Diophantine Approximations and Diophantine Equations

228 Pages·1991·6.58 MB·English

by  Wolfgang M. Schmidt (auth.)

Lecture Notes in Mathematics 1467
Editors:
A. Dold, Heidelberg
B. Eckmann, Ztirich
F. Takens, Groningen
Wolfgang M. Schmidt
Diophantine Approximations
and Diophantine Equations
Springer-Verlag
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Author
Wolfgang M. Schmidt
Department of Mathematics. University of Colorado
Boulder, Colorado, 80309-0426. USA
Mathematics Subject Classification (1991): 11J68, 11J69, 11057, 11D61
ISBN 3-540-54058-X Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-54058-X Springer-Verlag New York Berlin Heidelberg
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Preface
The present notes are the outcome of lectures I gave at Columbia University in the
fall of 1987, and at the University of Colorado 1988//1989. Although there is necessarily
some overlap with my earlier Lecture Notes on Diophantine Approximation (Springer
Lecture Notes 785, 1980), this overlap is small. In general, whereas in the earlier Notes
I gave a systematic exposition with all the proofs, the present notes present a varirety
of topics, and sometimes quote from the literature wihtout giving proofs. Nevertheless,
I believe that the pace is again leisurely.
Chapter I contains a fairly thorough discussion of Siegel's Lemma and of heights.
Chapter II is devoted to Roth's Theorem. Rather than Roth's Lemma, I use a general-
ization of Dyson's Lemma as given by Esnault and Viehweg. A proof of this generalized
lemma is not given; it is beyond the scope of the present notes. An advantage of the
lemma is that it leads to new bounds on the number of exceptional approximations in
Roth's Theorem, as given recently by Bombieri and Van der Poorten. These bounds
turn out to be best possible in some sense. Chapter III deals with the Thue equation.
Among the recent developments are bounds by Bombieri and author on the number
of solutions of such equations, and by Mueller and the author on the number of so-
lutions of Thue equations with few nonzero coefficients, say s such coefficients (apart
from the constant term). I give a proof of the former, but deal with the latter only
up to s = 3, i.e., to trinomial Thue equations. Chapter IV is about S-unit equations
and hyperelliptic equations. S-unit equations include equations such as 2 ~ + 3 y = 4 z. I
present Evertse's remarkable bounds for such equations. As for elliptic and hyperelliptic
equations, I mention a few basic facts, often without proofs, and proceed to counting
the number of solutions as in recent works of Evertse, and of Silverman, where the con-
nection with the Mordell-Weil rank is explored. Chapter V is on certain diophantine
equations in more than two variables. A tool here is my Subspace Theorem, of which I
quote several versions, but without proofs. I study generalized S-unit equations, such
as, e.g. -4-a~1 -4- a~ 2 + ... -4- a~" = 0 with given integers ai > 1, as well as norm form
equations. Recent advances permit to give explicit estimates on the number of solutions.
The notes end with an Epilogue on the abc-conjecture of Oesterl$ and Masser.
Hand written notes of my lectures were taken at Columbia University by Mr. Ag-
boola, and at the University of Colorado by MS. Deanna Caveny. The manuscript was
typed by Ms. Andrea Hennessy and Ms. Elizabeth Stimmel. My thanks are due to
them.
January 1991 Wolfgang M. Schmidt
Table of Contents
Chapter Page
I. Siegel's Lemmas and Heights ................................................ 1
1. Siegel's Lemma ......................................................... 1
2. Geometry of Numbers .................................................. 3
3. Lattice Packings ....................................................... 7
4. Siegel's Lemma Again .................................................. 9
5. Grassman Algebra .................................................... 11
6. Absolute Values ....................................................... 18
7. Heights in Number Fields ............................................. 22
8. Heights of Subspaces .................................................. 28
9. Another Version of Siegel's Lemma .................................... 32
II. Diophantine Approximation ................................................ 34
1. Dirichlet's Theorem and Liouvilte's Theorem .......................... 34
2. Roth's Theorem ....................................................... 38
3. Construction of a Polynomial .......................................... 42
4. Upper Bounds for the Index ........................................... 45
5. Estimation of Volumes ................................................ 48
6. A Version of Roth's Theorem .......................................... 49
7. Proof of the Main Theorems ........................................... 52
8. Counting Good Rational Approximations .............................. 57
9. The Number of Good Approximations to Algebraic Numbers .......... 63
10. A Generalization of Roth's Theorem ................................... 69
III. The Thue Equation ........................................................ 73
1. Main Result ........................................................... 73
2. Preliminaries .......................................................... 76
3. More on the connection between Thue's Equation and
Diophantine Approximation ....................................... 83
4. Large Solutions ....................................................... 85
5. Small Solutions ....................................................... 86
6. How to Go From F(x) = 1 to F(x) = m ............................... 91
7. Thue Equations with Few Coefficients ................................. 99
8. The Distribution of the Roots of Sparse Polynomials .................. 100
9. The Angular Distribution of Roots ................................... 106
10. On Trinomials ....................................................... 111
11. Roots of f close to ~ ................................................. 116
12. Proof of 7A .......................................................... 119
13. Generalizations of the Thue Equation ................................ 124
Viii
Table of Contents (cont.)
Chapter Page
IV. S-unit Equations and Hyperelliptic Equations ............................. 127
1. S-unit Equations and Hyperelliptic Equations ........................ 127
2. Evertse's Bound ..................................................... 129
3. More on S-unit Equations ............................................ 134
4. Elliptic, Hyperelliptic, and Superelliptic Equations .................... 137
5. The Number of Solutions of Elliptic, Hyperelliptic, and
Superelliptic Equations .......................................... 142
6. On Elliptic Curves ................................................... 147
7. The Rank of Cubic Thue Curves ..................................... 159
8. Lower Bounds for the Number of Solutions of Cubic Thue Equations .. 163
9. Upper Bounds for Rational Points on Certain Elliptic Equations
in terms of the Mordell-Weil Rank ................................... 165
10. Isogenies ............................................................. 169
11. Upper Bounds on Cubic Thue Equations in Terms of the
Mordell-Weil Rank .............................................. 173
12. More General Results ................................................ 175
V. Diophantine Equations in More Than Two Variables ...................... 176
1. The Subspace Theorem .............................................. 176
2. Generalized S-unit Equations ........................................ 180
3. Norm Form Equations ............................................... 182
4. A Reduction .......................................................... 186
5. An Application of the Geometry of Numbers ......................... 188
6. Products of Linear Forms ............................................ 192
7. A Generalized Gap Principle ......................................... 193
8. Small Solutions ...................................................... 199
9. Large Solutions ...................................................... 200
10. Proof of Theorem 3B ................................................. 203
Epilogue. The abc-conjecture ................................................ 205
Bibliography .................................................................. 209
Index of Symbols ............................................................. 216
I. Siegel's Lemma and Heights
§1. Siegel's Lemma.
Consider a system of homogeneous linear equations
allxl + "" + al,~xn = 0
• (1.1)
amlX 1 _.}_ . . . qt_ amnXn = 0
If m < n and the coefficients lie in a field, then there is a nontrivial solution with
components in the field. If m < n and the coefficients lie in Z (the integers), then there
is a nontrivial solution in integers. (Just take a solution with rational components and
multiply by the common denominator.) It is reasonable to believe that if the coefficients
are small integers, then there will also be a solution in small integers. This idea was
used by A. Thue (1909) and formalized by Siegel in (1929; on p. 213 of his Collected
Works).
LEMMA 1. Suppose that in (1.1) the coet~cients aij lie in Z and have [alj[ < A
(1 <__ i,j < n) where A is natural Then there is a nontrivial solution in Z with
Ixil < 1 + (nA) m/("-m) (i = 1,... ,n).
Proof. We follow Siegel. Let H be an integer parameter to be specified later. Let
C be the cube consisting of points
= (Xl,... , xn)
_•
with
Ix l =< H (4 = 1,..., n).
There are (2H + 1) n integer points in this cube, since there are 2H + 1 possibilities for
each coordinate. Let T be the linear map R n --* ~m with
T_x ---- (allX 1 "[- "" + alnXn,. .. ,amlxl W"" TamnXn).
Writing Tx = y = (Yl,.-. , ym), we observe that the image of C lies in the cube
C' : [yj] =< nAg (j = 1,... ,m)
of R m. The number of integer points in C' is (2nAH- + 1) ~. Suppose now that
(2nAg + 1) m < (2H + 1)'. (1.2)
Then T restricted to the integers in the original cube will not be one-to-one. So there
exist x',x" in C, with x' ~ x", such that Tx' = Tx". Put x = x I - x". Then Tx = O.
So _x is a solution to the system with integer coordinates. Note that [x~ I =< 2/~ (i =
Typeset by .AA4S-TEX
1,... ,n), bec=use Ix~l, Ix~l < H (i = 1,... ,n), so Ix~l = Ix: - ~71 < I~:1 + Ix"l _-< 2H.
Choose H to be the natural number satisfying
(nA)mlCn~m~ <= 2H I~ (nA)m/(n-m) + 1.
Then
(2H + 1) n = (2H + 1)m(2H + 1) n-m
~__ (2H + 1)m(nA) m
> (2nAH + 1) m.
So there exists an x= satisfying Ixil < 2H < 1 + (nA) m/("-m).
So the proof of Siegers Lemma uses the box principle.
Can the exponent be improved? The answer is "no".
Siegel's Lemma is almost best possible. Put k = n-m, and for large P pick distinct
primes p~j (i < i < k, 1 < j < m) with Pr/< pij < P, where 0 < 77 < 1 is given. Put
Pj = PljV2j'"Pkj (1 =< j =< m), P,j = P~lpis (1 <i _5<k, 1 =<j =<m),
Qi = plapi2""pim (1 < i < k), Qij = Qilpij (1 _<i _<k, 1 =< j =< m).
Consider the system of m equations in n variables:
PllXl -~- "'" -~- PklXk -- PlY1 = 0
P12xl + "'" + Pk2zk -P2Y2 = 0
Plmxl +"" + PkmZk -Pmym = O.
The maximum modulus A of its coefficients has A < pk. The following are solution
vectors:
z =(QI,0,.. ,O, Qii,Q12,...,Qlm)
----1
z k = (0,0,... ,Qk,Qkl,Qki,... ,Qkm).
It is clear that every solution of our system of equations is a linear combination
cl_z1 + ... -4- ck___kz . For integer solutions, ci is necessarily a rational number whose
denominator is Qi, and then every component of ciz/ has denominator Qi. Moreover,
if, say, cl is not integral, then (since Qll, Q12,... , Qlm are coprime), Cl__Z1 is not integral
and has some component whose denominator is a prime Plj. But since c2_~,... , ckz__k
don't have Plj in the denominator, cl_zI + c2z2 + -.. + ckz k cannot be integral--a
contradiction. Therefore the integer solutions are z = ClZ 1 + .-- + ck_zk with cl,... , ck
in Z. When z # _0, say Cl # 0, the first component xl of_z has
Ixll > Q1 > (r~P) m = rf"P m > rf"A m/k = rf~A ''/("-'~).
Therefore every integer solution (Xl,... , xk, yl , . . . , Ym ) # (0,... , O) has
max(Ix, l,..., Ixkl) ~TmA'~ 1("-').
>
3
Here 77 may be taken arbitrarily close to 1.
Another approach is as follows. When m = n - 1, consider the system of equations
Axi - Xi+l = 0 (i = 1,... , n - 1).
Every nontriviM solution, in fact every nontriviM complex solution, has x,/xl = A ~-1.
Thus if we set
q(x_) = max Ix,/x¢ t,
with the maximum over i, j in i < i, j __< n with xj # O, then q(_z_) = A "-1 = A ''/(n-'O .
But then for integer solutions, max(lxll,..., Ix,I) => A m/<'-m)
Exercise la. Suppose now that m = 1. For large A, construct an equation
al Xl 4-'" + a.xn = 0
with integral coefficients and la~l =< A (i = 1,..., n), such that every nontrivial solution
x with complex components has
q(x_) >= c(n)A 11("-1) = c,(n,m)A ml('~-'O > O.
This approach can be carried out for general n, m. See Schmidt (1985).
§2. Geometry of Numbers.
The subject was founded by Minkowski (1896 ~¢ 1910). Other references are Cassels
(1959), Gruber and Lekkerkerker (1987), and Schmidt (1980, Chapter IV).
A lattice A is a subgroup of R" which is generated by n linearly independent vectors
__bl,... ,b n (linearly independent over Rn). The elements of this lattice are clb 1 + .-. +
c,b with Ci E Z.
/ / / / / / / -
/ / J /
- / / / / ) / /
- ~
The set bl,.., b is called a basis. A basis is not uniquely determined. For example,
' =n
bl,b I + b2,=b3,... ,b is another basis.
How unique is a basis? Suppose __b'l, b' is another basis. Then
n
6' = Z cijb and cij C Z
=i j
j=l
and
n
=b j = E c'i jb ~ and ctjiEZ.
i=1
So the matrices (cij) and (c~i) are inverse to each other and cij, c~i E Z, so det (c/j) =
det (c~i) = +1. Thus the matrix (cij) is unimodular, where by definition a unimodular
matrix is a square matrix with integer entries and determinant 1 or -1.
LEMMA 2A. A necessary and sumcient condition/'or a subset A of R n to be a
lattice is the foI1owing:
(i) A is a group under addition.
(ii) A contains n lineaxly independent vectors.
(iii) A is discrete.
For a proof, see e.g., Schmidt (1980, Ch. IV, Theorem 8A).
Consider R n with the Euclidean metric and A a lattice with =ba,... ,b n as basis. Let
II be the set of linear combinations Alb 1 + ..- + A,b n with 0 =< A~ < 1 (i = 1,..., n).
Then H is called a fundamental parallelepiped of A.
h,
The fundamental parallelepiped does depend on which basis is chosen. The volume
of II is given by V(H) = [det (bl,... , bn) I where the right-hand side involves the matrix
whose rows are respectively made up of the coordinates of bl,... ,b n with respect to an
orthonormal bases of ~n. This volume is independent of the chosen basis of the lattice,
since different bases are connected by unimodular transformations. It is an invariant of
the lattice.
We define
det A = V(II).
Notice that when b. = (hi1,... , bin), then
__--|
bll b12 ..- bin ~ iz bll b21 "" bnl "~
b21 b22 "'" b2n I [ b12 b22 "" bn2 )
V 2 = det . ) det / .
\ bnl bn2 ... bnn \ bin b2n
° • , bnl. l
bib bib 2 .-. =bibn )
= det (2.1)
where the inner product of vectors x_, y is denoted by x y.