admits a natural faithful action on the projective line P1Z over Z,∼
hence also on the set of F -rational points (P1Z \ {0, 1, ∞})(F ) → F ⋔ ,and that the orbit S3 ·z of z coincides with the set A. In particular,
the action of S3 on A induces, via ϕ−1 , a transitive action of S3
on B.
(ii) For every δ = {a, b} ∈ B, writeDδ = {f ∈ F ⋔ | |a(f )| ≥ 1, |b(f )| ≥ 1} ⊆ F ⋔ .
def

We note that the action of S3 on B [cf. (i)] induces a transitive
action on the set [of subsets of F ⋔ ] {Dδ }δ∈B . Then we have
∪
F⋔ =
Dδ .
δ∈B

(iii) For any ϵ ∈ R≥0 , we have
2−2 · J0∞ (α) · J1∞ (α) · J01 (α) ≤ max{26+ϵ · J(α), 1}≤ 29+ϵ · J0∞ (α) · J1∞ (α) · J01 (α).
(iv) Suppose that | · | is nonarchimedean, i.e., that for any x ∈ F , itholds that |x + 1| ≤ max{|x|, 1}. [Thus, for any x ∈ F such that|x| < 1, it holds that |x + 1| = 1.] Then we havemax{J(α), 1} = J0∞ (α) · J1∞ (α) · J01 (α).
Proof. First, we consider assertion (i). To verify assertion (i), it suﬃces toshow that B = B ′ . [Indeed, it follows from this equality that ♯B = 6. Thus,
to verify that ϕ is bijective, it suﬃces to show that ϕ is surjective. But thissurjectivity follows immediately from the equality B = B ′ and the various
definitions involved.] The inclusion B ′ ⊆ B follows immediately from thevarious definitions involved. Thus, it suﬃces to verify the inclusion B ⊆ B ′ .
First, we observe that A is the [disjoint] union of the following sets:
A0∞ = {z, z −1 }, A1∞ = {1−z, (1−z)−1 }, A01 = {z·(z−1)−1 , (z−1)·z −1 }.
def

def

def

Let δ ∈ B. Note that [as is easily verified] δ ∈/ {A0∞ , A1∞ , A01 }. Thus, we
may writeδ = {a, b}

14

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI— where the pair (a, b) satisfies precisely one of the following three conditions:(1) a ∈ A0∞ , b ∈ A1∞ ,(2) a ∈ A1∞ , b ∈ A01 ,(3) a ∈ A01 , b ∈ A0∞ .

On the other hand, in each of these three cases, one verifies immediatelythat the condition
A = {a, a−1 , b, b−1 , −ab, −(ab)−1 }implies that there are precisely two possibilities for δ, and, moreover, thatthese two possibilities are ∈ B ′ , as desired. This completes the proof ofassertion (i).
Next, we consider assertion (ii). Assertion (ii) follows immediately fromthe following claim:
def

Claim 1.6A: For f ∈ F ⋔ , δ ∈ B, write δ(f ) = ϕ(δ)(f ).
Suppose that it holds that|δ(f )| = max{|ϵ(f )|}.
ϵ∈B

Then we have f ∈ Dδ .
Let us verify Claim 1.6A. Write δ = {a, b}. Suppose that f ∈/ Dδ . Then we
may assume without loss of generality that |a(f )| < 1. Thus, we have|δ(f )| = |a(f )| · |b(f )| < |b(f )|.
On the other hand, since we have |b(f )| ∈ {ϵ(f )}ϵ∈B [cf. the latter portionof assertion (i), i.e., the fact that ϕ is a bijection], we obtain a contradiction.
Therefore, we conclude that f ∈ Dδ . This completes the verification ofClaim 1.6A, hence also of assertion (ii).
Next, we consider assertions (iii) and (iv). First, we observe that, in
assertion (iii), we may assume without loss of generality, that ϵ = 0. Write
D = D{1−z,z·(z−1)−1 } = {f ∈ F ⋔ | |f | ≥ |f − 1| ≥ 1} ⊆ F ⋔ .
defThen we observe that

F⋔ =

∪

(σ · D)σ∈S3[cf. assertion (ii)], and that for α ∈ F ⋔ , σ ∈ S3 , we haveJ0∞ (α) · J1∞ (α) · J01 (α) = J0∞ (σ · α) · J1∞ (σ · α) · J01 (σ · α),J(α) = J(σ · α)[cf. the fact discussed in the proof of Lemma 1.3, (i); Definition 1.4; theequality “A = S3 · z” discussed in assertion (i); the manifest invarianceof J(α) with respect to the transformations α 7→ 1 − α, α 7→ α−1 , whichcorrespond to a pair of generators of S3 ]. Thus, to verify assertions (iii) and(iv), we may assume without loss of generality that α ∈ D. Then observethat J0∞ (α) = |α| ≥ 1, J1∞ (α) = |α − 1| ≥ 1, J01 (α) = |α| · |α − 1|−1 ≥ 1,
hence thatJ0∞ (α) · J1∞ (α) · J01 (α) = |α|2 (≥ 1).
Now let us verify assertion (iii). The inequality
J0∞ (α) · J1∞ (α) · J01 (α) ≤ 22 · max{26 · J(α), 1}

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

15

follows immediately from Lemma 1.5. On the other hand, the inequality
max{26 · J(α), 1} ≤ 29 · J0∞ (α) · J1∞ (α) · J01 (α)
follows immediately from the following computation:J(α) = |α(α − 1) + 1|3 · |α|−2 · |α − 1|−2
≤ 23 · |α|−2 · |α − 1|−2 · max{|α|3 · |α − 1|3 , 1}
= 23 · |α| · |α − 1| ≤ 23 · |α|2 = 23 · J0∞ (α) · J1∞ (α) · J01 (α).
— where we apply the easily verified fact that |x + 1|3 ≤ 23 · max{|x|3 , 1}
for x ∈ F . This completes the proof of assertion (iii).
Finally, let us verify assertion (iv). First, observe that it follows immediately from our assumption that | · | is nonarchimedean thatD = {f ∈ F ⋔ | |f | = |f − 1|}.
Suppose that |α| = |α − 1| = 1 (respectively, |α| = |α − 1| > 1). Then we
haveJ(α) = |α(α − 1) + 1|3 ≤ (max{|α| · |α − 1|, 1})3 = 1 = |α|2
(respectively, J(α) = |α(α − 1) + 1|3 · |α|−4 = |α|6 · |α|−4 = |α|2 ).
Thus, we conclude thatmax{J(α), 1} = |α|2 = J0∞ (α) · J1∞ (α) · J01 (α),□

as desired. This completes the proof of assertion (iv).

Definition 1.7. Let Q be an algebraic closure of Q, F ⊆ Q a number field,
E an elliptic curve over F . Recall that E is isomorphic over Q to the elliptic
curve defined by an equationy 2 = x(x − 1)(x − λ)⋔— where λ ∈ Q [cf. [Silv1], Chapter III, Proposition 1.7, (a)]. Recall
further that the j-invariant j(E) of E satisfies
j(E) = 28 (λλ2 −λ+1)(λ−1)2
2

3(∈ F )[cf. [Silv1], Chapter III, Proposition 1.7, (b)], and that the symmetric groupon 3 letters S3 admits a natural faithful action on the projective line P1Q∼⋔

over Q, hence also on the set of Q-rational points (P1Q \ {0, 1, ∞})(Q) → Q .
For □ ∈ {non, arc}, we shall write
∑
def
htor
hS-tor(E) =□ (σ · λ),
□
σ∈S3
def

S-tor
hS-tor (E) = hS-tor
non (E) + harc (E)[cf. Remark 1.1.1] and refer to hS-tor (E) as the symmetrized toric heightof E. We shall also write hS-tor(E) for hS-tor (E). One verifies easily that
⊚
S-tor
S-tor
S-tor
hnon (E), harc (E), h(E) do not depend on the choice of “λ” [cf. theproof of [Silv1], Chapter III, Proposition 1.7, (c)].

16

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

Remark 1.7.1. One verifies easily [cf. Remark 1.1.1] that for □ ∈ {non, arc,⊚}, the quantity hS-tor(E) is unaﬀected by passage to a finite extension of□
the base field F of E.

Remark 1.7.2. It follows immediately from Lemma 1.3, (i), that for □ ∈{non, arc, ⊚}, we have∑hS-tor(E) =h□ (σ · λ).
□σ∈S3Proposition 1.8. (Comparison between hS-tor(E) and h□ (j(E))) In□
the notation of Definition 1.7, the following hold:S-tor (E) − h(i) 0 ≤ hnonnon (j(E)) ≤ 8 log 2.
(ii) −11 log 2 ≤ hS-torarc (E) − harc (j(E)) ≤ 2 log 2.

Proof. If v ∈ V(F ), then it is well-known that || · ||v satisfies the conditions(i), (ii), and (iii) of Definition 1.4, and, moreover, that, if v ∈ V(F )non , then|| · ||v is nonarchimedean in the sense of Lemma 1.6, (iv). Observe that,in the remainder of the proof, we may assume without loss of generality
that, in the situation of Definition 1.7, λ ∈ F ⋔ [cf. Remark 1.7.1]. In thefollowing, for v ∈ V(F ), we shall write J(λ)v , J0∞ (λ)v , J1∞ (λ)v , J01 (λ)v forthe “J(α)”, “J0∞ (α)”, “J1∞ (α)”, “J01 (α)” of Definition 1.4, where we take· “F ” to be F ;· “α” to be λ;· “| · |” to be || · ||v .
Here, we observe that, for □ ∈ {non, arc}, we have
tor
tor
−1
hS-tor(E) = 2 · htor□
□ (λ) + 2 · h□ (1 − λ) + 2 · h□ (λ · (λ − 1) )[cf. Lemma 1.3, (i); the set “A” of Lemma 1.6, (i)].
First, we consider assertion (i). It follows from Lemma 1.6, (iv), that∑
[F : Q] · hS-tor[Fv : Qpv ] · log(J0∞ (λ)v · J1∞ (λ)v · J01 (λ)v )non (E) =v∈V(F )non=∑

[Fv : Qpv ] · log max{J(λ)v , 1}.

v∈V(F )nonThus, to verify assertion (i), it suﬃces to show that, for every v ∈ V(F )nonlying over 2, it holds that0 ≤ log max{J(λ)v , 1} − log max{2−8 · J(λ)v , 1} ≤ 8 log 2[cf. the equality 2−8 · J(λ)v = ||j(E)||v ]. The first inequality follows immediately from the inequality J(λ)v ≥ 2−8 · J(λ)v . Next, we verify the second
inequality. If J(λ)v ≤ 1, thenlog max{J(λ)v , 1} − log max{2−8 · J(λ)v , 1} = 0 − 0 ≤ 8 log 2

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

17

Thus, we may assume that J(λ)v > 1, hence that max{J(λ)v , 1} = J(λ)v .
In particular, if 2−8 · J(λ)v > 1 (respectively, 2−8 · J(λ)v ≤ 1), then we havelog J(λ)v − log max{2−8 · J(λ)v , 1} = − log(2−8 ) = 8 log 2(respectively,
log J(λ)v − log max{2−8 · J(λ)v , 1} = log J(λ)v ≤ 8 log 2).
This completes the verification of the second inequality, hence also of assertion (i).
Next, we consider assertion (ii). Observe that∑
[F : Q] · hS-tor[Fv : R] · log(J0∞ (λ)v · J1∞ (λ)v · J01 (λ)v ).
arc (E) =v∈V(F )arc

Assertion (ii) then follows immediately from Lemma 1.6, (iii) — where wetake the “ϵ” of Lemma 1.6, (iii), to be 2 [cf. the equality 28 · J(λ)v =||j(E)||v ].
□Proposition 1.9. (Comparison between hnon (j(E)) and harc (j(E))) In
the notation of Definition 1.7, suppose that Q(λ) is mono-complex. [Here,note that the fact that Q(λ) is mono-complex does not depend on the choiceof “λ” [cf. the set “A” of Lemma 1.6, (i)].] Then the following hold:S-tor(i) hS-torarc (E) ≤ hnon (E).
(ii) harc (j(E)) ≤ hnon (j(E)) + 19 log 2.
(iii) If C ∈ R, then the element j(E) ∈ Q is completely determined up
to a finite number of possibilities by the condition hnon (−) ≤ C.

Proof. Assertion (i) follows immediately from Lemma 1.3, (iii), and thevarious definitions involved. Assertion (ii) follows from assertion (i) and
Proposition 1.8, (i), (ii). Finally, we consider assertion (iii). It follows from
assertion (ii) [cf. also Definition 1.1, (i)] thath(j(E)) = hnon (j(E)) + harc (j(E)) ≤ 2hnon (j(E)) + 19 log 2.
Thus, assertion (iii) follows immediately from Northcott’s theorem, i.e., thewell-known fact that the set of algebraic numbers of bounded degree andbounded height is finite [cf. [BG], Theorem 1.6.8].
□
Proposition 1.10. (Comparison between h(j(E)) and hFal (E), I) Let
F be a number field; E an elliptic curve over F that has semi-stable reduction over OF . Then, in the notation of Definitions 1.1, (i); 1.7 [cf.
also the discussion entitled “Curves” in §0], we have
0 ≤

Fal
1
12 · h(j(E)) − h (E)≤12 · log(1 + h(j(E))) + 2.071.

Proof. This follows immediately from [Lbr], Proposition 3.1 [and the surrounding discussion].
□

18

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

Remark 1.10.1. In the notation of Proposition 1.10, we observe thatthe normalized degree [cf. [IUTchIV], Definition 1.9, (i)] of the [eﬀective]arithmetic divisor determined by the q-parameters of E at the elements ofV(F )non coincides with hnon (j(E)). Indeed, this follows immediately from
[Silv1], Chapter VII, Proposition 5.5; the discussion at the beginning of[Silv2], Chapter V, §5.
Lemma 1.11. (Linearization of logarithms) Let a ∈ R>0 be a positive
real number. Then we have0 ≤ a − log(a) − 1.
In particular, [by taking “a” to be a · (1 + x)] we havelog(1 + x) − a · x ≤ a − log(a) − 1
for all nonnegative real x ∈ R≥0 .
□Proof. Lemma 1.11 is well-known and entirely elementary.

Proposition 1.12. (Comparison between h(j(E)) and hFal (E), II) Let
ξ ∈ R>0 be a positive real number. Write
defC(ξ) =

ξξ
12 · { 6(1+ξ) − log 6(1+ξ) − 1} + 2.071.

Then, in the notation of Proposition 1.10, we haveFal
112(1+ξ) · h(j(E)) − h (E)≤ C(ξ).

Proof. Indeed, we haveFal
112(1+ξ) · h(j(E)) − h (E)

{
=
≤

}

ξ
− 12(1+ξ)· h(j(E))
{
}
ξ
1
·
log(1
+
h(j(E)))−·
h(j(E))+ 2.071 ≤ C(ξ)
2
6(1+ξ)
Fal
1
12 · h(j(E)) − h (E)— where the first (respectively, second) inequality follows from Propositionξ
1.10 (respectively, Lemma 1.11, where we take “a” to be 6(1+ξ)
and “x” tobe h(j(E))).
□
Definition 1.13. Let κ ≤ 1 be a positive real number; Σ a finite subsetof V(Q) such that V(Q)arc ⊆ Σ. Write Σ ⊆ V(Q) for the inverse image ofΣ ⊆ V(Q) via the natural restriction map V(Q) ↠ V(Q). Recall the set of
rational functions “A” of Lemma 1.6, (i). Then we shall write
def

⋔

⋔

KΣ (κ) = {x ∈ Q | min min{||a(x)||w } ≥ κ} ⊆ Qw∈Σ a∈A[cf. the discussion entitled “Numbers” in §0] and refer to KΣ (κ) as a com⋔⋔pactly bounded subset of Q . Thus, the subset KΣ (κ) ⊆ Q is stabilized by⋔
the natural action of S3 on Q [cf. Lemma 1.6, (i)].

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

19

Corollary 1.14. (Upper bounds for hnon (j(E))) In the notation of Proposition 1.12, let l be a prime number. Suppose that E admits an l-cyclic
subgroup scheme [cf. [GenEll], Lemma 3.5], and that l is prime to the local heights [cf. [GenEll], Definition 3.3] of E at each of its places of [bad]multiplicative reduction [a condition that is satisfied, for instance, if l is >these local heights]. Then the following hold:(i) We havel12(1+ξ) · hnon (j(E))≤ hFal (E) + 12 log(l) + C(ξ).

In particular, by applying the first inequality of Proposition 1.10, we
obtain thatl−(1+ξ)12(1+ξ) · hnon (j(E))≤

1
1
12 · harc (j(E)) + 2 log(l) + C(ξ).

(ii) In the notation of Definitions 1.7, 1.13, suppose thatλ ∈ KΣ (κ).
[Note that the issue of whether or not λ ∈ KΣ (κ) does not depend on
the choice of the particular element “λ” within the S3 -orbit of “λ”[cf. the final portion of Definition 1.13].] Then we havel−(1+ξ)12(1+ξ) · hnon (j(E))

≤

1
1
11
2 log(l) + C(ξ) − 4 log(κ) + 12 log(2).

Suppose, moreover, that l ≥ 1015 . Then, by taking ξ to be 1, we
obtain thathnon (j(E)) ≤
≤

1
11
24 1
l−2 { 2 log(l) + C(1) − 4 log(κ) + 12 log(2)}24 1
1l−2 { 2 log(l) + 2.86 − 4 log(κ) + 0.64}−13−15≤ 5 · 10

− 6.01 · 10

log(κ)−14 , 6 ≤ 6.01 ·— where we apply the estimates log(l)l−2 ≤ 3.46 · 10
l−2
10−15 , 11log(2)
≤
0.64,
and
C(1)
≤
2.86.
12
(iii) Suppose that Q(λ) is mono-complex. Then we havel−2(1+ξ)12(1+ξ) · hnon (j(E))≤

119
2 log(l) + C(ξ) + 12 log(2).

Suppose, moreover, that l ≥ 1015 . Then, by taking ξ to be 1, we
obtain thathnon (j(E)) ≤
≤

19
24 1
l−4 { 2 log(l) + C(1) + 12 log(2)}24 1
l−4 { 2 log(l) + 2.86 + 1.1}
−13
−13

≤ 4.16 · 10

+ 0.96 · 10

= 5.12 · 10−13−14 , 24 ≤ 2.41 ·— where we apply the estimates log(l)l−4 ≤ 3.46 · 10
l−4
10−14 , 19log(2)
≤
1.1,
and
C(1)
≤
2.86.
12

Proof. First, we consider assertion (i). Let H ⊆ E be an l-cyclic subgroupdef

scheme. Write EH = E/H. [In particular, EH is isogenous to E, hence
has semi-stable reduction at all v ∈ V(F )non .] Thus, by applying the same

20

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

arguments as those applied in the proof of [GenEll], Lemma 3.5, we obtainthe following equality:hnon (j(EH )) = l · hnon (j(E))[cf. also Remark 1.10.1]. On the other hand, it follows from the discussion entitled “Curves” in §0; [Falt], Lemma 5, that we have the following
inequality:
hFal (EH ) ≤ hFal (E) + 12 log(l).
In light of the above equality and inequality, assertion (i) follows from Proposition 1.12.
Next, we consider assertion (ii). Note that, since λ ∈ KΣ (κ), for eachv ∈ V(Q(λ))arc , we have:−1max{||λ||v , ||λ||−1v } ≤ κ ;−1max{||λ − 1||v , ||λ − 1||−1v } ≤ κ ;−1
−1
max{||λ − 1||v · ||λ||−1v , ||λ||v · ||λ − 1||v } ≤ κ .

Thus, we conclude from Proposition 1.8, (ii), thatharc (j(E)) − 11 log(2) ≤ hS-torarc (E)≤1[Q(λ):Q]∑[Q(λ)v : R] · log(κ−3 )v∈V(Q(λ))arc= log(κ−3 ).
Assertion (ii) then follows immediately from the second inequality of assertion (i). Finally, assertion (iii) follows immediately from the second inequality of assertion (i) and Proposition 1.9, (ii).
□

2. Auxiliary numerical results
In the present section, we recall• a numerical result concerning the j-invariants of certain special elliptic curves over fields of characteristic zero;• certain eﬀective versions of the prime number theorem.
These results will be applied in §5.
Proposition 2.1. (j-invariants of arithmetic elliptic curves) Let F
be a field of characteristic zero; E an elliptic curve over F . Suppose that thehyperbolic curve obtained by removing the origin from E is “arithmetic”,
i.e., fails to admit an F -core [cf. [CanLift], Remark 2.1.1]. Then the
j-invariant j(E) of E coincides with one of the following:• 488095744= 214 · 313 · 5−3 ,125• 1556068= 22 · 733 · 3−4 ,81• 1728 = 26 · 33 ,• 0.

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

21

Proof. Proposition 2.1 follows immediately from [Sijs], Table 4 [cf. also
[Sijs], Lemma 1.1.1; [CanLift], Proposition 2.7].
□

Proposition 2.2. (Eﬀective versions of the prime number theorem)
For x ∈ R≥2 , writedefπ(x) = ♯{p ∈ Primes | p ≤ x};∑def
θ(x) =log(p).
p∈Primes; p≤x

Set
def

ηprm = 5 · 1020 ;defξprm = 1015 .

Then the following hold:(i) For any real number x ≥ ηprm , it holds thatxπ(x) ≤ 1.022 · log(x)[cf. [IUTchIV], Proposition 1.6].
(ii) For any real number x ≥ ξprm , it holds that
|θ(x) − x| ≤ 0.00071 · x
In particular, if A is a finite subset of Primes, and we write
∑
def
θA =
log(p)
p∈A[where we take the sum to be 0 if A = ∅], then there exists a prime
number p ∈/ A such thatp ≤ (1 − 0.00071)−1 · (θA + ξprm ) ≤ 1.00072 · (θA + ξprm )[cf. [IUTchIV], Proposition 2.1, (ii)].
Proof. First, we consider assertion (i). Observe that log(x) ≥ log(ηprm ) ≥1.17
47.66 ≥ 0.0246. Thus, it holds that
π(x) ≤x
1.17
log(x)−1− log(x)≤xlog(x)−1.0246[cf. [Ax1], Corollary 3.4; [Ax2]]. Therefore, we conclude thatx
1.022 · log(x)≥xlog(x)−1.0246≥ π(x).

Next, we consider assertion (ii). Observe that log(x) ≥ log(ξprm ) ≥ 34.53.
Then since 0.0242269
log(x) ≤ 0.00071, assertion (ii) follows immediately from [RS],Theorem 7.
□

22

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

3. µ6 -Theory for [EtTh]
In the present section, we introduce a slightly modified version of thenotion of an étale theta function of standard type [cf. Definitions 3.3, 3.5], a
notion which plays a central role in the theory developed in [EtTh]. We thenproceed to discuss how the adoption of such a modified version of the notionof an étale theta function of standard type aﬀects the theory developed in[EtTh].
We begin with certain elementary observations concerning roots of unity
and theta functions.
Lemma 3.1. (Group actions on primitive roots of unity) Let n ≥ 2
be an even integer; k an algebraically closed field of characteristic zero.
× for the set of primitive 2n-th roots of unity in k;
Write µ×2n ⊆ k××Aut(µ2n ) for the group of automorphisms of the set µ×2n ; Γ− ⊆ Aut(µ2n )×(respectively, Γ− ⊆ Aut(µ2n )) for the subgroup of cardinality two generated
×by the automorphism of µ×
2n defined as follows: ∀ζ ∈ µ2n ,ζ 7→ −ζ(respectively, ζ 7→ ζ −1 ).

[Note that since n is even, it follows that −ζ ∈ µ×2n .] Then the following
conditions are equivalent:(1) n ∈ {2, 4, 6}.
(2) The action of Γ− × Γ− on µ×
2n is transitive.
Proof. The fact that (1) ⇒ (2) is immediate from the definitions. Thus, itremains to verify that (2) ⇒ (1). First, we observe that the transitivity of
the action of the group Γ− × Γ− [whose cardinality is four] on µ×
2n implies
×that ♯(µ2n ) ≤ 4. In light of this observation, one verifies easily thatn ∈ {1, 2, 3, 4, 5, 6}.
Since n is even, we thus conclude that n ∈ {2, 4, 6}. This completes theproof of Lemma 3.1.
□
Proposition 3.2. (Theta values at primitive 12-th roots of unity) In
the notation of [EtTh], Proposition 1.4: Suppose that K̈ contains a primitive
12-th root of unity ζ12 . Thus, we note that the set of primitive 12-th rootsof unity in K̈ coincides with the set
5711{ζ12 , ζ12
, ζ12
, ζ12
} ⊆ K̈.

Recall the theta function Θ̈ of [EtTh], Proposition 1.4,
1
−1 ∑(n+ 12 )2Θ̈(U ) = qX 8 ·(−1)n · qX2· Ü 2n+1 ,n∈Zwhich satisfies the relations Θ̈(Ü ) = −Θ̈(Ü −1 ) = −Θ̈(−Ü ) [cf. [EtTh],
Proposition 1.4, (ii)]. Then the following hold:EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

23(i) We have7115
Θ̈(ζ12 ), Θ̈(ζ12), Θ̈(ζ12), Θ̈(ζ12) ∈ {Θ̈(ζ12 ), −Θ̈(ζ12 )}.
×(ii) We have Θ̈(ζ12 ) ∈ OK̈
.

Proof. Assertion (i) follows immediately from Lemma 3.1 and [EtTh], Proposition 1.4, (ii). Assertion (ii) follows immediately from the fact that ζ12 −
−1
×
−1 2
ζ12
∈ OK̈
[cf. the equality −(ζ12 − ζ12) = 1].
□
Remark 3.2.1. Lemma 3.1 and Proposition 3.2 arose from observations
due to Porowski. These observations are, in some sense, the starting pointof the theory developed in the present paper.
In the remainder of the present §3, we consider a slightly modified versionof [EtTh] based on “étale theta functions of µ6 -standard type”.
Definition 3.3. In the notation of [EtTh], Definition 1.9, suppose that K
contains a primitive 12-th root of unity. Note that the primitive 12-th rootsof unity in K determine precisely four 12-torsion points{τ1 , τ2 , τ3 , τ4 }of [the underlying elliptic curve of] Ẋ whose restriction to the special fiber
lies in the interior of [i.e., avoids the nodes of] the unique irreducible component of the special fiber.
(i) We shall refer to either of the following four sets of values [cf. [EtTh],
Proposition 1.4, (iii)] of η̈ Θ,Zη̈ Θ,Z |τ1 , η̈ Θ,Z |τ2 , η̈ Θ,Z |τ3 , η̈ Θ,Z |τ4⊆ K×

as a µ6 -standard set of values of η̈ Θ,Z .
(ii) If η̈ Θ,Z satisfies the property that the unique value ∈ K × [cf. Proposition 3.2, (i); Remark 4.2.3, (vi); [IUTchII], Remark 2.5.1, (ii)] ofmaximal order [i.e., relative to the valuation on K] of some µ6 standard set of values of η̈ Θ,Z is equal to ±1, then we shall say thatη̈ Θ,Z is of µ6 -standard type.
Remark 3.3.1. By applying Definition 4.3, together with a similar argument to the argument applied in the proof of [EtTh], Theorem 1.10, one may
prove a “µ6 -version” of [EtTh], Theorem 1.10, i.e., the assertion obtainedby replacing, in [EtTh], Theorem 1.10, (iii),“odd” −→ “arbitrary”[cf. Proposition 3.2, (ii); [IUTchIV], Remark 1.10.6, (ii)]. Note that, in thenotation of [EtTh], Theorem 1.10,
the dual graphs of the special fibers of the various coveringsof Cα , Cβ are somewhat more complicated in the case wherep ∈ {2, 3}.

24

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

On the other hand, since one may still reconstruct the dual graphs grouptheoretically, this will not aﬀect the proof of the µ6 -version of [EtTh], Theorem 1.10, in any significant way.

Definition 3.4. Let l ≥ 1 be an integer coprime to 6. In the notation of[EtTh], §1, suppose that• the residue characteristic of K is arbitrary;• K = K̈;
• K contains a primitive 12-th root of unity ζ12[cf. [EtTh], Definition 1.7, and the preceding discussion; [EtTh], Definition
2.5].
(i) Suppose, in the situation of [EtTh], Definitions 2.1, 2.3, that theellquotient ΠX ↠ Q factors through the natural quotient ΠX ↠ Z
determined by the quotient Πtp
X ↠ Z discussed at the beginning of[EtTh], §1, and that the choice of a splitting of Dx → GK [cf. [EtTh],
Proposition 2.2, (ii)] that determined the covering X log → X log iscompatible with the “{±1}-structure” of the µ6 -version of Theorem
1.10, (iii), of Remark 3.3.1. Then we shall say that the orbicurve oftype (1, l-tors) (respectively, (1, l-torsΘ ); (1, l-tors)± ; (1, l-torsΘ )± )under consideration is of type (1, Z/lZ) (respectively, (1, (Z/lZ)Θ );(1, Z/lZ)± ; (1, (Z/lZ)Θ )± ).
(ii) In the notation of the above discussion and the discussion at the endof [EtTh], §1, we shall refer to a smooth log orbicurve isomorphic tothe smooth log orbicurveẊlog(respectively, Ẋ

log

; Ċ

log

; Ċ

log

)

obtained by taking the composite of the covering
X log (respectively, X log ; C log ; C log )of C log with the covering Ċ log → C log , as being of type (1, µ2 ×Z/lZ)(respectively, (1, µ2 ×(Z/lZ)Θ ); (1, µ2 ×Z/lZ)± ; (1, µ2 ×(Z/lZ)Θ )± ).

Remark 3.4.1. In the “µ6 -version” of [EtTh], Remark 2.5.1, the portion
concerning “Ċ” should be eliminated.

Definition 3.5. In the notation of Definition 3.3 and the discussion preceding of [EtTh], Definition 2.7, if η̈ Θ,Z is of µ6 -standard type, then we shall also
refer to η̈ Θ,l·Z , η̈ Θ,l·Z , η̈ Θ,l·Z×µ2 , η̈ Θ,l·Z×µ2 , η̈ Θ,Z×µ2 as being of µ6 -standard
type.

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

25

Remark 3.5.1. By applying Definitions 3.3, 3.4, 3.5 [cf. also Remarks
3.3.1, 3.4.1], the exposition of [EtTh], §1, §2, goes through without essentialchange under the assumptions stated in the first paragraph of Definition 3.4,
with the following exception: In the “µ6 -versions” of [EtTh], Proposition
2.12; [EtTh], Remark 2.12.1, the portions concerning the hyperbolic orbicurves whose notation contains a “ ˙ ” [i.e., a single “overline dot”] shouldbe eliminated.
Remark 3.5.2. By applying Definitions 3.3, 3.4, 3.5 [cf. also Remarks 3.3.1,
3.4.1, 3.5.1], the exposition of [EtTh], §3, §4, goes through without essentialchange under the assumptions stated in the first paragraph of Definition 3.4,
with the following exception: In the “µ6 -version” of [EtTh], Example 3.9,
the portions concerning the hyperbolic orbicurves whose notation contains
a “ ˙ ” [i.e., a single “overline dot”] should be eliminated.
Remark 3.5.3. By applying Definitions 3.3, 3.4, 3.5 [cf. also Remarks
3.3.1, 3.4.1, 3.5.1, 3.5.2], the exposition of [EtTh], §5, goes through without essential change under the assumptions stated in the first paragraph ofDefinition 3.4, with the following exceptions:(i) Throughout the “µ6 -version” of [EtTh], §5, the portions concerning thehyperbolic orbicurves whose notation contains a “ ˙ ” [i.e., a single “overlinedot”] should be eliminated.
(ii) In the “µ6 -version” of the statement and proof of [EtTh], Proposition
5.3, as well as the preceding discussion, the notation “Ÿ” should be replaced
by “Y”.
(iii) In the “µ6 -version” of [EtTh], Theorem 5.7, as well as the “µ6 version” of the remainder of [EtTh], §5, the following modification should
be made:√“Θ̈( −1)−1 · Θ̈” −→ “Θ̈(ζ12 )−1 · Θ̈”.

4. µ6 -Theory for [IUTchI-III]
In the present section, we introduce a slightly modified version of thenotion of initial Θ-data [cf. Definitions 4.1], a notion which plays a central
role in the theory developed in [IUTchI-IV]. We then proceed to discuss how
the adoption of such a modified version of the notion of initial Θ-data aﬀects
the theory developed in [IUTchI-III].
Definition 4.1. We shall refer to as µ6 -initial Θ-data any collection of data
(F /F, XF , l, C K , V, Vbad
mod , ϵ)satisfying the following conditions:

26

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI• [IUTchI], Definition 3.1, (a), (c), (d);• The “µ6 -version” of [IUTchI], Definition 3.1, (b), i.e., the condition
obtained by replacing, in [IUTchI], Definition 3.1, (b),“odd” −→ “arbitrary”;• The “µ6 -versions” of [IUTchI], Definition 3.1, (e), (f), i.e., the conditions obtained by replacing, in [IUTchI], Definition 3.1, (e), (f),“[EtTh], Definition 2.5, (i)” −→ “Definition 3.4, (i)”[cf. Remark 4.1.1 below].

Remark 4.1.1. In the notation of Definition 4.1, write EF for√ the elliptic
curve over F determined by XF [so XF ⊆ EF ]. Then since −1 ∈ F [cf.
[IUTchI], Definition 3.1, (a)], and, moreover, the 3-torsion points of EF
are rational over F [cf. [IUTchI], Definition 3.1, (b)], we conclude that F
contains a primitive 12-th root of unity ζ12 [cf. the conditions in the firstdisplay of Definition 3.4].

Remark 4.1.2. By applying Definitions 3.3, 3.4, 3.5, 4.1 [cf. also Remarks
3.3.1, 3.4.1, 3.5.1, 3.5.2, 3.5.3, 4.1.1], the exposition of [IUTchI], §3, goes
through without essential change, with the following exceptions: In the “µ6 version” of [IUTchI], Example 3.2, the following modifications should be
made:• In [IUTchI], Example 3.2, (ii),√“ −1” −→ “ζ12 ”;• In [IUTchI], Example 3.2, (iv),√√“ −qv ” −→ “ζ12 qv ”;“µ2l (TX )-multiple” −→ “µ6l (TX )-multiple”;vv“µ2l (−)-orbit” −→ “µ6l (−)-orbit”.
• In [IUTchI], Example 3.2, (v),“µ2l (−)-orbit” −→ “µ6l (−)-orbit”.

Remark 4.1.3. By applying Definitions 3.3, 3.4, 3.5, 4.1 [cf. also Remarks
3.3.1, 3.4.1, 3.5.1, 3.5.2, 3.5.3, 4.1.1, 4.1.2], the exposition of [IUTchI], §4,§5, §6, goes through without essential change, with the following exceptions:
In the “µ6 -version” of [IUTchI], Example 4.4, the following modifications
should be made:• In [IUTchI], Example 4.4, (i),“the unique torsion point of order 2”

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

27−→ “a torsion point of order 6”.
Thus, throughout the µ6 -version of [IUTchI], Example 4.4, (i) —and indeed throughout the remainder of the “µ6 -version” of [IUTchIIV] — “µ− ” is to be regarded as being allowed to vary among thetorsion points of order 6 that satisfy the condition stated in theinitial definition of “µ− ”, with the following exception: In [IUTchI],
Remark 5.2.3, the notation “µ− ” is to be understood in the original“non-µ6 ” sense, i.e., as the unique torsion point of order 2 discussed
in the original “non-µ6 ” version of [IUTchI], Example 4.4, (i).
• In [IUTchI], Example 4.4, (i),“evaluation points” −→ “µ6l -evaluation points”;“evaluation sections” −→ “µ6l -evaluation sections”;“µ2l -orbit” −→ “µ6l -orbit”.
Definition 4.2. Suppose that we are in the situation of [IUTchII], Remark
1.4.1 [cf. also Remark 4.2.2 below]. Writeτ ∈ X k (k)for a torsion point of order 6 whose closure in any stable model of X k over Ok
intersects the same irreducible component of the special fiber of the stablemodel as the zero cusp [cf. Remark 4.1.3]. Since k contains a primitive
12l-th root of unity [cf. Remark 4.2.2 below], it follows from the definitionof an “étale theta function of µ6 -standard type” [cf. Definitions 3.3, (ii);3.5] that there exists a rational pointτ Ÿ ∈ Ÿ k (k)such that τ Ÿ 7→ τ . Write
Dτ ⊆ ΠtpŸk

for the decomposition group of τ Ÿ [which is well-defined up to ∆tp-conjugacy].
Ÿk

We shall refer to either of the pairs(ιŸ ∈ Aut(Ÿ k ), τ Ÿ );(ιŸ ∈ Aut(Πtp)/Inn(∆tp), Dτ )ŸŸkk

as a µ6 -pointed inversion automorphism. Again, we recall from Definitions
3.3, (ii); 3.5, thatan “étale theta function of µ6 -standard type” is defined precisely by the condition that its restriction to Dτ be a 2l-throot of unity.

Remark 4.2.1. By applying Definitions 3.3, 3.4, 3.5, 4.1, 4.2 [cf. also
Remarks 3.3.1, 3.4.1, 3.5.1, 3.5.2, 3.5.3, 4.1.1, 4.1.2, 4.1.3], the exposition of
the Abstract and Introduction of [IUTchII] goes through without essentialchange, with the following exceptions:

28

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI(i) In the “µ6 -version” of the Abstract of [IUTchII], the following modification should be made:“2-torsion point” −→ “6-torsion point”.
(ii) In the “µ6 -version” of the first display and the discussion immediately
following the first display of the Introduction of [IUTchII], the following
modifications should be made:(√(√)∑ 1 (m+ 1 )2 )∑1
(m+ 21 )2
2“ −1 ·qv2
ζ3m+2 qv2
” −→ “ −1 ·
”;
m∈Z

m∈Z

“2-torsion point” −→ “6-torsion point −ζ3 ”;“2l-th root of unity” −→ “6l-th root of unity”.
(iii) In the “µ6 -version” of the paragraph of the Introduction of [IUTchII]that begins “Constant multiple rigidity”, the following modifications
should be made:“[2-]torsion point” −→ “[6-]torsion point”;“2l-th roots of unity” −→ “6l-th roots of unity”.
Remark 4.2.2. By applying Definitions 3.3, 3.4, 3.5, 4.1, 4.2 [cf. also
Remarks 3.3.1, 3.4.1, 3.5.1, 3.5.2, 3.5.3, 4.1.1, 4.1.2, 4.1.3], the expositionof [IUTchII], §1, goes through without essential change, with the following
exceptions:(i) In the “µ6 -version” of the discussion preceding [IUTchII], Definition
1.1, the following modifications should be made:“odd prime number” −→ “prime number ≥ 5”;“of odd residue” −→ “of arbitrary residue”;“4l-th root” −→ “12l-th root”.
(ii) In the “µ6 -version” of [IUTchII], Remark 1.12.2, (ii), the following
modifications should be made:“the 2-torsion point “µ− ” of [IUTchI], Example 4.4, (i)”−→ “a 6-torsion point “τ ” as in Definition 4.2”;“the 2-torsion point “µ− ” are reconstructed”−→ “6-torsion points “τ ” are reconstructed”.
(iii) In the “µ6 -version” of [IUTchII], Remark 1.12.2, (iii), the following
modification should be made:“where we recall that . . . is assumed to be”−→ “where we assume, for simplicity, that . . . is”.

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

29(iv) In the “µ6 -version” of [IUTchII], Remark 1.12.4, the following modification should be made:“ the point “µ− ” ” −→ “ 6-torsion points “τ ” ”.

Remark 4.2.3. By applying Definitions 3.3, 3.4, 3.5, 4.1, 4.2 [cf. also Remarks 3.3.1, 3.4.1, 3.5.1, 3.5.2, 3.5.3, 4.1.1, 4.1.2, 4.1.3, 4.2.2], the expositionof [IUTchII], §2, goes through without essential change, with the following
exceptions:(i) In the “µ6 -version” of [IUTchII], Corollary 2.4, (ii), (b), the following
modifications should be made:“Dµδ − ” −→ “Dτδ ”;“the torsion point “µ− ” of Remark 1.4.1, (i), (ii)”−→ “some torsion point “τ ” as in Definition 4.2”.
(ii) In the “µ6 -version” of [IUTchII], Corollary 2.4, (ii), (c), the following
modifications should be made:δ ”;
δ” −→ “Dt,τ“Dt,µ−“[IUTchI], Example 4.4, (i)” −→ “Remark 4.1.3”;“µ− -translate” −→ “τ -translate”.
(iii) In the “µ6 -version” of [IUTchII], Corollary 2.5, (ii), the following
modifications should be made:
δ
δ ”;
“Dt,µ
” −→ “Dt,τ−“yield µ2l -,” −→ “yield µ6l -,”.
(iv) In the “µ6 -version” of [IUTchII], Corollary 2.5, (iii), the following
modification should be made:“µ2l ” −→ “µ6l ”.
(v) In the “µ6 -version” of [IUTchII], Remark 2.5.1, (i), the following
modification should be made:“µ2l ” −→ “µ6l ”.
(vi) In the “µ6 -version” of [IUTchII], Remark 2.5.1, (ii), the following
modification should be made:√±1“± −1” −→ “±ζ12”.
(vii) In the “µ6 -version” of [IUTchII], Remark 2.5.1, (iii), the following
modification should be made:“µ2l -orbit” −→ “µ6l -orbit”.

30

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI(viii) In the “µ6 -version” of [IUTchII], Remark 2.5.2, (i), the following
modification should be made:
γ1
γ1
“Dt,µ
” −→ “Dt,τ”.
−(ix) In the “µ6 -version” of [IUTchII], Corollary 2.6, (ii), the following
modification should be made:“µ2l ” −→ “µ6l ”;
δ
δ ”.
“Dt,µ
” −→ “Dt,τ−(x) In the “µ6 -version” of [IUTchII], Remark 2.6.3, (i), the following
modification should be made:“µ− -translates” −→ “τ -translates”.
(xi) In the “µ6 -version” of [IUTchII], Corollary 2.8, (i), the following
modifications should be made:
δ
δ ”;
“Dt,µ
” −→ “Dt,τ−“yield µ2l -,” −→ “yield µ6l -,”.
(xii) In the “µ6 -version” of [IUTchII], Corollary 2.8, (ii), the following
modification should be made:“µ2l ” −→ “µ6l ”.
(xiii) In the “µ6 -version” of [IUTchII], Corollary 2.9, (i), the following
modifications should be made:
δ
δ ”;
“Dt,µ
” −→ “Dt,τ−“yield µ2l -,” −→ “yield µ6l -,”.
(xiv) In the “µ6 -version” of [IUTchII], Corollary 2.9, (ii), the following
modification should be made:“µ2l ” −→ “µ6l ”.
Remark 4.2.4. By applying Definitions 3.3, 3.4, 3.5, 4.1, 4.2 [cf. also
Remarks 3.3.1, 3.4.1, 3.5.1, 3.5.2, 3.5.3, 4.1.1, 4.1.2, 4.1.3, 4.2.2, 4.2.3], theexposition of [IUTchII], §3, goes through without essential change, with thefollowing exceptions:(i) In the “µ6 -version” of [IUTchII], Corollary 3.5, (i), the following modification should be made:
δ
δ ”.
“Dt,µ
” −→ “Dt,τ−(ii) In the “µ6 -version” of [IUTchII], Corollary 3.5, (ii), the following
modifications should be made:⋇⋇“(2l)l ” −→ “(6l)l ”;

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY31Θ“Ψ2l·ξ (MΘ∗ )” −→ “Ψ6l·ξ (M∗ )”;“ξ 2l·N ” −→ “ξ 6l·N ”;
δ
δ ”;
“Dt,µ
” −→ “Dt,τ−ΘΘΘ“Ψ2l·ξ1 (MΘ∗ ) = Ψ2l·ξ2 (M∗ )” −→ “Ψ6l·ξ1 (M∗ ) = Ψ6l·ξ2 (M∗ )”.

(iii) In the “µ6 -version” of [IUTchII], Remark 3.5.1, (i), the following
modification should be made:
δ
δ ”.
“Dt,µ
” −→ “Dt,τ−(iv) In the “µ6 -version” of [IUTchII], Corollary 3.6, (ii), the following
modification should be made:“Ψ2l·ξ (−)” −→ “Ψ6l·ξ (−)”.
(v) In the “µ6 -version” of [IUTchII], Definition 3.8, (ii), (iii), the following
modifications should be made:Θ
“F2l·ξ (MΘ∗ )” −→ “F6l·ξ (M∗ )”;“FF2l·ξ († F v )” −→ “FF6l·ξ († F v )”;“Ψ2l·ξ (−)” −→ “Ψ6l·ξ (−)”.
Remark 4.2.5. By applying Definitions 3.3, 3.4, 3.5, 4.1, 4.2 [cf. also
Remarks 3.3.1, 3.4.1, 3.5.1, 3.5.2, 3.5.3, 4.1.1, 4.1.2, 4.1.3, 4.2.2, 4.2.3, 4.2.4],
the exposition of [IUTchII], §4, goes through without essential change, withthe following exceptions: In the “µ6 -version” of [IUTchII], Definition 4.9,(ii), the following modifications should be made:“2l-torsion subgroup” −→ “6l-torsion subgroup”;“µ2l (‡ A)” −→ “µ6l (‡ A)”.
Remark 4.2.6. By applying Definitions 3.3, 3.4, 3.5, 4.1, 4.2 [cf. also
Remarks 3.3.1, 3.4.1, 3.5.1, 3.5.2, 3.5.3, 4.1.1, 4.1.2, 4.1.3, 4.2.2, 4.2.3, 4.2.4,
4.2.5], the exposition of [IUTchIII], §1, §2, §3, goes through without essentialchange, with the following exceptions:(i) In the “µ6 -version” of [IUTchIII], Proposition 3.5, (ii), (c), the following modification should be made:“2l-torsion subgroup” −→ “6l-torsion subgroup”.
(ii) In the “µ6 -version” of [IUTchIII], Remark 3.11.4, (i), the following
modification should be made:“2l-th roots of unity” −→ “6l-th roots of unity”.
(iii) In the “µ6 -version” of [IUTchIII], Fig. 3.4, the following modification
should be made:“µ2l ” −→ “µ6l ”.

32

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

5. µ6 -Theory for [IUTchIV]In the present section, we first give explicit log-volume estimates for the“µ6 -version” of Θ-pilot objects [cf. Theorem 5.1; Corollary 5.2]. [We refer to[IUTchIV], Theorem 1.10; [IUTchIV], Corollary 2.2, (ii), (iii), for the original“non-µ6 ” versions of these results.] Theorem 5.1 follows directly from themodified version of [IUTchI-III] discussed in §4 [cf. also §3], together withcertain estimates from §2, while Corollary 5.2 is obtained by combining
Theorem 5.1 with the theory of §1, §2. We then examine Corollary 5.2
in more detail in the case of mono-complex number fields; this yields aneﬀective version of the ABC inequality over mono-complex number fields [cf.
Theorem 5.3], as well as an eﬀective version of a conjecture of Szpiro over thefield of rational numbers [cf. Theorem 5.4]. As an application, we computean explicit integer n0 > 0 such that for any prime number p ≥ n0 , theFermat equation xp + y p = z p does not have any positive integer solutions[cf. Corollary 5.8], i.e., we give an alternative approach, via fundamentallydiﬀerent techniques, to verifying an eﬀective asymptotic version of “Fermat’s
Last Theorem”, as proven in [Wls]. We also apply the eﬀective version of
the ABC inequality that we obtain to a generalized version of the Fermat
equation [cf. Corollary 5.9].
Theorem 5.1. (Log-volume estimates for the “µ6 -version” of Θpilot objects) Fix a collection of µ6 -initial Θ-data [cf. Definition 4.1].
Suppose that we are in the situation of the “µ6 -version” of [IUTchIII], Corollary 3.12 [cf. Remark 4.2.6], and that the elliptic curve EF has good reduction at every place ∈ V(F )good ∩V(F )non that does not divide 2·3·5·l. In
def

the notation of Definition 4.1, let us write dmod = [Fmod : Q], (1 ≤) emod (≤dmod ) for the maximal ramification index of Fmod [i.e., of places ∈ Vnon
mod ]
def

def

over Q, d∗mod = 212 · 33 · 5 · dmod , e∗mod = 212 · 33 · 5 · emod (≤ d∗mod ), anddef

Fmod ⊆ Ftpd = Fmod (EFmod [2]) ⊆ F
for the “tripodal” intermediate field obtained from Fmod by adjoining thefields of definition of the 2-torsion points of any model of EF ×F F over
Fmod [cf. [IUTchIV], Proposition 1.8, (ii), (iii)]. Moreover, we assume thatthe (3 · 5)-torsion points of EF are defined over F , and that√√def
F = Fmod ( −1, EFmod [2 · 3 · 5]) = Ftpd ( −1, EFtpd [3 · 5])√— i.e., that F is obtained from Ftpd by adjoining −1, together with thefields of definition of the (3 · 5)-torsion points of a model EFtpd of the elliptic
curve EF ×F F over Ftpd determined by the Legendre form of the Weierstrass equation [cf., e.g., the statement of Corollary 5.2, below; [IUTchIV],
Proposition 1.8, (vi)]. [Thus, it follows from [IUTchIV], Proposition 1.8,(iv), that EF ∼= EFtpd ×Ftpd F over F , and from Definition 4.1 that l ̸= 5.]
If Fmod ⊆ F□ ⊆ K is any intermediate extension which is Galois over Fmod ,then we shall writeF□
dADiv∈ ADivR (F□ )EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

33

for the eﬀective arithmetic divisor determined by the diﬀerent ideal of F□
over Q,F□qADiv∈ ADivR (F□ )for the eﬀective arithmetic divisor determined by the q-parameters of thedefelliptic curve EF at the elements of V(F□ )bad = Vbad
mod ×Vmod V(F□ ) (̸= ∅)[cf. [GenEll], Remark 3.3.1],
F

□
fADiv
∈ ADivR (F□ )F□
for the eﬀective arithmetic divisor whose support coincides with Supp(qADiv),
but all of whose coeﬃcients are equal to 1 — i.e., the conductor — and

def

def

F

F

□
□
log(dFv □ ) = degV(F□ )v (dADiv) ∈ R≥0 ; log(dFvQ□ ) = degV(F□ )v (dADiv) ∈ R≥0
Q

def

F

□
log(dF□ ) = deg(dADiv) ∈ R≥0

def

def

F

F

□
□
) ∈ R≥0 ; log(qvQ ) = degV(F□ )v (qADiv) ∈ R≥0log(qv ) = degV(F□ )v (qADiv
Q

def

F

□
log(q) = deg(qADiv) ∈ R≥0

def

def

F

F

□
□
) ∈ R≥0 ; log(fFvQ□ ) = degV(F□ )v (fADiv) ∈ R≥0log(fFv □ ) = degV(F□ )v (fADiv
Q

def

F

□
log(fF□ ) = deg(fADiv) ∈ R≥0def

def— where v ∈ Vmod = V(Fmod ), vQ ∈ VQ = V(Q), V(F□ )v = V(F□ ) ×Vmoddef{v}, V(F□ )vQ = V(F□ )×VQ {vQ } [cf. also [IUTchIV], Definition 1.9]. Here,
we observe that the various “log(q(−) )’s” are independent of the choice ofF□ , and that the quantity “| log(q)| ∈ R>0 ” defined in the µ6 -version of[IUTchIII], Corollary 3.12 [cf. Remark 4.2.6], is equal to 2l1 · log(q) ∈ R [cf.
the definition of “q ” in [IUTchI], Example 3.2, (iv)]. Moreover, supposevthatl ≥ 1015 .
Then one may take the constant “CΘ ∈ R” of the µ6 -version of [IUTchIII],
Corollary 3.12 [cf. Remark 4.2.6], to be{l+1·(1 + 12·dlmod ) · (log(dFtpd ) + log(fFtpd )) + 4.08803 · e∗mod · l
4·| log(q)|}− 61 · (1 − 12)·log(q)−1l2
and hence, by applying the inequality “CΘ ≥ −1” of the µ6 -version of[IUTchIII], Corollary 3.12 [cf. Remark 4.2.6], conclude that1
6 · log(q)≤ (1 + 20·dlmod ) · (log(dFtpd ) + log(fFtpd )) + 4.0881 · e∗mod · l≤ (1 + 20·dlmod ) · (log(dF ) + log(fF )) + 4.0881 · e∗mod · l.

Proof. Theorem 5.1 follows by applying a similar argument to the argument
applied in the proof of [IUTchIV], Theorem 1.10. In the present paper,
however, we replace some of the estimates applied in the proof of [IUTchIV],
Theorem 1.10, as follows:

34

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI• We replace the estimate “ 4(l+5)≤ 20
l+1
3 ” appearing in the final portionof Step (v) of the proof of [IUTchIV], Theorem 1.10, by the estimate
4(l+5)
l+1

16= 4 + l+1
≤ 4 + 100
≤ 4 + 10−13l— cf. our assumption that l ≥ 1015 .
∗• We replace the estimate “lmod·log(s≤ ) ≤ 34 ·(e∗mod ·l +ηprm )” appearing
in Step (viii) of the proof of [IUTchIV], Theorem 1.10, by the estimate∗
lmod· log(s≤ ) ≤ 1.022 · e∗mod · l— cf. Proposition 2.2, (i); our assumption that l ≥ 1015 , which implies theestimate e∗mod · l ≥ 212 · 33 · 5 · 1015 ≥ ηprm = 5 · 1020 .
• We replace the estimate“ 13 · 43 · e∗mod · l ≥ 2 · 2 · 212 · 3 · 5 · l ≥ 2 · log(l) + 56”
appearing in Step (viii) of the proof of [IUTchIV], Theorem 1.10, by theestimate
10−5 · 1.022 · e∗mod · l ≥ (10−5 · 212 · 32 · 5) · 3l ≥ 3l ≥ 2 · log(l) + 56
— where the first (respectively, second; third) inequality follows from theestimate 1.022 ≥ 1 (respectively, 212 · 32 · 5 ≥ 105 ; l ≥ max{56, log(l)}[which is a consequence of our assumption that l ≥ 1015 ]).
In light of these [three] modifications, together with the estimate(4 + 10−13 + 10−5 ) · 1.022 ≤ (4 + 2 · 10−5 ) · 1.022 ≤ 4.08803,
we conclude that one may take the constant “CΘ ∈ R” to be the constantstated in Theorem 5.1.
)−1 ≤ 2” appearing in the
Finally, by replacing the estimate “(1 − 12l2
final portion of Step (viii) of the proof of [IUTchIV], Theorem 1.10, by theestimate(1 − 12)−1 = 1 + l2 12≤ 1 + 100
≤ 1 + 10−13ll2−12[where we apply the estimates l ≥ 1015 , l2 − 12 ≥ l], we obtain [by applying
the estimate 4.08803 · (1 + 10−13 ) ≤ 4.0881] the final inequality of Theorem
5.1.
□
Corollary 5.2. (Construction of suitable µ6 -initial Θ-data) Write X
for the projective line over Q; D ⊆ X for the divisor consisting of thethree points “0”, “1”, and “∞”; (Mell )Q for the moduli stack of elliptic
curves over Q. We shall regard X as the “λ-line” — i.e., we shall regard
the standard coordinate on X as the “λ” in the Legendre form “y 2 =x(x − 1)(x − λ)” of the Weierstrass equation defining an elliptic curve —defand hence as being equipped with a natural classifying morphism UX =X \ D → (Mell )Q [cf. the discussion preceding [IUTchIV], Proposition 1.8].
Let κ ∈ R>0 ∩ R≤1 ;
def

∼

⋔

K = KV(Q)arc (κ) ⊆ UX (Q) (→ Q )EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

35

a compactly bounded subset [cf. Definition 1.13]; d ∈ Z>0 [cf. [IUTchIV],
Corollary 2.2, (ii), (iii)]; ϵ ∈ R>0 ∩R≤1 [cf. [IUTchIV], Corollary 2.2, (iii)].
Writelog(q∀(−) )for the R-valued function on (Mell )Q (Q), hence on UX (Q), obtained byforming the normalized degree “deg(−)” of the eﬀective arithmetic divisor
determined by the q-parameters of an elliptic curve over a number fieldat arbitrary nonarchimedean places [cf. [IUTchIV], Corollary 2.2, (i)];UX (Q)≤d ⊆ UX (Q) for the subset of Q-rational points defined over a finite
extension field of Q of degree ≤ d; UX (Q)mcx ⊆ UX (Q) for the subset of Qrational points defined over a mono-complex number field [cf. Definition
1.2]. Setdef

δ = 212 · 33 · 5 · d = 552960 · d;κlog = 5 · 10−13 − 6.01 · 10−15 log(κ)[cf. the term “HK ” in the first display of [IUTchIV], Corollary 2.2, (iii)];
3.4 · 1030 · ϵ−166/81(d = 1)
def
hd (ϵ) =6 · 1031 · ϵ−174/85(d = 2)
3.4 · 1030 · ϵ−166/81 · d5 (d ≥ 3)
def

4+ϵd ” in the first display of [IUTchIV],[cf. the term “Hunif · ϵ−3 · ϵ−3d · dCorollary 2.2, (iii), where we take “ϵd ” to be 1]. Then there exists a finite
subsetExcκ,d,ϵ ⊆ UX (Q)≤d⊆ UX (Q)≤d ∩ UX (Q)mcx )(respectively, Excmcxd,ϵ— which depends only on κ, d, ϵ (respectively, d, ϵ) and contains all points
corresponding to elliptic curves that admit automorphisms of order > 2 —satisfying the following properties:• The function log(q∀(−) ) is
≤ max{κlog , hd (ϵ)}(respectively, ≤ hd (ϵ))

on Excκ,d,ϵ (respectively, Excmcxd,ϵ ).
• Let EF be an elliptic curve over a number field F ⊆ Q that determinesa Q-valued point of (Mell )Q which lifts [not necessarily uniquely!] to a pointxE ∈ UX (F ) ∩ UX (Q)≤d ∩ K(respectively, xE ∈ UX (F ) ∩ UX (Q)≤d ∩ UX (Q)mcx )such thatxE ∈/ Excκ,d,ϵ (respectively, xE ∈/ Excmcxd,ϵ ).
Write Fmod for the minimal field of definition of the corresponding point∈ (Mell )Q (Q) anddef

Fmod ⊆ Ftpd = Fmod (EFmod [2]) ⊆ F
for the “tripodal” intermediate field obtained from Fmod by adjoining thefields of definition of the 2-torsion points of any model of EF ×F Q over

36

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

Fmod [cf. [IUTchIV], Proposition 1.8, (ii), (iii)]. Moreover, we assume thatthe (3 · 5)-torsion points of EF are defined over F , and that√√def
F = Fmod ( −1, EFmod [2 · 3 · 5]) = Ftpd ( −1, EFtpd [3 · 5])√— i.e., that F is obtained from Ftpd by adjoining −1, together with thefields of definition of the (3 · 5)-torsion points of a model EFtpd of the elliptic
curve EF ×F Q over Ftpd determined by the Legendre form of the Weierstrass equation discussed above [cf. [IUTchIV], Proposition 1.8, (vi)]. [Thus,
it follows from [IUTchIV], Proposition 1.8, (iv), that EF ∼= EFtpd ×Ftpd F
over F , so xE ∈ UX (Ftpd ) ⊆ UX (F ); it follows from [IUTchIV], Proposition 1.8, (v), that EF has stable reduction at every element of V(F )non .]Writelog(q∀ )for the result of applying the function “log(q∀(−) )” to xE . Then EF and Fmod
arise as the “EF ” and “Fmod ” for a collection of µ6 -initial Θ-data as in
Theorem 5.1 that satisfies the following conditions:11(C1) (1015 ·d ≤) (log(q∀ )) 2 ≤ l ≤ 1.464δ·(log(q∀ )) 2 ·log(1.45δ·log(q∀ ));(C2) we have an inequality∀
1
6 · log(q )≤ (1 + ϵ) · (log-diﬀX (xE ) + log-condD (xE ))— where we write log-diﬀX for the [normalized] log-diﬀerent function on
UX (Q) [cf. [GenEll], Definition 1.5, (iii)]; log-condD for the [normalized]
log-conductor function on UX (Q) [cf. [GenEll], Definition 1.5, (iv)].
Proof. First, let us recall that if the once-punctured elliptic curve associatedto EF fails to admit an F -core, then it holds thatj(EF ) ∈ {214 · 313 · 5−3 , 22 · 733 · 3−4 , 26 · 33 , 0}[cf. Proposition 2.1]. Thus, if we take the set Excκ,d,ϵ (respectively, Excmcxd,ϵ ) tobe the [finite!] collection of points corresponding to these four j-invariants,
then we may assume that the once-punctured elliptic curve associated to EF
admits an F -core — hence, in particular, does not have any automorphismsof order > 2 over Q — and that it holds thatlog(q∀(−) ) ≤ max{log(53 ), log(34 )} = log(53 )on Excκ,d,ϵ (respectively, Excmcxd,ϵ ) [cf. Remark 1.10.1]. [In the discussion tofollow, it will in fact be necessary to enlarge the finite set Excκ,d,ϵ (respectively, Excmcxd,ϵ ) several times.]
Next, let us write∑defh = log(q∀ ) = [F1:Q] ·hv · fv · log(pv )v∈V(F )non— that is to say, hv = 0 for those v at which EF has good reduction; hv ∈ N≥1
is the local height of EF [cf. [GenEll], Definition 3.3] for those v at which
EF has bad multiplicative reduction. Now it follows [from [GenEll], Proposition 1.4, (iv) [cf. also the proof of [IUTchIV], Corollary 2.2, (i)] (respectively, from Proposition 1.9, (iii), of the present paper)] that the inequality

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY37

h1/2 < 1015 · d implies that there is only a finite number of possibilities forthe j-invariant of EF . Thus, by possibly enlarging the finite set Excκ,d,ϵ(respectively, Excmcxd,ϵ ), we may assume without loss of generality thath1/2 ≥ 1015 · d (≥ ξprm ),[cf. the notation of Proposition 2.2], and that it holds thatlog(q∀(−) ) ≤ max{log(53 ), 1030 · d2 } = 1030 · d2
on Excκ,d,ϵ (respectively, Excmcxd,ϵ ).
Thus, since [F : Q] ≤ δ [cf. the properties (E3), (E4), (E5) in the proofof [IUTchIV], Theorem 1.10], it follows that∑
δ · h1/2 ≥ [F : Q] · h1/2 =h−1/2 · hv · fv · log(pv )
≥

∑
v

≥

v
−1/2

h

∑

∑

· hv · log(pv ) ≥h−1/2 · hv · log(pv )hv ≥ h1/2

log(pv )hv ≥ h1/2

and1.45δ · h1/2 · log(1.45δ · h) ≥ 1.45 · [F : Q] · h1/2 · log(1.45 · [F : Q] · h)∑≥
1.45 · h−1/2 · log(1.45 · hv · fv · log(pv )) · hv · fv · log(pv )hv ̸= 0≥∑h−1/2 · log(hv ) · hv ≥hv ̸= 0≥∑∑h−1/2 · log(hv ) · hv

hv ≥ h1/2

log(hv )hv ≥ h1/2— where the sums are all over v ∈ V(F )non [possibly subject to various conditions, as indicated], and we apply the elementary estimate 1.45 · log(pv ) ≥1.45 · log(2) ≥ 1.
Thus, in summary, we conclude from the estimates made above that if wetakeA
to be the [finite!] set of prime numbers p such that p either(S1) is ≤ h1/2 ,(S2) divides a nonzero hv for some v ∈ V(F )non , or(S3) is equal to pv for some v ∈ V(F )non for which hv ≥ h1/2 ,then it follows from Proposition 2.2, (ii), together with our assumption thath1/2 ≥ ξprm , that, in the notation of Proposition 2.2, (ii),θA ≤ 2 · h1/2 + δ · h1/2 + 1.45δ · h1/2 · log(1.45δ · h)
= (2 + δ + 1.45δ · log(1.45δ · h)) · h1/2
≤ 1.4621δ · h1/2 · log(1.45δ · h)

38

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI— where we apply the estimates 1 + 0.00071 ≤ 2;
2 ≤ 0.0121δ · log(1.45δ · 1030 ) − δ ≤ 0.0121δ · log(1.45δ · h) − δ[cf. the fact that the function
0.0121x · log(1.45x · 1030 ) − x
is monotonically increasing for x ∈ R≥552960 ]. On the other hand, since wehaveξprm ≤ h1/2 ≤ 0.0001δ · h1/2 · log(1.45δ · h)[cf. the estimates 1 ≤ 0.0001δ and 1 ≤ log(1.45δ · h)], we obtain that
1.00072 · (θA + ξprm ) ≤ 1.464δ · h1/2 · log(1.45δ · h)[cf. the estimate 1.00072·(1.4621+0.0001) ≤ 1.464]. In particular, it follows
from Proposition 2.2, (ii), that there exists a prime number l such that
(P1) (1015 ·d ≤) h1/2 ≤ l ≤ 1.464δ ·h1/2 ·log(1.45δ ·h) [cf. the condition(C1) in the statement of Corollary 5.2];(P2) l does not divide any nonzero hv for v ∈ V(F )non ;(P3) if l = pv for some v ∈ V(F )non , then hv < h1/2 .
Next, let us observe that, again by possibly enlarging the finite set Excκ,d,ϵ(respectively, Excmcxd,ϵ ), we may assume without loss of generality that, in theterminology of [GenEll], Lemma 3.5,
(P4) EF does not admit an l-cyclic subgroup scheme,
and that it holds thatlog(q∀(−) ) ≤ max{1030 · d2 , κlog }(respectively, log(q∀(−) ) ≤ max{1030 · d2 , 5.12 · 10−13 } = 1030 · d2 )on Excκ,d,ϵ (respectively, Excmcxd,ϵ ). Indeed, the existence of an l-cyclic subgroup scheme of EF , together with the fact that l ≥ 1015 [cf. (P1)], would
imply thath ≤ κlog(respectively, h ≤ 5.12 · 10−13 )[cf. (P2); Remark 1.10.1; Corollary 1.14, (ii) (respectively, Corollary 1.14,(iii))]. On the other hand, [by [GenEll], Proposition 1.4, (iv) [cf. also
the proof of [IUTchIV], Corollary 2.2, (i)] (respectively, Proposition 1.9,(iii))] this implies that there is only a finite number of possibilities for the
j-invariant of EF . This completes the proof of the above observation.
Next, we observe that(P5) if we write Vbad
mod for the set of nonarchimedean places ∈ Vmod that
do not divide l and at which EF has bad multiplicative reduction,then Vbadmod ̸= ∅.

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

39

Indeed, if Vbad
mod = ∅, then it follows, in light of the definition of h, from (P3)thath ≤ h1/2 · log(l).
In particular, we haveh1/2 ≤ log(l) ≤≤=≤log(1.464δ) + 0.5 · log(h) + log(log(1.45δ · h))log(1.464δ) + 0.5 · log(h) + log(1.45δ · h)
1.5 · log(h) + log(1.464δ) + log(1.45δ)
1.5 · log(h) + 2 · log(2δ)— where the second inequality follows from (P1); the third inequality follows
from the fact that log(x) ≤ x for all x ∈ R≥1 ; the fourth inequality follows
from the estimate 1.464 · 1.45 ≤ 4. Thus, if we write f (x) for the function
x1/2 − 1.5 · log(x) − 2 · log(2δ),
then it holds that f (h) ≤ 0. On the other hand, since [as is easily verified]
f (x) is monotonically increasing for x ∈ R≥9 , we obtain thatf (h) ≥ f (1030 · d2 )= 1015 · d − 3 · log(1015 · d) − 2 · log(213 · 33 · 5 · d)
≥ 1015 · d − 5 · log(1015 · d) > 0— where we apply the estimate 213 · 33 · 5 ≤ 1015 ; the fact that 5 · log(x) < x
for all x ∈ R≥13 — a contradiction. This completes the proof of the above
observation. This property (P5) implies that(P6) the image of the outer homomorphism Gal(Q/F ) → GL2 (Fl ) determined by the l-torsion points of EF contains the subgroup SL2 (Fl ) ⊆GL2 (Fl ).
Indeed, since, by (P5), EF has bad multiplicative reduction at some place∈ Vbadmod ̸= ∅, (P6) follows formally from (P2), (P4), and [GenEll], Lemma
3.1, (iii) [cf. the proof of the final portion of [GenEll], Theorem 3.8].
Now it follows formally from (P1), (P2), (P5), and (P6) that, if one takes“F ” to be Q, “F ” to be the number field F of the above discussion, “XF ” tobe the once-punctured elliptic curve associated to EF , “l” to be the prime
bad
number l of the above discussion, and “Vbadmod ”to be the set Vmod of (P5),then there exist data “C K ”, “V”, and “ϵ” such that all of the conditions ofDefinition 4.1 are satisfied, and, moreover, that
(P7) the resulting µ6 -initial Θ-data(F /F, XF , l, C K , V, Vbad
mod , ϵ)satisfies the various conditions in the statement of Theorem 5.1.
Here, we note in passing that the crucial existence of data “V” and “ϵ”
satisfying the requisite conditions follows, in essence, as a consequence ofthe fact [i.e., (P6)] that the Galois action on l-torsion points contains thefull special linear group SL2 (Fl ).

40

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

In light of (P7), we may apply Theorem 5.1 [cf. also the fact that e∗mod ≤d∗mod ] to conclude that1
6 · log(q)≤ (1 + 20·dlmod ) · (log(dFtpd ) + log(fFtpd )) + 4.0881 · d∗mod · l≤ (1 + δ · h−1/2 ) · (log(dFtpd ) + log(fFtpd ))+ 5.985 · δ 2 · h1/2 · log(1.45δ · h)

— where we apply (P1), as well as the estimates 20 · dmod ≤ d∗mod ≤ δ
and 4.0881 · 1.464 ≤ 5.985.
Next, let us observe that it follows from (P3) that1
1
6 · h − 6 · log(q)

≤

1/2
1
· log(l).
6 ·hThus, we conclude that1
6 ·h

≤ (1 + δ · h−1/2 ) · (log(dFtpd ) + log(fFtpd )) + 61 · h1/2 · log(l)+ 5.985 · δ 2 · h1/2 · log(1.45δ · h)

and hence that
(P8) the following equality holds:
1
6 ·h· (1 − h−1/2 · log(l) − 35.91 · δ 2 · h−1/2 · log(1.45δ · h))≤ (1 + δ · h−1/2 ) · (log(dFtpd ) + log(fFtpd )).

Now we claim that
Claim 5.2A: If h ≥ hd (ϵ), then it holds that71.82 · δ 2 · h−1/2 · log(1.45δ · h) ≤ ϵ · (1 − 10−7 ).
Indeed, since [as is easily verified] the function x−1/2 · log(1.45δ · x) is monotonically decreasing for x ∈ R>1 , to verify Claim 5.2A, it suﬃces to showthat
71.82 · δ 2 · hd (ϵ)−1/2 · log(1.45δ · hd (ϵ)) ≤ ϵ · (1 − 10−7 ).
def

Let us prove this inequality. First, suppose that d ∈ {1, 2}. Write δ1 =def

212 · 33 · 5, δ2 = 213 · 33 · 5. Then one verifies easily that71.82 · δ 2 · hd (1)−1/2 · log(1.45δ · hd (1)) ≤ 1 − 10−7 .
Thus, if d = 1, then we have71.82 · δ 2 · hd (ϵ)−1/2 · log(1.45δ · hd (ϵ))}{
log(ϵ2/81 )·83
= 71.82 · δ 2 · hd (1)−1/2 · log(1.45δ · hd (1)) · ϵ83/81 · 1 − log(1.45δ·hd (1))≤ ϵ · (1 − 10−7 ) · ϵ2/81 · (1 − log(ϵ2/81 )) ≤ ϵ · (1 − 10−7 );if d = 2, then we have71.82 · δ 2 · hd (ϵ)−1/2 · log(1.45δ · hd (ϵ))}{
log(ϵ2/85 )·87
= 71.82 · δ 2 · hd (1)−1/2 · log(1.45δ · hd (1)) · ϵ87/85 · 1 − log(1.45δ·hd (1))≤ ϵ · (1 − 10−7 ) · ϵ2/85 · (1 − log(ϵ2/85 )) ≤ ϵ · (1 − 10−7 ).

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

41

Here, we apply the estimate log(1.45δ1 ·h1 (1)) ≥ 83; the estimate log(1.45δ2 ·h2 (1)) ≥ 87; our assumption that 0 < ϵ ≤ 1; the fact that x · (1 − log(x)) ≤ 1
for all x ∈ R>0 .
Next, suppose that d ≥ 3. Then we have71.82 · δ 2 · hd (ϵ)−1/2 · log(1.45δ · hd (ϵ)){}
6·log(d)= 71.82 · δ12 · h1 (ϵ)−1/2 · log(1.45δ1 · h1 (ϵ)) · d−1/2 · 1 + log(1.45δ
1 ·h1 (ϵ))

6≤ ϵ · (1 − 10−7 ) · d−1/2 · (1 + 83
· log(d)) ≤ ϵ · (1 − 10−7 )— where we apply the estimate log(1.45δ1 · h1 (ϵ)) ≥ log(1.45δ1 · h1 (1)) ≥ 83;
6
the fact that x−1/2 ·(1+ 83·log(x)) ≤ 1 for all x ∈ R≥3 ; the estimate obtained
above in the case where d = 1. This completes the proof of Claim 5.2A.
Next, we claim that
Claim 5.2B: If h ≥ hd (ϵ), then it holds that2 · h−1/2 · log(l) + 71.82 · δ 2 · h−1/2 · log(1.45δ · h) + δ · h−1/2 ≤ ϵ.
Indeed, since [cf. (P1)] it holds that71.82 · δ · log(1.45δ · h) ≥ 71.82 · 212 · 33 · 5 · log(1.45 · 212 · 33 · 5 · 1030 ) ≥ 108 ,we have

71.82 · δ 2 · h−1/2 · log(1.45δ · h) ≥ 108 · δ · h−1/2 .
Moreover, since it holds that71.82 · δ 2 ≥ 71.82 · (212 · 33 · 5)2 ≥ 4 · 1012 ,we have [cf. (P1)]71.82 · δ 2 · h−1/2 · log(1.45δ · h)
≥ 1012 · 2 · h−1/2 · log((1.45δ · h)2 )≥ 1012 · 2 · h−1/2 · log(1.464δ · h1/2 · 1.45δ · h)
≥ 1012 · 2 · h−1/2 · log(1.464δ · h1/2 · log(1.45δ · h))≥ 1012 · 2 · h−1/2 · log(l)— where the second inequality follows from the estimate 1.45 · h1/2 ≥ 1.464;
the third inequality follows from the fact that x ≥ log(x) for all x ∈ R≥1 .
Thus, it follows from Claim 5.2A that2 · h−1/2 · log(l) + 71.82 · δ 2 · h−1/2 · log(1.45δ · h) + δ · h−1/2≤ (10−12 + 1 + 10−8 ) · 71.82 · δ 2 · h−1/2 · log(1.45δ · h)
≤ (10−12 + 1 + 10−8 ) · ϵ · (1 − 10−7 ) ≤ ϵ— where we apply the estimate (10−12 + 1 + 10−8 ) · (1 − 10−7 ) ≤ 1. This
completes the proof of Claim 5.2B.
Here, note that the inequality h < hd (ϵ) implies [by [GenEll], Proposition
1.4, (iv) [cf. also the proof of [IUTchIV], Corollary 2.2, (i)] (respectively,
Proposition 1.9, (iii))] that there is only a finite number of possibilities forthe j-invariant of EF . Thus, by possibly enlarging the finite set Excκ,d,ϵ(respectively, Excmcxd,ϵ ), we may assume without loss of generality thath ≥ hd (ϵ),

42

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

and that it holds thatlog(q∀(−) ) ≤ max{1030 · d2 , κlog , hd (ϵ)}= max{κlog , hd (ϵ)}(respectively,log(q∀(−) ) ≤ max{1030 · d2 , hd (ϵ)}= hd (ϵ))[cf. the estimate hd (ϵ) ≥ 1030 · d2 ] on Excκ,d,ϵ (respectively, Excmcxd,ϵ ).
Thus, in light of Claim 5.2B, it follows from (P8) [cf. also (P1)] that1
6 ·h

≤ (1 + ϵ) · (log(dFtpd ) + log(fFtpd ))≤ (1 + ϵ) · (log-diﬀX (xE ) + log-condD (xE ))— where we apply the fact that for any x, y ∈ R>0 such that 2x + y ≤ ϵ, itholds that
(1 − x)−1 · (1 + y) ≤ 1 + ϵ[which is a consequence of the fact that 0 < ϵ ≤ 1]. This completes the proofof (C2), hence [cf. (P1), (P7)] of Corollary 5.2.
□

Theorem 5.3. (Eﬀective version of the ABC inequality over monocomplex number fields) Let L be a mono-complex number field [cf.
Definition 1.2]; a, b, c ∈ L× nonzero elements of L such thata + b + c = 0;ϵ a positive real number ≤ 1. Write ∆L for the absolute value of the disdef
criminant of L; d = [L : Q];∏
def
HL (a, b, c) =max{|a|v , |b|v , |c|v };v∈V(L)def

IL (a, b, c) = {v ∈ V(L)non | ♯{|a|v , |b|v , |c|v } ≥ 2} ⊆ V(L)non ;defradL (a, b, c) =∏♯(OL /pv );v∈IL (a,b,c){def

hd (ϵ) =3.4 · 1030 · ϵ−166/81 (d = 1)
6 · 1031 · ϵ−174/85(d = 2).

Then we have
HL (a, b, c) ≤ 25d/2 · max{exp( d4 · hd (ϵ)), (∆L · radL (a, b, c))3(1+ϵ)/2 }≤ 25d/2 · exp( d4 · hd (ϵ)) · (∆L · radL (a, b, c))3(1+ϵ)/2 .
Proof. Write w ∈ V(L)arc for the unique element of V(L)arc [cf. Definition
1.2]. First, we claim the following:

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

Claim 5.3A: It holds thatbd · htornon ( c ) =∑12 log |bc|w +43

log max{|a|v , |b|v , |c|v }.

v∈V(L)non

Indeed, we compute:bd · htornon ( c ) =∑1
2

log max{| cb |v , | cb |v }v∈V(L)non

=

∑

1
2

1
log( |bc|· max{|b|2v , |c|2v })v

v∈V(L)non=1
2 log |bc|w +∑log max{|b|v , |c|v }v∈V(L)non=1
2 log |bc|w +∑log max{|a|v , |b|v , |c|v }v∈V(L)non— where the third equality (respectively, the fourth equality) follows fromthe product formula (respectively, the fact that for v ∈ V(L)non ,|a|v = |b + c|v ≤ max{|b|v , |c|v }).
This completes the proof of Claim 5.3A.
Next, we observe that, to verify Theorem 5.3, we may assume without
loss of generality that||a||w ≤ ||b||w ≤ ||c||w .
Then we observe the following:
Claim 5.3B: It holds that || cb ||w ≥ 12 , hence that | cb |w ≥ ( 12 )d .
Indeed, since a + b + c = 0, we have||c||w = ||a + b||w ≤ ||a||w + ||b||w ≤ 2 · ||b||w .
This completes the proof of Claim 5.3B.
Next, in the notation of the portion of Corollary 5.2 that does not concern
κ/κlog /K, we consider the element∼⋔− ac ∈ UX (F ) ∩ UX (Q)≤d ∩ UX (Q)mcx ⊆ UX (Q) (→ Q ).
Thus, by applying Corollary 5.2 [cf. the bound on the restriction of thefunction “log(q∀(−) )” to “Excmcxd,ϵ ”; the displayed inequality of (C2)], we obtain
that
∀
1
( )
6 · log(q − a )c≤ max{(1 + ϵ) · (log-diﬀX (− ac ) + log-condD (− ac )), 16 · hd (ϵ)}≤ max{(1 + ϵ) · ( d1 · log(∆L ) + d1 · log(radL (a, b, c))), 16 · hd (ϵ)}.

Now we claim the following:
Claim 5.3C: It holds that∀1
( )6 · log(q − a )
c

≥

2
5
3d · log(HL (a, b, c)) − 3 log 2.

44

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

Indeed, it follows from Lemma 1.3, (i), (iv); Proposition 1.8, (i); Remark
1.10.1, that
(∗1 )

∀
1
4
( )
6 · log(q − a ) + 3 log 2
c

≥

tor a
tor b
tor b
1
3 · (hnon ( c ) + hnon ( c ) + hnon ( a ))

≥

tor b
2
3 · hnon ( c ).

On the other hand, we have(∗2 )bd · htornon ( c ) =∑12 log |bc|w +log max{|a|v , |b|v , |c|v }v∈V(L)non=1b
2 log | c |w + log(HL (a, b, c))≥d1
2 log 2 + log(HL (a, b, c))— where the first equality (respectively, the second equality; the final inequality) follows from Claim 5.3A (respectively, the fact that |c|w ≥ |b|w ≥|a|w ; Claim 5.3B). The inequality of Claim 5.3C follows immediately from(∗1 ) and (∗2 ).
Finally, in light of Claim 5.3C, we obtain thatlog(HL (a, b, c)) ≤ max{ d4 · hd (ϵ), 32 (1 + ϵ) · log(∆L · radL (a, b, c))} + 5d
2 log 2.
This completes the proof of Theorem 5.3.

□

Remark 5.3.1. The astronomically large constants in the inequalities established in Theorem 5.3 reflect the explicit [i.e., “non-conjectural”] natureof inter-universal Teichmüller theory. Their size may seem quite unexpected,
especially from the point of view of the classical [“conjectural”] literature onsuch inequalities, where sometimes it is even naively assumed that these
constant may be taken to be as small as 1.

Remark 5.3.2. The approach to applying the version of the ABC inequality established in Theorem 5.3 to “Fermat’s Last Theorem” in the present
paper [cf. Corollary 5.8 below] extends to other diophantine equations [cf.
Corollary 5.9 below]. Namely, in view of the very large constants [cf. Remark 5.3.1] that appear, in order to apply such an inequality to a concrete
diophantine equation of the form u + v = w with polynomial functions u,v, w which involve, respectively, positive integers l, m, n as exponents, one
needs first to establish a lower bound on potential solutions of this equation[cf. Lemma 5.7 below; the second to last display of the proof of Corollary5.9]. One then applies a suitable version of the ABC inequality to obtain an
upper bound on l, m, n, under the condition that the diophantine equation
admits a solution of the desired type [cf. the portion of the proof of Corollary 5.8 subsequent to the application of Lemma 5.7; the final display of theproof of Corollary 5.9]. Finally, the existence of solutions to the diophantine equation for l, m, n satisfying the upper bound may be investigated bymeans of computer calculations.

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

45

Theorem 5.4. (Eﬀective version of a conjecture of Szpiro) Let a, b,
c be nonzero coprime integers such thata + b + c = 0;ϵ a positive real number ≤ 1. Then we have|abc| = ||abc||C ≤ 24 · max{exp(1.7 · 1030 · ϵ−166/81 ), (rad(abc))3(1+ϵ) }≤ 24 · exp(1.7 · 1030 · ϵ−166/81 ) · (rad(abc))3(1+ϵ)— which may be regarded as an explicit version of the inequality( ∏ )3+ϵ”“|abc| ≤ C(ϵ)
pp|abcconjectured in [Szp], §2 [i.e., the “forme
forte” of loc. cit., where we note∏that the “p” to the right of the “ ” in the above display was apparently
unintentionally omitted in loc. cit.].
Proof. First, we claim the following:
Claim 5.4A: In the notation of Theorem 5.3, suppose that||a||w ≤ ||b||w ≤ ||c||w— where w ∈ V(L)arc denotes the unique element of V(L)arc[cf. Definition 1.2]. Then it holds that.
HL (a, b, c) ≤ 24d/3 · max{exp( d6 · hd (ϵ)), (∆L · radL (a, b, c))1+ϵ } · |abc−2 |−1/3w
Indeed, it follows from the discussion immediately preceding Claim 5.3C inthe proof of Theorem 5.3 that we have
Claim 5.4B: It holds that∀1
( )6 · log(q − a )c≤ max{ d1 · (1 + ϵ) · log(∆L · radL (a, b, c)), 61 · hd (ϵ)}.

Now we claim the following:
Claim 5.4C: It holds that∀1
( )6 · log(q − a )
c

≥

−2
1
1
4
d · log(HL (a, b, c)) + 3d · log |abc |w − 3 log 2.

Let us verify Claim 5.4C. First, let us recall the inequality (∗1 ) in the proofof Theorem 5.3
tor b
tor b
a
(†1 ) 1 · log(q∀( a ) ) + 4 log 2 ≥ 1 · (htornon ( ) + hnon ( ) + hnon ( )).
6

3

−c

3

c

c

a

On the other hand, we compute:
tor b
tor b
a
d · (htor
non ( c ) + hnon ( c ) + hnon ( a ))

(†2 )
=

1
2 · (log |ac|w + log |bc|w + log |ba|w ) + 3= log |abc|w + 3∑∑v∈V(L)nonlog max{|a|v , |b|v , |c|v }v∈V(L)non= log |abc−2 |w + 3 · log(HL (a, b, c))log max{|a|v , |b|v , |c|v }46

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI— where the first equality (respectively, the third equality) follows from
Claim 5.3A (respectively, the fact that |c|w ≥ |b|w ≥ |a|w ). The inequalityof Claim 5.4C follows immediately from (†1 ) and (†2 ). The inequality ofClaim 5.4A then follows immediately from the inequalities of Claims 5.4B,
5.4C.
Next, we observe that, to verify Theorem 5.4, we may assume without
loss of generality that||a||C ≤ ||b||C ≤ ||c||C .
Now we apply the inequality in Claim 5.4A to the present situation, by
taking “L” to be Q. Then we have
−1/3||c||C ≤ 24/3 · max{exp( 61 · 3.4 · 1030 · ϵ−166/81 ), (rad(abc))1+ϵ } · ||abc−2 ||C.

Therefore, by raising this inequality to the 3-rd power, we conclude that||abc||C ≤ 24 · max{exp(1.7 · 1030 · ϵ−166/81 ), (rad(abc))3(1+ϵ) }.
□This completes the proof of Theorem 5.4.

In the following, we give an alternative approach to proving an eﬀective
asymptotic version of “Fermat’s Last Theorem”, as proven in [Wls]. The
following Lemmas 5.5, 5.6, 5.7 are entirely elementary, but their statements
and proofs are given in full detail for lack of a suitable reference.
Lemma 5.5. (Elementary identity) Let p ≥ 3 be an odd integer; r, s
integers such that r + s ̸= 0. Then we have
p

−1

p

(r + s )(r + s)

= ps

p−1

− (r + s)p−2∑(−1)i+1 (i + 1)rp−2−i si .

i=0

Proof. One verifies immediately that we may assume without loss of generality that r ̸= 0. Then, to verify Lemma 5.5, it suﬃces to show [by dividingby rp−1 ] the following equality of elements ∈ Q(x):p(1 + x )(1 + x)

−1

= px

p−1

p−2
∑
− (1 + x)(−1)i+1 (i + 1)xi .
i=0

Write ∂ for the derivation d/dx on Q(x). Then:pxp−1 = ∂(1 + xp ) = ∂{(1 + xp )(1 + x)−1 · (1 + x)}= (1 + xp )(1 + x)−1 + (1 + x)∂{(1 + xp )(1 + x)−1 }= (1 + xp )(1 + x)−1 + (1 + x)∂p−2(∑(−1)i+1 xi+1

)

i=−1

= (1 + xp )(1 + x)−1 + (1 + x)p−2∑(−1)i+1 (i + 1)xi .

i=0This completes the verification of Lemma 5.5.

□

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

47

Lemma 5.6. (Elementary properties of possible solutions of the
Fermat equation) Let p ≥ 3 be a prime number; r, s, t nonzero coprime
integers such thatrp + sp + tp = 0.
Then the following hold:(i) Let l be a prime number which divides r + s, (rp + sp )(r + s)−1 ∈ Z.
Then it holds that l = p.
(ii) Suppose that p does not divide t. Then r + s and (rp + sp )(r + s)−1
are coprime. In particular, [since (r +s)·(rp +sp )(r +s)−1 = (−t)p ]there exist integers u and ũ such thatr + s = up ,(rp + sp )(r + s)−1 = ũpt = −uũ.

(iii) Suppose that p divides t. Then it holds thatr + s ∈ pZ,(rp + sp )(r + s)−1 ∈ pZ \ p2 Z.

In particular, if we write t = pk v, where k ∈ Z>0 , v ∈ Z \ pZ, then[since (r + s) · (rp + sp )(r + s)−1 = (−t)p ] there exist integers w ∈/ pZ
and w̃ ∈/ pZ such thatr + s = pkp−1 wp ,(rp + sp )(r + s)−1 = pw̃p ,v = −ww̃[cf. (i)].

Proof. First, we consider assertion (i). Let l be a prime number which
divides r + s and (rp + sp )(r + s)−1 . In particular, it follows from Lemma
5.5 that l divides psp−1 . Thus, if l ̸= p, then we conclude that l divides s,hence that l divides r = (r + s) − s — a contradiction. This completes theproof of assertion (i).
Next, we consider assertion (ii). Suppose that r + s and (rp + sp )(r + s)−1
are not coprime. Then it follows from assertion (i) that p divides r + s and(rp + sp )(r + s)−1 , hence that p divides rp + sp = (−t)p — a contradiction.
Therefore, we conclude that r + s and (rp + sp )(r + s)−1 are coprime. This
completes the proof of assertion (ii).
Finally, we consider assertion (iii). We begin by observing that(r + s)p ≡ rp + sp ≡ −tp ≡ 0 (mod p),
hence that r + s ≡ 0 (mod p). In particular, it follows from Lemma 5.5 that(rp + sp )(r + s)−1 ≡ 0 (mod p). Thus, to verify assertion (iii), it suﬃces toprove the following claim:
Claim 5.6A: It holds that (rp + sp )(r + s)−1 ∈/ p2 Z.

48

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

Indeed, suppose that (rp + sp )(r + s)−1 ∈ p2 Z. Write r + s = pm, wherem ∈ Z. Then since we have(rp + sp )(r + s)−1 = {rp + (pm − r)p }(pm)−1
p ( )
{
}
∑
p
p
=
r +
(pm)p−i (−r)i (pm)−1ii=0
p−1 ( )∑
p=(pm)p−i−1 (−r)i ,
i

i=0
p
p
our assumption that (r + s )(r + s)−1 ∈ p2 Z implies that prp−1 ∈ p2 Z.

Thus, we conclude that r ∈ pZ, hence that s = (r + s) − r ∈ pZ — acontradiction. Therefore, we conclude that (rp + sp )(r + s)−1 ∈/ p2 Z. This
completes the proof of Claim 5.6A, hence also of assertion (iii).
□

Lemma 5.7. (Elementary estimate for possible solutions of the Fermat equation) Let p ≥ 3 be a prime number; x, y, z coprime positive
integers such thatxp + y p = z p .
Then it holds thatp.
z > (p+1)
2
Proof. First, we consider the case where p divides xy. [In particular, p does
not divide z.] In this case, to verify Lemma 5.7, we may assume without lossof generality that p divides x. [In particular, p does not divide y.] Then it
follows by applying Lemma 5.6, (ii), first in the case where we take “(r, s, t)”
to be (x, y, −z), then in the case where we take “(r, s, t)” to be (z, −x, −y),that there exist positive integers a and b such thatx + y = ap ,z − x = bp .

Here, observe that(z − y)p ≡ z p − y p ≡ xp ≡ 0 (mod p),
hence that z − y ≡ 0 (mod p). Thus, we obtain that(b − a)p ≡ bp − ap ≡ (z − y) − 2x ≡ 0 (mod p),
hence that b − a ≡ 0 (mod p). Now we claim the following:
Claim 5.7A: It holds that max{a, b} ≥ p + 1.
Indeed, suppose that max{a, b} ≤ p. Then it follows from the fact thatb − a ≡ 0 (mod p) that a = b, hence that z = 2x + y. In particular, weconclude that z p = (2x + y)p > xp + y p — a contradiction.
In light of Claim 5.7A, we have2z > z + y = ap + bp > (p + 1)p .
This completes the proof of Lemma 5.7 in the case where p divides xy.
Next, we consider the case where p does not divide xyz. Then it follows
by applying Lemma 5.6, (ii), first in the case where we take “(r, s, t)” to be(x, y, −z), then in the case where we take “(r, s, t)” to be (z, −x, −y), andEXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

49

finally in the case where we take “(r, s, t)” to be (z, −y, −x), that there exist
positive integers a, b, and c such thatx + y = ap ,z − x = bp ,z − y = cp .

Here, observe that(z − x − y)p ≡ z p − xp − y p ≡ 0 (mod p),
hence that z − x − y ≡ 0 (mod p). Thus, we obtain that(b + c − a)p ≡ bp + cp − ap ≡ 2(z − x − y) ≡ 0 (mod p),
hence that b + c − a ≡ 0 (mod p). Now we claim the following:
Claim 5.7B: It holds that a ≥ p + 1.
Indeed, suppose that a ≤ p. Observe that since (2x + y)p > xp + y p = z p ,(x + 2y)p > xp + y p = z p , it holds that x + y > z − x, x + y > z − y, hence
that a > b, a > c. Thus, we conclude that−p ≤ −a < b + c − a < a + a − a ≤ p,hence that b + c − a = 0. Next, we claim thatdef

Claim 5.7C: Write E = {□ ∈ {a, b, c} | □ is even}. Then it
holds that ♯E = 1.
Indeed, it follows immediately from the equality b + c − a = 0 that ♯E ≥ 1.
Suppose that ♯E ≥ 2. Then it follows from the equality b + c − a = 0 thata, b, and c are even. In particular, since ap (= x + y) divides z p (= xp + y p ),we conclude that z is even. On the other hand, this implies that x and y
are even [cf. the equalities z − x = bp and z − y = cp ] — a contradiction.
This completes the proof of Claim 5.7C.
Now suppose that E = {a} [cf. Claim 5.7C]. Here, note that it follows by
applying Lemma 5.6, (ii), in the case where we take “(r, s, t)” to be (x, y, −z),that there exists a positive integer ã such that (xp + y p )(x + y)−1 = ãp . [Inparticular, we have z = aã.] Then since(bp + cp )(b + c)−1 = (bp + cp )a−1 = (2z − x − y)a−1 = 2ã − ap−1 ,we conclude that (bp + cp )(b + c)−1 is an even integer. On the other hand,
since
p−1
∑
(bp + cp )(b + c)−1 =(−1)i bp−i−1 ci ,i=0and, moreover, each term “bp−i−1 ci ” is odd, we conclude that (bp +cp )(b+c)−1
is odd — a contradiction.
Thus, it follows from Claim 5.7C that E ∈ {{b}, {c}}. Moreover, to verify
Claim 5.7B, we may assume without loss of generality that E = {b}. Next,
observe that it follows by applying Lemma 5.6, (ii), in the case where wetake “(r, s, t)” to be (z, −x, −y), that there exists a positive integer b̃ suchthat (z p − xp )(z − x)−1 = b̃p . [In particular, we have y = bb̃.] Then since(ap − cp )(a − c)−1 = (ap − cp )b−1 = (2y + x − z)b−1 = 2b̃ − bp−1 ,50

MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

we conclude that (ap − cp )(a − c)−1 is an even integer. On the other hand,
since
p−1
∑
ap−i−1 ci ,(ap − cp )(a − c)−1 =
i=0
p−i−1
i
and, moreover, each term “ac ” is odd, we conclude that (ap − cp )(a −c)−1 is odd — a contradiction. This completes the proof of Claim 5.7B.
In light of Claim 5.7B, we have2z > x + y = ap ≥ (p + 1)p .

This completes the proof of Lemma 5.7 in the case where p does not divide
xyz.
Finally, we consider the case where p divides z. [In particular, p does notdivide xy.] Then it follows by applying Lemma 5.6, (ii), first in the casewhere we take “(r, s, t)” to be (z, −x, −y), then in the case where we take“(r, s, t)” to be (z, −y, −x), that there exist positive integers b and c suchthatz − x = bp ,z − y = cp .
Moreover, it follows by applying Lemma 5.6, (iii), in the case where we take“(r, s, t)” to be (x, y, −z), that there exist positive integers w ∈/ pZ and k,
together with a negative integer v ∈/ pZ, such thatz = −pk v.

x + y = pkp−1 wp ,Next, observe that(b + c)p ≡ bp + cp ≡ 2z − x − y ≡ 0 (mod p),
hence that b + c ≡ 0 (mod p). In particular, it follows from the equality
p

p

b +c

= pc

p−1

(b + c) − (b + c)

2

p−2
∑

(−1)i+1 (i + 1)bp−2−i ci

i=0[cf. Lemma 5.5] that bp + cp ∈ p2 Z. Thus, since we have
−2pk v = 2z = (z − x) + (z − y) + (x + y) = bp + cp + pkp−1 wp ∈ p2 Z,
we conclude that k ≥ 2 [cf. the fact that kp−1 ≥ 2]. Therefore, we concludethat2z > x + y ≥ p2p−1 wp > (p + 1)p[cf. the fact that γ 2γ−1 > (γ + 1)γ for all γ ∈ R≥3 ]. This completes theproof of Lemma 5.7 in the case where p divides z.
□

Remark 5.7.1.
(i) In the notation of Lemma 5.7, we observe that a stronger estimatez > z − x > (2p20/7 )p
may be obtained by means of techniques of classical algebraic number theory
that are somewhat more involved than the argument given above in the proof
of Lemma 5.7 [cf. [Ink1]; [Ink2], Theorem 2].

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

51(ii) In fact, it follows from [Ink1], (4), that, in the situation of (i), if we
assume further that p divides xyz, the an even stronger estimate
z >

p3p−1
2

may be obtained.
Corollary 5.8. (Application to “Fermat’s Last Theorem”) Let
p > 1.615 · 1014
be a prime number. Then there does not exist any triple (x, y, z) of positive
integers that satisfies the Fermat equation
xp + y p = z p .
Proof. Suppose that there exists a triple (x, y, z) of positive integers suchthat xp + y p = z p . Here, we may assume without loss of generality that x,y, z are coprime. Then it follows from Lemma 5.7 thatz >(p+1)p.
2

Now we apply Theorem 5.3 to the present situation, by taking• “L” to be Q;• “(a, b, c)” to be (xp , y p , −z p );• “ϵ” to be 1.
Then, in the notation of Theorem 5.3, we havez p ≤ 25/2 · exp( 41 · h1 (1)) · (radQ (xp , y p , −z p ))3≤ 25/2 · exp( 41 · h1 (1)) · (xyz)3≤ 25/2 · exp( 14 · h1 (1)) · (z 3 )3— where we apply the fact that z ≥ max{x, y}. Thus, we obtain that{}p−9(p+1)p< z p−9 ≤ 25/2 · exp( 41 · h1 (1)).
2
In particular, we conclude that(p − 9)(−1 + p · log2 (p + 1)) <5
1
2 + log2 (e) · 4 · h1 (1)< 1.227 · 1030 .

On the other hand, since [as is easily verified] the function(x − 9)(−1 + 1.44x · log(x + 1))
is monotonically increasing for x ∈ R≥9 , we have1(p − 9)(−1 + p · log2 (p + 1)) = (p − 9)(−1 + log(2)· p · log(p + 1))> (p − 9)(−1 + 1.44 · p · log(p + 1))> 1.227 · 1030
— where the first (respectively, second) inequality follows from the esti1
mate log(2)> 1.44 (respectively, our assumption that p > 1.615 · 1014 ) — acontradiction. This completes the proof of Corollary 5.8.
□

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MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

Remark 5.8.1. By combining Corollary 5.8 with the numerical estimateof [Cop] [cf. [Cop], Abstract; the discussion following the first display of[Cop], §3], we obtain [by applying the estimate 7.568 · 1017 > 1.615 · 1014 ]the following result:
Let p be an odd prime number. Then there does not exist
any triple (x, y, z) of positive integers such that p does notdivide xyz, and, moreover, the Fermat equation
xp + y p = z p
is satisfied.
This assertion is often called the first case of Fermat’s Last Theorem.

Remark 5.8.2.
(i) If we apply the estimatez > (2p20/7 )pof Remark 5.7.1, (i), instead of Lemma 5.7 in the proof of Corollary 5.8,
then the quantity “1.615 · 1014 ” in Corollary 5.8 may be replaced by 9.58 ·1013 . Indeed, by applying this estimate of Remark 5.7.1, (i), we obtain theestimate(2p20/7 )p(p−9) < z p−9 ≤ 25/2 · exp( 41 · h1 (1)).
In particular, we conclude thatp(p − 9)(1 + 20
7 · log2 (p)) <5
1
2 + log2 (e) · 4 · h1 (1)< 1.227 · 1030 .

Thus, it suﬃces to observe that the manifestly monotonically increasing
functionx(x − 9)(1 + 20
7 · log2 (x))
satisfies the inequality > 1.227 · 1030 for x > 9.58 · 1013 .
(ii) Suppose that p divides xyz. That is to say, we suppose that weare in the situation of what is often called the second case of Fermat’s Last
Theorem. Then if we apply the estimate
z >

p3p−1
2

of Remark 5.7.1, (ii), instead of Lemma 5.7 in the proof of Corollary 5.8,
then the quantity “1.615 · 1014 ” in Corollary 5.8 may be replaced by 9.39 ·1013 . Indeed, by applying this estimate of Remark 5.7.1, (ii), we obtain theestimate
3p−1( p 2 )(p−9) < z p−9 ≤ 25/2 · exp( 41 · h1 (1)).
In particular, we conclude that(p − 9)((3p − 1) log2 (p) − 1) <5
1
2 + log2 (e) · 4 · h1 (1)< 1.227 · 1030 .

Thus, it suﬃces to observe that the manifestly monotonically increasing
function(x − 9)((3x − 1) log2 (x) − 1)satisfies the inequality > 1.227 · 1030 for x > 9.39 · 1013 .

EXPLICIT ESTIMATES IN INTER-UNIVERSAL TEICHMÜLLER THEORY

53

Finally, we give an application of the ABC inequality of Theorem 5.4 to a
generalized version of Fermat’s Last Theorem, which does not appear to beaccessible via the techniques involving modularity of elliptic curves over Q
and deformations of Galois representations that play a central role in [Wls].
Corollary 5.9. (Application to a generalized version of “Fermat’s
Last Theorem”) Let r, s, t be nonzero integers every two of which are
coprime. Writedef

S = {(X, Y, Z) ∈ Z3 | ||XY Z||C ≥ 2}.
Let l, m, n be positive integers such thatmin{l, m, n} > max{2.453 · 1030 , log2 ||rst||C , 10 + 5 log2 (rad(rst))}.
Then there does not exist any triple (x, y, z) ∈ S of coprime [i.e., the set ofprime numbers which divide x, y, and z is empty] integers that satisfies theequationrxl + sy m + tz n = 0.
def

Proof. Write k = min{l, m, n}. Suppose that there exists a triple (x, y, z) ∈S of coprime integers such that rxl + sy m + tz n = 0. Then we claim thefollowing:
Claim 5.9A: rxl , sy m , and tz n are coprime.
Indeed, suppose that a prime number p divides rxl , sy m , and tz n . Let us
consider the setdef

E = {□ ∈ {x, y, z} | p divides □}.
Then, by applying our assumption that (x, y, z) are coprime (respectively,
every two of (r, s, t) are coprime), we conclude that ♯E ≤ 2 (respectively,♯E ≥ 2), hence that ♯E = 2. Thus, to verify Claim 5.9A, we may assume
without loss of generality that p divides x and y. [In particular, p does notdivide z.] Then observe that pk divides rxl and sy m , hence also tz n . In
particular, since p does not divide z, we conclude that pk divides t. Thus,
we havelog2 ||rst||C ≥ log2 ||t||C ≥ log2 pk ≥ k— a contradiction. This completes the proof of Claim 5.9A.
Now we apply Theorem 5.4 to the present situation, by taking• “(a, b, c)” to be (rxl , sy m , tz n ) [cf. Claim 5.9A];• “ϵ” to be 1.
Then we have||rst||C · ||xyz||kC ≤ ||rstxl y m z n ||C≤ 24 · max{exp(1.7 · 1030 ), (rad(rstxl y m z n ))6 }= 24 · max{exp(1.7 · 1030 ), (rad(rstxyz))6 }.

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MOCHIZUKI, FESENKO, HOSHI, MINAMIDE, AND POROWSKI

On the other hand, since||rst||C · ||xyz||kC ≥ rad(rst) · ||xyz||kC> 24 · (rad(rst))6 · ||xyz||6C≥ 24 · (rad(rstxyz))6[cf. our assumptions that k > 4+(6−1)·log2 (rad(rst))+6 and ||xyz||C ≥ 2],
we obtain that2k ≤ ||rst||C · ||xyz||kC ≤ 24 · exp(1.7 · 1030 ),hence that k ≤ 2.453 · 1030 — a contradiction. This completes the proof ofCorollary 5.9.
□

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Updated versions of [IUTchI-IV] are available at the following webpage:

http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html(Shinichi Mochizuki) Research Institute for Mathematical Sciences, Kyoto
University, Kyoto 606-8502, JAPANE-mail address: motizuki@kurims.kyoto-u.ac.jp(Ivan Fesenko) School of Mathematics, University of Nottingham, Nottingham NG7 2RD, England
E-mail address: ivan.b.fesenko@gmail.com(Yuichiro Hoshi) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, JAPANE-mail address: yuichiro@kurims.kyoto-u.ac.jp(Arata Minamide) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, JAPANE-mail address: minamide@kurims.kyoto-u.ac.jp(Wojciech Porowski) School of Mathematics, University of Nottingham, Nottingham NG7 2RD, England
E-mail address: wo.porowski@gmail.com