DIOPHANTINE APPROXIMATION FOR ALGEBRAIC POINTS ON CURVES

By

Thomas Plante

A DISSERTATION

to Michigan State University

in partial fulfillment of the requirements

Submitted

for the degree of

Mathematics – Doctor of Philosophy

2020

DIOPHANTINE APPROXIMATION FOR ALGEBRAIC POINTS ON CURVES

ABSTRACT

By

Thomas Plante

A foundational result in Diophantine approximation is Roth’s theorem, which asserts that for
a given algebraic number (cid:11) and " > 0, the inequality j(cid:11) (cid:0) p(cid:157)qj < 1(cid:157)q2+" has only finitely many
solutions in rational numbers p(cid:157)q 2 Q. Ridout and Lang subsequently proved a general form
of Roth’s theorem allowing for arbitrary absolute values (including p-adic absolute values) and
permitting arbitrary (fixed) number fields k in place of the rational numbers. Wirsing proved a
generalization of Roth’s theorem, where the approximating elements are taken from varying number
fields of degree d, and the quantity 2 + " in Roth’s theorem is replaced by 2d + ". As a consequence
of a deep inequality of Vojta, Wirsing’s theorem, appropriately formulated, may be extended to
a Diophantine approximation result for algebraic points of degree d on a nonsingular projective
curve.

The main theorem of this thesis improves on this general form of Wirsing’s theorem further, but
only in the cases where d = 2 and d = 3. More specifically, if we let C be a nonsingular projective
curve over a number field k, S a finite set of places of k and P1; : : : ; Pn 2 C„k” be distinct and
define the divisor D :=

i=1 Pi, then letting R 2 C„k” and " > 0, we get

∑n

mD;S„P” (cid:20) „Nd„D” + "”hR„P” + O„1”

for all P 2 C„k” with »k„P” : k… = d 2 f2; 3g, where mD;S„P” is a sum of local heights associated
to D and S, hR is a global height associated to R, and

{(cid:12)(cid:12)(cid:12)(

Nd„D” := max

)

(cid:12)(cid:12)(cid:12)}

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

\ Supp„D”

taken over all finite k-morphisms (cid:27) : C ! P1 of degree d. As Nd„D” (cid:20) 2d, the main result gives
a refinement of Wirsing’s theorem depending on the divisor D.

In much the same way that Roth’s theorem can be used to prove Siegel’s Theorem about integral
points on a genus-0 affine curve, the main theorem of this dissertation implies Levin’s generalization
of Siegel’s theorem for algebraic points of degree d in the cases d = 2; 3.

After a review of some properties of global heights and local heights, we also show that the
main theorem is sharp, in the sense that counterexamples exist when Nd„D” + " is replaced with
Nd„D” (cid:0) ".

To prove the main theorem, we first show that for curves of large enough genus, the number of
morphisms (cid:27) : C ! P1 of degree d = 2; 3 defined over k is finite. We then prove that curves for
which the number of such morphisms is finite satisfy the main theorem. Finally, we prove the main
theorem for the remaining small genus curves on a case by case basis.

We transfer the Diophantine approximation problem for points of degree d on C to a Diophan-
tine approximation problem for rational points on Symd„C” (the dth symmetric power of C). We
exploit the map from Symd„C” to Jac„C”, the Jacobian of C, and its associated geometry. We
use Diophantine approximation results for abelian varieties (when we’re on Jac„C”), Diophantine
approximation results for projective spaces (fibers of Symd„C” 7! Jac„C”), and Diophantine ap-
proximation results on Symd„C” directly, plus algebraic geometry to connect all of these. In the
course of the proof we make use of Schmidt’s subspace theorem and its generalizations due to
Ru-Wong and Evertse-Ferretti as well as a version of Roth’s theorem for nonreduced divisors and
Faltings’ approximation theorem for rational points on abelian varieties.

TABLE OF CONTENTS

CHAPTER 1 AN INTRODUCTION TO DIOPHANTINE APPROXIMATION . . .

.
1.1 The Road to Roth’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Further Approximation Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Integral Points on Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
4
7
CHAPTER 2 REVIEW OF HEIGHTS . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Heights on Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Heights on Projective Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Heights on Closed Subschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
CHAPTER 3 PREPARATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
. 22
3.1 Useful Results
3.2 Sharpness of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Facts on Trigonal Maps .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
CHAPTER 4 PROOF OF THE MAIN THEOREM . . . . . . . . . . . . . . . . . . . 31
4.1 Algebraic Points on Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Quadratic Points on Elliptic Curves
. . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Cubic Points on Curves of Low Genus . . . . . . . . . . . . . . . . . . . . . . . 45

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

BIBLIOGRAPHY .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

iv

CHAPTER 1

AN INTRODUCTION TO DIOPHANTINE APPROXIMATION

1.1 The Road to Roth’s Theorem

An important question in Diophantine approximation is how closely a given real number (cid:11) 2 R
can be approximated by rational numbers p(cid:157)q 2 Q; in particular, that to approximate (cid:11) closely both
the numerator, p, and denominator, q, must be large. A typical way of expressing this is to ask, for
a given (cid:11) 2 R and e > 0, whether or not the inequality

has infinitely many solutions p(cid:157)q 2 Q. In 1842, it was shown by Dirichlet that
Theorem 1.1.1. For each (cid:11) 2 R n Q the inequality

Theorem 1.1.2 (Liouville). If (cid:11) is an algebraic number of degree d and if e > d, then the inequality

In fact, given an algebraic number (cid:11) of degree d, you can find an effective (explicitly described)

constant C = C„(cid:11)” > 0 such that the inequality

q

(cid:12)(cid:12)(cid:12)(cid:12) p
(cid:12)(cid:12)(cid:12)(cid:12) p

q

(cid:12)(cid:12)(cid:12)(cid:12) p

(cid:12)(cid:12)(cid:12)(cid:12) p

q

jqje

(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) 1
(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) 1

q2

(cid:0) (cid:11)

(cid:0) (cid:11)

(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) 1

jqje

(cid:0) (cid:11)

(cid:12)(cid:12)(cid:12)(cid:12) (cid:21) C

jqjd

(cid:0) (cid:11)

has infinitely many solutions p(cid:157)q 2 Q.

Meanwhile, in 1844, Liouville showed that

q
has at most finitely many solutions p(cid:157)q 2 Q.

holds for all p(cid:157)q 2 Q n f(cid:11)g.

1

∑1
i=1 10(cid:0)i!. Then the partial sums

∑n

Example 1.1.3. Let (cid:11) =
1jqje as long as n (cid:21) e. By Liouville’s theorem, (cid:11) is transcendental.

i=1 10(cid:0)i! are solutions to

(cid:12)(cid:12)(cid:12) (cid:20)

(cid:0) (cid:11)

(cid:12)(cid:12)(cid:12) p

q

In order to start getting nice applications to Diophantine equations, however, one needs an

exponent of d (cid:0) ".
Theorem 1.1.4 (Thue, 1909). Let (cid:11) 2 Q, »Q„(cid:11)” : Q… = d. Let " > 0. Then there are only finitely
many rational numbers p
q

(cid:12)(cid:12)(cid:12)(cid:12) <

2 Q satisfying(cid:12)(cid:12)(cid:12)(cid:12)(cid:11) (cid:0) p
(
)3 (cid:0) 2 =
(

q

x
y

1
d
2 +1+"

q

)((

:

)2

(cid:0) 3p

2

x
y

3p
2 + 3p

4

x
y

+

x
y

)

Example 1.1.5. Consider integer solutions to the equation x3 (cid:0) 2y3 = 1. Then

(cid:12)(cid:12)(cid:12) x

=

1
y3

(cid:12)(cid:12)(cid:12) < C

(cid:0) 3p
Which implies
2
many solutions x; y 2 Z.

y

y3 for some constant C > 0. By Thue’s theorem, there are only finitely

More generally, Thue proved that if f 2 Z»x; y… is an irreducible binary form with deg„ f ” (cid:21) 3,

and r 2 Z, f „x; y” = r has only finitely many integer solutions x and y.

p

The exponent d
p

d by Siegel (1921)
2d by Dyson and Gelfond (independently) (1947). In 1955 Roth managed to eliminate the

2 + 1 in Thue’s theorem was subsequently improved to 2

and
dependence on d altogether.

Theorem 1.1.6 (Roth’s Theorem). For every algebraic number (cid:11) and every " > 0, the inequality

q
has only finitely many solutions p(cid:157)q 2 Q.

C = C„(cid:11); "” > 0 such that for all p(cid:157)q 2 Q n f(cid:11)g
(cid:0) (cid:11)

(cid:12)(cid:12)(cid:12)(cid:12) p

(cid:12)(cid:12)(cid:12)(cid:12) p

q

(cid:12)(cid:12)(cid:12)(cid:12) (cid:20)

(cid:0) (cid:11)

1
jqj2+"

(cid:12)(cid:12)(cid:12)(cid:12) (cid:21) C

:

jqj2+"

2

Equivalently, for every real algebraic number (cid:11) and every " > 0, there exists a constant

note here that Roth’s Theorem only asserts the existence of such a constant C and provides no way
of calculating it, that is, C is not effective. Another equivalent expression of Roth’s Theorem is that
for every real algebraic number (cid:11) and every " > 0, there exists a constant C = C„(cid:11); "” > 0 such
that for all p(cid:157)q 2 Q n f(cid:11)g

log max

(cid:0) (cid:11)

(cid:20) „2 + "” log max fjpj; jqjg + C:

; 1

{(cid:12)(cid:12)(cid:12)(cid:12) p

q

}

(cid:12)(cid:12)(cid:12)(cid:12)(cid:0)1

This form is significant, because it is written in terms of functions called heights. The left hand side
of the inequality is what’s known as a local height of p(cid:157)q, relative to the algebraic number (cid:11), which
for now we denote (cid:21)(cid:11)„p(cid:157)q”. note that the closer p(cid:157)q gets to (cid:11), the larger the local height becomes,
so we think of local height like an inverse distance function. The height of a rational number p(cid:157)q,
with gcd„p; q” = 1, is h„p(cid:157)q” := log max fjpj; jqjg. Height can be thought of as a measure of the
arithmetic complexity of a number. Thus, using big-O notation, we may restate Roth’s theorem:
For every algebraic number (cid:11) and every " > 0, the inequality

(cid:21)(cid:11)„(cid:12)” (cid:20) „2 + "”h„(cid:12)” + O„1”

holds for all (cid:12) 2 Q.

Generalizing to an arbitrary absolute value j(cid:1)jv on a number field k (normalized as in Definition

2.1.3), we define the following:
Definition 1.1.7. Let k be a number field. Given (cid:11) 2 k and an absolute value j (cid:1) jv on k (with some
fixed extension to k), define the local height associated to (cid:11) at v by

(cid:21)(cid:11);v„(cid:12)” := log max

for all (cid:12) 2 k.

}

{(cid:12)(cid:12)(cid:12)(cid:12) 1

(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12) (cid:0) (cid:11)

; 1

v

Similarly, we may define the (global) height of an algebraic number (see Definition 2.1.6).

3

1.2 Further Approximation Results

In 1958, Ridout generalized Roth’s theorem to p-adic absolute values on Q and soon after Lang

generalized the result to an arbitrary number field.

Theorem 1.2.1 (Roth’s Theorem, General Form). Let S be a finite set of absolute values on a
number field k. For each v 2 S, let (cid:11)v 2 k and fix an extension of j (cid:1) jv to k. Let " > 0. Then for
all (cid:12) 2 k n f(cid:11)vjv 2 Sg

∑

v2S

(cid:21)(cid:11)v;v„(cid:12)” (cid:20) „2 + "”h„(cid:12)” + O„1”:

We can also view Roth’s Theorem as a statement about k-rational points on the projective line
by defining local heights (cid:21)D;v with respect to a divisor D on P1„k” (see Definition 2.2.1). Then we
get the following:

Theorem 1.2.2 (Roth’s Theorem on P1). Let S be a finite set of places of a number field k. Let
i=1 Pi, and " > 0. Then for all P 2 P1„k”nfP1; : : : ; Pqg
P1; : : : ; Pq 2 P1„k” be distinct points, D =
(cid:21)D;v„P” (cid:20) „2 + "”h„P” + O„1”:

mD;S„P” :=

∑

∑q

v2S

We will study throughout inequalities of this form, where a sum of local heights, with respect
to some divisor, is bounded by a constant multiple of a global height (where the constant may
depend on the divisor). For curves of genus greater than one, such inequalities are uninteresting
(for rational points) due to Faltings’ theorem:
Theorem 1.2.3 (Faltings’ Theorem). Let C be a smooth projective curve of genus g (cid:21) 2, defined
over a number field k. Then the number of k-rational points of C is finite.

Proof. See [8, Part E].

□

This is exactly the statement that, given any finite set S of absolute values on a number field
k and distinct points P1; : : : ; Pq 2 C„k”, mD;S„P” (cid:20) O„1” for all P 2 C„k” n fP1; : : : ; Pqg, where
D =

∑q

i=1 Pi.

4

It remains to ask what happens on curves of genus one, i.e. elliptic curves. The answer for
this case is due to Siegel.
It requires the notion of a global height associated to a divisor on
a curve C (see Definition 3.8).
In the case of curves, we will typically take the divisor D to
consist of a single point R 2 C„k”. By Theorem 2.2.12, if R; R′ 2 C„k”, then for any " > 0,
jhR„P” (cid:0) hR′„P”j (cid:20) "hR„P” + O„1”, so that the choice of the point R will not be significant in our
inequalities.

∑q

Theorem 1.2.4 (Siegel). Let C be a smooth projective curve of genus one, defined over a number
field k. Let S be a finite set of places of k. Let P1; : : : ; Pq 2 C„k” be distinct points and let
D =

i=1 Pi. Let R 2 C„k” and let " > 0. Then for all P 2 C„k” n fP1; : : : ; Pqg

mD;S„P” (cid:20) "hR„P” + O„1”:

Another way to generalize Roth’s theorem is to consider, instead of k-rational points, algebraic

points in C„k” of degree d over k. We denote the support of a divisor D by Supp„D”.

∑q

Theorem 1.2.5 (Wirsing’s Theorem). Let S be a finite set of places of a number field k. Let
P1; : : : ; Pq 2 P1„k” be distinct points and let D =
i=1 Pi. Let " > 0 and let d be a positive integer.
Then for all points P 2 P1„k” n Supp„D” satisfying »k„P” : k… (cid:20) d,
mD;S„P” (cid:20) „2d + "”h„P” + O„1”:

Theorem 1.2.6 (Vojta’s Inequality). Let C be a nonsingular projective curve defined over a number
field k with canonical divisor K. Let S be a finite set of places of k. Let P1; : : : ; Pq 2 C„k” be
distinct points and let D =
i=1 Pi. Let A be an ample divisor on C. Let " > 0. If r is a positive
integer then

∑q

mD;S„P” + hK„P” (cid:20) da„P” + "hA„P” + O„1”

for all points P 2 C„k”n Supp„D” for which »k„P” : k… (cid:20) r, where da is the arithmetic discriminant
(a value defined by Vojta using Arakelov theory).

Proof. See [17].

□

5

Vojta’s inequality implies many important results in Diophantine Geometry, including Falt-
ing’s theorem and Wirsing’s theorem. Vojta also conjectured that the arithmetic discriminant
in the inequality could be replaced by the smaller geometric (logarithmic) discriminant d„P” :=
log jDk„p”j(cid:157)»k : Q…. The relationship between these discriminants is similar to that between the
arithmetic and geometric genus. Vojta’s conjecture implies many other famous conjectures, such
as the famous ABC conjecture.

We can use Vojta’s inequality to generalize Wirsing’s theorem to curves.

Theorem 1.2.7. Let C be a nonsingular projective curve over a number field k. Let S be a finite
set of places of k and let R 2 C„k”. Let P1; : : : ; Pn 2 C„k” be distinct and define the divisor
D :=

i=1 Pi. Let " > 0 and let d be a positive integer. Then

∑n

mD;S„P” (cid:20) „2d + "”hR„P” + O„1”

for all P 2 C„k” with »k„P” : k… = d.

Proof. Follows from Vojta’s inequality. See [14, Equation (2.0.3)].

□

Song and Tucker show that this result is sharp in the following sense.

Theorem 1.2.8. Let C be a nonsingular projective curve over a number field k and let φ : C ! P1
be a nonconstant morphism of degree d. Let S be a finite set of places of k and let R 2 C„k”. Let
" > 0. Then there exists a choice of D such that there is an infinite set of points P 2 C„k” with
»k„P” : k… (cid:20) d satisfying

mD;S„P” (cid:21) „2d (cid:0) "”hR„P” + O„1”:

Proof. See [14, Theorem 2.3].

□

However, by replacing 2d with a constant depending on the divisor D, we found we could

improve on Theorem 1.2.7 as follows:

6

Theorem 1.2.9 (The Main Theorem). Let C be a nonsingular projective curve over a number field
k. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and define the divisor
D :=

i=1 Pi 2 Div„C”. Let R 2 C„k” and let " > 0. Then if d 2 f2; 3g,

∑n

mD;S„Q” (cid:20) „Nd„D” + "”hR„Q” + O„1”

for all Q 2 C„k” with »k„Q” : k… = d, where

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”
taken over all finite k-morphisms (cid:27) : C ! P1 of degree d.

Nd„D” := max

{(cid:12)(cid:12)(cid:12)(

)

(cid:12)(cid:12)(cid:12)}

\ Supp„D”

note that the most that Nd„D” could ever be is 2d, which agrees with Theorem 1.2.7. These
diophantine approximation inequalities are closely related to qualitative results for integral points
on curves.

1.3 Integral Points on Curves

Let k be a number field and let S be a finite set of places of k containing all archimedean places.
The ring of S-integers, denoted Ok;S, is defined to be the set of all (cid:12) 2 k such that j (cid:12)jv (cid:20) 1 for all
v < S.
Theorem 1.3.1 (Siegel’s Theorem, 1929). Let C (cid:26) An be a nonsingular affine curve over a number
field k and let S be a finite set of places of k containing the archimedean places. An S-integral
point of C is a point whose affine coordinates are all in the ring Ok;S. If C has genus g > 0, or if
C has at least three distinct points at infinity, then C has only finitely many S-integral points.

Proof. See [1, Theorem 7.3.9].

□

Although this qualitative result preceded the above diophantine approximation inequalities on

projective curves, they are related in the following way:

7

Let C (cid:26) An be a nonsingular affine curve over a number field k. Let ˜C be the nonsingular
projective completion of C and let D be the very ample divisor on ˜C consisting of the points at
infinity of C. Let R 2 ˜C„k” and let " > 0. By Lemma 3.1.7, the set of S-integral points on C
corresponds (up to a finite number of points) to the set of points P 2 ˜C„k” n Supp„D” satisfying
mD;S„P” = hD„P” + O„1”. If C has genus g = 0 and deg„D” (cid:21) 3, then ˜C is birationally equivalent
to P1 and

deg„D”hR„P” = hdeg„D”R„P”

(cid:20) „1 + "”hD„P” + O„1” by Theorem 2.2.12
= „1 + "”mD;S„P” + O„1”
(cid:20) „1 + "”„2 + "”hR„P” + O„1” by Theorem 1.2.2,

implying hR„P” (cid:20) O„1”. By Theorem 2.2.11, there are only finitely many such points. If C has
genus g = 1, then Theorem 1.2.4 says mD;S„P” (cid:20) "hD„P” + O„1” for all P 2 ˜C n Supp„D”. But if
mD;S„P” = hD„P” + O„1”, then this implies hD„P” (cid:20) O„1” and once again Theorem 2.2.11 implies
there are only finitely many such points. And of course if C has genus g (cid:21) 2 then Theorem 1.2.3
implies mD;S„P” (cid:20) O„1”. So if mD;S„P” = hD„P” + O„1”, then hD„P” (cid:20) O„1” and there are only
finitely such points. Thus these three results together imply Siegel’s theorem.

In this way diophantine approximation inequalities for rational points on projective varieties can
be thought of as more precise statements of qualitative results for integral points on affine varieties.
Similar to how Roth’s theorem implies the genus-0 case of Siegel’s theorem, the main theorem

can be used to show the d = 2; 3 cases of the following qualitative result on integral points.
Theorem 1.3.2 (Levin). Let C (cid:26) An be a nonsingular affine curve over a number field k and let S
be a finite set of places of k containing the archimedean places. Let ˜C be a nonsingular projective
completion of C and let „ ˜C n C”„k” = fP1; : : : ; Pqg. Let d be a positive integer. Let Ok;S denote the
integral closure of Ok;S in k. Then there exists a finite extension L of k such that the set

fP 2 C„OL;S” j »L„P” : L… (cid:20) dg

8

is infinite if and only if there exists a morphism φ :
φ„fP1; : : : ; Pqg” (cid:26) f0; 1g.

˜C ! P1, over k with deg„φ” (cid:20) d and

Proof. See [10, Theorem 1.9]

□
To see how this is implied by the main theorem, let C (cid:26) An be a nonsingular affine curve over
a number field k and let S be a finite set of places of k containing the archimedean places. Let ˜C
be a nonsingular projective completion of C and let „ ˜C n C”„k” = fP1; : : : ; Pqg. Let d 2 f2; 3g. Let
D = P1 + : : : + Pq be a divisor on ˜C and let R 2 ˜C„k”. Then for all P 2 ˜C„OL;S” n fP1; : : : ; Pqg
with »k„P” : k… = d,

deg„D”hR„P” = hdeg„D”R„P”

(cid:20) „1 + "”hD„P” + O„1” by Theorem 2.2.12
= „1 + "”mD;S„P” + O„1”
(cid:20) „1 + "”„Nd„D” + "”hR„P” + O„1” by Theorem 1.2.2,

{(cid:12)(cid:12)(cid:12)(

)

(cid:12)(cid:12)(cid:12)}

where

Nd„D” := max

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

\ Supp„D”

taken over all finite k-morphisms (cid:27) : C ! P1 of degree d. But if there does not exist a morphism
φ : ˜C ! P1, over k with deg„φ” (cid:20) d and φ„fP1; : : : ; Pqg” (cid:26) f0; 1g then N < deg„D”, and so the
above inequality implies hR„P” (cid:20) O„1” and by Theorem 2.2.11, there are only finitely many such
points P.

The main idea behind this result and the main theorem is that we expect almost all of the degree
d points on a curve are coming from pulling back rational points on P1 by maps of degree d. The
following are two other results that are closely related to this heuristic approach.

C is called hyperelliptic (respectively bielliptic) if it admits a map φ : C ! X of degree 2 onto

a curve X of genus zero (respectively one).

9

Theorem 1.3.3 (Harris-Silverman). Let C be a nonsingular projective curve over a number field
k. Suppose C has genus g (cid:21) 2. Assume that C is neither hyperelliptic nor bielliptic. Then the set
of points P 2 C„k” with »k„P” : k… (cid:20) 2 is finite.

Proof. See [6, Corollary 3]
Theorem 1.3.4 (Corvaja-Zannier). Let C (cid:26) An be a nonsingular affine curve over a number field
k and let S be a finite set of places of k containing the archimedean places. Let ˜C be a nonsingular
projective completion of C and let „ ˜C n C”„k” = fP1; : : : ; Pqg.
(a) If q (cid:21) 5, then C contains only finitely many quadratic (over k) S-integral points.
(b) If q (cid:21) 4, then there exists a finite set of rational maps φ :

□

all but finitely many of the quadratic S-integral points on C are sent to P1
mentioned maps.

˜C ! P1 of degree 2 such that
k by some of the

Proof. See [2, Corollary 1]

□

10

CHAPTER 2

REVIEW OF HEIGHTS

2.1 Heights on Projective Spaces

The height of a rational number p(cid:157)q is defined to be h„p(cid:157)q” = log max fjpj; jqjg. Similarly the
height of a point P = „x0; : : : ; xn” 2 Pn„Q”, with coordinates chosen so that x0; : : : ; xn 2 Z and
gcd„x0; : : : ; xn” = 1, is defined to be h„P” = log max fjx0j; : : : ; jxnjg. One of the key features of
heights is that, given a bound, the number of points of bounded height is finite, that is the set

{

}

P 2 Pn„Q” j h„P” (cid:20) B

is finite for any B > 0.

In order to generalize the concept of heights to number fields other than Q, we recall some
properties of absolute values on fields. An absolute value on a field k is a map j (cid:1) j : k ! R
satisfying the following.

• j(cid:11)j (cid:21) 0 for all (cid:11) 2 k.
• j(cid:11)j = 0 if and only if (cid:11) = 0.
• j(cid:11)(cid:12)j = j(cid:11)jj (cid:12)j for all (cid:11); (cid:12) 2 k.
• j(cid:11) + (cid:12)j (cid:20) j(cid:11)j + j (cid:12)j for all (cid:11); (cid:12) 2 k.

We say two absolute values on a field k are equivalent if they induce the same topology on k.

It can be shown that
Proposition 2.1.1. Two absolute values j (cid:1) j1; j (cid:1) j2 on a field k are equivalent if and only if there is
a positive real number s such that

for all x 2 k.

jxj1 = jxjs
2

11

Proof. See [11, Proposition 3.3].

□

We denote the set of equivalence classes of non-trivial absolute values on a field k by Mk and
call its elements places. If k′ is an extension of k and v is a place of k, we say a place w of k′
extends v, denoted wjv, if the restriction to k of any representative of w is a representative of v.

Definition 2.1.2. The completion of k with respect to the place v is an extension field kv, with a
place w such that:
(a) wjv

(b) The topology of kv induced by w is complete.

(c) k is a dense subset of kv in the above topology.

The completion exists and is unique up to isometric isomorphisms. By abuse of notation, we shall
denote the unique place w also by v.

The standard absolute values on Q are the (usual) archimedean absolute value j (cid:1) j1 = j (cid:1) j and
(cid:0)ordp„(cid:1)” for each prime p. By Ostrowski’s Theorem, this is a

the p-adic absolute values j (cid:1) jp = p
complete list of all absolute values on Q up to equivalence.

Definition 2.1.3. Let p be a place on Q. We consider a number field k and a place v of k extending
p. For any x 2 k, we define

jxjv = jNkv(cid:157)Qp

where Nkv(cid:157)Qp is the norm from kv to Qp.

„x”j1(cid:157)»k:Q…

p

note that j (cid:1) jv is an extension to k of j (cid:1) j»kv:Qp…(cid:157)»k:Q…

p

2.1 is equivalent to j (cid:1) jp. That is vjp.

on Q, an absolute value that by Proposition

These normalized absolute values satisfy the following.

12

Lemma 2.1.4. Let k be a number field with a place v 2 Mk. We consider a finite-dimensional

extension field k′ of k. For any x 2 k(cid:3), ∏

Proof. See [1, Lemma 1.3.7].
Proposition 2.1.5 (The Product Formula). Let k be a number field. For any x 2 k(cid:3),

□

jxjw = jxjv:

jxjv = 1:

w2Mk′
wjv

∏

v2Mk

∑

v2Mk

∑

v2Mk

Proof. See [1, Proposition 1.4.4].
Definition 2.1.6. The (absolute) height of a point P 2 Pn„Q” with homogeneous coordinates
„x0; : : : ; xn” is defined to be

□

h„P” =

log jxijv

max
0(cid:20)i(cid:20)n

where k is some number field containing x0; : : : ; xn. We also define the height of an algebraic
number (cid:11) 2 Q to be the height of the point „(cid:11); 1” 2 P1„Q”

h„(cid:11)” := h„„(cid:11); 1”” =

log max fj(cid:11)jv; 1g:

Proposition 2.1.7. The height, so defined, is independent of the choice of homogeneous coordinates
for P and the choice of number field k containing them.

Proof. See [1, Lemmas 1.5.2 and 1.5.3].

□

One reason for studying heights is because they satisfy the following.
Theorem 2.1.8 (Northcott’s Theorem). For any numbers B; d (cid:21) 0, the set

fP 2 Pn„Q” j h„P” (cid:20) B and »Q„P” : Q… (cid:20) dg

13

is finite. In particular, for any fixed number field k, the set
j h„P” (cid:20) Bg

fP 2 Pn
k

is finite.

Proof. See [1, Theorem 1.6.8].

□

Another important property of heights is preservation under Galois action.

Proposition 2.1.9. Let (cid:27) 2 Gal„Q(cid:157)Q”. Then h„P” = h„(cid:27)„P”” for all P 2 Pn„Q”, where

(

)

(cid:27)

„xi”n
i=1

:= „(cid:27)„xi””n
i=1

Proof. See [1, Proposition 1.5.17].

□

2.2 Heights on Projective Varieties

Let X be a projective variety over a number field k. We denote the set of divisors on X by
Div„X”. We consider a divisor D on X with associated line bundle O„D” and rational section
sD. There are base-point-free line bundles L, M on X such that O„D” (cid:27) L (cid:10) M(cid:0)1. Now choose
generating global sections s0; : : : ; sm of L and t0; : : : ; tn of M, and call the data

D = „sD; L; s; M; t”

a presentation of the divisor D. We denote the support of the divisor D by Supp„D”.
Definition 2.2.1. Let v 2 Mk. For P 2 X„k” n Supp„D”, we define the local height of P relative to
the presentation D at the place v to be

(cid:12)(cid:12)(cid:12)(cid:12) sk

tlsD

(cid:12)(cid:12)(cid:12)(cid:12)

„P”

:

v

(cid:21)D;v„P” := log max

k

min
l

Definition 2.2.2. If D1 and D2 are divisors with presentations Di = „sDi; Li; si; Mi; ti”, then we
define the presentation D1 + D2 of the divisor D1 + D2 to be

D1 + D2 := „sD1sD2; L1 (cid:10) L2; s1s2; M1 (cid:10) M2; t1t2”:

14

Similarly, if a divisor D has presentation D = „sD; L; s; M; t”, we define the presentation (cid:0)D of
the divisor (cid:0)D to be

(cid:0)D := „s

(cid:0)1
D ; M; t; L; s”:

Furthermore, if (cid:25) : Y ! X is a morphism of projective varieties such that (cid:25)„Y” is not contained in
Supp„D”, then we define the presentation (cid:25)(cid:3)D of the divisor (cid:25)(cid:3)D on Y to be

(cid:25)(cid:3)D := „(cid:25)(cid:3)

sD; (cid:25)(cid:3)

L; (cid:25)(cid:3)s; (cid:25)(cid:3)

M; (cid:25)(cid:3)t”:

With these definitions, it is immediate that the corresponding local heights satisfy

(cid:21)D1+D2;v = (cid:21)D1;v + (cid:21)D2;v;

(cid:21)(cid:0)D;v = (cid:0)(cid:21)D;v;

and

(cid:21)(cid:25)(cid:3)D;v

= (cid:21)D;v ◦ (cid:25)

for all v 2 Mk.
Definition 2.2.3. An Mk-constant is a map (cid:13) : Mk ! R with the property that (cid:13)v = 0 for all but
finitely many v 2 Mk.
Theorem 2.2.4. Let D and D′ be two presentations of the divisor D. Then there is an Mk-constant
(cid:13) such that

j(cid:21)D;v (cid:0) (cid:21)D′;v

j (cid:20) (cid:13)v:

Proof. See [1, Theorem 2.2.11].

□

For this reason, whenever an inequality holds only up to the addition of a bounded function,
we will omit the presentation D and, by abuse of notation, denote a local height relative to D at a
place v by (cid:21)D;v.
Proposition 2.2.5. Let D be an effective divisor on X. Then there is a presentation D of D such
that for any P < Supp„D” and for any field extension k (cid:26) k′ (cid:26) k such that P 2 X„k′” and any place
v 2 Mk′, it holds that (cid:21)D;v„P” (cid:21) 0.

Proof. See [1, Proposition 2.3.9].

□

15

Example 2.2.6. Let (cid:11) 2 k. The point „(cid:11); 1” in P1

A = „x0 (cid:0) (cid:11)x1; O

For (cid:12) 2 k n f(cid:11)g and v 2 Mk the corresponding local height is

(cid:21)(cid:11);v„(cid:12)” := (cid:21)A;v„„(cid:12); 1”” = log max

P1„1”; „x1; x0 (cid:0) (cid:11)x1”; O

k has the presentation
P1; 1”:
}

{(cid:12)(cid:12)(cid:12)(cid:12) 1

(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12) (cid:0) (cid:11)

; 1

v

which is the local height we described in Definition 1.1.7.
Example 2.2.7. The hyperplane fx0 = 0g in Pn

k has the presentation
D = „x0; OPn„1”; „x0; x1; : : : ; xn”; OPn; 1”:
k with x0„P” , 0 and v 2 Mk the corresponding local height is

For P 2 Pn

{(cid:12)(cid:12)(cid:12)(cid:12) xi

x0

}

(cid:12)(cid:12)(cid:12)(cid:12)

v

(cid:21)D;v„P” = log max

„P”

and the product formula becomes

h„P” =

(cid:21)D;v„P”:

v2Mk
For this reason we make the following definition.

i

∑

∑

v2Mk′

Definition 2.2.8. Let D = „sD; L; s; M; t” be a presentation of a divisor D on X. For P 2 X„k”
there are si and tj such that si„P” , 0, tj„P” , 0 by definition of base point free line bundles. Thus
we can find a non-zero rational section s of O„D” such that P is not contained in the support of
the divisor D„s”. Then D„s” = „s; L; s; M; t” is a presentation of D„s”. If k′ is a finite extension
k (cid:26) k′ (cid:26) k such that P 2 X„k′”, then we define the global height of P relative to D by

hD„P” =

(cid:21)D„s”;v

„P”:

Proposition 2.2.9. The global height hD is independent of the choices of k′ and of the section s.

Proof. See [1, Proposition 2.3.4].

□

16

It is immediately clear by the definition and the above discussion that, given presentations D1,
D2, D of divisors D1, D2, D respectively and a morphism of projective varieties (cid:25) : X ! Y, the
corresponding global heights also satisfy

hD1+D2 = hD1 + hD2

; h(cid:0)D = (cid:0)hD; and h(cid:25)(cid:3)D = hD ◦ (cid:25):

Furthermore, by Proposition 3.2, if D 2 Div„X” is effective, then there exists a presentation D of
D such that hD„P” (cid:21) 0 for all P 2 X„k”.

For two divisors D1 and D2, we write D1 (cid:24) D2 if D1 is linearly equivalent to D2, i.e.

difference D2 (cid:0) D1 is a principal divisor.
Theorem 2.2.10. Let D1, D2 be presentations of divisors D1, D2 2 Div„X” with D1 (cid:24) D2. Then
there is a constant (cid:13) > 0 such that

the

jhD2

(cid:0) hD1

j (cid:20) (cid:13):

Proof. See [1, Theorem 2.3.6].

□

Similarly to local heights, whenever an inequality holds only up to the addition of a bounded
function, we will omit the presentation D and, by abuse of notation, denote a global height relative
to D by hD. Furthermore, in the special case where the divisor D is an effective divisor of degree
1 on a curve and Supp„D” = fRg, we define hR := hD.

Global heights also satisfy Northcott’s theorem.

Theorem 2.2.11 (Northcott’s Theorem). Let D be an ample divisor on X and let D be a presentation
of D. Then the set

fP 2 X„k” j hD„P” (cid:20) B;

»k„P” : k… (cid:20) dg

is finite for any constants B; d 2 R.

Proof. See [1, Theorem 2.4.9].

□
Recall that two divisors D1; D2 2 Div„X” are algebraically equivalent if there exists a
connected algebraic set T, two points t1; t2 2 T„k” and a line bundle L on X (cid:2) T such that

17

O„D1” (cid:27) LjX(cid:2)ft1g and O„D2” (cid:27) LjX(cid:2)ft2g. In particular if X is a curve, then any two points on X
are algebraically equivalent.
Theorem 2.2.12. Let A; B 2 Div„X” be algebraically equivalent divisors with A ample. Let " > 0
and d a positive integer. Then for all P 2 X„k” with »k„P” : k… (cid:20) d we have

jhA„P” (cid:0) hB„P”j (cid:20) "hA„P” + O„1”:

Proof. Follows from [8, Theorem B.5.9] and Theorem 2.2.11.

□

Given a presentation D = „sD; L; s; M; t” of a divisor D 2 Div„X”, we have the morphisms

and

φ : X ! Pm
k

;

P 7! „s0„P”; s1„P”; : : : ; sm„P””

  : X ! Pn
k

;

P 7! „t0„P”; t1„P”; : : : ; tn„P””:

By the product formula we see that hD„P” = h„φ„P”” (cid:0) h„ „P””. Therefore, by Proposition 2.1.9,
for any (cid:27) 2 Gal„k(cid:157)k” we have hD„(cid:27)„P”” = hD„P”.

2.3 Heights on Closed Subschemes

We identify a closed subscheme X of a nonsingular projective variety V with its ideal sheaf IX.

Then for closed subschemes X and Y we define the following:

IX+Y = IXIY
IX\Y = IX + IY
IX[Y = IX \ IY :
We say X (cid:26) Y if IY (cid:26) IX. Let
: V ! W be a morphism of projective varieties. Then
for any closed subscheme X (cid:26) V and Z (cid:26) W we define the scheme-theoretic image f „X” by
If „X” = Ker„OW ! f(cid:3)OX” and the scheme theoretic inverse image f (cid:3)Z by If (cid:3)Z = f (cid:0)1IZ (cid:1) OV.

f

18

Lemma 2.3.1. Let f : V ! W be a morphism of projective varieties and let Z (cid:26) W be a closed
subscheme. The scheme theoretic inverse image f (cid:3)Z is the fibered product

f (cid:3)Z

Z
of Z and V over W, that is f (cid:3)Z = Z (cid:2)W V.

V

f

W

Proof. See Tag 01JU on The Stacks Project [15, Lemma 01JU].
Lemma 2.3.2. Let f : V ! W be a morphism of projective varieties. Then for any Zariski-closed
subschemes X (cid:26) V and Z (cid:26) W,
(a) X (cid:26) f (cid:3)„ f „X””
(b) f „ f (cid:3)Z” (cid:26) Z

□

Proof.

(a) By definition of f „X” we get a commutative diagram of sheaves,

f(cid:3)OX

OW(cid:157)If „X”

f(cid:3)OV
f #
OW

to which the corresponding commutative diagram of schemes is as follows.

X

f „X”

V

f

W

By the universal property of the fibered product, there exists a unique morphism X !
f „X” (cid:2)W V such that the following diagram commutes.
f „X” (cid:2)W V

X

φ

V

19

By Lemma 2.3.1,
immersion. Thus we also get the commutative diagram of structure sheaves,

f „X” (cid:2)W V is the inverse image f (cid:3)„ f „X”” and the map φ is a closed

OV(cid:157)IX

OV(cid:157)If (cid:3)„ f „X””

φ#
OV
which implies that If (cid:3)„ f „X”” (cid:26) IX, that is X (cid:26) f (cid:3)„ f „X””.

(b) By Lemma 2.3.1, we know that f (cid:3)Z = Z (cid:2)W V. Thus we get the corresponding commutative

diagram of sheaves,

f(cid:3)O f (cid:3)Z

f(cid:3)OV
f #
OW

OW(cid:157)IZ

from which it follows that IZ (cid:26) Ker„OW ! f(cid:3)O f (cid:3)Z” = If „ f (cid:3)Z”, that is f „ f (cid:3)Z” (cid:26) Z.
∩

□
Lemma 2.3.3. Let X (cid:26) V be a closed subscheme. There exist effective divisors D1; : : : ; Dr such
that X =

Di.

Proof. See [13, Lemma 2.2].

□

Given such D1; : : : ; Dr, we define the local heights corresponding to the closed subscheme X
to be (cid:21)X;v := minif(cid:21)Di;vg, for all places v 2 Mk. note that on Supp„Di” we may think of (cid:21)Di;v as
having value 1 so that (cid:21)X;v is defined (up to a bounded function) outside
Supp„Di” = Supp„X”.
Theorem 2.3.4. Let V be a projective variety defined over a number field k and let v 2 Mk. Then
up to a bounded function we have the following:
(a) For all closed subschemes X (cid:26) V and places v 2 Mk, the local height (cid:21)X;v is well-defined,

∩

up to a bounded function, independent of the choice of effective divisors D1; : : : ; Dr.

(b) For all closed subschemes X; Y (cid:26) V,

(cid:21)X\Y;v = minf(cid:21)X;v; (cid:21)Y;vg:

20

(c) For all closed subschemes X; Y (cid:26) V,

(cid:21)X+Y;v = (cid:21)X;v + (cid:21)Y;v:

(d) If closed subschemes X; Y (cid:26) V satisfy X (cid:26) Y, then
(cid:21)X;v (cid:20) (cid:21)Y;v:

(e) For all closed subschemes X; Y (cid:26) V,

maxf(cid:21)X;v; (cid:21)Y;vg (cid:20) (cid:21)X[Y;v (cid:20) (cid:21)X;v + (cid:21)Y;v:

(f) If closed subschemes X; Y (cid:26) V satisfy Supp„X” (cid:26) Supp„Y”, then there exists a constant c (cid:21) 0

such that

(cid:21)X;v (cid:20) c(cid:21)Y;v:

(g) Let φ : W ! V be a morphism of varieties, and let X (cid:26) V be a closed subscheme. Then

(cid:21)W;φ(cid:3)X;v

= (cid:21)V;X;v ◦ φ:

Proof. See [13, Theorem 2.1].

□

21

CHAPTER 3

PREPARATION

3.1 Useful Results

∩r

Given a nonsingular variety V defined over a number field k and a closed subscheme X, and
i=1 Di, we define the local heights corresponding to the closed

given D1; : : : ; Dr such that X =
subscheme X to be

for all points P 2 V n X and all places v 2 Mk. These are well-defined by Theorem 2.3.4. For a set
of places S (cid:26) Mk we similarly define

(cid:21)X;v„P” := min
1(cid:20)i(cid:20)r

mX;S„P” :=

f(cid:21)Di;v„P”g
∑

(cid:21)X;v„P”:

v2S

Faltings generalized Siegel’s approximation result on elliptic curves to abelian varieties as follows.

Theorem 3.1.1. Let V be an abelian variety over a number field k. Let S be a finite set of places
of k. Let X (cid:26) V be a closed subscheme and A an ample divisor on V. Let " > 0. Then for any
P 2 V„k” n Supp„X”

mX;S„P” (cid:20) "hA„P” + O„1”:

Proof. See [5, Theorem 2].

□

Roth’s theorem was also generalized to higher dimensional projective spaces.

Theorem 3.1.2 (Schmidt’s Subspace Theorem). Let k be a number field. Let S be a finite set of
places of k. For each v 2 S let fL0v; : : : ; Lnvg be a linearly independent set of linear forms in the
variables x0; : : : ; xn, with coefficients in k and let Hiv be the hyperplane in Pn associated with the
linear form Liv. Let " > 0. Then there exists a finite set H of hyperplanes of Pn
k such that for all
P 2 Pn
k

n∪

H2H H

∑

n∑

(cid:21)Hiv

v2S

i=0

;v„P” (cid:20) „Nd„D” + 1 + "”h„P”:

22

Proof. See [1, Chapter 7].

□
We say a set of m +1 hyperplanes in Pn is in j-subgeneral position if n (cid:20) j (cid:20) m and any subset
of size j + 1 of the hyperplanes will have empty intersection. In this case we get the following
result of Ru and Wong.
Theorem 3.1.3. Let k be a number field. Let S be a finite set of places of k. Let fH0; : : : ; Hmg be
a set of hyperplanes in Pn in d-subgeneral position. Let " > 0. Then there exists a finite set H of
hyperplanes of Pn

n∪

k such that for all P 2 Pn

k

H2H H

i=0
Proof. See [12, Theorem 3.5].

mHi;S„P” (cid:20) „2d (cid:0) Nd„D” + 1 + "”h„P”:

□

m∑

Generalizations of the subspace theorem to projective varieties have been given, independently,

by Corvaja and Zannier [3, Theorem 3] and by Evertse and Ferretti, whose version we state.
Theorem 3.1.4 (Evertse-Ferretti). Let X be a projective subvariety of PN of dimension n (cid:21) 1 defined
over a number field k. Let S be a finite set of places of k. For all v 2 S, let H0;v; : : : ; Hn;v 2 PN be
hypersurfaces over k such that

X \ H0;v \ (cid:1) (cid:1) (cid:1) \ Hn;v = ∅

Let " > 0. Then there exists a proper Zariski-closed subscheme Z (cid:26) X such that, for all points
P 2 X„k” n Z,

∑

n∑

;v„P”
(cid:21)Hi;v
degHi;v

< „Nd„D” + 1 + "”h„P”:

□

v2S

i=0

Proof. See [4, Theorem 1.1]

This may be reformulated in terms of divisors as follows.

Theorem 3.1.5 (Evertse-Ferretti, reformulated). Let X be a projective variety of dimension n
defined over a number field k. Let S be a finite set of places of k. For each v 2 S let D0;v; : : : ; Dn;v
be effective divisors on X, defined over k, in general position. Suppose there exists an ample divisor

23

A and positive integers di;v such that Di;v is linearly equivalent to di;v A for all i and for all v 2 S.
Let " > 0. Then there exists a proper Zariski-closed subscheme Z (cid:26) X such that for all points
P 2 X„k” n Z,

∑

n∑

;v„P”
(cid:21)Di;v
di;v

(cid:20) „Nd„D” + 1 + "”hA„P” + O„1”:

□

i=0
Proof. See [9, Theorem 3.1]

v2S

Levin proved that the linear equivalence in this reformulation may be replaced by numerical

equivalence.

Theorem 3.1.6. Let X be a projective variety of dimension n defined over a number field k. Let S
be a finite set of places of k. For each v 2 S let D0;v; : : : ; Dn;v be effective divisors on X, defined
over k, in general position. Suppose there exists an ample divisor A and positive integers di;v such
that Di;v is numerically equivalent to di;v A for all i and for all v 2 S. Let " > 0. Then there exists
a proper Zariski-closed subscheme Z (cid:26) X such that for all points P 2 X„k” n Z,

i=0
Proof. See [9, Theorem 3.2].

v2S

;v„P”
(cid:21)Di;v
di;v

(cid:20) „Nd„D” + 1 + "”hA„P” + O„1”:

□

∑

n∑

Let k be a number field and let S be a finite set of places of k containing all archimedean places.
The ring of S-integers, denoted OS, is defined to be the set of all x 2 k such that jxjv (cid:20) 1 for
all v < S. If S consists of only the archimedean places, we call these the ring of integers of k,
denoted Ok. If V is a projective variety over k and D a very ample effective divisor on V with
x = „x0 = 1; x1; : : : ; xn) a basis for L„D”, then the map P 7! „x1„P”; x2„P”; : : : ; xn„P”” defines an
embedding of V n Supp„D” into An. We say a point P 2 V„k” n Supp„D” is „D; x; S”-integral if
xi„P” 2 OS for 0 (cid:20) i (cid:20) n. note that for any point P 2 V„k” n Supp„D”, we may choose a b 2 Ok
which clears the denominators of xi„P”, 1 (cid:20) i (cid:20) n, so that x′ = „x0 = 1; bx1; : : : ; bxn” is a basis of
O„D” such that P is „D; x′; S”-integral. In order to find a more intrinsic definition, we need to look
at sets of points.

24

Lemma 3.1.7. Let D be a very ample effective divisor on V. Let R be a subset of V„k” n Supp„D”.
Then the following are equivalent.
(a) There exists a basis x = „x0 = 1; x1; : : : ; xn” of O„D” such that R is a set of „D; x; S”-integral

points.

(b) There exists a presentation D of D and an Mk-constant (cid:13) such that for all P 2 R and all

v 2 Mk n S,

(cid:21)D;v„P” (cid:20) (cid:13)v.

Proof. See [16, Lemma 1.4.1].

□

Going back to Roth’s theorem, if the points Pi are not distinct we can still say the following.

Theorem 3.1.8 (Roth’s Theorem with Multiplicities). Let S be a finite set of places of a number
field k. Let P1; : : : ; Pn 2 P1
k be distinct points and let c1; c2; : : : cn be positive integers with
c1 (cid:21) c2 (cid:21) (cid:1) (cid:1) (cid:1) (cid:21) cn. Let D =
i=1 ciPi and let " > 0. Then for P 2 P1
mD;S„P” < „c1 + c2 + "”h„P” + O„1”:

∑n

k

Proof. See [10, Theorem 2.1].

□

3.2 Sharpness of the Main Result

Let C be a nonsingular projective curve defined over a number field k. Fix a point R 2 C„k”.
For P 2 C„k” , let k„P” be the field of definition of P (the field extension of k generated by its local
coordinates).
Theorem 3.2.1. Let P1; : : : ; Pn 2 C„k” be distinct and define the divisor D :=
i=1 Pi. Let " > 0.
Denote the set of archimedean places of k by Sk;1. Then there exists a finite extension k′ of k and

∑n

25

taken over all finite k-morphisms (cid:27) :

an infinite collection of points P 2 C„k” with »k′„P” : k′… (cid:20) d satisfying

{(cid:12)(cid:12)(cid:12)(

mD;Sk′;1„P” (cid:21) „N (cid:0) "”hR„P” + O„1”

)

(cid:12)(cid:12)(cid:12)}

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

where Nd„D” := max
C ! P1 of degree d.
Proof. By finitely extending k to k′, we may assume O(cid:3)

\ Supp„D”

k′ is an infinite set.

Since D is effective, we know mD;Sk′;1 (cid:21) O„1”, so we may assume N (cid:21) 1. For each k-
morphism (cid:27) : C ! P1 of degree d, let (cid:27)„Supp„D”” = fT1; T2; : : : ; Trg and define ni = ni„(cid:27)” :=
j(cid:27)(cid:0)1„Ti” \ Supp„D”j for i = 1; 2; : : : ; r. Reorder indices so that n1 (cid:21) n2 (cid:21) : : : (cid:21) nr. Then choose
a morphism (cid:27) with T1 = 0 and T2 = 1 such that n1„(cid:27)” + n2„(cid:27)” = N.

We consider the infinite set of points P 2 C„k” satisfying (cid:27)„P” = „(cid:11); 1” for some (cid:11) 2 O(cid:3)

k′. For

each v 2 Mk′ define the local height (cid:21)0+1;v on P1 by

∑m
∑
For any effective divisor E on C and v 2 Mk′, we have (cid:21)E;v„P” (cid:21) 0. Thus we have mE;Sk′;1„P” =
i=1 Qi then mQi;Sk′;1„P” (cid:20)

v2Sk′;1 (cid:21)E;v„P” (cid:20)∑

(cid:21)E;v„P” = hE„P”. So if (cid:27)(cid:3)„0 + 1” =

v2M′
k

26

(cid:21)0+1;v„„a; b”” := log max
0(cid:20)i(cid:20)1

{

Since (cid:21)0+1;v„(cid:27)„P”” = log max
v 2 Mk′ n Sk′;1 by definition of O(cid:3)

(cid:11)

j(cid:11)jv;
k′. Thus
m(cid:27)(cid:3)„0+1”;Sk′;1„P” =

= log max

v

x0x1
for all v 2 Mk′, we see that (cid:21)0+1;v„(cid:27)„P”” = 0 for

b

a

v

v

{(cid:12)(cid:12)(cid:12)a

(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12) b

(cid:12)(cid:12)(cid:12)(cid:12)

;

}

:

}(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)„a;b”

i

{(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x2
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) 1
(cid:12)(cid:12)(cid:12)
}
∑
∑
∑

=

v

=

v2Sk′;1

v2Sk′;1

(cid:21)(cid:27)(cid:3)„0+1”;v

„P” + O„1”

(cid:21)0+1;v„(cid:27)„P”” + O„1”

(cid:21)0+1;v„(cid:27)„P”” + O„1”

v2Mk′
= h0+1„(cid:27)„P”” + O„1”
= h(cid:27)(cid:3)„0+1”„P” + O„1”

„P” and

hQi

m∑

i=1

mQi;Sk′;1„P” = m(cid:27)(cid:3)„0+1”;Sk′;1„P” + O„1”

m∑
= h(cid:27)(cid:3)„0+1”„P” + O„1”
„P” + O„1”:

=

hQi

i=1

Therefore in fact mQ;Sk′;1„P” = hQ„P” for all Q 2 Supp„(cid:27)(cid:3)„0 + 1””. So

∑

mD;Sk′;1„P” =
(cid:21)

Q2Supp„D”

∑
mQ;Sk′;1„P” + O„1”
∑

Q2Supp„D”\Supp„(cid:27)(cid:3)„0+1””

mQ;Sk′;1„P” + O„1”
hQ„P” + O„1”

=

Q2Supp„D”\Supp„(cid:27)(cid:3)„0+1””
(cid:21) „n1„(cid:27)” + n2„(cid:27)” (cid:0) "”hR„P” + O„1” by Theorem 2.2.12
= „N (cid:0) "”hR„P” + O„1”:

□

3.3 Facts on Trigonal Maps

From this point forward the symbol (cid:24) will refer to linear equivalence of divisors on a variety.
We will make frequent use of the following theorem about divisors on curves. For a divisor D

on a nonsingular projective curve C, let l„D” := dimk H0„C; O„D””.

Theorem 3.3.1 (Riemann-Roch Theorem). Let D be a divisor on a curve C of genus g. Let K be a
canonical divisor on C. Then

l„D” (cid:0) l„K (cid:0) D” = deg D + 1 (cid:0) g:

Proof. See [7, Theorem IV.1.3].

□

27

Lemma 3.3.2. Let C be a curve of genus g (cid:21) 3 defined over k. Then C cannot be both hyperelliptic
and trigonal.
Proof. Suppose C is both hyperelliptic and trigonal. Then there exist finite morphisms φ : C ! P1
and   : C ! P1 of degrees 2 and 3 respectively. By the universal property of fibered products
there exists a unique morphism φ : C ! P1 (cid:2) P1 such that the following diagram commutes.

C

φ

φ

 

P1

P1 (cid:2) P1

P1

By the commutative diagram we see the degree of φ divides the degrees of φ and  , that is 2
and 3, so the degree of φ is 1, meaning φ is a birational morphism. Thus φ„C” is an irreducible
curve on P1 (cid:2) P1 of type „2; 3”. It follows that the genus of C is bounded above by the formula
„d1 (cid:0) 1”„d2 (cid:0) 2” for the genus of a nonsingular curve of type „d1; d2” on P1 (cid:2) P1. That is to say
g (cid:20) „2 (cid:0) 1”„3 (cid:0) 1” = 2.
□
Lemma 3.3.3. Let C be a curve of genus g (cid:21) 5 defined over k. Then C can be trigonal in at most
one way (up to k-automorphism).

Proof. Suppose C is trigonal in more than one way. Then there exist two finite morphisms
φ : C ! P1 and   : C ! P1 of degree 3, distinct even up to an automorphism of P1. By the
universal property of fibered products there exists a unique morphism φ : C ! P1 (cid:2) P1 such that
the following diagram commutes.

C

φ

φ

P1

 

(cid:25)1

P1 (cid:2) P1
(cid:25)2

P1

By the commutative diagram we see the degree of φ divides the degrees of φ and  , so the degree
of φ is either 1 or 3. However if the degree of φ is 3 then the projections (cid:25)1 and (cid:25)2 each have
degree 1, so (cid:25)1jφ„C” and (cid:25)2jφ„C” are birational maps. Thus (cid:25)1jφ„C” ◦ (cid:25)2j(cid:0)1
φ„C” is a birational map

28

from P1 to P1, which necessarily extends to a k-automorphism. But then φ = (cid:25)1jφ„C” ◦ (cid:25)2j(cid:0)1
a contradiction since φ and   were distinct trigonal maps up to k-automorphism of P1.

φ„C” ◦  ,

Thus we conclude the degree of φ is 1, meaning φ is a birational morphism. Then φ„C” is an
irreducible curve on P1 (cid:2) P1 of type „3; 3”. It follows that the genus of C is bounded above by the
formula „d1 (cid:0) 1”„d2 (cid:0) 2” for the genus of a nonsingular curve of type „d1; d2” on P1 (cid:2) P1. That is
to say g (cid:20) „3 (cid:0) 1”„3 (cid:0) 1” = 4.
□

Lemma 3.3.4. Let C be a curve of genus g = 3 defined over k. Then C has only finitely many
distinct trigonal k-morphisms up to k-automorphism of P1.
Proof. Let φ : C ! P1 be a finite k-morphism of degree 3. Let D be an effective divisor defined
over k in the corresponding base-point free linear system. Then deg„D” = 3, l„D” = 2. By
Riemann-Roch, l„K (cid:0) D” = 1, which is to say there exists a unique point P 2 C„k” such that
K (cid:0) D (cid:24) P. Since K (cid:0) D is defined over k it follows that P is k-rational.

If   : C ! P1 is another trigonal morphism and E a corresponding divisor such that K (cid:0) E (cid:24) P
then D (cid:24) E, implying φ and   correspond to the same linear system. Thus φ and   are equivalent
up to automorphism of P1.

By Falting’s Theorem, there exists at most finitely many k-rational points on C. Since each
trigonal k-morphism has a corresponding k-rational point and no two distinct trigonal k-morphisms
correspond to the same point, it follows that there are finitely many trigonal k-morphisms up to
automorphism of P1.
□

Lemma 3.3.5. Let C be a curve of genus g = 4 defined over k. Then C can be trigonal in at most
two distinct ways up to a k-automorphism of P1.

Proof. Suppose C is trigonal in more than one way. Then there exist two finite morphisms
φ : C ! P1 and   : C ! P1 of degree 3, distinct even up to a k-automorphism of P1. Let D and
E be corresponding effective divisors of the respective base-point free linear systems. Let „1; f ”
and „1; g” be local equations for φ and  . Because D + E + „1”, D + E + „ f ”, D + E + „g”, and

29

D + E +„ f (cid:1) g” are all effective divisors, we can think of the functions 1, f , g, and f (cid:1) g as elements
of L„D + E”.
Assume 1,

f , g, and f (cid:1) g are k-linearly dependent. Consider the map φ : C ! P1 (cid:2) P1
given by „φ;  ” = „1; f ” (cid:2) „1; g”. Composing with the Segre embedding gives a map C ! P3 with
coordinates „1; f ; g; f (cid:1) g”. Since these are k-linearly dependent, the image lies in a hyperplane in
P3. A hyperplane in P3 cuts out a curve of type „1; 1” in P1 (cid:2) P1. Thus the image of φ lies in a curve
H of type „1; 1”. Since neither φ nor   is constant, H must be a copy of P1 embedded in P1 (cid:2) P1.
Thus we get an induced morphism φ′ : C ! H satisfying the following commutative diagram.

C

φ′

φ

 

(cid:25)1

H

P1

P1 (cid:2) P1
(cid:25)2

P1

Since H has type „1; 1” it follows that the projections (cid:25)1jH and (cid:25)2jH restricted to H are morphisms
of degree 1 from P1 to itself, that is automorphisms of P1. Thus φ = (cid:25)1jH ◦ (cid:25)2j(cid:0)1
◦  , a
contradiction since φ and   were distinct trigonal maps up to k-automorphism of P1. Thus
f , g, and f (cid:1) g must in fact be k-linearly independent. Therefore
we see the four functions 1,
l„D + E” = dimk L„D + E” (cid:21) 4.

By Riemann-Roch we have l„K (cid:0) D (cid:0) E” = l„D + E” (cid:0) 3 (cid:21) 1. Since deg„K (cid:0) D (cid:0) E” = 0 we

H

conclude K (cid:0) D (cid:0) E (cid:24) 0, that is D + E is a canonical divisor.

Let (cid:24) : C ! P1 be a finite morphism of degree 3 with corresponding effective divisor F. If (cid:24)
is not equivalent to φ (up to a k-automorphism of P1) then the same argument shows that D + F
is a canonical divisor. Thus E (cid:24) F, implying   and (cid:24) have the same corresponding linear system,
□
which is to say (cid:24) is equivalent to  .

30

CHAPTER 4

PROOF OF THE MAIN THEOREM

4.1 Algebraic Points on Curves

A divisor B on a projective variety X of dimension n is called big if there exists an a > 0 and

j0 > 0 such that H0„V; OV„jB”” (cid:21) a jn for all j > j0.

Lemma 4.1.1 (Kodaira’s Lemma). Let B be a big divisor and D an arbitrary divisor on a nonsin-
gular projective variety V. Then for m ≫ 0,

H0„V; OV„mB (cid:0) D”” , 0:

Corollary 4.1.2. Let A be an ample divisor and D an arbitrary divisor on a nonsingular projective
variety V. Then for m ≫ 0, hD„P” (cid:20) mhA„P” + O„1” for all P 2 V„k”.

Proof. Since ample divisors are big, by Kodaira’s Lemma we have

H0„V; OV„mA (cid:0) D”” , 0

for m ≫ 0. So there exists an effective divisor E such that mA (cid:24) D + E. Choose an n 2 Z so that
E + nA is very ample.

„m + n”hA„P” = h„m+n”A„P” + O„1”
= hD+E+nA„P” + O„1”
= hD„P” + hE+nA„P” + O„1”
(cid:21) hD„P” + O„1”:

□
Let C be a nonsingular projective curve of genus g (cid:21) 1 defined over a number field k.
Fix a point R 2 C„k”. For Q 2 fP 2 C : »k„P” : k… = dg let
˜Q1; : : : ; ˜Qd be the Galois
conjugates of Q, in some order, and define ¯Q = „ ˜Q1; : : : ; ˜Qd” 2 Cd and φ : Cd ! Symd„C” by

31

„P1; : : : ; Pd” 7! P1 + : : : + Pd. We identify Symd„C” with the set of effective divisors of degree d
on C. Let (cid:25)1 : Cd ! C be the first projection and let (cid:22) : Symd„C” ! Jac„C” be the map given by
P1 + : : : + Pd 7! OC„P1 + : : : + Pd (cid:0) dR”.

We’ll need the following Lemma, as shown in [9].

Lemma 4.1.3. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and define the
divisor D :=

i=1 Pi. Then for all Q 2 C„k” with »k„Q” : k… = d
1
1D;S„φ„ ¯Q”” + O„1” and hR„Q” =
d

mφ(cid:3)(cid:25)(cid:3)

1
d

mD;S„Q” =

hφ(cid:3)(cid:25)(cid:3)

1R„φ„ ¯Q”” + O„1”:

∑n

Proof. Since local heights are invariant under Galois maps,

i=1

i=1

d∑
d∑
d∑
m∑d
i=1 (cid:25)(cid:3)
mφ(cid:3)φ(cid:3)(cid:25)(cid:3)
mφ(cid:3)(cid:25)(cid:3)

i=1

mD;S„ ˜Qi” + O„1”

mD;S„(cid:25)i„ ¯Q”” + O„1”

i D;S„ ¯Q” + O„1”
m(cid:25)(cid:3)
„ ¯Q” + O„1”
i D;S
1D;S„ ¯Q” + O„1”
1D;S„φ„ ¯Q”” + O„1”:

mD;S„Q” =

=

=

=

=

=

1
d

1
d

1
d
1
d
1
d
1
d

□
The proof for global heights is identical.
Theorem 4.1.4. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and
i=1 Pi. Let " > 0. Then for all Q 2 C„k” with »k„Q” : k… = d and
define the divisor D :=
(cid:22)„φ„ ¯Q”” < (cid:22)„φ(cid:3)(cid:25)(cid:3)
„D””,
1

∑n

mD;S„Q” (cid:20) "hR„Q” + O„1”:

Proof. note that φ(cid:3)(cid:25)(cid:3)
1
sion d (cid:0) 1. By Lemma 2.3.2(a), φ(cid:3)(cid:25)(cid:3)
1

„D” is an effective divisor on Symd„C” and so a closed subscheme of dimen-
„D”””. Let A be an ample divisor on Jac„C”.

„D” (cid:26) (cid:22)(cid:3)„(cid:22)„φ(cid:3)(cid:25)(cid:3)
1

32

note that φ(cid:3)(cid:25)(cid:3)

1R is also ample. By Corollary 4.1.2 there exists an integer m > 0 such that

h(cid:22)(cid:3) A„E” (cid:20) mhφ(cid:3)(cid:25)(cid:3)

1R„E” + O„1”

for all E 2 Symd„C”. Furthermore, for all Q 2 C„k” with »k„Q” : k… = d and (cid:22)„φ„ ¯Q”” <
(cid:22)„φ(cid:3)(cid:25)(cid:3)
1

„D””,

mφ(cid:3)(cid:25)(cid:3)

1D;S„φ„ ¯Q”” (cid:20) m(cid:22)(cid:3)„(cid:22)„φ(cid:3)(cid:25)(cid:3)
= m(cid:22)„φ(cid:3)(cid:25)(cid:3)
(cid:20) "
m
"
m
(cid:20) "hφ(cid:3)(cid:25)(cid:3)

=

1D””;S„φ„ ¯Q”” + O„1” by Theorem 2.3.4(d)
1D”;S„(cid:22) ◦ φ„ ¯Q”” + O„1”

hA„(cid:22) ◦ φ„ ¯Q”” + O„1” by Theorem 3.1.1
h(cid:22)(cid:3) A„φ„ ¯Q”” + O„1”

1R„φ„ ¯Q”” + O„1” by Corollary 4.1.2.

By Lemma 4.1.3, the desired result holds on C.

□

Conversely, we see the following lemma.

Lemma 4.1.5. Let C be a nonsingular projective curve of genus g defined over a number field k.
Let P 2 C„k” and let Q 2 C„k” with »k„Q” : k… = d. Suppose that (cid:22)„φ„ ¯Q”” 2 (cid:22)„φ(cid:3)(cid:25)(cid:3)
„D””, that is
1

for some points Rj 2 C„k”. If

g (cid:21)

˜Q1 + (cid:1) (cid:1) (cid:1) + ˜Qd (cid:24) P + R1 + (cid:1) (cid:1) (cid:1) + Rd(cid:0)1

8>>>><>>>>:1 when d = 2

2 when d = 3

then Rj , ˜Qk for any j and k. In particular, there exists a k-morphism (cid:27) : C ! P1 of degree d
such that (cid:27)(cid:3)„(cid:27)„Q”” = ˜Q1 + (cid:1) (cid:1) (cid:1) + ˜Qd.

Furthermore if d = 4 and g (cid:21) 5, then we also get such a k-morphism, provided ˜Q1+ ˜Q2+ ˜Q3+ ˜Q4

is not a sum of two hyperelliptic divisors on C.

33

Proof. note that because »k„P” : k… = 1 and »k„ ˜Q j” : k… = d > 1, we know ˜Q j , P for all
1 (cid:20) j (cid:20) d.

Without loss of generality, assume Rd(cid:0)1 = ˜Qd, so that cancelling like terms would leave us

with

˜Q1 + (cid:1) (cid:1) (cid:1) + ˜Qd(cid:0)1 (cid:24) P + R1 + (cid:1) (cid:1) (cid:1) + Rd(cid:0)2:

˜Q1 = P or g = 0. But

˜Q1 , P and by

If d = 2, then this says ˜Q1 (cid:24) P, implying either

hypothesis g (cid:21) 1. Thus we reach a contradiction.
If d = 3, then this says ˜Q1 + ˜Q2 (cid:24) P + R1.

If one of

˜Q2, this would imply R1 is equal to either

˜Q1 and ˜Q2 is equal to one of P
˜Q1 or
and R1, then since P is not equal to ˜Q1 or
˜Q2, so P is linearly equivalent to one of
˜Q1 and ˜Q2 further implying g = 0, a contradiction.
On the other hand, if the points on the left side of the linear equivalence are distinct from the
points on the right side then there exists a morphism (cid:27) : C ! P1 of degree two such that
(cid:27)„P” = (cid:27)„R1”. Since g (cid:21) 2, this says C is hyperelliptic. Since P is a k-rational point and the
k(cid:157)k
unique hyperelliptic map (cid:27) is a k-morphism, R1 must be k-rational as well. Let (cid:24) 2 Gal
.
Then (cid:24)„R2” (cid:24) (cid:24)„ ˜Q1 + ˜Q2 + ˜Q3”(cid:0) P (cid:0) R1 = ˜Q1 + ˜Q2 + ˜Q3 (cid:0) P (cid:0) R1 (cid:24) R2. Since g > 0, (cid:24)„R2” = R2.
Since (cid:24) was arbitrary, R2 is k-rational, contradicting the assumption R2 = ˜Q3 since ˜Q3 is a cubic
point over k.

(

)

If d = 4, then this says ˜Q1 + ˜Q2 + ˜Q3 (cid:24) P +R1 +R2. Since g (cid:21) 5, there exists at most one trigonal
map (up to automorphism of P1), at most one hyperelliptic map (up to automorphism of P1), and not
both. If C has a unique trigonal map (cid:27) with (cid:27)„P” = (cid:27)„R1” = (cid:27)„R2”, by uniqueness, (cid:27) is k-rational,
else we could use Galois conjugation to get a new hyperelliptic morphism. Since P is a k-rational
point, we see that R1 and R2 are at worst quadratic conjugates. Thus for any (cid:24) 2 Gal
, we have
(cid:24)„R3” (cid:24) (cid:24)„ ˜Q1 + ˜Q2 + ˜Q3 + ˜Q4”(cid:0) P (cid:0) (cid:24)„R1 + R2” = ˜Q1 + ˜Q2 + ˜Q3 + ˜Q4 (cid:0) P (cid:0) R1 (cid:0) R2 (cid:24) R3. Since
g > 0, (cid:24)„R3” = R3. Since (cid:24) was arbitrary, R3 is a k-rational point, contradicting the assumption
R3 = ˜Q4, since ˜Q4 is a quartic point over k.

k(cid:157)k

(

)

If this does not give a trigonal map, say R2 = ˜Q3 as well, then C has a unique hyperelliptic
map (cid:27) with (cid:27)„ ˜Q1” = (cid:27)„ ˜Q2” and (cid:27)„P” = (cid:27)„R1” and by the same argument in the d = 3 case, R1 is

34

(

)

k(cid:157)k

k-rational. Thus for any (cid:24) 2 Gal
, we have (cid:24)„R2 + R3” (cid:24) (cid:24)„ ˜Q1 + ˜Q2 + ˜Q3 + ˜Q4” (cid:0) P (cid:0) R1 =
˜Q1 + ˜Q2 + ˜Q3 + ˜Q4 (cid:0) P (cid:0) R1 (cid:24) R2 + R3. So (cid:27)„R2” = (cid:27)„R3”. But R2 = ˜Q3 and R3 = ˜Q4,
so (cid:27)„ ˜Q3” = (cid:27)„ ˜Q4”, meaning ˜Q1 + ˜Q2 + ˜Q3 + ˜Q4 is a sum of two hyperelliptic divisors on C,
□
contradicting the hypothesis.
Theorem 4.1.6. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and define
i=1 Pi. Let (cid:27) : C ! P1 be a k-morphism of degree d. Let (cid:27)„Supp„D”” =
the divisor D :=
fT1; T2; : : : ; Trg and define ni = ni„(cid:27)” := j(cid:27)(cid:0)1„Ti” \ Supp„D”j for i = 1; 2; : : : ; r. Reorder indices
so that n1 (cid:21) n2 (cid:21) : : : (cid:21) nr. Let " > 0. Then for all Q 2 C n Supp„D” with »k„Q” : k… = d and
(cid:27)„Q” 2 P1„k”

∑n

mD;S„Q” (cid:20) „n1 + n2 + "”hR„Q” + O„1”

Proof. Let (cid:19) : P1 ! Symd„C” (cid:26) Div+„C” be given by (cid:19)„T” = (cid:27)(cid:3)„T”. Then (cid:19) is an embedding with
(cid:19)(cid:3)φ(cid:3)(cid:25)(cid:3)
1 = (cid:27)(cid:3). Since (cid:27)„Q” 2 P1„k”, any Galois map over k will only permute the set (cid:27)(cid:0)1„(cid:27)„Q””
implying (cid:19) ◦ (cid:27)„Q” = (cid:27)(cid:3)„(cid:27)„Q”” = ˜Q1 + (cid:1) (cid:1) (cid:1) + ˜Qd. Thus up to O„1”
„D”;S„φ„ ¯Q”” by Lemma 4.1.3
„D”;S„i ◦ (cid:27)„Q””
„D”;S„(cid:27)„Q””

mD;S„Q” =

=

=

mφ(cid:3)(cid:25)(cid:3)
1
mφ(cid:3)(cid:25)(cid:3)
1
mi(cid:3)φ(cid:3)(cid:25)(cid:3)
1
m(cid:27)(cid:3)„D”;S„(cid:27)„Q””
„n1 + n2 + "”h(cid:27)„Q”„(cid:27)„Q”” by Theorem 3.1.8
„n1 + n2 + "”h(cid:27)(cid:3)(cid:27)„Q”„Q”
„n1 + n2 + "”

1
d
1
d
1
d
1
=
d
(cid:20) 1
d
1
d
1
d
(cid:20) „n1 + n2 + "”„1 + "”hR„Q”
= „n1 + n2 + "„n1 + n2 + 2””hR„Q”:

„Q” + (cid:1) (cid:1) (cid:1) + h ˜Qd

h ˜Q1

„Q”

]

[

=

=

And replacing " with "(cid:157)„2d + 2” gives the desired result.

□

35

∑n

Lemma 4.1.7. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and define the
i=1 Pi. Let " > 0. Suppose there exist only a finite number of k-morphisms C ! P1
divisor D :=
of degree d (modulo automorphisms of P1). Then for all Q 2 C„k” with »k„Q” : k… = d such that
(cid:27)„Q” 2 P1„k” for some k-morphism (cid:27) : C ! P1 of degree d, we have
mD;S„Q” (cid:20) „Nd„D” + "”hR„Q” + O„1”

\ Supp„D”

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

where Nd„D” := max
C ! P1 of degree d.
Proof. Let f(cid:27)jg be the unique set of representatives of the equivalence classes of k-morphisms
C ! P1 of degree d (modulo automorphisms of P1) such that

taken over all finite k-morphisms (cid:27) :

n1„(cid:27)j” =

„0” \ Supp„D”

„1” \ Supp„D”

(cid:12)(cid:12)(cid:12) and n2„(cid:27)j” =

(cid:12)(cid:12)(cid:12)(cid:27)(cid:0)1

j

(cid:12)(cid:12)(cid:12)

{(cid:12)(cid:12)(cid:12)(

(cid:12)(cid:12)(cid:12)(cid:27)(cid:0)1

j

for all j, where ni is as defined in Theorem 4.1.6. By definition it is clear that N = n1„(cid:27)j” + n2„(cid:27)j”
for some j. Let Q 2 C„k” with »k„Q” : k… = d such that (cid:27)„Q” 2 P1„k” for some k-morphism
(cid:27) : C ! P1 of degree d. After composing with an automorphism of P1 we may assume (cid:27) = (cid:27)j
for some j. Thus by Theorem 4.1.6, for all Q 2 C„k” with »k„Q” : k… = d such that (cid:27)„Q” 2 P1„k”
for some k-morphism (cid:27) : C ! P1 of degree d, we have

mD;S„Q” (cid:20) max

j

f„n1„(cid:27)j” + n2„(cid:27)j” + "”hR„Q” + Oj„1”g

(cid:20) „Nd„D” + "”hR„Q” + O„1”:

□
Corollary 4.1.8. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and define
the divisor D :=

i=1 Pi. Let " > 0. If g (cid:21) 2, then for all Q 2 C„k” with »k„Q” : k… = 2,

∑n

mD;S„Q” (cid:20) „Nd„D” + "”hR„Q” + O„1”

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

\ Supp„D”

taken over all finite k-morphisms (cid:27) :

{(cid:12)(cid:12)(cid:12)(

where Nd„D” := max
C ! P1 of degree 2.

(cid:12)(cid:12)(cid:12)}

(cid:12)(cid:12)(cid:12)}

)

)

36

Proof. For those Q 2 C„k” with (cid:22)„φ„ ¯Q”” < (cid:22)„φ(cid:3)(cid:25)(cid:3)
„D”” the result follows from Theorem 4.1.4,
1
regardless of the value of Nd„D”. For those Q 2 C„k” with »k„Q” : k… = 2 and (cid:22)„φ„ ¯Q”” 2
(cid:22)„φ(cid:3)(cid:25)(cid:3)
„D””, Lemma 4.1.5 implies there exists a morphism (cid:27) : C ! P1 of degree 2 such that
1
(cid:27)(cid:3)„(cid:27)„Q”” = φ„ ¯Q”, that is (cid:27)„Q” 2 P1„k”. Since g (cid:21) 2, such a hyperelliptic map is unique up to an
automorphism of P1, so the result follows from Lemma 4.1.7.
□
Corollary 4.1.9. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and define
the divisor D :=

i=1 Pi. Let " > 0. If g (cid:21) 3, then for all Q 2 C„k” with »k„Q” : k… = 3,

∑n

{(cid:12)(cid:12)(cid:12)(

(cid:12)(cid:12)(cid:12)}

\ Supp„D”

mD;S„Q” (cid:20) „Nd„D” + "”hR„Q” + O„1”

)

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

taken over all finite k-morphisms (cid:27) :

where Nd„D” := max
C ! P1 of degree 3.
Proof. For those Q 2 C„k” with (cid:22)„φ„ ¯Q”” < (cid:22)„φ(cid:3)(cid:25)(cid:3)
„D”” the result follows from Theorem 4.1.4,
1
regardless of the value of Nd„D”. For those Q 2 C„k” with »k„Q” : k… = 3 and (cid:22)„φ„ ¯Q”” 2
(cid:22)„φ(cid:3)(cid:25)(cid:3)
„D””, Lemma 4.1.5 implies there exists a morphism (cid:27) : C ! P1 of degree 3 such that
1
(cid:27)(cid:3)„(cid:27)„Q”” = φ„ ¯Q”, that is (cid:27)„Q” 2 P1„k”. If g (cid:21) 5, by Theorem 3.3.2 such a trigonal morphism
is unique up to an automorphism of P1. If g = 4, then by Theorem 3.3.5 there exist at most two
such trigonal morphisms. If g = 3, then by Theorem 3.3.4 there are only finitely many trigonal
□
k-morphisms. So the result follows from Lemma 4.1.7.

4.2 Quadratic Points on Elliptic Curves

First, a general lemma.

∑n

Lemma 4.2.1. Let C be a projective nonsingular curve of genus g defined over a number field
k. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and define the divisor
i=1 Pi. Let R 2 C„k” and let " > 0. Let d be a positive integer and let Z be a finite union of
D :=
irreducible curves in Symd„C”.

37

(a) If g = 1 and d = 2 and there does not exist a subset fi1; : : : ; i4g (cid:26) f1; : : : ; ng such that

Pi1 + Pi2

(cid:24) Pi3 + Pi4

then for all Q 2 C„k” with »k„Q” : k… = 2 and φ„ ¯Q” 2 Z we have
mD;S„Q” (cid:20) „3 + "”hR„Q” + O„1”:

(b) If g = 1 and d = 3 and there does not exist a subset fi1; : : : ; i6g (cid:26) f1; : : : ; ng such that

Pi1 + Pi2 + Pi3

(cid:24) Pi4 + Pi5 + Pi6

then for all Q 2 C„k” with »k„Q” : k… = 3 such that φ„ ¯Q” 2 Z we have

mD;S„Q” (cid:20) „5 + "”hR„Q” + O„1”:

(c) If g = 2 and d = 3 and there does not exist a subset fi1; : : : ; i6g (cid:26) f1; : : : ; ng such that

Pi1 + Pi2 + Pi3

(cid:24) Pi4 + Pi5 + Pi6

then for all Q 2 C„k” with »k„Q” : k… = 2 such that φ„ ¯Q” 2 Z we have

mD;S„Q” (cid:20) „5 + "”hR„Q” + O„1”:

Furthermore if there does not exist a subset fi1; : : : ; i5g (cid:26) f1; : : : ; ng and a point T 2 C„k”
distinct from Pi1

; Pi3 such that

; Pi2

Pi1 + Pi2 + Pi3

(cid:24) Pi4 + Pi5 + T

then for all Q 2 C„k” with »k„Q” : k… = 3 such that φ„ ¯Q” 2 Z we have

mD;S„Q” (cid:20) „4 + "”hR„Q” + O„1”:

∑r

Proof. Let Z =
Supp„φ(cid:3)(cid:25)(cid:3)

j=1 Z j be the decomposition of Z into irreducible curves. Since φ„ ¯Q” <
1D” is nonempty, that is Z j is not a subset of

1D”, we may assume Z j n Supp„φ(cid:3)(cid:25)(cid:3)

38

Supp„φ(cid:3)(cid:25)(cid:3)
1Pi” for i = 1; : : : ; n. We think of Z j with the reduced induced closed subscheme
structure and let (cid:26)j : Z j ! X be the closed immersion. If there are only finitely many points
Q 2 C„k” such that »k„Q” : k… = d and φ„ ¯Q” 2 Z j, then we are done. Else Z j has infinitely many
k-rational points, thus by Falting’s Theorem Z j has genus 0 or 1. If Z j has genus 1 then for those
Q 2 C„k” with »k„Q” : k… = d and φ„ ¯Q” 2 (cid:26)j„Z j”,

mD;S„Q” =

=

„φ„ ¯Q””” + O„1”
j
„φ„ ¯Q””” + O„1” by Lemma 3.1.1

1D;S„φ„ ¯Q”” + O„1” by Lemma 4.1.3
mφ(cid:3)(cid:25)(cid:3)
1D;S„(cid:26)(cid:0)1
φ(cid:3)(cid:25)(cid:3)
m(cid:26)(cid:3)
j
1R„(cid:26)(cid:0)1
h(cid:26)(cid:3)
φ(cid:3)(cid:25)(cid:3)
j
1R„φ„ ¯Q”” + O„1”
hφ(cid:3)(cid:25)(cid:3)

1
d
1
d
(cid:20) "
d
"
d
= "hR„Q” + O„1” by Lemma 4.1.3.

=

j

Alternatively if Z j has genus 0 then, since abelian varieties admit no rational subvarieties,
the image (cid:22)„Z j” 2 Jac„C” = C is a point. Thus Z j is contained in a fiber Y of (cid:22). Let E 2 Y.
Then the subvariety Y (cid:26) Symd„C” may be thought of as the complete linear system jEj of the
degree d divisor E on C. Furthermore the divisor φ(cid:3)(cid:25)(cid:3)
1PijY may be thought of as the linear system
jE (cid:0) Pij + Pi, and thus as a linear subspace of the projective space Y.

If d = g + 1, where g = 1 or 2, then by Riemann-Roch the complete linear systems of divisors
of degree d all have dimension 1, so Y = P1. Since φ(cid:3)(cid:25)(cid:3)
1PijY is a linear subspace, it is either a
single point, or all of Y. Furthermore, because Z j has dimension 1, it follows that Z j = Y. So
either Z j (cid:26) Supp„φ(cid:3)(cid:25)(cid:3)

1Pi”, a contradiction, or (cid:26)(cid:3)

1PijY is a degree 1 divisor.

1Pi = φ(cid:3)(cid:25)(cid:3)

φ(cid:3)(cid:25)(cid:3)

j

If g = 1 and d = 3, then by Riemann-Roch, the complete linear systems of divisors of degree
1PijY
1Pi are in 3-subgeneral position, thus so
1PijY. By Theorem 3:1:3, there exists a finite set H of lines in Y such that for all

d all have dimension 2. Riemann-Roch also tells us that jE (cid:0) Pij has dimension 1 and so φ(cid:3)(cid:25)(cid:3)
is a line in the projective plane Y. note the divisors φ(cid:3)(cid:25)(cid:3)
are the lines φ(cid:3)(cid:25)(cid:3)

39

P 2 Y n∪

H2H H

n∑

mφ(cid:3)(cid:25)(cid:3)

1DjY;S„P” =

mφ(cid:3)(cid:25)(cid:3)

1PijY;S„P” + O„1”

i=1
(cid:20) „5 + "”h„P” + O„1” by Theorem 3.1.3,

where h is the canonical height on Y (cid:27) P2. Thus if Z j < H, then for all Q 2 C„k” with
»k„Q” : k… = d and φ„ ¯Q” 2 (cid:26)j„Z j”
1
mD;S„Q” =
1D;S„φ„ ¯Q”” + O„1” by Lemma 4.1.3
3mφ(cid:3)(cid:25)(cid:3)
1D;S„(cid:26)(cid:0)1
1
„φ„ ¯Q””” + O„1”
3m(cid:26)(cid:3)
φ(cid:3)(cid:25)(cid:3)
=
j
1R„(cid:26)(cid:0)1
(cid:20) 1
„5 + "”h(cid:26)(cid:3)
φ(cid:3)(cid:25)(cid:3)
3
1
„5 + "”hφ(cid:3)(cid:25)(cid:3)
1R„φ„ ¯Q”” + O„1”
3
= „5 + "”hR„Q” + O„1” by Lemma 4.1.3.

„φ„ ¯Q””” + O„1”

=

j

j

j

Since this would satisfy every inequality in part (b), we need not consider this case any further. If
1Pi”, a contradiction, or (cid:26)(cid:3)
Z j 2 H, then either Z j (cid:26) Supp„φ(cid:3)(cid:25)(cid:3)
φ(cid:3)(cid:25)(cid:3)
„Pi” can coincide, and if c divisors
1
φ(cid:3)(cid:25)(cid:3)
(cid:26)(cid:3)
j

At most d of the divisors (cid:26)(cid:3)

1Pi has degree 1.

”; : : : ; (cid:26)(cid:3)
j

1„Pic”

φ(cid:3)(cid:25)(cid:3)

φ(cid:3)(cid:25)(cid:3)

1„Pi1

j

j

coincide, then

Pi1 + : : : + Pic + Rc+1 + : : : + Rd (cid:24) E

∑p

for some Rc+1; : : : ; Rd 2 C„k”. Let (cid:26)(cid:3)
c1; c2; (cid:1) (cid:1) (cid:1) ; cp with c1 (cid:21) c2 (cid:21) (cid:1) (cid:1) (cid:1) (cid:21) cp. Then for Q 2 C„k” with »k„Q” : k… = d and φ„ ¯Q” 2 Z j

m=1 cmEm for some positive integers

„D” =

φ(cid:3)(cid:25)(cid:3)
1

j

mD;S„Q” =

=

„D”;S„φ„ ¯Q”” + O„1” by Lemma 4.1.3
„D”jYjk

mφ(cid:3)(cid:25)(cid:3)
1
;S„φ„ ¯Q”” + O„1”
mφ(cid:3)(cid:25)(cid:3)
1
„c1 + c2 + "”hφ(cid:3)(cid:25)1(cid:3)„R”jYjk
„c1 + c2 + "”hφ(cid:3)(cid:25)1(cid:3)„R”„φ„ ¯Q”” + O„1”

1
d
1
d
(cid:20) 1
d
1
d
= „c1 + c2 + "”hR„Q” + O„1” by Lemma 4.1.3:

=

„φ„ ¯Q”” + O„1” by Lemma 3.1.8

40

If there does not exist a subset fi1; : : : ; i2dg (cid:26) f1; : : : ; ng such that
(cid:24) Pid+1 + : : : + Pi2d

Pi1 + : : : + Pid

then for all Q 2 C„k” with »k„Q” : k… = d such that φ„ ¯Q” 2 Z j we have
mD;S„Q” (cid:20) „2d (cid:0) 1 + "”hR„Q” + O„1”:

Finally, in the special case where g = 2 and d = 3, if there does not exist a subset fi1; : : : ; i5g (cid:26)

f1; : : : ; ng and a point T 2 C„k” distinct from Pi1
Pi1 + Pi2 + Pi3

; Pi3 such that
; Pi2
(cid:24) Pi4 + Pi5 + T

then either such a linear equivalence does not exist for any T 2 C„k” , in which case, for all Q 2 C„k”
with »k„Q” : k… = 3 such that φ„ ¯Q” 2 Z, we have

mD;S„Q” (cid:20) „4 + "”hR„Q” + O„1”:

Else, there does exist a subset fi1; : : : ; i5g (cid:26) f1; : : : ; ng and a point T 2 C„k” such that

Pi1 + Pi2 + Pi3

(cid:24) Pi4 + Pi5 + T

; Pi2

; Pi3

; Pi2

; Pi3

1PijY has degree 1 as shown above.

g. However since every other point in the linear
with either T < C„k” or T 2 fPi1
equivalence is k-rational and thus fixed under Galois maps over k, it follows that T is also fixed,
g, say T = Pi3. But that would imply
and so T 2 C„k” automatically. Thus T 2 fPi1
φ(cid:3)(cid:25)(cid:3)
1Pi3 intersects the projective line Y = Z j in more than one point, contradicting the fact that
φ(cid:3)(cid:25)(cid:3)
1Pi = φ(cid:3)(cid:25)(cid:3)
(cid:26)(cid:3)
j
Since there are only finitely many irreducible components Z j in Z, we may take the maximum
of the constants O„1” in each of the above inequalities to prove the same inequalities holds for all
Q 2 C„k” with »k„Q” : k… = d such that φ„ ¯Q” 2 Z.
□
Proposition 4.2.2. Suppose g = 1. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be
i=1 Pi. Let " > 0. Then for all Q 2 C„k” with »k„Q” : k… = 2
distinct and define the divisor D :=

∑n

mD;S„Q” (cid:20) „4 + "”hR„Q” + O„1”

41

and if there does not exist a subset fi1; : : : ; i4g (cid:26) f1; : : : ; ng such that

Pi1 + Pi2
then for all Q 2 C„k” with »k„Q” : k… = 2 we have

(cid:24) Pi3 + Pi4

mD;S„Q” (cid:20) „3 + "”hR„Q” + O„1”:

„R” + (cid:25)(cid:3)
2

„R” + (cid:25)(cid:3)
2

„R”” = 2φ(cid:3)(cid:25)(cid:3)
1

„R” is an ample divisor on Sym2„C” and therefore so is φ(cid:3)(cid:25)(cid:3)
1

Proof. We may assume n > 2. Since (cid:25)(cid:3)
1
φ(cid:3)„(cid:25)(cid:3)
1
Since any two points on a curve are numerically equivalent, we also have φ(cid:3)(cid:25)(cid:3)
1
equivalent to φ(cid:3)(cid:25)(cid:3)
1
of φ(cid:3)(cid:25)(cid:3)
1

„R” is an ample divisor on C2, we see that
„R”.
„Pi” is numerically
„Pi” are in general position since the intersection

” in Sym2„C” is the point Pi1 + Pi2.

„R” for all i. The divisors φ(cid:3)(cid:25)(cid:3)
1

For each v 2 S, choose two of the divisors φ(cid:3)(cid:25)(cid:3)
1

„Pi” to be D1;v and D2;v. Then applying Theorem
3.1.6 with X = Sym2„C”, A = φ(cid:3)(cid:25)(cid:3)
„R” and di = 1, for 1 (cid:20) i (cid:20) 2, we see there exists a proper
1
Zariski-closed subscheme Z (cid:26) Sym2„C” such that for all points Q 2 C„k” with »k„Q” : k… = 2 and

„Pi1

” and φ(cid:3)(cid:25)(cid:3)
1

„Pi2

φ„ ¯Q” 2 Sym2„C”„k” n Z,∑

2∑

;v„φ„ ¯Q”” < „3 + "”hφ(cid:3)(cid:25)(cid:3)
1

„P1”„φ„ ¯Q””:

i=1

v2S

(cid:21)Di;v
Since at most two of the divisors φ(cid:3)(cid:25)(cid:3)
„Pi” intersect at a point, we see that a point in Sym2„C”„k”n Z
1
can be v-adically close to at most two of these divisors for each v 2 S. Points v-adically closest
to D1;v and D2;v cannot be (arbitrarily) v-adically close to any other of the divisors φ(cid:3)(cid:25)(cid:3)
„Pi”, that
1
„Pi” are bounded for such points. Thus for all points Q 2 C„k” with
is those local heights (cid:21)φ(cid:3)(cid:25)(cid:3)
1
»k„Q” : k… = 2 and φ„ ¯Q” 2 Sym2„C”„k” n Z with φ„ ¯Q” v-adically closest to D1;v and D2;v for each

42

v 2 S,

mD;S„Q” =

=

„φ„ ¯Q”” + O„1”

∑
n∑
1
„D”;S„φ„ ¯Q”” + O„1” by Lemma 4.1.3
2mφ(cid:3)(cid:25)(cid:3)
2∑
∑
1
(cid:21)φ(cid:3)(cid:25)(cid:3)
„Pi”;v
2
1
;v„φ„ ¯Q”” + O„1”
„R”„φ„ ¯Q”” + O„1”

1
2
< 1
2
= „3 + "”hR„Q” + O„1” by Lemma 4.1.3.

(cid:21)Di;v
v2S
„3 + "”hφ(cid:3)(cid:25)(cid:3)
1

i=1

1

v2S

i=1

=

Since this inequality does not depend upon the choices of D1;v and D2;v for each place v 2 S, the
result follows for all Q 2 C„k” with »k„Q” : k… = 2 and φ„ ¯Q” 2 Sym2„C”„k” n Z.

Since φ„ ¯Q” 2 Sym2„C”„k” it only remains to consider the case where φ„ ¯Q” 2 Z. By Lemma

4.2.1, then for all Q 2 C„k” with »k„Q” : k… = 2 such that φ„ ¯Q” 2 Z we have

mD;S„Q” (cid:20) „4 + "”hR„Q” + O„1”

and if there does not exist a subset fi1; i2; i3; i4g (cid:26) f1; : : : ; ng such that

Pi1 + Pi2

(cid:24) Pi3 + Pi4

then for all Q 2 C„k” with »k„Q” : k… = 2 such that φ„ ¯Q” 2 Z we have

mD;S„Q” (cid:20) „3 + "”hR„Q” + O„1”:

Combining the above inequalities, we get that the proposition holds for all Q 2 C„k” with »k„Q” :
k… = 2.
□

Corollary 4.2.3. Let C be a nonsingular projective curve of genus-1 defined over a number field k
and let R 2 C„k”. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and define
the divisor D :=

i=1 Pi. Let " > 0. Then for all Q 2 C„k” with »k„Q” : k… = 2,

∑n

mD;S„Q” (cid:20) „N2„D” + "”hR„Q” + O„1”

43

{(cid:12)(cid:12)(cid:12)(

)

(cid:12)(cid:12)(cid:12)}

\ Supp„D”

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

taken over all k-morphisms (cid:27) : C ! P1

where N2„D” := max
of degree 2. If no such k-morphism exists, we say N2„D” = 0.
Proof. First note that since degree 2 morphisms are at most 2-to-1, we have N2„D” (cid:20) 4. By
Riemann-Roch, the dimension of the complete linear system jEj of any divisor E on C of positive
degree is given by dimjEj = deg„E” (cid:0) 1. If there are only finitely many points Q 2 C„k” with
»k„Q” : k… = 2, then mD;S„Q” (cid:20) O„1”, so we assume there are infinitely many points Q 2 C„k” with
»k„Q” : k… = 2. In particular, there exists a pair of conjugate k-quadratic points Q1; Q2 2 C„k”.
Since jQ1 + Q2j=1 and jQ1 + Q2 (cid:0) Tj = 0 for all T 2 C„k”, it follows that Q1 + Q2 is a degree 2
divisor defined over k, whose complete linear system, jQ1 + Q2j, is base-point-free of dimension
1. Thus it induces a degree 2 morphism ! : C ! P1 defined over k.

If deg„D” = 1, then after composing with a k-automorphism of P1, we may assume !„P1” = 0,
so N2„D” = 1. If deg„D” = 2, then after composing with a k-automorphism of P1, we may assume
!„P1” = 0 and either !„P2” = 0 or !„P2” = 1, so N2„D” = 2. In both these cases N = deg„D”, so
we have

mD;S„Q” (cid:20) hD„Q” by Definition 2.2.8

(cid:20) „deg„D” + "”hR„Q” + O„1” by Theorem 2.2.12
= „Nd„D” + "”hR„Q” + O„1”:

Suppose deg„D” (cid:21) 3. Then since jP1 + P2j=1 and jP1 + P2 (cid:0) Tj = 0 for all T 2 C„k”, it
follows that P1 + P2 is a degree 2 divisor defined over k, whose complete linear system, jP1 + P2j,
is base-point-free of dimension 1. Thus it induces a degree 2 morphism (cid:27) : C ! P1 defined
over k, such that (cid:27)„P1” = (cid:27)„P2”. By composing with an automorphism of P1, we may assume
(cid:27)„P1” = (cid:27)„P2” = 0 and (cid:27)„P3” = 1. Thus we see N2„D” (cid:21) 3.

If N2„D” = 4, then the inequality follows from Proposition 4.2.2.
Conversely, if there exists a subset fi1; i2; i3; i4g (cid:26) f1; : : : ; ng such that Pi1 + Pi2
” = (cid:27)„Pi2

then there exists a k-morphism (cid:27) : C ! P1 of degree 2 such that (cid:27)„Pi1
(cid:27)„Pi3

” = 1, so that N2„D” = 4.

” = (cid:27)„Pi4

(cid:24) Pi3 + Pi4,
” = 0 and

44

Thus if N2„D” = 3, then such a subset of f1; : : : ; ng does not exist and the inequality again
□

follows from Proposition 4.2.2.

4.3 Cubic Points on Curves of Low Genus

It remains to show the main theorem holds for points of degree 3 on curves of genus 1 and 2.

Theorem 4.3.1. Let X be an irreducible projective surface defined over a number field k. Let S be
a finite set of places of k and let n 2 N. For each v 2 S, let D1;v; : : : ; Dn;v be effective divisors on
X, defined over k having no irreducible components in common in their supports. Suppose there
exists an ample divisor A and positive integers di;v such that Di;v is numerically equivalent to di;v A
for all i and for all v 2 S. Suppose there exists a nonconstant k-morphism (cid:27) : X ! E for some
elliptic curve E defined over k. Let " > 0. Then there exists a proper closed subscheme Z (cid:26) X

such that for all points P 2 X„k” n Z,∑
n∑

(cid:20) „3 + "”hA„P” + O„1”:
Proof. Let R 2 E„k”. By Theorem 3.1.1, for any Q 2 E„k” we have

v2S

;v„P”
(cid:21)Di;v
di;v

i=1

mQ;S„P” (cid:20) "hR„P” + O„1”

for all P 2 E„k” n fQg. Thus for any Q 2 E„k”

m(cid:27)(cid:3)Q;S„P” = mQ;S„(cid:27)„P”” + O„1”
(cid:20) "hR„(cid:27)„P”” + O„1”
= "h(cid:27)(cid:3)R„P” + O„1”

for all P 2 X„k” n (cid:27)(cid:3)„Q”. By Corollary 4.1.2, there exists an M 2 N, depending only on R and A,
such that for all P 2 X„k”

"h(cid:27)(cid:3)R„P” (cid:20) "M (cid:1) hA„P” + O„1”:

By replacing " with "(cid:157)M we can say that for any Q 2 E„k”

m(cid:27)(cid:3)Q;S„P” (cid:20) "hA„P” + O„1”

45

for all P 2 X„k” n (cid:27)(cid:3)„Q”. Therefore, by Lemma 2.3.2(a) and Theorem 2.3.4(d), for any Q 2 X„k”
we have

mQ;S„P” (cid:20) m(cid:27)(cid:3)„(cid:27)„Q””;S„P” + O„1”

(cid:20) "hA„P” + O„1”

for all P 2 X„k” n (cid:27)(cid:3)„(cid:27)„Q””.

For each v 2 S, let

∪

Yv =

1(cid:20)i< j(cid:20)n

Supp„Di;v” \ Supp„Dj;v”:

Since no two of the divisors share irreducible components in common, it follows that Yv is a finite
set of points. Partition the set X by which of the points of Yv each point is v-adically closest to.
note that if a point is v-adically closest to two points of Yv, then it cannot be v-adically close to any
of the divisors Di;v, that is all of the local heights (cid:21)Di;v
;v will be bounded. Consider those points
P that are v-adically closest to Q 2 Yv.

For each pair of indices i and j we have, by definition,

(cid:21)Di;v

\Dj;v

;v„P” = minf(cid:21)Di;v

;v„P”; (cid:21)Dj;v

;v„P”g

for all P 2 X„k” n Supp„Di;v” \ Supp„Dj;v”. But if Q 2 Supp„Di;v” \ Supp„Dj;v”, we have

(cid:21)Q;v„P” = minf(cid:21)Di;v

;v„P”; (cid:21)Dj;v

;v„P”g + O„1”

for those points P 2 X„k” n fQg such that Q is the v-adically closest point to P out of all of the
points in Yv. Thus we have

minf(cid:21)Di;v

;v„P”; (cid:21)Dj;v

;v„P”g (cid:20) "hA„P” + O„1”

for all such P 2 X„k”n (cid:27)(cid:3)„(cid:27)„Q””. Since we can repeat this for each pair of indices i and j, it follows
;v„P”, can be greater than " (cid:1) hA„P” for
that at most one of the local heights (cid:21)Di;v
infinitely many of the points P 2 X„k” n (cid:27)(cid:3)„(cid:27)„Q”” that are v-adically closest to Q. As such

;v„P”, say (cid:21)DiQ;v

;v„P”
(cid:21)Di;v
di;v

(cid:20)

;v„P”
(cid:21)DiQ;v
di;v

+ "„n (cid:0) 1”hA„P” + O„1”

∑

i

46

(∪n

)

that are v-adically closest to Q.

for all P 2 X„k” n

i=1 Supp„Di;v” [ (cid:27)(cid:3)„(cid:27)„Q””

Repeating this process for each v 2 S gives a collection of partitions, which together make a
finer partition of X into those Qv 2 Yv that the points are closest to for each v 2 S together with a
proper (since (cid:27) is nonconstant) closed subscheme of X,

Then we have

∑
n∑
for all P 2 X„k” n∪

v2S
v2S

i=0

∪

∪

v2S

Q2Yv

(cid:27)(cid:3)„(cid:27)„Q””:

Z1 =

∑

)

;v„P”

(cid:20)

v2S

(∪n

(cid:21)DiQv
;v
di;v

;v„P”
(cid:21)Di;v
di;v
i=1 Supp„Di;v” [ (cid:27)(cid:3)„(cid:27)„Qv””
∑
∪

;v„P”

+ "„n (cid:0) 1”jSjhA„P” + O„1”

v 2 S. But by Theorem 3.1.6, there exists a proper closed subscheme Z2 (cid:26) X such that

that are v-adically closest to Qv for each

v2S
for all P 2 X„k” n Z2. Let Z =
v2S
;v„P”
(cid:21)Di;v
di;v

∑

n∑

v2S

i=0

∪n

(cid:20) „3 + "”hA„P”

(cid:21)DiQv
;v
di;v
i=1 Supp„Di;v” [ Z1 [ Z2. Then for all P 2 X„K” n Z

(cid:20) 3hA„P” + " „1 + „n (cid:0) 1”jSj” hA„P” + O„1”:

Thus by replacing " with "(cid:157)„1 + „n (cid:0) 1”jSj”, we obtain the desired result.

□

Proposition 4.3.2. Let C be an elliptic curve defined over k. Let S be a finite set of places of k.
Let P1; : : : ; Pn 2 C„k” be distinct and define the divisor D :=
i=1 Pi. Let R 2 C„k” and let " > 0.
Then for all Q 2 C„k” with »k„Q” : k… = 3

∑n

mD;S„Q” (cid:20) „6 + "”hR„Q” + O„1”

and if there does not exist a subset fi1; : : : ; i6g (cid:26) f1; : : : ; ng such that

Pi1 + Pi2 + Pi3
then for all Q 2 C„k” with »k„Q” : k… = 3 we have

(cid:24) Pi4 + Pi5 + Pi6

mD;S„Q” (cid:20) „5 + "”hR„Q” + O„1”:

47

Proof. By Theorem 3.1.6, there exists a proper closed subscheme Z (cid:26) Sym3„C” such that for all
Q 2 C„k” such that »k„Q” : k… = 3 and φ„ ¯Q” < Z we have

1
mD;S„Q” =
3mφ(cid:3)(cid:25)(cid:3)
(cid:20) 1
„4 + "”hφ(cid:3)(cid:25)(cid:3)
3
= „4 + "”hR„Q” + O„1” by Lemma 4.1.3.

1D;S„φ„ ¯Q”” + O„1” by Lemma 4.1.3
1R„Q” + O„1” by Theorem 3.1.6

Let Z j be an irreducible component of Z. Then (cid:22)jZ j is a k-morphism from the irreducible surface
Z j to an elliptic curve Jac„C” = C.

If (cid:22)jZ j is a constant map then Z j is contained in a fiber of (cid:22). But the fibers of (cid:22) are 2-dimensional
by Riemann-Roch, so it follows that Z j is a fiber of (cid:22), so there exists an E 2 Div„C” such that
Z j = jEj (cid:27) P2. Since φ(cid:3)(cid:25)(cid:3)
1Pi is the set of effective divisors on C with Pi in their support, we
see that it cuts out the 1-dimensional linear system jE (cid:0) Pij + Pi in jEj. Furthermore the divisors
„φ(cid:3)(cid:25)(cid:3)
1Pi”jZ j on Z j are in 3-subgeneral position. Thus we may apply Theorem 3.1.3 with r = 2 and
Li = „φ(cid:3)(cid:25)(cid:3)

1Pi”jZ j to get

mD;S„Q” =

=

1D;S„φ„ ¯Q”” + O„1” by Lemma 4.1.3
mLi;S„φ„ ¯Q”” + O„1”

n∑
1
3mφ(cid:3)(cid:25)(cid:3)
1
3
(cid:20) 1
3
= „5 + "”hR„Q” + O„1” by Lemma 4.1.3.

1R„Q” + O„1” by Theorem 3.1.3

i=1
„5 + "”hφ(cid:3)(cid:25)(cid:3)

for all Q 2 C„k” such that »k„Q” : k… = 3 and φ„ ¯Q” 2 Z j n Yj, where Yj is a union of hyperplanes
Yjk (cid:26) Z j (cid:27) P2. By Lemma 4.2.1, if there does not exist a subset fi1; i2; i3; i4; i5; i6g (cid:26) f1; : : : ; ng
(cid:24) Pi4 + Pi5 + Pi6, then for all Q 2 C„k” with »k„Q” : k… = 3 and φ„ ¯Q” 2 Yj
such that Pi1 + Pi2 + Pi3

otherwise

mD;S„Q” (cid:20) „5 + "”hR„Q” + O„1”

mD;S„Q” (cid:20) „6 + "”hR„Q” + O„1”:

48

Combining the above inequalities we get the result on all of Z j. Since there are only finitely many
irreducible components Z j in Z, we may take the maximum of all of the above constants in O„1” to
get the result for all Q 2 C„k” with »k„Q” : k… = 3.
□

Lemma 4.3.3. Let C be a nonsingular projective curve of genus 1 over an algebraic number field
k. Let d (cid:21) 3 be an integer and let P1; : : : ; P2d(cid:0)1 2 C„k” be distinct. Let P2d(cid:0)1 be the identity
element of the elliptic curve C. Suppose that for every morphism (cid:27) : C ! P1 over k of degree d
we have

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

\ fP1; : : : ; P2d(cid:0)1g

(cid:12)(cid:12)(cid:12) (cid:20) 2d (cid:0) 2:

Then for every i 2 f1; : : : ; 2d (cid:0) 2g, the inverse of Pi in the elliptic curve C is Pj for some
j 2 f1; : : : ; 2d (cid:0) 2g n fig.

Proof. We proceed by induction on the number of pairs of inverse points identified. Trivially,
we have already identified zero such pairs. Suppose we have identified m pairs already and that
m < d (cid:0) 1. Let M (cid:26) f1; : : : ; 2d (cid:0) 2g be a set of representative indices, each pair of inverse points
being represented by exactly one index, so that jMj = m. For each i 2 M, define i(cid:3) to be the index
of the inverse of Pi, that is to say Pi + Pi(cid:3) (cid:24) 2P2d(cid:0)1. Let I (cid:26) f1; : : : ; 2d (cid:0) 2g be a subset disjoint
from M with jIj = d (cid:0) 2(cid:0) m and let J = f1; : : : ; 2d (cid:0) 2g n „M [ I”. By Riemann-Roch, there exists
a unique R 2 C„k” such that

∑

∑

R + P2d(cid:0)1 +

Pi (cid:24)

Pj :

j2J

i2M[I
j j 2 Jg, then this linear equivalence defines a
Furthermore R 2 C„k” by uniqueness. If R < fPj
base-point-free 1-dimensional linear system over k of degree d. Thus there exists a corresponding
morphism (cid:27) : C ! P1 over k with (cid:27)(cid:0)1„0” = fR; P2d(cid:0)1g [ fPi j i 2 M [ Ig and (cid:27)(cid:0)1„1” = fPj j
j 2 Jg, so that

(cid:12)(cid:12)(cid:12) = 2d (cid:0) 1:

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

(cid:12)(cid:12)(cid:12)(

(cid:12)(cid:12)(cid:12)(

)

)

\ fP1; : : : ; P2d(cid:0)1g

49

As this would contradict the hypothesis, we conclude there exists an r 2 J such that R = Pr. That
is to say

∑

∑

Pj :
Once again by Riemann-Roch, there exists a T 2 C„k” such that

j2Jnfrg

i2M[I

P2d(cid:0)1 +

Pi (cid:24)

(4.1)

Pi:

∑

i2M[I

j j 2 J n frgg. But if T = P2d(cid:0)1,

P2d(cid:0)1 +

j2Jnfrg

Pj (cid:24) T + Pr +

By the same reasoning as above, either T = P2d(cid:0)1 or T 2 fPj
then

∑

Pj (cid:24) Pr +

Pi:

i2M[I

∑

∑

j2Jnfrg

Comparing this to equivalence (4.1) reveals that Pr (cid:24) P2d(cid:0)1. But according to Riemann-Roch, if
two points on C are linearly equivalent, then they are equal, that is Pr = P2d(cid:0)1, a contradiction
since r , 2d (cid:0) 1 and the Pi are distinct. Therefore there exists a t 2 J n frg such that T = Pt. That
is to say

∑
∑

∑
∑

i2M[I

P2d(cid:0)1 +

j2Jnfr;tg

Pj (cid:24) Pr +

Pi:

2P2d(cid:0)1 +

Pi (cid:24)

i2f1;:::;2d(cid:0)2gnfr;tg

i2f1;:::;2d(cid:0)2g

(4.2)

Pi:

Adding equivalences (4.1) and (4.2) together we get

And after cancelling, we get Pr + Pt (cid:24) 2P2d(cid:0)1, which is exactly the condition that Pr and Pt are
inverse points in the elliptic curve C with identity P2d(cid:0)1. Thus we have determined m + 1 distinct
pairs of inverse points on C. Therefore, by induction we conclude that we can find d (cid:0) 1 distinct
pairs of inverse points, which must consist of P1; : : : ; P2d(cid:0)2 as desired.

□

Proposition 4.3.4. Let C be a nonsingular projective curve of genus 1 over an algebraic number
field k. Let P1; : : : ; P5 2 C„k” be distinct. Suppose that for every morphism (cid:27) : C ! P1 over k of
degree 3 we have

(cid:12)(cid:12)(cid:12)(

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

\ fP1; : : : ; P5g

(cid:12)(cid:12)(cid:12) (cid:20) 4:

)

50

Then if one of the five given points is chosen to be the identity element of the elliptic curve C, the
five given points then form a subgroup of C„k” of order 5.

Proof. Without loss of generality, let P5 be the identity element of the elliptic curve C. By Lemma
4.3.3, for every k 2 f1; : : : ; 4g there exists a k(cid:3) 2 f1; : : : ; 4g n fkg such that Pk + Pk(cid:3) (cid:24) 2P5. Let
these four non-identity points be called Pi; Pi(cid:3); Pj; Pj(cid:3).
By Riemann-Roch, there exists an R 2 C„k” such that

R + P5 + Pi(cid:3) (cid:24) Pi + Pj + Pj(cid:3) :

(4.3)

Since we assume the above morphism (cid:27) does not exist, it must be the case that either R = Pi,
R = Pj, or R = Pj(cid:3). But if R = Pi, then equivalence (4.3) simplifies to

P1(cid:3) + P5 + Pi(cid:3) (cid:24) Pi + 2P5:

So Pi(cid:3) (cid:24) P5. By Riemann-Roch, this implies Pi(cid:3) = P5, a contradiction.

Therefore R = Pr with r 2 f j; j(cid:3)g. In this case, equivalence (4.3) simplifies to

Pr + P5 + Pi(cid:3) (cid:24) Pi + 2P5:

Adding Pi to both sides and cancelling like terms gives Pr + P5 (cid:24) 2Pi. As elements of the elliptic
curve C, this says 2Pi = Pr. Taking inverses, we also get 2Pi(cid:3) = Pr(cid:3). Repeating this process for
the linear equivalence

T + P5 + Pj(cid:3) (cid:24) Pj + Pi + Pi(cid:3)

gives T = Pt for some t 2 fi; i(cid:3)g and so Pt + P5 (cid:24) 2Pj. As elements of the elliptic curve C, this
says 2Pj = Pt and 2Pj(cid:3) = Pt(cid:3). Since 2P5 = P5, we see that these five points are closed under
doubling.

Once again by Riemann-Roch, there exists a point A 2 C„k” such that

A + Pi + Pj (cid:24) Pi(cid:3) + Pj(cid:3) + P5:

Since we assume the above morphism (cid:27) does not exist, it must be the case that either A = Pi(cid:3),
A = Pj(cid:3), or A = P5. If A = P5 however, then Pi + Pj = Pi(cid:3) + Pj(cid:3) and adding Pi + Pj to both sides

51

gives Pr + Pt = 0, which is not possible, given r 2 f j; j(cid:3)g and t 2 fi; i(cid:3)g. Thus A 2 fPi(cid:3); Pj(cid:3)g and
Pi + Pj = Pu for some u 2 fi(cid:3); j(cid:3)g. Taking inverses gives Pi(cid:3) + Pj(cid:3) = Pu(cid:3). Similarly, Pi + Pj(cid:3) = Pv
for some v 2 fi(cid:3); jg and so Pi(cid:3) + Pj = Pv
(cid:3). Thus the set fP1; : : : ; P5g is closed under addition, and
□
therefore forms a subgroup of order 5 in the elliptic curve C.

It is worth noting that this obstruction does not appear in higher degree examples, as demon-

strated in the following.

Proposition 4.3.5. Let C be a nonsingular projective curve of genus 1 over an algebraic number
field k. Let d (cid:21) 4 be an integer and let P1; : : : ; P2d(cid:0)1 2 C„k” be distinct. Then there exists a
morphism (cid:27) : C ! P1 over k of degree d such that

(cid:12)(cid:12)(cid:12)(

)

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

\ fP1; : : : ; P2d(cid:0)1g

(cid:12)(cid:12)(cid:12) = 2d (cid:0) 1:

Proof. By way of contradiction, assume such a morphism does not exist. Let P2d(cid:0)1 be the identity
element of the elliptic curve C. Then by Lemma 4.3.3, for every i 2 f1; : : : ; 2d (cid:0) 2g there exists an
i(cid:3) 2 f1; : : : ; 2d (cid:0) 2g n fig such that Pi + Pi(cid:3) (cid:24) 2P2d(cid:0)1. Let M be a set of representative indices
of these pairs, that is for each i 2 f1; : : : ; 2d (cid:0) 2g, either i 2 M or i(cid:3) 2 M and not both, so that
jMj = d (cid:0) 1. Let M1 = M n f1; 1(cid:3)g and note jM1j = d (cid:0) 2.

The proof is broken into two cases, but those cases are treated very similarly. First suppose d

∑
is even. Let K (cid:26) M1 with jKj = d(cid:0)2
2 . By Riemann-Roch, there exists an R 2 C„k” such that
„Pi + Pi(cid:3)” (cid:24) P2d(cid:0)1 + P1(cid:3) +

„Pi + Pi(cid:3)”:

R + P1 +

(4.4)

∑

i2K

i2M1nK

Since we assume the desired morphism does not exist, it must be the case that either R = P2d(cid:0)1,
R = P1(cid:3), or R 2 fPi j i 2 K or i(cid:3) 2 K}. But if R = P2d(cid:0)1, then equivalence (4.4) simplifies to

P2d(cid:0)1 + P1 + „d (cid:0) 2”P2d(cid:0)1 (cid:24) P2d(cid:0)1 + P1(cid:3) + „d (cid:0) 2”P2d(cid:0)1:

So P1 (cid:24) P1(cid:3). By Riemann-Roch, this implies P1 = P1(cid:3), a contradiction to Lemma 4.3.3. And if
R = P1(cid:3), then equivalence (4.4) simplifies to

P1(cid:3) + P1 + „d (cid:0) 2”P2d(cid:0)1 (cid:24) P2d(cid:0)1 + P1(cid:3) + „d (cid:0) 2”P2d(cid:0)1:

52

So P1 (cid:24) P2d(cid:0)1. By Riemann-Roch, this implies P1 = P2d(cid:0)1, a contradiction to the hypothesis.

Therefore R = Pr for some r 2 f1; : : : ; 2d (cid:0) 2g with either r 2 K or r(cid:3) 2 K. In this case

equivalence (4.4) simplifies to

Pr + P1 + „d (cid:0) 2”P2d(cid:0)1 (cid:24) P2d(cid:0)1 + P1(cid:3) + „d (cid:0) 2”P2d(cid:0)1:

Adding P1(cid:3) to both sides and cancelling like terms gives
Pr + P2d(cid:0)1 (cid:24) 2P1(cid:3) :

Now choose K′ (cid:26) M1 n fr; r(cid:3)g with jK′j = d(cid:0)2

(4.5)
2 . note this subset exists because jM1 n fr; r(cid:3)gj =

d (cid:0) 3 and d(cid:0)2
2

(cid:20) d (cid:0) 3 () d (cid:21) 4. By Riemann-Roch, there exists an T 2 C„k” such that

„Pi + Pi(cid:3)” (cid:24) P2d(cid:0)1 + P1(cid:3) +

„Pi + Pi(cid:3)”:

(4.6)

∑

T + P1 +

i2M1nK′

∑

i2K′

Once again this implies that T = Pt for some t 2 f1; : : : ; 2d (cid:0) 2g with either t 2 K′ or t(cid:3) 2 K′.
Therefore equivalence (4.6) simplifies to

Pt + P1 + „d (cid:0) 2”P2d(cid:0)1 (cid:24) P2d(cid:0)1 + P1(cid:3) + „d (cid:0) 2”P2d(cid:0)1:

Adding P1(cid:3) to both sides and cancelling like terms gives
Pt + P2d(cid:0)1 (cid:24) 2P1(cid:3) :

(4.7)
Comparing equivalences (4.5) and (4.7) reveals that Pr (cid:24) Pt, which implies that Pr = Pt, but that
implies r = t, a contradiction to the construction of K′. Therefore our initial assumption must have
been false and there exists a morphism (cid:27) : C ! P1 over k of degree d such that

(cid:12)(cid:12)(cid:12)(

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

\ fP1; : : : ; P2d(cid:0)1g

Now suppose d is odd. Let K (cid:26) M1 with jKj = d(cid:0)1

2 . By Riemann-Roch, there exists an

R 2 C„k” such that

R + P2d(cid:0)1 + P1 +

„Pi + Pi(cid:3)” (cid:24) P1(cid:3) +

„Pi + Pi(cid:3)”:

(4.8)

)

∑

i2M1nK

53

(cid:12)(cid:12)(cid:12) = 2d (cid:0) 1:
∑

i2K

Since we assume the desired morphism does not exist, it must be the case that either R = P1(cid:3) or
R 2 fPi j i 2 K or i(cid:3) 2 K}. But if R = P1(cid:3), then equivalence (4.8) simplifies to
P1(cid:3) + P2d(cid:0)1 + P1 + „d (cid:0) 3”P2d(cid:0)1 (cid:24) P1(cid:3) + „d (cid:0) 1”P2d(cid:0)1:
So P1 (cid:24) P2d(cid:0)1. By Riemann-Roch, this implies P1 = P2d(cid:0)1, a contradiction.

Therefore R = Pr for some r 2 f1; : : : ; 2d (cid:0) 2g with either r 2 K or r(cid:3) 2 K. In this case

equivalence (4.8) simplifies to

Pr + P2d(cid:0)1 + P1 + „d (cid:0) 3”P2d(cid:0)1 (cid:24) P1(cid:3) + „d (cid:0) 1”P2d(cid:0)1:

Adding P1(cid:3) to both sides and cancelling like terms gives
Pr + P2d(cid:0)1 (cid:24) 2P1(cid:3) :
Now choose K′ (cid:26) M1nfr; r(cid:3)g with jK′j = d(cid:0)1
∑
and d(cid:0)1
2

(4.9)
2 . note this subset exists because jM1nfr; r(cid:3)gj = d(cid:0)3

(cid:20) d (cid:0) 3 () d (cid:21) 5. By Riemann-Roch, there exists an T 2 C„k” such that

T + P2d(cid:0)1 + P1 +

„Pi + Pi(cid:3)” (cid:24) P1(cid:3) +

„Pi + Pi(cid:3)”:

(4.10)

i2M1nK′

Once again this implies that T = Pt for some t 2 f1; : : : ; 2d (cid:0) 2g with either t 2 K′ or t(cid:3) 2 K′.
Therefore equivalence (4.10) simplifies to

∑

i2K′

Pt + P2d(cid:0)1 + P1 + „d (cid:0) 3”P2d(cid:0)1 (cid:24) P1(cid:3) + „d (cid:0) 1”P2d(cid:0)1:

Adding P1(cid:3) to both sides and cancelling like terms gives
Pt + P2d(cid:0)1 (cid:24) 2P1(cid:3) :

(4.11)
Comparing equivalences (4.9) and (4.11) reveals that Pr (cid:24) Pt, which implies that Pr = Pt, but that
implies r = t, a contradiction to the construction of K′. Therefore our initial assumption must have
been false and there exists a morphism (cid:27) : C ! P1 over k of degree d such that

(cid:12)(cid:12)(cid:12)(

)

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

(cid:12)(cid:12)(cid:12) = 2d (cid:0) 1:

□

\ fP1; : : : ; P2d(cid:0)1g

54

Now we prove the desired inequality holds for cubic points on elliptic curves, excluding the
special case when the support of the divisor consists of exactly five points forming a subgroup of
the elliptic curve.

Corollary 4.3.6. Let C be a nonsingular projective curve of genus 1 defined over a number field
k and let R 2 C„k”. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct, such
that the set fP1; : : : ; Png is not a subgroup of order five of the elliptic curve „C„k”; Pn” (which is
i=1 Pi. Let " > 0. Then for all Q 2 C„k”
automatically true if n , 5). Define the divisor D :=
with »k„Q” : k… = 3,

∑n
(cid:12)(cid:12)(cid:12)}

{(cid:12)(cid:12)(cid:12)(

)

mD;S„Q” (cid:20) „N3„D” + "”hR„Q” + O„1”

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

\ Supp„D”

taken over all k-morphisms (cid:27) : C ! P1

where N3„D” := max
of degree 3. If no such k-morphism exists, we say N3„D” = 0.
Proof. First note that since degree 3 morphisms are at most 3-to-1, we have N3„D” (cid:20) 6. We also
clearly have N3„D” (cid:20) deg„D”. By Riemann-Roch, the dimension of the complete linear system jEj
of any divisor E on C of positive degree is given by dimjEj = deg„E” (cid:0) 1. If there are only finitely
many points Q 2 C„k” with »k„Q” : k… = 3, then mD;S„Q” (cid:20) O„1”, so we assume there are infinitely
many points Q 2 C„k” with »k„Q” : k… = 3. In particular, there exists a triple of conjugate k-cubic
points Q1; Q2; Q3 2 C„k”. Since jQ1 + Q2 + Q3j = 2 and jQ1 + Q2 + Q3 (cid:0) T (cid:0) Uj = 0 for all
T; U 2 C„k”, it follows that Q1 + Q2 + Q3 is a degree 3, very ample divisor defined over k, whose
complete linear system jQ1 +Q2 +Q3j has dimension 2. Thus it induces an embedding F : C ! P2
defined over k, whose image is a curve over k of degree 3 such that the points F„Q1”; F„Q2”; F„Q3”
lie on a line H over k. Projecting from any k-rational point A not on the curve F„C” in P2 produces
a degree 3 morphism fA : C ! P1 over k.

If deg„D” = 1, then choose any line L over k in P2 containing F„P1”. As the intersection of two
lines over k, the point A := L \ H is k-rational, thus different from F„Q1”; F„Q2”; F„Q3”, thus not
a point on F„C”. So fA is a degree 3 morphism over k and after composing with a k-automorphism
of P1, we may assume fA„P1” = 0, so that N3„D” = 1.

55

If deg„D” = 2, then let L be the line in P2 containing F„P1” and F„P2”, thus a line over k.
As the intersection of two lines over k, the point B := L \ H is k-rational, thus different from
F„Q1”; F„Q2”; F„Q3”, thus not a point on F„C”. So fB is a morphism of degree 3 over k and after
composing with a k-automorphism of P1, we may assume fB„P1” = fB„P2” = 0, so that N3„D” = 2.
If deg„D” = 3, then after composing with a k-automorphism of P1, we may assume fB„P1” =

fB„P2” = 0 and fB„P3” = 1, so that N3„D” = 3.

If deg„D” = 4, then let H2 be the line in P2 containing F„P1”; F„P2”; F„P3”, and let gA :
C ! P1 be the morphism of degree 3 over k induced by the projection from a point A 2 H2„k”.
Then gA„P1” = gA„P2” = gA„P3”, so composing with a k-automorphism of P1, we may assume
gA„P1” = gA„P2” = gA„P3” = 0 and gA„P4” = 1, so that N3„D” = 4.

If deg„D” = 5, then by the hypothesis and by Proposition 4.3.4, we have N3„D” = 5.
In these cases N3„D” = deg„D”, so we have

mD;S„Q” (cid:20) hD„Q” by Definition 2.2.8

(cid:20) „deg„D” + "”hR„Q” + O„1” by Theorem 2.2.12
= „N3„D” + "”hR„Q” + O„1”:

Now suppose deg„D” (cid:21) 6. If we assume that N3„D” (cid:20) 4, then by Proposition 4.3.4, we know
that the points fP1; : : : ; P5g form a subgroup of order 5 of the elliptic curve C with identity element
P5. In particular there exist i; j 2 f2; 3; 4g such that Pi + Pj = P1. Replacing P1 with P6 and
applying Proposition 4.3.4 again to show that the points fP2; : : : ; P6g form a subgroup of order
5 in the elliptic curve C with identity element P5. But these points obey the same group law as
before, and Pi + Pj = P1, meaning the set is not closed under addition, a contradiction. Therefore
we conclude that N3„D” (cid:21) 5.

If N3„D” = 6, then the inequality follows from Proposition 4.3.2.
(cid:24)
Conversely, if there exists a subset fi1; i2; i3; i4; i5; i6g (cid:26) f1; : : : ; ng such that Pi1 + Pi2 + Pi3
” =
Pi4 + Pi5 + Pi6, then there exists a k-morphism (cid:27) : C ! P1 of degree 3 such that (cid:27)„Pi1
”. After composing with a k-automorphism of
(cid:27)„Pi2

” and (cid:27)„Pi4

” = (cid:27)„Pi3

” = (cid:27)„Pi5

” = (cid:27)„Pi6

56

P1, we may assume (cid:27)„Pi1
N3„D” = 6.

” = (cid:27)„Pi2

” = (cid:27)„Pi3

” = 0 and (cid:27)„Pi4

” = (cid:27)„Pi5

” = (cid:27)„Pi6

” = 1, so that

Thus if N3„D” = 5, then such a subset of f1; : : : ; ng does not exist and the inequality again
□

follows from Proposition 4.3.2.

Proposition 4.3.7. Let C be a nonsingular projective curve of genus 2 defined over a number field
k. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and define the divisor
D :=

i=1 Pi. Let R 2 C„k” and let " > 0.

∑n

If there exists a subset fi1; i2; i3; i4; i5; i6g (cid:26) f1; : : : ; ng such that Pi1 +Pi2 +Pi3

(cid:24) Pi4 +Pi5 +Pi6,

then for all Q 2 C„k” with »k„Q” : k… = 3

mD;S„Q” (cid:20) „6 + "”hR„Q” + O„1”:

If such a linear equivalence does not exist, but there exists a subset fi1; i2; i3; i4; i5g (cid:26) f1; : : : ; ng
(cid:24) Pi4 + Pi5 +T,

; Pi3 such that Pi1 + Pi2 + Pi3

and a point T 2 C„k” distinct from the points Pi1
then for all Q 2 C„k” with »k„Q” : k… = 3

; Pi2

mD;S„Q” (cid:20) „5 + "”hR„Q” + O„1”:

And if neither of these hold, then for all Q 2 C„k” with »k„Q” : k… = 3

mD;S„Q” (cid:20) „4 + "”hR„Q” + O„1”

Proof. By Theorem 3.1.6, there exists a proper closed subscheme Z (cid:26) Sym3„C” such that for all
P 2 Sym3„C”„k” n Z,

mD;S„P” (cid:20) „4 + "”hφ(cid:3)(cid:25)(cid:3)

1R„P”:

Let Z j be an irreducible component of Z. Then (cid:22)jZ j is a k-morphism from the irreducible surface
Z j to the abelian variety Jac„C”.

If (cid:22)jZ j were a constant map then Z j would be contained in a fiber of (cid:22). But the fibers of (cid:22) are

1-dimensional by Riemann-Roch, a contradiction, so (cid:22)jZ j is nonconstant.

57

If the image of (cid:22)jZ j is a curve, then Theorem 4.3.1 says there exists a proper closed subscheme

Yj of Z j such that for all P 2 Z j„k” n Yj

mDjZi

;S„P” (cid:20) „3 + "”h„φ(cid:3)(cid:25)(cid:3)

1R”jZi

„P” + O„1”:

Since Yj is a proper closed subscheme we may assume it is 1-dimensional, and is thus a union
of curves. By Lemma 4.2.1, if there does not exist a subset fi1; i2; i3; i4; i5g (cid:26) f1; : : : ; ng and a
point T 2 C„k” distinct from the points Pi1

; Pi2

; Pi3 such that
(cid:24) Pi4 + Pi5 + T;

Pi1 + Pi2 + Pi3

then for all Q 2 C„k” with »k„Q” : k… = 3 and φ„ ¯Q” 2 Yj

mD;S„Q” (cid:20) „4 + "”hR„Q” + O„1”:

And if there does not exist a subset fi1; i2; i3; i4; i5; i6g (cid:26) f1; : : : ; ng such that Pi1 + Pi2 + Pi3
Pi4 + Pi5 + Pi6, then for all Q 2 C„k” with »k„Q” : k… = 3 and φ„ ¯Q” 2 Yj

(cid:24)

mD;S„Q” (cid:20) „5 + "”hR„Q” + O„1”;

else

(

)

mD;S„Q” (cid:20) „6 + "”hR„Q” + O„1”:
Finally, if the image of (cid:22)jZ j is all of Jac„C”, then Yj = Supp

is a finite union
of irreducible curves Yjk on Z j. By Theorem 4.1.4, it is enough to show the inequality holds on
these curves, which of course follows once again from Lemma 4.2.1.

(cid:22)(cid:3) (cid:22)(cid:3)φ(cid:3)(cid:25)(cid:3)

1DjZ j

Combining the above inequalities we get the result on all of Z j. Since there are only finitely
many irreducible components Z j in Z, we make take the maximum of all of the above constants
O„1” to get the result for all Q 2 C„k” with »k„Q” : k… = 3.

□

Lemma 4.3.8. Let C be a nonsingular projective curve of genus 2. Let D be an effective divisor
on C of degree 3. Then the complete linear system jDj has a base point if and only if K (cid:20) D for a
canonical divisor K on C.

58

Proof. By Riemann-Roch, dimjDj = 1. Suppose D has a base point P 2 C„k”. Then there exists
an effective divisor E of degree 2 such that D = E + P. Furthermore dimjEj = dimjD (cid:0) Pj =
dimjDj = 1. By Riemann-Roch, dimjK (cid:0) Ej = 0, and since deg„K (cid:0) E” = 0, this implies E (cid:24) K.
That is to say, E is a canonical divisor with E (cid:20) D, as was to be shown.

Conversely, suppose K (cid:20) D for a canonical divisor K. Then D = K + P for some P 2 C„k”.
□

Thus dimjD (cid:0) Pj = dimjKj = 1 = dimjDj, so P is a base point of D.

{(cid:12)(cid:12)(cid:12)(

)

(cid:12)(cid:12)(cid:12)}

∑n

Corollary 4.3.9. Let C be a nonsingular projective curve of genus 2 defined over a number field
k. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and define the divisor
i=1 Pi. Suppose that either n (cid:20) 2 or else that there exist i1; i2; i3 2 f1; : : : ; ng such that the
D :=
set fPi1
g does not contain a pair of hyperelliptic conjugates (which is automatically true
; Pi2
when n (cid:21) 5). Let " > 0. Then for all Q 2 C„k” with »k„Q” : k… = 3,
mD;S„Q” (cid:20) „N3„D” + "”hR„Q” + O„1”

; Pi3

\ Supp„D”

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

taken over all finite k-morphisms (cid:27) :

where N3„D” := max
C ! P1 of degree 3. If no such k-morphism exists, we say N3„D” = 0.
Proof. First note that since degree 3 morphisms are at most 3-to-1, we have N3„D” (cid:20) 6. We also
clearly have N3„D” (cid:20) deg„D”. If there are only finitely many points Q 2 C„k” with »k„Q” : k… = 3,
then mD;S„Q” (cid:20) O„1”, so we assume there are infinitely many points Q 2 C„k” with »k„Q” : k… = 3.
In particular, there exists a triple of conjugate k-cubic points Q1; Q2; Q3 2 C„k”. By Riemann-Roch,
dimjQ1 + Q2 + Q3j = 1.

If the linear system jQ1 + Q2 + Q3j has a base point, then by Lemma 4.3.8, K (cid:20) Q1 + Q2 + Q3
for a canonical divisor K on C. Without loss of generality, say K = Q1 + Q2. By uniqueness, jKj
is fixed under Galois action. Thus if (cid:24) is any Galois action that takes Q2 to Q3, then (cid:24)„Q1” = Qi
for some i 2 f1; 2g, so Q1 + Q2 (cid:24) (cid:24)„Q1 + Q2” = Qi + Q3 and subtracting Qi from both sides
leaves Q j (cid:24) Q3 for some j 2 f1; 2g, implying Q j = Q3, a contradiction. Therefore jQ1 + Q2 + Q3j
is a 1-dimensional base point free linear system of degree 3 defined over k. It follows that the

59

corresponding k-morphism ! : C ! P1 has degree 3 with !„Q1” = !„Q2” = !„Q3”. Therefore if
deg„D” = 1 or 2, then N3„D” = deg„D”.

Alternatively, if there exist i1; i2; i3 2 f1; : : : ; ng such that fPi1

of hyperelliptic conjugates, then the complete linear system jPi1 + Pi2 + Pi3
by Lemma 4.3.8. So there exists a k-morphism (cid:13) : C ! P1 of degree 3 with (cid:13)„Pi1
(cid:13)„Pi3

” = 0. If deg„D” = 3, then this shows N3„D” = 3.
In these cases, where N3„D” = deg„D”, we have

; Pi2

; Pi3

g does not contain a pair
j is base point free
” =

” = (cid:13)„Pi2

mD;S„Q” (cid:20) hD„Q” by Definition 2.2.8

(cid:20) „deg„D” + "”hR„Q” + O„1” by Theorem 2.2.12
= „N3„D” + "”hR„Q” + O„1”:

From here on, suppose deg„D” (cid:21) 4. Then there exists a fourth point in the support of D and
the morphism (cid:13) must map it somewhere. After a k-automorphism of P1, we may assume it is sent
to infinity, and so N3„D” (cid:21) 4.

If N3„D” = 6, then the inequality follows from Proposition 4.3.7.
(cid:24)
Conversely, if there exists a subset fi1; i2; i3; i4; i5; i6g (cid:26) f1; : : : ; ng such that Pi1 + Pi2 + Pi3
” =
Pi4 + Pi5 + Pi6, then there exists a k-morphism (cid:27) : C ! P1 of degree 3 such that (cid:27)„Pi1
(cid:27)„Pi2

” = (cid:27)„Pi3
Thus if N3„D” = 5, then such a subset of f1; : : : ; ng does not exist and the inequality again

” = 1, so that N3„D” = 6.

” = 0 and (cid:27)„Pi4

” = (cid:27)„Pi6

” = (cid:27)„Pi5

follows from Proposition 4.3.7.

” = (cid:14)„Pi2

” = (cid:14)„Pi3

If N3„D” (cid:21) 5 then there exists a subset fi1; i2; i3; i4; i5g (cid:26) f1; : : : ; ng and a k-morphism
(cid:14) : C ! P1 of degree 3 such that (cid:14)„Pi1
” = 1.
Since 0 (cid:24) 1 in P1, it follows that (cid:14)(cid:3)0 (cid:24) (cid:14)(cid:3)1. That is Pi1 + Pi2 + Pi3
(cid:24) Pi4 + Pi5 + T for some
T 2 C„k”. Since (cid:14)„T” = 1, it is clear that T is distinct from the points Pi1
; Pi3. Since (cid:14)
is defined over k, for any Galois action (cid:24) 2 G„k(cid:157)k” we have (cid:24)„(cid:14)(cid:3)„1”” = (cid:14)(cid:3)„(cid:24)„1”” = (cid:14)(cid:3)„1” so
Pi4 + Pi5 + (cid:24)„T” = (cid:24)„Pi4 + Pi5 + T” = Pi4 + Pi5 + T, implying (cid:24)„T” = T. Since (cid:24) was arbitrary,
we have T 2 C„k”. Ergo if such a subset and point do not exist, then N3„D” < 5, so N3„D” = 4 and

” = 0 and (cid:14)„Pi4

” = (cid:14)„Pi5

; Pi2

60

the result follows from Proposition 4.3.7.

□

Corollary 4.3.10. Let C be a nonsingular projective curve of genus 2 defined over a number field
k. Let S be a finite set of places of k. Let P1; : : : ; Pn 2 C„k” be distinct and define the divisor
D :=

∑n

i=1 Pi. Let " > 0. Then for all Q 2 C„k” with »k„Q” : k… = 3,
mD;S„Q” (cid:20) „ ˜N3„D” + "”hR„Q” + O„1”
\ Supp„D”

(cid:27)(cid:0)1„0” [ (cid:27)(cid:0)1„1”

{(cid:12)(cid:12)(cid:12)(

)

(cid:12)(cid:12)(cid:12)}

where ˜N3„D” = N3„C; k; D” := max
phisms (cid:27) : C ! P1 of degree 3. If no such morphism exists, we say ˜N3„D” = 0.
Proof. note that N3„D” (cid:20) ˜N3„D”, so the inequality is implied by Corollary 4.3.9 except in cases
where deg„D” = 3 or 4.

taken over all finite mor-

Consider the morphism »(cid:1); (cid:1)… : C2 ! Jac„C” defined by »T; U… = O„U (cid:0) T”. If »T; U… = »T′; U′…,
then T + U′ (cid:24) T′ + U, and since the hyperelliptic morphism is unique, it follows that »(cid:1); (cid:1)… is 2-to-1,
and comparing dimensions, we see it is surjective.

Thus if deg„D” = 4, there exist T; U 2 C„k” such that

Pi1 + Pi2 + T (cid:24) Pi3 + Pi4 + U:

We can always arrange the points so that Pi1 and Pi2 are not hyperelliptic conjugates and Pi3 and
Pi4 are not hyperelliptic conjugates, so this defines a base point free linear system of dimension 1
and degree 3. Thus there is a morphism (cid:27) : C ! P1 of degree 3 with (cid:27)„Pi1
” = 0 and
(cid:27)„Pi3

” = (cid:27)„Pi4
Similarly if deg„D” = 3, then for any V 2 C„k” there exist T; U 2 C„k” such that

” = 1, that is ˜N3„D” = 4.

” = (cid:27)„Pi2

Pi1 + Pi2 + T (cid:24) Pi3 + V + U:

We can always arrange the points so that Pi1 and Pi2 are not hyperelliptic conjugates and choose
V so that V is not the hyperelliptic conjugate of Pi3 and V is not equal to Pi1 or Pi2, so that this
defines a base point free linear system of dimension 1 and degree 3. Thus there is a morphism
(cid:27) : C ! P1 of degree 3 with (cid:27)„Pi1

” = 1, that is ˜N3„D” = 3.

” = 0 and (cid:27)„Pi3

” = (cid:27)„Pi2

61

In these cases, where ˜N3„D” = deg„D”, we have

mD;S„Q” (cid:20) hD„Q” by Definition 2.2.8

(cid:20) „deg„D” + "”hR„Q” + O„1” by Theorem 2.2.12
= „ ˜N3„D” + "”hR„Q” + O„1”:

□

62

BIBLIOGRAPHY

63

BIBLIOGRAPHY

[1] Enrico Bombieri and Walter Gubler. Heights in Diophantine geometry, volume 4 of New

Mathematical Monographs. Cambridge University Press, Cambridge, 2006.

[2] P. Corvaja and U. Zannier. On integral points on surfaces. Ann. of Math. (2), 160(2):705–726,

2004.

[3] Pietro Corvaja and Umberto Zannier. On a general Thue’s equation. Amer. J. Math.,

126(5):1033–1055, 2004.

[4]

Jan-Hendrik Evertse and Roberto G. Ferretti. A generalization of the Subspace Theorem with
polynomials of higher degree. In Diophantine approximation, volume 16 of Dev. Math., pages
175–198. SpringerWienNewYork, Vienna, 2008.

[5] Gerd Faltings. Diophantine approximation on abelian varieties. Ann. of Math. (2), 133(3):549–

576, 1991.
Joe Harris and Joe Silverman. Bielliptic curves and symmetric products. Proc. Amer. Math.
Soc., 112(2):347–356, 1991.

[6]

[7] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York-Heidelberg, 1977. Grad-

uate Texts in Mathematics, No. 52.

[8] Marc Hindry and Joseph H. Silverman. Diophantine geometry, volume 201 of Graduate Texts

in Mathematics. Springer-Verlag, New York, 2000. An introduction.

[9] Aaron Levin. On the Schmidt subspace theorem for algebraic points. Duke Math. J.,

163(15):2841–2885, 2014.

[10] Aaron Levin. Wirsing-type inequalities. Bull. Inst. Math. Acad. Sin. (N.S.), 9(4):685–710,

2014.

[11] Jürgen Neukirch. Algebraic number theory, volume 322 of Grundlehren der Mathematischen
Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin,
1999. Translated from the 1992 German original and with a note by Norbert Schappacher,
With a foreword by G. Harder.

[12] Min Ru and Pit-Mann Wong.

planes in general positiong. Invent. Math., 106(1):195–216, 1991.

Integral points of Pn (cid:0) f2n + 1

hyper-

[13] Joseph H. Silverman. Arithmetic distance functions and height functions in Diophantine

geometry. Math. Ann., 279(2):193–216, 1987.

[14] Xiangjun Song and Thomas J. Tucker. Dirichlet’s theorem, Vojta’s inequality, and Vojta’s

conjecture. Compositio Math., 116(2):219–238, 1999.

[15] The Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu, 2018.

64

[16] Paul Vojta. Diophantine approximations and value distribution theory, volume 1239 of

Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1987.

[17] Paul Vojta. A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing. J. Amer.

Math. Soc., 5(4):763–804, 1992.

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