Math. J. Okayama Univ. 52 (2010), 1–28

ARITHMETIC ELLIPTIC CURVESIN GENERAL POSITION
Shinichi MOCHIZUKI
Abstract. We combine various well-known techniques from the theoryof heights, the theory of “noncritical Belyi maps”, and classical analytic
number theory to conclude that the “ABC Conjecture”, or, equivalently,
the so-called “Effective Mordell Conjecture”, holds for arbitrary rational
points of the projective line minus three points if and only if it holds forrational points which are in “sufficiently general position” in the sensethat the following properties are satisfied: (a) the rational point underconsideration is bounded away from the three points at infinity at agiven finite set of primes; (b) the Galois action on the l-power torsion
points of the corresponding elliptic curve determines a surjection onto
GL2 (Zl ), for some prime number l which is roughly of the order ofthe sum of the height of the elliptic curve and the logarithm of thediscriminant of the minimal field of definition of the elliptic curve, butdoes not divide the conductor of the elliptic curve, the rational primes
that are absolutely ramified in the minimal field of definition of theelliptic curve, or the local heights [i.e., the orders of the q-parameter atprimes of [bad] multiplicative reduction] of the elliptic curve.

Introduction
In the classical intersection theory of subvarieties, or cycles, on algebraic
varieties, various versions of the “moving lemma” allow one to replace agiven cycle by another cycle which is equivalent, from the point of viewof intersection theory, to the given cycle, but is supported on subvarieties
which are in a “more convenient” position — i.e., typically, a “more general”
position, which is free of inessential, exceptional pathologies — within theambient variety.
In the present paper, after reviewing, partly for the purpose of establishingnotation and terminology, the general theory of heights in §1, we proceed in§2 to apply the theory of “noncritical Belyi maps” of [7], together with atechnique developed by Elkies [cf. [2]; [3]] — which, in fact, may be traced
back to the work of Moret-Bailly and Szpiro [cf. the discussion at the top of[2], p. 106] — for relating Belyi maps to the ABC Conjecture, to show that
the ABC, Effective Mordell, and Vojta Conjectures for arbitrary rational
points of hyperbolic curves over number fields are equivalent to the ABC
Conjecture for rational points over a number field of the projective line
Mathematics Subject Classification. Primary 11G05; Secondary 11G50.
Key words and phrases. elliptic curve, number field, Belyi map, ABC Conjecture,
Mordell Conjecture, Vojta Conjecture.
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SHINICHI MOCHIZUKI

minus three points which lie in some “compactly bounded subset” of the setof all rational points [cf. Theorem 2.1]. Here, “compactly bounded” meansthat at some finite collection S of primes of the number field that includes
the archimedean primes, the rational points lie in some given compact subsetof the set of rational points of the projective line minus three points overthe completion of the number field at the given prime [cf. Example 1.3, (ii)].
Such compactly bounded subsets of the set of rational points are of interest
since the height of a rational point that lies in such a compactly bounded
subset may be computed, up to a bounded discrepancy, by considering the
distance of the rational point from the divisor at infinity solely at the primes
that do not lie over primes of S.
Thus, in summary, we use the noncritical Belyi maps of [7] to “movethe rational points over number fields of interest away from the divisor at
infinity at the primes lying over [typically archimedean!] primes of S”.
This use of Belyi maps is interesting in that it is reminiscent of the use of
Belyi maps in the technique of “Belyi cuspidalization” [cf. [8], §2, §3; [9], §2;[10], §3]. That is to say, in the technique of Belyi cuspidalization, “critical”[i.e., more precisely, “not necessarily noncritical”!] Belyi maps are used to“move rational points closer to the divisor at infinity at nonarchimedean
primes” [cf., especially, the theory of [8], §3].

Theorem 2.1

Belyi cuspidalization

uses noncritical Belyi maps uses critical Belyi maps
moves rational points ofmoves rational points ofinterest away frominterest closer todivisor at infinity
divisor at infinityof interest primarily atof interest atarchimedean primes
nonarchimedean primesof a number fieldof a number field
In particular, the use of Belyi maps in Theorem 2.1 of the present paper
proceeds, so to speak, in the “opposite direction” to their use in the theoryof Belyi cuspidalization.
Finally, in §3, §4, we examine another example of the notion of being“arithmetically in general position”, namely, the phenomenon of “full Galois
actions” — i.e., Galois actions whose images are equal to either SL2 (Zl ) or
GL2 (Zl ) — on the l-power torsion points of an elliptic curve over a number
field for some prime number l. More precisely, we combine the techniques of[11], [4] relating isogenies and heights of elliptic curves over number fields [cf.

ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

3§3] with some classical analytic number theory involving the prime number
theorem [cf. §4] to show [cf. Corollaries 4.3, 4.4] that if one considers elliptic
curves over number fields of bounded degree which are either “degenerating”[i.e., admit at least one prime of potentially multiplicative reduction] or“compactly bounded away from infinity” [cf. Theorem 2.1], then, with finitely
many possible exceptions, one obtains a “full Galois action” for some prime
number l which is roughly of the order of the sum of the height of the elliptic
curve and logarithm of the discriminant of the minimal field of definition of
the elliptic curve and, moreover, is prime to various numbers that are, insome sense, characteristic to the elliptic curve, such as the conductor of theelliptic curve, the rational primes that are absolutely ramified in the minimal
field of definition of the elliptic curve, and the local heights [i.e., the orders of
the q-parameter at primes of multiplicative reduction] of the elliptic curve.
1. Generalities on Heights
In the present §1, we review, partly for the purpose of establishing notation and terminology, various well-known facts concerning arithmetic line
bundles and heights.
Let X be a normal scheme which is proper and flat over Spec(Z) [wheredef

Z denotes the ring of rational integers]. Write XQ = X ×Z Q for the genericfiber of X; X arc for the compact normal complex analytic space determinedby X. Thus, the underlying topological space of X arc may be identified withthe set of complex points X(C), equipped with the topology induced by the
topology of C. Note, moreover, that the complex conjugation automorphism
ιC of C induces a complex conjugation automorphism ιX of X arc which iscompatible with ιC .
Definition 1.1.
(i) We shall refer to as an arithmetic line bundle L = (L, | − |L ) on X any
pair consisting of a line bundle L on X and a hermitian metric | − |L on theline bundle Larc determined by L on X arc that is compatible [in the evident
sense] with the complex conjugation automorphism ιX . If L = (L, | − |L ),M = (M, |−|M ) are arithmetic line bundles on X, then a morphism L → M
is defined to be a morphism of line bundles L → M such that locally on
X arc , sections of L with | −|L ≤ 1 map to sections of M with | −|M ≤ 1. We
define the set Γ(L) of global sections of L = (L, | − |L ) over X to be the setof morphisms OX → L, where we write OX for the trivial line bundle OX
equipped with the trivial hermitian metric. There is an evident notion of
tensor product of arithmetic line bundles on X. The isomorphism classes ofarithmetic line bundles on X, together with the operation of tensor product,

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SHINICHI MOCHIZUKI

thus determine a group APic(X).
(ii) Let φ : Y → X be a morphism of normal, Z-proper, Z-flat schemes.
Then there is an evident notion of pull-back of arithmetic line bundles by φ.
If L is an arithmetic line bundle on X, then we shall denote its pull-back to
Y by the notation φ∗ L or, when there is no fear of confusion, L|Y .
Let F be a number field [i.e., a finite extension of the rational number field
Q], whose ring of integers we denote by OF , and whose set of valuations
we denoteSby V(F ). Thus, V(F ) decomposes as a disjoint union V(F ) =V(F )nonV(F )arc of nonarchimedean and archimedean valuations. If v ∈V(F ), then we shall write Fv for the completion of F at v and | − |v : Fv → R
for the real-valued valuation map determined by v. If v ∈ V(F )non , thenwe shall write ordv (−) : Fv → Z for the order defined by v and qv for the
cardinality of the residue field of Fv .
An arithmetic divisor on F is defined to be a finite formal sum
X
cv · vv∈V(F )

— where cv ∈ Z if v ∈ V(F )non and cv ∈ R if v ∈ V(F )arc [cf. [14], §1.1].
Here, if all of the cv are ≥ 0, then we shall say that the arithmetic divisor
is effective. Thus, the arithmetic divisors on F naturally form a group
ADiv(F ). If f ∈ F , then the assignment
XX
def[Fv : R] · log(|f |v ) · v
ordv (f ) · v −f 7→ ADiv(f ) =v∈V(F )arcv∈V(F )non[where log denotes the natural logarithm] determines an element ∈ ADiv(F ),which we shall refer to as the principal arithmetic divisor associated to f .
Thus, the principal arithmetic divisors determine a subgroup APrc(F ) ⊆ADiv(F ). Moreover, as is well-known, there is a natural isomorphism∼

ADiv(F )/APrc(F ) → APic(Spec(OF ))— cf. [14], Proposition 1.1. In particular, the degree map
degF : ADiv(F )/APrc(F ) → R
defined by sending
V(F )non ∋ v 7→ log(qv );V(F )arc ∋ v 7→ 1

[cf. [14], §1.1] determines a homomorphism APic(Spec(OF )) → R, which
we shall also denote by degF . When there is no fear of confusion, we shallalso regard [by abuse of notation] degF as a map defined on ADiv(F ). NoteARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

5

defthat if we set degF = [F1:Q] · degF , then for any finite extension K of F , itfollows that
degK (L|Spec(OK ) ) = degF (L)for any arithmetic line bundle L on Spec(OF ). In particular, if Q is analgebraic closure of Q, then for any[
x ∈ X(Q) =X(F )Q⊇F, [F :Q]<∞and any arithmetic line bundle M on X, it makes sense to definedefhtM (x) = degF (x∗F M) ∈ R— where xF : Spec(OF ) → X is any morphism that gives rise to x.
Definition 1.2.
(i) We shall refer to the function
htM : X(Q) → R
as the height function associated to the arithmetic line bundle M.
(ii) Fix a subset F ⊆ X(Q). If α, β : F → R are functions, then we shallwrite
α .F β (respectively, α &F β; α ≈F β)
if there exists a [“constant”] C ∈ R such that β(x) − α(x) ≤ C (respectively,
α(x) − β(x) ≤ C; |α(x) − β(x)| ≤ C) for all x ∈ F ; we shall omit thesubscript F when there is no fear of confusion. [Thus, α ≈ β if and only if
α . β and α & β.] The relation “≈” clearly defines an equivalence relationon the set of functions F → R; we shall refer to an equivalence class relative
to this equivalence relation as a(n) [F -]BD-class [i.e., “bounded discrepancy
class”]. The BD-class of α will be denoted by [α]F , or simply [α], when there
is no fear of confusion. Finally, we observe that it makes sense to apply thenotation “&”, “.”, “≈” to BD-classes.
Example 1.3. Various Natural Subsets of the Set of Points. In thenotation of the above discussion, we consider various natural examples ofsubsets “F ⊆ X(Q)” as in Definition 1.2, (ii).
S(i) If d ∈ N {∞}, then we shall denote by
X(Q)≤d ⊆ X(Q)6

SHINICHI MOCHIZUKI

the union of the subsets X(F ) ⊆ X(Q) as F ranges over the number fields
such that [F : Q] ≤ d [so X(Q)≤∞ = X(Q)];def

X(Q)=d = X(Q)≤d \X(Q)≤d−1 ⊆ X(Q)[cf. the discussion in Definition 1.5, (i), below of “minimal fields of definition”]. More generally, if E ⊆ X(Q) is any subset, then we shall write
def T
def T
E ≤d = E X(Q)≤d , E =d = E X(Q)=d . If E ⊆ X(Q) is a subset such thateach E ≤d [where d ranges over the positive integers] is finite, then we shallsay that E is Galois-finite.
(ii) Let us refer to a compact subset of a topological space which is equal
to the closure of its interior as a compact domain. Let V ⊆ V(Q) be a finiteTdefsubset that contains V(Q)arc . For each v ∈ V arc = V V(Q)arc (respecT
deftively, v ∈ V non = V V(Q)non ), let
Kv ⊆ X arc (respectively, Kv ⊆ X(Qv ))be a nonempty ιX -stable compact domain (respectively, a nonempty
Gal(Qv /Qv )-stable subset whose intersection with each X(K) ⊆ X(Qv ),for K ⊆ Qv a finite extension of Qv , is a compact domain in X(K)) suchthat Kv 6= X arc (respectively, Kv 6= X(Qv )). Then let us write
KV ⊆ X(Q)for the subset of points x ∈ X(F ) ⊆ X(Q), where [F : Q] < ∞, such thatfor each v ∈ V arc (respectively, v ∈ V non ), the set of [F : Q] points of X arc(respectively, X(Qv )) determined by x is contained in Kv . We shall refer toa subset of X(Q) obtained [i.e., as a “KV ”] in this fashion as a compactly
bounded subset, to V as the support of the compactly bounded subset, and to
the “Kv ” as the bounding domains of the compactly bounded subset. Notethat by applying well-known approximation results in elementary number
theory, it follows immediately [cf. the definition given above of the term“compact domain”!] that the bounding domains of a compactly bounded
subset of X(Q), hence also the support of a compactly bounded subset of
X(Q), are completely determined by the compactly bounded subset itself.
Proposition 1.4 (Basic Properties of Heights). In the notation of theabove discussion, let L = (L, | − |L ), M = (M, | − |M ) be arithmetic line
def

def

bundles on X. Write LQ = L|XQ , MQ = M|XQ . Then:(i) We have
htL⊗M (x) = htL (x) + htM (x)ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

7

for x ∈ X(Q).
(ii) If some positive tensor power of the line bundle LQ on XQ is generated by global sections [for instance, if the line bundle LQ is ample],thenhtL & 0[i.e., where “&” denotes “&X(Q) ”].
(iii) The BD-class of htL depends only on the isomorphism class of
the line bundle LQ on XQ . In particular, it makes sense to write [htL ] or
[htLQ ] for the BD-class [htL ] of htL .
(iv) Let d be a positive integer, C ∈ R. Suppose further that the line bundle
LQ is ample on XQ . Then the set of points x ∈ X(Q)≤d [cf. Example 1.3,(i)] such that htL (x) ≤ C is finite.
Proof. Assertion (i) follows immediately from the definitions. Next, weconsider assertion (ii). To verify assertion (ii), it suffices to show that ifs ∈ Γ(XQ , LQ ), and we write F ⊆ X(Q) for the subset of points at whichs is nonzero, then htL &F 0. Now observe that if M is an arithmetic line
bundle that arises [by pull-back to X] from an arithmetic line bundle on
Spec(Z), thenhtL⊗M ≈X(Q) htL[cf. assertion (i)]. Moreover, for a suitable choice of M arising from anarithmetic line bundle on Spec(Z), the section s determines a section t of
LQ ⊗ MQ that extends to a section of L ⊗ M over X such that |t|L⊗M ≤ 1
on X arc [where we recall that X arc is compact!]. Thus, by replacing L by
L ⊗ M and s by t, we may assume that s determines a section of L over X
and that |s|L ≤ 1 on X arc . Now the fact that htL (x) ≥ 0 for all x ∈ F is
immediate from the definitions. This completes the proof of assertion (ii).
Next, to verify assertion (iii), it suffices to apply assertions (i), (ii) to the−1arithmetic line bundle L ⊗ M in the situation where LQ ∼= MQ .
Finally, we consider assertion (iv). A proof of this well-known fact may,essentially, be found in [12] [cf. [12], Theorem 2.1]. From the point ofview of the present discussion, an argument may be given as follows: First,
we observe that [some positive tensor power of] the ample line bundle LQ
yields an embedding ǫQ : XQ ⊆ PQ , where P = PnZ is a projective spaceover Z. Next, by blowing up X in an appropriate fashion, we conclude that
there exists a normal, Z-proper, Z-flat scheme Y , together with morphisms
φ : Y → X, ψ : Y → P such that φQ is an isomorphism, and ψQ = ǫQ ◦ φQ .
Thus, to verify assertion (iv) for the pair (X, L), it suffices to verify assertion(iv) for the pair (Y, φ∗ L), which, by assertion (iii), is equivalent to assertion

8

SHINICHI MOCHIZUKI(iv) for the pair (Y, ψ ∗ LP ), where we write LP for the line bundle OP (1)equipped with the standard Fubini-Study metric. But assertion (iv) for thepair (Y, ψ ∗ LP ) follows from assertion (iv) for the pair (P, LP ). Thus, we
may assume that (X, L) = (P, LP ).
Next, observe that since any finite set of points of a fiber of P → Spec(Z)is contained in the complement of a hypersurface, it follows immediately
that, for any positive integer e ≤ d, the product P × . . . × P of e copies of P
[over Z] admits a [necessarily normal, Z-proper, and Z-flat] quotient Q by
the action of the symmetric group on e letters, and that some positive tensor
power of the tensor product of the pull-backs of LP by the various projections
P ×. . .×P → P descends to an arithmetic line bundle LQ = (LQ , |−|LQ ) on
Q such that (LQ )Q is ample on QQ . Since each point xP ∈ P (F ) for F suchthat [F : Q] = e determines [by considering the various Q-conjugates of xP ]a point xQ ∈ Q(Q) [which, in turn, determines xP , up to a finite number ofpossibilities], we thus conclude [by varying e] that it suffices to verify thatthe set of points of y ∈ Q(Q) such that htLQ (y) ≤ C is finite. Moreover,by embedding Q via (LQ )Q into some projective space Z = Pm
Z as in theargument of the preceding paragraph, we thus conclude that it suffices toverify that the set of points of z ∈ Z(Q) such that htLZ (z) ≤ C is finite,where LZ is given by OZ (1) equipped the standard Fubini-Study metric.
Next, let us observe that [as is easily verified] there exists a unique arithdef

metic line bundle LS on S = Spec(Z), up to isomorphism, of degree [i.e.,“degQ ”] C; moreover, any arithmetic line bundle on S of degree ≤ C embeds into LS . In particular, by applying such an embedding, we conclude
that given any point z ∈ Z(Q) = Z(S) such that htLZ (z) ≤ C, the m + 1
standard generating sections of LZ restrict, via the morphism S → Z determined by z, to sections ∈ Γ(LS ). Since [as is easily verified] Γ(LS ) is a finite
set, it thus follows that there is only a finite number of possibilities for the
projective coordinates of z. This completes the proof of assertion (iv). Remark 1.4.1. Observe that it follows immediately from the definitions,
together with Proposition 1.4, (iii), that the theory of “BD-classes of heightfunctions htL (−) on X(Q)” in fact depends only on the scheme XQ . In
particular, this theory may be applied to any normal, projective scheme
Y over Q [i.e., by regarding Y as the “XQ ” determined by some Z-flat,
Z-projective model “X” of Y that arises from a projective embedding of Y ].
Definition 1.5.
(i) Note if x ∈ X(F ) ⊆ X(Q), where [F : Q] < ∞, then by considering

ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

9

the scheme-theoretic image of the corresponding morphism Spec(F ) → X,
one obtains a well-defined minimal field of definition Fmin ⊆ F of x. In
particular, it makes sense to say that F “is a minimal field of definition ofx” [i.e., that F = Fmin ].
(ii) Let E ⊆ Z be an effective Cartier divisor contained in the regular locus
of a normal noetherian scheme Z. Then observe that the closed subscheme
Ered ⊆ Z is also an effective Cartier divisor. We shall say that E is reduced
if E = Ered .
(iii) Let x ∈ X(F ) ⊆ X(Q), where F is a minimal field of definition of x.
Then the different ideal of F determines an effective arithmetic divisorδx ∈ ADiv(F )which is supported in V(F )non . In particular, the assignmentdef

X(Q) ∋ x 7→ log-diffX (x) = degF (δx ) ∈ R
determines a well-defined log-different function log-diffX on X(Q).
def(iv) Fix an effective Cartier divisor D ⊆ X; write UX = X\D. Letx ∈ UX (F ) ⊆ UX (Q), where F is a minimal field of definition of x [regarded
as a point of X(Q)]. Then the morphism Spec(OF ) → X determined by x
[where we recall that X is proper!] allows one to pull-back the divisor D toSpec(OF ) so as to obtain an effective divisor Dx on Spec(OF ). Note that
Dx , (Dx )red may also be regarded as arithmetic divisors ∈ ADiv(F ) that aresupported in V(F )non . We shall refer todeffD
x = (Dx )red ∈ ADiv(F )as the conductor of x. Thus, the assignmentdef

X(Q) ∋ x 7→ log-condD (x) = degF (fDx)∈R
determines a well-defined log-conductor function log-condD on UX (Q) ⊆X(Q).
Remark 1.5.1. In the spirit of Remark 1.4.1, we observe that the log-different
function log-diffX on X(Q) [cf. Definition 1.5, (iii)] depends only on thescheme XQ . On the other hand, although the log-conductor function
log-condD on UX (Q) [cf. Definition 1.5, (iv)] may depend on the pair ofZ-schemes (X, D), one verifies immediately that the BD-class of log-condD
on UX (Q) depends only on the pair of Q-schemes (XQ , DQ ). Indeed, thisfollows immediately from the observation that if (X ′ , D′ ) is another pair[i.e., consisting of an effective Cartier divisor D ′ in a normal, Z-proper,

10

SHINICHI MOCHIZUKI∼

Z-flat scheme X ′ ], then any isomorphism XQ′ → XQ that induces an iso∼′ →DQ extends, for some finite set of prime numbers Σ, tomorphism DQ∼
an isomorphism X ′ ×Z Z[Σ−1 ] → X ×Z Z[Σ−1 ] that induces an isomorphism
def∼

D′ ×Z Z[Σ−1 ] → D ×Z Z[Σ−1 ] — where we write Z[Σ−1 ] = Z[{p−1 }p∈Σ ].
Proposition 1.6 (Conductor Bounded by the Height). Let D ⊆ X bean effective Cartier divisor, L = (L, | − |L ) an arithmetic line bundle
def

def

on X such that L = OX (D). Write U = X\D, htD = htL [cf. Proposition
1.4, (iii)]. Then
log-condD . htD
on U (Q).
Proof. Write s ∈ Γ(X, L) for the section determined by the tautologicalinclusion OX ֒→ OX (D). Then the asserted inequality log-condD . htL
follows, for the contributions at the nonarchimedean primes, from the definition of log-condD [i.e., involving “(−)red ”] in Definition 1.5, (iv), and, for thecontributions at the archimedean primes, from the fact that the continuousfunction |s|L on the compact topological space X arc is bounded.
Proposition 1.7 (Conductors and Log Differents). Let
φ:Y →Zbe a generically finite morphism of normal, Z-proper, Z-flat schemes of dimension two. [Thus, the induced morphism on generic fibers φQ : YQ → ZQ
is a finite [possibly] ramified covering of smooth, proper curves overfinite extensions of Q.] Let e be a positive integer;
D ⊆Y;

E⊆Z

effective, Z-flat Cartier divisors such that the generic fibers DQ , EQ satisfy
the following conditions: (a) DQ , EQ are reduced; (b) DQ = φ−1
Q (EQ )red ;
def

def

(c) if we write UY = Y \D, UZ = Z\E, then φQ restricts to a finite étalemorphism (UY )Q → (UZ )Q ; (d) the ramification index of φQ at each pointof DQ divides e. Then:(i) If we restrict functions on Z(Q), UZ (Q) to UY (Q) via φ, then1· log-condE
log-condE − log-condD . log-diffY − log-diffZ . 1 −e
on UY (Q).
(ii) Suppose that the ramification index of φQ at each point of DQ isARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

11

equal to e. Then the sheaves of differentials “ω(−) ” on YQ , ZQ satisfy the
relation(1.1)

deg(ωYQ ) = deg(ωYQ (DQ )) − deg(OYQ (DQ ))ndeg(OZQ (EQ )) o
= deg(ωZQ (EQ )|YQ ) 1 −e · deg(ωZQ (EQ ))— where we use the notation “deg(−)” to denote the degree of a line bundle
on YQ or ZQ .
Proof. First, we consider assertion (i). We begin by observing that there
exists a finite set of prime numbers Σ such that the restriction of Y → Z todef
the spectrum of Z[Σ−1 ] = Z[{p−1 }p∈Σ ] is a finite tamely ramified morphism of smooth, proper families of curves over finite étale coverings ofSpec(Z[Σ−1 ]). In particular, the “prime-to-Σ portion” of the inequality“log-condE − log-condD = log-diffY − log-diffZ ≤ (1 − e−1 ) · log-condE ”[i.e., with “=” and “≤”, not “.”!] follows immediately from the elementary
theory of differents. Note that the “portion over Σ” of log-condE , log-condDis ≈ 0 [cf. Remark 1.5.1], while [again by the elementary theory of differents]
the “portion over Σ” of log-diffY − log-diffZ is ≥ 0 [i.e., with “≥”, not “&”!].
Thus, to complete the proof of assertion (i), it suffices to show that the“portion over Σ” of the quantity log-diffY − log-diffZ is bounded in UY (Q).
Moreover, by working locally, we reduce immediately to the following elementary claim:Fix a prime number p and a positive integer d. Then there
exists a positive integer n such that for any finite Galois
extension L/K of finite extensions of Qp with [L : K] ≤ d,
the different ideal of L/K contains pn · OL [where we write
OL for the ring of integers of L].
To verify this claim, we reason as follows: By separating the extension
L[ζ]/K, where ζ is a primitive p-th root of unity, into a composite of wildly
ramified and tamely ramified extensions [and observing that if we restrict totamely ramified L/K, then it suffices to take n = 1], we reduce immediately
to the case of wildly ramified L/K such that K contains a primitive p-th rootof unity ζ. Moreover, since Gal(L/K) is a [necessarily solvable!] p-group oforder ≤ d, it suffices to consider the case where [L : K] = p. Since ζ ∈ K,
it follows immediately from elementary Kummer theory that L = K(λ) fordef

some λ ∈ L such that κ = λp ∈ K. Moreover, by multiplying κ by ana , where
element of (K × )p , we may assume that κ is a unit multiple of πK
πK is a uniformizer of K and a is a nonnegative integer < p. In particular,
it follows that κ ∈ OK , but κ ∈/ pp · OK , hence that OL ⊇ λ · OL ⊇ p · OL .

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On the other hand, since in this case, we have an inclusion of OK -algebras
OK [X]/(X p − κ) ֒→ OL , one computes easily that the different ideal of L/K
contains p · λp−1 · OL ⊇ p1+(p−1) · OL = pp · OL . This completes the proofof the claim, and hence of assertion (i). Assertion (ii) follows immediatelyfrom the Riemann-Hurwitz formula for ramified coverings of smooth, proper
curves.

2. Bounds on Heights
In the present §2, we discuss our first main result [cf. Theorem 2.1],
to the effect that the so-called Effective Mordell or ABC Conjectures areequivalent to the ABC Conjecture for rational points [over number fields ofbounded degree] contained in a fixed compactly bounded subset of the set of
all rational points [cf. Example 1.3, (ii)]. The technique used in Corollary
2.1 goes back to work of Elkies [cf. [2]; [3]] and Moret-Bailly and Szpiro[cf. the discussion at the top of [2], p. 106], except that instead of using“arbitrary” Belyi maps, we use noncritical Belyi maps, as discussed in [7].
Theorem 2.1 (Compactly Bounded Subsets and the ABC Conjecture). Let Σ be a finite set of prime numbers. Then in the terminology of§1 [cf., especially, Definitions 1.2, 1.5; Example 1.3; Remarks 1.4.1, 1.5.1],
the following two statements are equivalent:(i) (Effective Mordell/ABC/Vojta Conjecture) Let X be a smooth,proper, geometrically connected curve over a number field; D ⊆ X a reduced
def
divisor; UX = X\D; d a positive integer; ǫ ∈ R>0 a positive real number.
Write ωX for the canonical sheaf on X. Suppose that UX is a hyperbolic
curve — i.e., that the degree of the line bundle ωX (D) is positive. Thenthe inequality of BD-classes of functions
htωX (D) . (1 + ǫ)(log-diffX + log-condD )holds on UX (Q)≤d .
(ii) (ABC Conjecture for Σ-Supported Compactly Bounded Subdef

sets) Let P = P1Q be the projective line over Q; C ⊆ P the divisor
defconsisting of the three points “0”, “1”, and “∞”; UP = P \C; d a positive integer; ǫ ∈ R>0 a positive real number; KV ⊆ UP (Q) a compactly
bounded subset [i.e., regarded as a subset of P (Q) — cf. Example 1.3,(ii)] whose support contains Σ. Write ωP for the canonical sheaf on P .
Then the inequality of BD-classes of functions
htωP (C) . (1 + ǫ)(log-diffP + log-condC )ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

T

holds on KV

13

UP (Q)≤d .

Proof. The fact that (i) =⇒ (ii) is immediate from the definitions. Thus,
it suffices to verify that (ii) =⇒ (i). Let X, D, UX , d, ǫ be as in (i). Now
it follows immediately from the well-known structure of étale fundamentalgroups of hyperbolic curves over algebraically closed fields of characteristic
zero that for any positive integer e, there exists a connected finite étale
Galois covering UY → UX , such that if we write Y for the normalization ofdef
X in UY and E = (D ×X Y )red ⊆ Y , then Y is a hyperbolic curve, and,moreover, Y → X is ramified at each point of E with ramification index
equal to e. Here, we think of e as a fixed number that will be chosen below.
def
Write d′ = d · deg(Y /X). Then I claim that to complete the proof of theimplication “(ii) =⇒ (i)”, it suffices to verify the inequality
htωY . (1 + ǫ′ ) · log-diffY′

on UY (Q)≤d for arbitrary ǫ′ ∈ R>0 . Indeed, suppose that this inequality
is satisfied for some ǫ′ ∈ R>0 such that (1 + ǫ′ )2 ≤ 1 + ǫ. Then by choosing appropriate normal, Z-proper, Z-flat models for Y , X, we may apply
Proposition 1.7, (i), to conclude that log-diffY . log-diffX + log-condD on′
UY (Q)≤d ; moreover, by Proposition 1.7, (ii), it follows that by choosing e
to be sufficiently large, we may assume that deg(ωX (D)|Y ) = deg(ωY (E)) ≤(1 + ǫ′ ) · deg(ωY ). But, by Proposition 1.4, (i), (ii), (iii), this implies that′htωX (D) . (1 + ǫ′ ) · htωY on UY (Q)≤d . Thus, we conclude thathtωX (D) . (1 + ǫ′ ) · htωY . (1 + ǫ′ )2 · log-diffY. (1 + ǫ′ )2 (log-diffX + log-condD ). (1 + ǫ)(log-diffX + log-condD )′on UY (Q)≤d . Since every point of UX (Q)≤d clearly lifts to a point of′
UY (Q)≤d , this completes the verification of the claim.
Thus, in summary, to complete the proof of Theorem 2.1, it suffices toverify that (ii) implies (i) for data X, D, UX , d, ǫ as in (i) such that D = ∅.
To this end, let us suppose that the inequality
htωX . (1 + ǫ) · log-diffX
is false on X(Q)=d . Let V ⊆ V(Q) be a finite subset that contains V(Q)arcand [the subset of V (Q)non determined by] Σ. Then it follows immediatelyfrom the compactness of the set of rational points of X over any finite extension of Qv for v ∈ V that there exists a subset Ξ ⊆ X(Q)=d , together witha(n) [unordered] d-tuple of points Ξv of X(Qv ) for each v ∈ V , such thatthe inequality htωX . (1 + ǫ) · log-diffX is false on Ξ, and, moreover, the14

SHINICHI MOCHIZUKI[unordered] d-tuples of Q-conjugates of points ∈ Ξ converge, as [unordered]d-tuples of points of X(Qv ), to Ξv . Moreover, by the main result of [7] [cf.
[7], Theorem 2.5], there exists a “noncritical Belyi map”
φ:X →P
which is unramified over UP , and, moreover, “noncritical” at the pointsof each Ξv [i.e., maps the points of each Ξv into UP (Qv )]. In particular,[after possibly eliminating finitely many elements from Ξ] it follows that
there exists a compactly bounded subset KV ⊆ UP (Q) [i.e., whose bounding
domains are the unions of Galois-conjugates of images via φ of sufficiently
small compact neighborhoods of the points of Ξv in X(Qv ), for v ∈ V ] suchthat φ(Ξ) ⊆ KV .
def
Now set E = φ−1 (C)red ⊆ X [so φ∗ ωP (C) ∼= ωX (E)]. Let ǫ′ ∈ R>0 besuch that the inequality
1 + ǫ′ ≤ (1 + ǫ) · 1 − ǫ′ · deg(E)/deg(ωX )is satisfied. Then by (ii), it follows that htωP (C) . (1 + ǫ′ )(log-diffP +log-condC ) on φ(Ξ). On the other hand, by choosing appropriate normal,
Z-proper, Z-flat models for P , X, we may apply Proposition 1.7, (i) [where
we take “e” to be 1], to conclude that log-diffP + log-condC . log-diffX +log-condE on Ξ. Also, we recall from Proposition 1.6 that log-condE .
def

htE = htOX (E) on Ξ. Thus, by Proposition 1.4, (i), (iii), we compute to findthat
htωX ≈ htωX (E) − htE ≈ htωP (C) − htE. (1 + ǫ′ )(log-diffP + log-condC ) − htE. (1 + ǫ′ )(log-diffX + log-condE ) − htE. (1 + ǫ′ )(log-diffX + htE ) − htE ≈ (1 + ǫ′ ) · log-diffX + ǫ′ · htE≈ (1 + ǫ′ ) · log-diffX + ǫ′ · (deg(E)/deg(ωX )) · htωX
on Ξ, i.e., that htωX . (1 + ǫ) · log-diffX on Ξ — in contradiction to ourhypothesis on Ξ. This completes the proof of Theorem 2.1.

3. Full Special Linear Galois Actions on Torsion Points
In the present §3, we give various conditions on a prime l that ensurethat the image of the Galois representation on the l-power torsion pointsof an elliptic curve over a number field is “rather large” [cf. Theorem 3.8].
In particular, we show that if one considers elliptic curves over number
fields of bounded degree that have at least one [nonarchimedean] prime ofpotentially [bad] multiplicative reduction, then, if one excludes finitely many

ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

15

exceptional elliptic curves, it holds that for any prime number l of the orderof the height of the elliptic curve, the image of the associated Galois representation in GL2 (Zl ) contains SL2 (Zl ). Here, the condition of having atleast one prime of potentially multiplicative reduction holds automatically,
with finitely many exceptions, if one restricts oneself to elliptic curves arisingfrom points of the moduli stack of elliptic curves that are contained in somecompactly bounded subset of the set of all points. In particular, for such a
prime number l, the elliptic curve will not have any rational torsion pointsof order l. Thus, this result may be regarded as a sort of “poor man’s uniform boundedness conjecture” — now Merel’s theorem [cf. [6]] – although,in fact, it is not strictly implied by Merel’s theorem. Alternatively, it may
be regarded as an effective version of a theorem of Serre [cf. [11], Chapter
IV, Theorem 3.2]. The proof is similar to that of Faltings’ proof of the Tate
Conjecture [cf. [4]], only technically much simpler. That is to say, the maintechnique is essentially the standard one, going back to Tate, for proving“Tate conjecture-type results”.
We begin by reviewing various well-known facts concerning the structureof SL2 (Fl ), SL2 (Zl ), for l a prime number ≥ 5.
Lemma 3.1 (The Structure of SL2 ). Let l ≥ 5 be a prime number.
Then:

def
(i) Let G ⊆ SL2 (Fl ) be the subgroup generated by the matrices α = 10 11 ,def
β = 11 01 . Then G = SL2 (Fl ).
(ii) The finite group SL2 (Fl ) has no nontrivial abelian quotients.

def(iii) Let H ⊆ GL2 (Fl ) be a subgroup that contains the matrix α = 10 11 ,as well as at least one matrix that is not upper triangular. Then SL2 (Fl ) ⊆H.
(iv) Let J ⊆ GL2 (Zl ) be a closed subgroup whose image HJ in GL2 (Fl )
defcontains the matrix α = 10 11 , as well as a matrix which is not upper
triangular. Then SL2 (Zl ) ⊆ J.
Proof. First, we consider assertion (i). Note that if µ, λ ∈ Fl , then β µ · αλ[where we observe that this expression makes sense since both αl and β l are


def
λ
. Thus, if
equal to the identity matrix] takes the vector v = 01 to µ·λ+1
def 0 1we let γ = 1 0 , then for any λ ∈ F×l , there exists a gλ ∈ G such thatλλ · γ · v = 0 = gλ · v. In particular, we have λ · g1 · v = λ · γ · v = gλ · v, so
λ· v ∈ G· v. Thus, in summary, we have proven that (Fl ×Fl ) −{ 00 } ⊆ G· v.
Now let us prove that an arbitrary element δ ∈ SL2 (Fl ) is contained in G.

16

SHINICHI MOCHIZUKI

By the conclusion of the above argument, we may assume that δ · v = v. But
this implies that δ is a lower triangular matrix all of whose diagonal entries
are equal to 1. Thus, δ is a power of β, hence contained in G, as desired.
This completes the proof of assertion (i).
2
Next, we consider assertion (ii). Let λ ∈ F×l be such that λ 6= 1. [Notedefthat such a λ exists since l ≥ 5.] Let ǫ = λ0 λ0−1 . Then ǫ · α · ǫ−1 · α−1 =2
αλ −1 . Thus, α (and, similarly, β) is contained in the commutator subgroupof SL2 (Fl ), so assertion (ii) follows from assertion (i). This completes theproof of assertion (ii).
Next, we consider assertion (iii). Note that α generates an l-Sylow subgroup S of GL2 (Fl ), and that the number of l-Sylow subgroups of GL2 (Fl )is precisely l + 1. Since the normalizer of S in GL2 (Fl ) is the set of upper
triangular matrices, and we have assumed that H contains at least one nonupper triangular matrix, it follows that the number nH of l-Sylow subgroupsof H is ≥ 2. On the other hand, by the general theory of Sylow subgroups,
it follows that nH is congruent to 1 modulo l. Since 2 ≤ nH ≤ l + 1, we
thus obtain that nH = l + 1. In particular, in the notation of assertion (i),we conclude that α, β ∈ H. Thus, by assertion (i), we have SL2 (Fl ) ⊆ H,
as desired. This completes the proof of assertion (iii).
Finally, we consider assertion (iv). By assertion (iii), we have that SL2 (Fl )⊆ HJ . Let J ′ ⊆ SL2 (Zl ) be the closure of the commutator subgroup of J.
Thus, by assertion (ii), J ′ surjects onto SL2 (Fl ). Now by [11], ChapterIV, §3.4, Lemma 3, this implies that SL2 (Zl ) = J ′ ⊆ J, as desired. This
completes the proof of assertion (iv).

Next, we consider the local theory at nonarchimedean primes. Let K bea finite extension of Qp [where p is a prime number] with residue field k,
maximal ideal mK ⊆ OK , and valuation map vK : K × → Z [which we
defnormalize so that vK is surjective]; K an algebraic closure of K; GK =Gal(K/K); E → Spec(OK ) a one-dimensional semi-abelian scheme overOK such that the generic fiber EK of E is proper, while the special fiber Ekof E is isomorphic to (Gm )k , the multiplicative group over k.
Let l be a prime number [possibly equal to p]; writedef

Ml (E) = Hom(Z/l · Z, E(K))
for the “mod l” Tate module of EK . Thus, Ml (E) is [noncanonically!] isomorphic as an Fl -module to Fl × Fl , and, moreover, is equipped with acontinuous action by GK [induced by the natural action of GK on K]. Also,
it is well-known [cf., e.g., [5], Chapter III, Corollary 7.3] that Ml (E) fits into

ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

17

a natural exact sequence of GK -modules
0 → Fl (1) → Ml (E) → Fl → 0— where the “(1)” is a Tate twist, and “Fl ” is equipped with the trivial Galois action. Moreover, the extension class associated to this exact sequenceis precisely that obtained by extracting an l-th root of the Tate parameterqE ∈ mK . The Tate parameter is an element of mK that is naturally associated to E and has the property that the subscheme OK /(qE ) is equal to thepull-back via the classifying morphism Spec(OK ) → Mell — where we write
Mell for the moduli stack of one-dimensional semi-abelian varieties over Z— of the divisor at infinity ∞M ⊆ Mell . Thus, the above exact sequence
splits if and only if qE has an l-th root in K. Note that in order for this tohappen, it is necessary that vK (qE ) be divisible by l. In particular, we havethe following well-known result.
Lemma 3.2 (Local Rank One Subgroups of l-Torsion).
(i) LetN ⊆ Ml (E)be a one-dimensional Fl -subspace which is stabilized by GK . Then eithervK (qE ) ∈ l · Z, or N is equal to the submodule Fl (1) ⊆ Ml (E) of the above
exact sequence.
(ii) The submodule Fl (1) ⊆ Ml (E) of the above exact sequence defines
a finite, flat subgroup scheme µl ⊆ E over OK , whose quotient we denotedefby E ′ = E/µl . [Thus, E ′ → Spec(OK ) is a one-dimensional semi-abelianscheme over OK whose generic fiber is proper, and whose special fiber isisomorphic to (Gm )k .] The Tate parameter qE ′ of E ′ satisfies the relationl ; in particular, we haveqE ′ = qE
deg∞ (E ′ ) = l · deg∞ (E)def— where we write deg∞ (E) = log(#(OK /(qE ))) ∈ R.
Definition 3.3. We shall refer to the positive integer vK (qE ) ∈ Z>0 as the
local height of E [or EK ].
Remark 3.3.1. Note that even if EK only has potentially multiplicative reduction, one may define the local height of EK as the element ∈ Q computedby dividing the local height of EK ×K L for some finite extension L of K overwhich EK has multiplicative reduction by the ramification index of L/K [soone verifies immediately that this definition is independent of the choice of
L].

18

SHINICHI MOCHIZUKI

Next, let F ⊆ Q be a number field, E → Spec(OF ) a one-dimensionalsemi-abelian variety whose generic fiber EF is proper. Also, for simplicity,
we assume that F is totally imaginary. Thus, EF is an elliptic curve over F .
Let us write ωE for the finite, flat OF -module of rank one consisting of theinvariant differentials on E. If v ∈ V(F )arc , then we get a natural metric ondef(ωE )v = (ωE ) ⊗F Fv by integration: if α ∈ (ωE )v , then
Z
2 def
α∧α
|α| =
Ev

def

— where Ev = E ×F Fv , and α is the complex conjugate of α. Thus, by
equipping ωE with this metric at the archimedean places of F , we obtain anarithmetic line bundle ω E on Spec(OF ). Let us write
def

htFalt (E) = degF (ω E ) ∈ R
for the so-called Faltings height of the elliptic curve E.
Next, let us observe that E → Spec(OF ) defines a classifying morphism
φ : Spec(OF ) → Mell
— where we write Mell for the moduli stack of one-dimensional semi-abelianvarieties over Z. As is well-known, this stack has a “divisor at infinity”∞M ⊆ Mell , whose complement Mell ⊆ Mell is the moduli stack of elliptic
curves over Z. Set
def
deg∞ (E) = degF (∞E ) ∈ R
def— where we write ∞E = φ−1 (∞M ) ⊆ Spec(OF ).
Although the algebraic stack Mell is not a scheme, it is nevertheless normal, Z-proper, and Z-flat. Thus, one verifies immediately that the theory of§1 extends immediately to the case of Mell ; in particular, one may consider
functions on subsets of Mell (Q), BD-classes of such functions, and heightfunctions on Mell (Q) associated to arithmetic line bundles on Mell . Writeht∞
for the BD-class of height functions determined by the line bundle
OMell (∞M ). Also, we observe that if we write [E] ∈ Mell (Q) for the isodef

morphism class determined by E, then we may regard [E] 7→ htFalt ([E]) =def

htFalt (E), [E] 7→ deg∞ ([E]) = deg∞ (E) as functions on Mell (Q). Now wehave the following well-known result.
Proposition 3.4 (Faltings Heights and the Divisor at Infinity). For
any ǫ ∈ R>0 , we havedeg∞ . ht∞ . 12(1 + ǫ) · htFalt . (1 + ǫ) · ht∞

ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

19

on Mell (Q). In particular, if C ∈ R, then the set of points [E] ∈ Mell (Q)≤dsuch that htFalt ([E]) ≤ C is finite.
Proof. The first “.” follows immediately from the definitions [cf. the proofof Proposition 1.6]. The remaining “.’s” follows from [13], Proposition 2.1[cf. also the discussion in the proof of [5], Chapter V, Proposition 4.5, of thelogarithmic singularities at infinity of the metric defined on “ω E ”, as one
allows “E” to vary in Mell (Q); the proof of [12], Proposition 8.2]. Finally,
the finiteness assertion follows immediately from the inequalities alreadyshown, together with Proposition 1.4, (iv).
Lemma 3.5 (Global Rank One Subgroups of l-Torsion). Let ǫ ∈ R>0 ;l a prime number; HF ⊆ EF a subgroup scheme such that HF ×F Q is
isomorphic to the constant group scheme determined by Z/l · Z. We shalldef

call such a subgroup scheme HF l-cyclic. Write (EH )F = EF /HF . [Thus,(EH )F is isogenous to EF , hence has semi-stable reduction at all thefinite primes of F and extends to a one-dimensional semi-abelian scheme
EH → Spec(OF ).] Suppose further that l is prime to the local heights of
E at all of its primes of [bad] multiplicative reduction [a condition that is
satisfied, for instance, if l is > these local heights]. Then we have1l · deg∞ (E) ≤ htFalt (E) + 2 log(l) + C
12(1 + ǫ)
for some constant C ∈ R which [may depend on ǫ but] is independent of
E, F , HF , and l.
Proof. Note that the assumption on l implies, by Lemma 3.2, (i), that at all
the primes of multiplicative reduction, HF corresponds to the subspace Fl (1)of Lemma 3.2, (i). Thus, at primes of multiplicative reduction, EH may be
identified with the elliptic curve “E ′ ” of Lemma 3.2, (ii). In particular, it
follows that deg∞ (EH ) = l · deg∞ (E). On the other hand, the degree l covering morphism EF → (EH )F extends [cf., e.g., [5], Chapter I, Proposition
2.7] to a morphism E → EH . Thus, we have a natural inclusion ωEH ⊆ ωE
whose cokernel is annihilated by l. Moreover, since integrating a (1, 1)-form
over Ev differs from integrating over (EH )v by a factor of l, we conclude
that htFalt (EH ) = deg(ω EH ) ≤ deg(ω E ) + 2 · log(l) = htFalt (E) + 2 · log(l).
Thus, Lemma 3.5 follows from Proposition 3.4.

Before continuing, we observe the following result.

20

SHINICHI MOCHIZUKI

Lemma 3.6 (An Elementary Estimate). Let ǫ ∈ R>0 . Then there exists
a constant C0 ∈ R>0 such that for all x, y ∈ R such that y ≥ 1 andx ≥ C0 y 1+ǫ , it holds that x ≥ y · log(x).
Proof. This follows immediately from the well-known elementary fact thatx1/(1+ǫ) · log(x)/x = log(x) · x−ǫ/(1+ǫ) → 0 as x → ∞.
Lemma 3.7 (Finite Exceptional Sets). Let KV ⊆ Mell (Q) be a compactly bounded subset, ǫ ∈ R>0 . Then there exists a constant C ∈ R>0
and a Galois-finite [cf. Example 1.3, (i)] subset Exc ⊆ Mell (Q) that satisfy
the following property: Let EL be an elliptic curve over a number field L
defwith semi-stable reduction at all the finite primes of L; d = [L : Q];[EL ] ∈ Mell (Q) the point determined by EL ; l a prime number. Consider
the following two conditions on the above data:(a) l ≥ 100d · (htFalt ([EL ]) + C · dǫ ), and EL has at least one prime of[bad] multiplicative reduction;(b) [EL ] ∈ KV , and l is prime to the local heights of EL at all of its
primes of multiplicative reduction.
Then if (a) is satisfied, then l is > the local heights of EL at all the primesof multiplicative reduction. If (b) is satisfied, and [EL ] 6∈ Exc, then EL has atleast one prime of multiplicative reduction. If either (a) or (b) is satisfied,and, moreover, EL admits an l-cyclic subgroup scheme HL ⊆ EL , then[EL ] ∈ Exc.
Proof. First, observe that if v is any local height of EL , then d·deg∞ ([EL ]) ≥v · log(2). Next, let us observe that by Proposition 3.4, for an appropriate
choice of C, we may assume that htFalt ([EL ]) + C · dǫ ≥ htFalt ([EL ]) + C ≥1
14 · deg∞ ([EL ]) [i.e., where we take “12(1 + ǫ)” to be 14]. Thus, condition(a) implies that 100 · log(2) 100d· deg∞ ([EL ]) ≥
·v >v
l ≥
14
14
— i.e., that l is > the local heights of EL at all the primes of multiplicative
reduction. On the other hand, if condition (b) is satisfied, and, moreover EL
has no primes of multiplicative reduction [so deg∞ ([EL ]) = 0], then the factthat [EL ] ∈ KV implies that ht∞ ([EL ]) is bounded [independently of L, EL ,l], so, by Proposition 1.4, (iv), [EL ]Tbelongs to some [fixed] finite exceptional
set Excd [which we think of as Exc Mell (Q)≤d ], as desired.
For the remainder of the present proof, let us assume that either (a) or(b) is satisfied [so l is prime to the local heights of EL ], and that EL admits

ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

21

an l-cyclic subgroup scheme HL ⊆ EL . Thus, by Lemma 3.5, we conclude
that
l
(†)
· deg∞ ([EL ]) ≤ htFalt ([EL ]) + 2 · log(l) + C ′14
for some constant C ′ ∈ R [i.e., “C”] as in Lemma 3.5.
Now suppose that condition (a) is satisfied. Let C0 be as in Lemma 3.6.
Then let us observe that if we choose C sufficiently large [cf. Proposition
56d 1+ǫ
3.4] that the inequality of condition (a) implies that l ≥ C0 ( log(2)) , then56d], we conclude
by Lemma 3.6 [where we take “x” to be l, and “y” to be log(2)that l ≥ 56d·log(l)log(2) , hence thatll · log(2)· deg∞ ([EL ]) ≤ htFalt ([EL ]) +
+ C′
(‡)
14
28d
[i.e., by substituting into (†)]. Now since EL has at least one prime ofbad reduction, it follows that log(2) ≤ d · deg∞ ([EL ]). Thus, substituting
into (‡), we obtain that l·log(2)≤ htFalt ([EL ]) + C ′ . On the other hand,28dlog(2)
Falt
2([EL ]), so, by substituting,28 ≥ 100 , and, by assumption, l ≥ 100d · ht
Falt
Falt
we obtain that 2 ht ([EL ]) ≤ ht ([EL ]) + C ′ , i.e., that htFalt ([EL ]) ≤ C ′ .
But this implies, by Proposition 3.4, that [EL ]Tbelongs to some [fixed] finite
exceptional set Excd [which we think of as Exc Mell (Q)≤d ], as desired.
Now suppose that condition (b) is satisfied. Recall that by definition,ht∞ ([EL ]) may be computed as the sum of deg∞ ([EL ]) and an archimedean
term which, in the present situation, is bounded, since [EL ] ∈ KV . In particular, by modifying C ′ — in a fashion that depends on KV ! — and applying Proposition 3.4, we may assume that deg∞ ([EL ]) ≥ ht∞ ([EL ]) − C ′ ≥7 · htFalt ([EL ]) − 2C ′ . Thus, by substituting into (†) [and perhaps modifying
C ′ again], we obtain thatll
· htFalt ([EL ]) ≤· deg∞ ([EL ]) + l · C ′ ≤ htFalt ([EL ]) + 3l · C ′214— i.e., [since we may assume that C was chosen so that l ≥ 5] that l ·htFalt ([EL ]) ≤ 2(l − 2)htFalt ([EL ]) ≤ 12l · C ′ , so htFalt ([EL ]) ≤ 12 · C ′ . Thus,
again we conclude by Proposition 3.4 that [EL ]Tbelongs to some [fixed] finite
exceptional set Excd [which we think of as Exc Mell (Q)≤d ], as desired. 
Theorem 3.8 (Full Special Linear Galois Actions). Let Q be an algebraic closure of Q, KV ⊆ Mell (Q) a compactly bounded subset [cf.
Example 1.3, (ii)], ǫ ∈ R>0 . Then there exist a constant C ∈ R>0 and a
Galois-finite [cf. Example 1.3, (i)] subset Exc ⊆ Mell (Q) that satisfy thefollowing property:

22

SHINICHI MOCHIZUKILet EL be an elliptic curve over a number field L ⊆ Q such that the isodef
morphism class [EL ] ∈ Mell (Q) of EL does not belong to Exc; d = [L : Q];l a prime number such that [at least] one of the following two conditions
is satisfied:(a) l ≥ 23040 · 100d · (htFalt ([EL ]) + C · dǫ ), and EL has at least one
prime of potentially multiplicative reduction;(b) [EL ] ∈ KV , and l is prime to the local heights of EL at all of its
primes of potentially multiplicative reduction [cf. Remark 3.3.1], as
well as to the number 2 · 3 · 5 = 30.
Then the image of the Galois representation Gal(Q/L) → GL2 (Zl ) associated to EL contains SL2 (Zl ).
Proof. First, let us observe that for an EL as in the statement of Theodef
rem 3.8, there exists a Galois extension L′ of L of degree that divides d0 =(32 −1)(32 −3)(52 −1)(52 −5) = 23040 [i.e., the order of GL2 (F3 )×GL2 (F5 )],so as to render the 3- and 5-torsion points of EL rational over L′ [which has
the effect of eliminating automorphisms of elliptic curves in all characterdef
istics], we may assume that EL′ = EL ×L L′ has semi-stable reduction at
all of the finite primes of L. [Here, we note that passing to such a Galois
extension of L only affects the prime decomposition of the local heights via
the primes that divide d0 , of which there are only finitely many, namely, 2,
3, and 5.] Next, let us observe that for a suitable choice of C ∈ R>0 andExc ⊆ Mell (Q), if EL and l satisfy condition (a) (respectively, (b)) of thestatement of Theorem 3.8, then EL′ and l satisfy condition (a) (respectively,(b)) of Lemma 3.7 [perhaps for a different “C”]. Thus, [cf. the portion ofLemma 3.7 that asserts that “l is > the local heights”] for EL , l as in thestatement of Theorem 3.8, the local height of EL′ at a finite prime of L′ ofmultiplicative reduction is not divisible by l. Note, moreover, that at least
one such finite prime exists [cf. the content of condition (a); Lemma 3.7 inthe case of condition (b)]. Thus, it follows from the discussion of the local
theory preceding Lemma 3.2 that the image of Galois in GL2 (Fl ) contains
1 1
the element “α” of Lemma 3.1, (iv), i.e., 0 1 . On the other hand, by the
portion of Lemma 3.7 concerning l-cyclic subgroup schemes, it follows thatthe image of Galois in GL2 (Fl ) contains at least one matrix which is not
upper triangular. Thus, we conclude from Lemma 3.1, (iv), that the imageof Galois in GL2 (Zl ) contains SL2 (Zl ), as desired.
ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

23

4. Primes of Prescribed Size
In the present §4, we combine the full Galois action result given in Theorem 3.8 with some classical analytic number theory to give versions of Theorem 3.8 [cf. Corollaries 4.3, 4.4] that assert the existence of bounded prime
numbers l satisfying various properties, such as the property of being distinct from various prime numbers that are, in some sense, “characteristic”
to the elliptic curve under consideration.
We begin by reviewing the following result in classical analytic number
theory.
Lemma 4.1 (The Existence of Primes of Prescribed Size). Write
R′>0 ⊆ R>0 for the complement in R>0 of the set of prime numbers;X
def
θ(x) =log(p)
p<x— where the sum is over prime numbers p < x — for x ∈ R′>0 . Let M bea positive integer; ǫ, xǫ , Cǫ ∈ R>0 such that 0 < ǫ < 41 , ǫ · xǫ > Cǫ , and,moreover, we have:(i) 45 · x + Cǫ > θ(x), for all x ∈ R′>0 ; θ(x) > (1 − ǫ)x, for all x ∈ R′>0
such that x ≥ xǫ ;(ii) M · log(x) ≤ ǫ · x, for all x ∈ R>0 such that x ≥ xǫ .
Also, let us write
def

xA =

X

log(p)

p∈A

for any finite set of prime numbers A. Then for any nonnegative h ∈ R
and any finite set of prime numbers A such that xA > xǫ , there exist M
distinct prime numbers p1 , . . . , pM such that pj 6∈ A, and h ≤ pj ≤(1 + 6ǫ) · xA + 8h, for j = 1, . . . , M .
def

def

Proof. Indeed, write δ = 6ǫ, yA = (1 + δ) · xA + 8h, and suppose thatthe conclusion of Lemma 4.1 is false. Then it follows that there exists
an offending finite set of primes A such that xA > xǫ , and, moreover, allprime numbers p such that h ≤ p ≤ yA belong — with M − 1 possible
exceptions — to A. But then it follows from the definitions that xA ≥−(M −1) · log(yA ) −θ((1+δ)h) +θ(yA) ≥ −M · log(yA ) −θ((1+δ)h) +θ(yA).
Thus, we compute:xA ≥ −M · log(yA ) − θ((1 + δ)h) + θ(yA )≥ −ǫ · yA − 5(1 + δ)h/4 − Cǫ + (1 − ǫ) · yA

24

SHINICHI MOCHIZUKI= (1 − 2ǫ) · yA − 5(1 + δ)h/4 − Cǫ
= (1 + δ − 2ǫ − 2δ · ǫ) · xA + 8(1 − 2ǫ)h − 5(1 + δ)h/4 − Cǫ
≥ (1 + δ/2 − 2ǫ) · xA + 4h − 5(1 + δ)h/4 − Cǫ≥ xA + (ǫ · xA − Cǫ ) + (4h − 5(1 + δ)h/4)— a contradiction, since ǫ · xA > ǫ · xǫ > Cǫ , 5(1 + δ)/4 < 4. Note that
here we may assume without loss of generality that yA , (1 + δ)h ∈ R′>0 , forinstance by replacing δ, h by real numbers slightly greater than the given δ,h — which does not affect the above argument in any substantive way. 
Remark 4.1.1. The issue of finding ǫ, xǫ , Cǫ which satisfy condition (ii) of
Lemma 4.1 is entirely elementary. On the other hand, with regard to condition (i), the fact that θ(x)/x → 1 as x → ∞ is a well-known consequenceof the prime number theorem of classical analytic number theory — cf., e.g.,[1], p. 76.
Lemma 4.2 (Some Elementary Estimates). Let n be a positive integer;
p1 , . . . , pn prime numbers; h1 , . . . , hn positive integers. Then we havePnlog(pj ) ≤ h
j=1P
Pn
n
j=1 log(hj ) ≤j=1 log(hj + 1) ≤ 3h/2
def Pnj=1— where h =hj · log(pj ).

Proof. Indeed, the first and second inequalities are immediate; the third
inequality follows from the easily verified fact that log(H + 1) ≤ (3H/2) ·log(2) for any positive integer H.
Corollary 4.3 (Full Galois Actions for Degenerating Elliptic Curves).
Let Q an algebraic closure of Q; ǫ ∈ R>0 . Then there exists a constantC ∈ R>0 and a Galois-finite [cf. Example 1.3, (i)] subset Exc ⊆ Mell (Q)which satisfy the following property:
Let EL be an elliptic curve over a number field L ⊆ Q, where L is aminimal field of definition of the point [EL ] ∈ Mell (Q), and [EL ] 6∈ Exc;
S a finite set of prime numbers. Suppose that EL has at least one
def
prime of potentially multiplicative reduction. Write d = [L : Q];
def P
xS =
p∈S log(p). Then there exist prime numbers l◦ , l• 6∈ S which satisfy the following conditions:

ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

25(a) l◦ , l• are prime to the primes of potentially multiplicative reduction, as well as to the local heights, of EL . Moreover, l• isprime to the primes of Q that ramify in L, as well as to the ramification indices of primes of Q in L.
(b) The image of the Galois representation Gal(Q/L) → GL2 (Zl◦ )associated to EL contains SL2 (Zl◦ ). The Galois representation
Gal(Q/L) → GL2 (Zl• ) associated to EL is surjective.
(c) The inequalitiesl◦ ≤ 23040 · 900d · htFalt ([EL ]) + 2xS + C · d1+ǫl• ≤ 23040 · 900d · htFalt ([EL ]) + 6d · log-diffMell ([EL ])) + 2xS + C · d1+ǫhold.
Proof. First, let us observe that if EL is as in the statement of Corollary
4.3, and l is any prime number that is unramified in L, then the image ofthe Galois representation Gal(Q/L) → GL2 (Zl ) associated to EL contains
SL2 (Zl ) if and only if the Galois representation Gal(Q/L) → GL2 (Zl ) issurjective. [Indeed, this follows immediately from the well-known fact thatthe field extension Q(ζl∞ )/Q obtained by adjoining the l-th power roots ofunity to Q is totally ramified over the prime l, hence linearly disjoint fromthe extension L/Q.] Thus, by applying Theorem 3.8 — relative to condition(a) of Theorem 3.8 — we conclude that to complete the proof of Corollary
4.3, it suffices to show the existence of prime numbers l◦ , l• that satisfy theconditions (a), (c), and, moreover, are ≥ 23040 · 100d · (htFalt ([EL ]) + C ′ · dǫ ),where C ′ ∈ R>0 is a constant [i.e., the “C” of Theorem 3.8].
Now let us write S◦ for the union of S, the primes of Q that lie underprimes of potentially multiplicative reduction of EL , and the primes thatappear in the prime decomposition of the local heights of EL ; write S• forthe union of S◦ , the primes of Q that ramify in L, and the primes that
divide the ramification indices of primes of Q in L. Thus, [in the notationof Lemma 4.1] we conclude from Lemma 4.2 and Proposition 3.4 thatxS◦ ≤ xS + (1 + 3/2) · 23040d · deg∞ ([EL ])≤ xS + 3 · 12 · 23040d · htFalt ([EL ]) + d · C ′′— where we take the “1 + ǫ” of Proposition 3.4 to be 6/5; C ′′ arises fromthe constant implicit in the inequalities of BD-classes in Proposition 3.4; the“h” of Lemma 4.2 corresponds to 23040d · deg∞ ([EL ]) [cf. the meaning of“d0 = 23040” in the proof of Theorem 3.8]. In a similar vein, since [as is

26

SHINICHI MOCHIZUKI

easily verified, by considering the trace of an extension of number fields] theprimes appearing in the arithmetic divisor that gives rise to “log-diffMell ” [cf.
Definition 1.5, (iii)] appear with multiplicity ≥ one less than the ramification
indices of L/Q, we conclude that xS• ≤ xS + 3 · 12 · 23040d · htFalt ([EL ]) + 3d ·log-diffMell ([EL ])) + d · C ′′ . Now we apply Lemma 4.1 by taking “M ” to be1, “1 + 6ǫ” to be 2, “h” to be 23040 · 100d · (htFalt ([EL ]) + C ′ · dǫ ), and “A” tobe S◦ or S• [and applying the estimate 2 · 3 · 12 + 8 · 100 ≤ 100 + 800 = 900];
here, we observe that by enlarging S [and possibly increasing the “C” ofcondition (c)], we may always assume that xǫ ≤ xS (≤ xS◦ ≤ xS◦ ). Thus,
the existence of l◦ , l• as desired follows immediately from Lemma 4.1 [cf.
also Remark 4.1.1].
Corollary 4.4 (Full Galois Actions for Compactly Bounded Subsets). Let Q an algebraic closure of Q; KV ⊆ Mell (Q) a compactly
bounded subset [cf. Example 1.3, (ii)]. Then there exists a constantC ∈ R>0 and a Galois-finite [cf. Example 1.3, (i)] subset Exc ⊆ Mell (Q)which satisfy the following property:
Let EL be an elliptic curve over a number field L ⊆ Q, where L isa minimal field of definition of the point [EL ] ∈ Mell (Q), [EL ] ∈ KV ,defand [EL ] 6∈ Exc; S a finite set of prime numbers. Write d = [L : Q];
def P
xS = p∈S log(p). Then there exist prime numbers l◦ , l• 6∈ S which satisfy
the following conditions:(a) l◦ , l• are prime to the primes of potentially multiplicative reduction, as well as to the local heights, of EL . Moreover, l• isprime to the primes of Q that ramify in L, as well as to the ramification indices of primes of Q in L.
(b) The image of the Galois representation Gal(Q/L) → GL2 (Zl◦ )associated to EL contains SL2 (Zl◦ ). The Galois representation
Gal(Q/L) → GL2 (Zl• ) associated to EL is surjective.
(c) The inequalitiesl◦ ≤ 23040 · 100d · htFalt ([EL ]) + 2xS + C · dl• ≤ 23040 · 100d · htFalt ([EL ]) + 6d · log-diffMell ([EL ])) + 2xS + C · dhold.

ARITHMETIC ELLIPTIC CURVES IN GENERAL POSITION

27

Proof. The proof is entirely similar to [but slightly easier than] the proof ofCorollary 4.3, except that instead of applying condition (a) of Theorem 3.8,
we apply condition (b) of Theorem 3.8. Also, when applying Lemma 4.1,
we take “h” to be 0.
Remark 4.4.1. Typically, in computations of arithmetic degrees, such asheights [i.e., “htFalt (−)”, etc.], the terms that appear in a sum that computes the arithmetic degree are not terms that are “polynomial in l”, butrather terms that are linear in log(l). Thus, in the context of Corollaries
4.3, 4.4, if l is equal to l◦ or l• , then log(l) is of the order oflog(htFalt (−)) + (1 + ǫ) log(d)orFaltlog(ht (−)) + log(log-diffMell (−)) + (1 + ǫ) log(d)— hence asymptotically bounded byǫ · (htFalt (−) + d)orFalt
ǫ · (ht (−) + log-diffMell (−) + d)
— for ǫ > 0.
Remark 4.4.2. Let UP ⊆ P be as in Theorem 2.1, (ii). Thus, by regarding
UP as the “λ-line” [i.e., regarding the standard coordinate on P as the “λ” inthe Legendre form “y 2 = x(x−1)(x−λ)” of the Weierstrass equation defining
an elliptic curve], one obtains a natural finite étale [classifying] morphism
UP → Mell ×Z Q. Then one verifies immediately that [relative to the elliptic
curves obtained from points ∈ UP (Q) via this classifying morphism] one
obtains a result entirely similar to Corollary 4.4 by replacing “Mell ⊆ Mell ”
by UP ⊆ P . In particular, in the context of Theorem 2.1, (ii), one may
always assume the existence of prime numbers l◦ , l• as in [this UP ⊆ P
version of] Corollary 4.4.
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28

SHINICHI MOCHIZUKI[8] S. Mochizuki, Galois Sections in Absolute Anabelian Geometry, Nagoya Math. J. 179(2005), pp. 17-45.
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Shinichi Mochizuki
Research Institute for Mathematical Sciences
Kyoto University
Kyoto, 606-8502 Japane-mail address: motizuki@kurims.kyoto-u.ac.jp(Received March 26, 2008 )(Revised February 17, 2009 )