Thijs Vorselen
On Kronecker’s Theorem
over the adèles
Master’s thesis, defended on April 27, 2010
Thesis advisor: Jan-Hendrik Evertse
Mathematisch Instituut
Universiteit Leiden
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Kronecker’s Theorem
1.1 An introduction to the geometry of numbers . . . . . . .
1.2 Kronecker’s approximation theorem . . . . . . . . . . . .
1.3 Systems of linear equations over principal ideal domains .
1.4 A more general Kronecker’s Theorem . . . . . . . . . . .
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1
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4
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12
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15
17
19
effective Kronecker’s Theorem
Uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
An effective Kronecker’s Theorem . . . . . . . . . . . . . . . . . . . .
Subspace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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31
2 Valuations
2.1 Some algebraic preliminaries . . . .
2.2 An introduction to absolute values
2.3 Completions . . . . . . . . . . . . .
2.4 Absolute values on number fields .
3 An
3.1
3.2
3.3
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4 Geometry of numbers over the adèles
4.1 Adèles . . . . . . . . . . . . . . . . . .
4.2 Strong Approximation Theorem . . . .
4.3 Fundamental domain . . . . . . . . . .
4.4 The Haar measure on the adèle space .
4.5 Minkowski’s Theorem for adèle spaces .
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34
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5 Kronecker’s Theorem over the adèles
5.1 Kronecker’s theorem for adèle spaces . . . . . . . . . . . . . . . . . .
5.2 An effective adèlic Kronecker’s Theorem . . . . . . . . . . . . . . . .
5.3 A more general adèlic Kronecker’s Theorem . . . . . . . . . . . . . .
45
45
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55
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
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iii
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Introduction
In this thesis we state and prove “effective” versions and adèlic generalizations
of Kronecker’s Theorem. Kronecker’s Theorem takes an important place in the field
of mathematics called Diophantine approximation. This field of mathematics is concerned with approximating real numbers by rational numbers. Kronecker’s Theorem
deals with inhomogeneous Diophantine inequalities and is published in 1884 by Kronecker in a paper [10] called “Näherungsweise ganzzahlige Auflösung linearer Gleichungen”.
Let Li (q) = αi1 q1 + · · · + αin qn (i = 1, . . . , m) be m linear forms with real
coefficients αij and let A be the m × n matrix with elements αij . The following
theorem is a special case of Kronecker’s Theorem.
Theorem 1. (Kronecker) Assume that
{z ∈ Qm : AT z ∈ Qn } = {0}.
(1)
Then for every ε > 0, b = (b1 , . . . , bm ) ∈ Rm , there exist p = (p1 , . . . , pm ) ∈ Zm ,
q ∈ Zn such that
|Li (q) − pi − bi | ≤ ε
for i = 1, . . . , m.
Condition (1) of this theorem is a necessary condition. Another important theorem
from Diophantine approximation is Dirichlet’s Theorem. This theorem is proven by
Dirichlet in 1840 and deals with homogeneous Diophantine inequalities. If we compare
Kronecker’s Theorem with Dirichlet’s Theorem, then we come across an interesting
difference.
Theorem 2. (Dirichlet) For every ε with 0 < ε < 1, there exist p ∈ Zm , q ∈ Zn
with q 6= 0 such that
|Li (q) − pi | ≤ ε for i = 1, . . . , m,
kqk ≤ ε−m/n .
In contrast to Kronecker’s Theorem this theorem gives an upper bound for kqk
that is easy to calculate in terms of ε. We call such an upper bound an effective
upper bound. Motivated by this observation we ask ourselves if there also exists an
effective upper bound in the case of Kronecker’s Theorem. In this thesis we answer
this question in the affirmative for a matrix A with algebraic elements αij .
In 1966 Mahler [15] published a proof of Kronecker’s Theorem with methods from
the geometry of numbers. Geometry of numbers is concerned with convex bodies and
integer vectors in n-dimensional Euclidean space. In 1988 R. Kannan and L. Lovász
iv
also use geometry of numbers to prove a quantitative version of Theorem 1 for the
case n = 1, published in [9]. Their theorem gives an ineffective upper bound for kqk
and is the starting point of our proof of an effective Kronecker’s Theorem.
In the first chapter we give an introduction to geometry of numbers and generalize
the proof of R. Kannan and L. Lovász to arbitrary m. We also give a more general
quantitative version of Kronecker’s Theorem.
In the second chapter we recall some results from algebraic number theory and
more specifically the theory of valuations. These results are used in Chapter 3 to
prove an effective version of Kronecker’s Theorem in the case that the elements of A
are algebraic.
In Chapter 4 we introduce the adèle space. This space was introduced by Chevalley
in 1940 and is useful to solve number theoretic problems. Many of the concepts and
theorems in geometry of numbers have been generalized over the adèles. An important
result is the adèlic version of the Second Theorem of Minkowki. This theorem is
published by R. McFeat [16] in 1971. Unaware of McFeat’s result E. Bombieri and
J. Vaaler [3] proved the same theorem in 1983. Motivated by the results in geometry
of numbers over the adèles we wondered if it would be possible to generalize a version
of Kronecker’s Theorem over the adèles.
In Chapter 5 we prove some adèlic versions of Kronecker’s Theorem. Again the
article by R. Kannan and L. Lovász forms the starting point of our proof. The adèlic
Second Theorem of Minkowski also plays a crucial role.
I would like to thank Jan-Hendrik Evertse for the idea of this thesis and for many
suggestions and corrections.
v
Chapter 1
Kronecker’s Theorem
In this chapter we give an introduction to the geometry of numbers and prove some
versions of Kronecker’s Theorem using geometry of numbers.
1.1
An introduction to the geometry of numbers
Let n be an integer with n ≥ 1. We denote by Rn the vector space of n-dimensional
column vectors. We use shorthand x, y ∈ Rn by (x1 , . . . , xn )T , (y1 , . . . , yn )T ∈ Rn ,
respectively. A body is a closed, bounded, connected subset of Rn with inner points.
A body C is called convex if for all x, y ∈ C and all t ∈ [0, 1] the point (1 − t)x + ty
lies in C. A body C in Rn is called symmetric, if x ∈ C implies that −x ∈ C.
Henceforth, let C be a bounded symmetric convex body in Rn . Such a body C is
Lebesgue measurable, so it has a finite volume, which we denote by V (C). For λ ∈ R
we define
λC = {x ∈ Rn : x = λy, y ∈ C}.
Let h , i denote the standard inner product on Rn , given by
hx, yi =
n
X
for x, y ∈ Rn .
xi y i
i=1
The polar of C, denoted by C ∗ , is given by
C ∗ := {x ∈ Rn : hx, yi ≤ 1 for all y ∈ C}.
This is again a bounded symmetric convex body. See Gruber and Lekkerkerker [7],
Chapter 2, Section 14, Theorem 1 for a proof of this fact.
A lattice in Rn is a discrete subgroup of Rn that spans Rn as a real vector space.
If L is a lattice and {x1 , . . . , xn } is a basis of L, then | det(x1 , . . . , xn )| is called the
determinant of L and is denoted by det L. Define the n × n matrix
A := [x1 , . . . , xn ],
where x1 , . . . , xn are the columns of A. This matrix A is invertible, because x1 , . . . , xn
are linearly independent. We have
L = {Ax : x ∈ Zn }.
1
The dual of L is given by
L∗ = {x ∈ Rn : hx, yi ∈ Z for all y ∈ L}.
It is a corollary of the next lemma that L∗ is again a lattice in Rn . We use the following
notation. Let GLn (R) denote the set of invertible n × n matrices over a ring R.
Lemma 1.1. Let A ∈ GLn (R) and let L = {Ax : x ∈ Zn } be the associated lattice in
Rn . Then its dual is given by L∗ = {(A−1 )T x : x ∈ Zn }.
Proof. We have
h(A−1 )T x, Ayi = xT A−1 Ay = xT y = hx, yi.
Hence, h(A−1 )T x, Ayi ∈ Z if and only if hx, yi ∈ Z. Further, it is easy to see that
{x ∈ Rn : hx, yi ∈ Z for all y ∈ Zn } = Zn .
We conclude that
L∗ = {(A−1 )T x : x ∈ Zn }.
This proves the lemma.
We define the n successive minima λ1 (C, L), . . . , λn (C, L) of C with respect to L
by
λi (C, L) = inf{λ ∈ R>0 : dim(λC ∩ L) ≥ i}.
If it is not necessary to specify C and L, we will denote λi (C, L) by λi . It is easy to
see that 0 < λ1 ≤ · · · ≤ λn < ∞. We defined successive minima as infima, but they
are in fact minima, as suggested by their name.
Theorem 1.2. (Second Minkowski’s Convex Body Theorem) Let C be a
bounded symmetric convex body and L a lattice in Rn . Then the successive minima
λ1 , . . . , λn of C with respect to L satisfy
2n
V (C)
≤ λ1 . . . λn
≤ 2n .
n!
det L
Proof. See Minkowski [17], Kapitel V, for the original proof. We refer to Gruber
and Lekkerkerker [7], Chapter 2, Section 9, Theorem 1 for a shorter proof by Estermann.
2
Let C be a bounded symmetric convex body and L a lattice in Rn . For any u ∈ L
we define the set
λC + u = {x + u : x ∈ λC}.
Then µ(C, L), defined by
µ(C, L) = inf{λ ∈ R>0 :
[
(λC + u) = Rn },
u∈L
is called the covering radius or inhomogeneous minimum of C with respect to L.
For many problems in the geometry of numbers, it is useful to have an upper
bound for the product of µ(C, L) and λ1 (C ∗ , L∗ ).
Lemma 1.3. Let C be a symmetric convex body in Rn and L a lattice in Rn . Then
µ(C, L)λ1 (C ∗ , L∗ ) ≤ 21 n2 .
Proof. See Lagarias, Lenstra, and Schnorr [12], Theorem 2.9.
Theorem 1.4. Let C be a convex body in Rn . Its successive minima λ1 , . . . , λn and
covering radius µ satisfy
1
λ
2 n
≤ µ ≤ 12 (λ1 + · · · + λn ).
Proof. First we prove the lower bound by contradiction. Let t be the number of
linearly independent vectors in (µ + 12 λn )C ∩ L. Suppose that µ + 21 λn < λn , implying
that t < n. Let x1 , . . . , xt ∈ (µ + 12 λn )C ∩ L be linearly independent. Choose x ∈
λn C ∩ L such that x 6∈ Span{x1 , . . . , xt }. There exists u ∈ L such that 12 x − u ∈ µC
by the definition of the covering radius. By symmetry and convexity of C we find
that both u = u − 12 x + 12 x and x − u = 21 x − u + 12 x are in (µ + 21 λn )x. Hence,
u, x − u ∈ Span{x1 , . . . , xt }, which contradicts x 6∈ Span{x1 , . . . , xt }. This proves the
lower bound.
Now, we prove the upper bound. Choose linearly independent x1 , . . . , xn ∈ L
such that xi ∈ λi C for i = 1, . . . , n. Let us take an arbitrary x ∈ Rn . There exist
α1 , . . . , αn ∈ R such that x = α1 x1 + · · · + αn xn , because x1 , . . . , xn span Rn . Choose
a1 , . . . , an ∈ Z such that |αi − ai | ≤ 21 for i = 1, . . . , n. Then x − (a1 x1 + · · · + an xn ) ∈
1
(λ1 + · · · + λn )C. This proves the theorem.
2
3
1.2
Kronecker’s approximation theorem
The aim of this section is to obtain a quantitative version of Kronecker’s approximation theorem for linear forms. The proof of this version is based on geometry of
numbers. This idea comes originally from K. Mahler. He published a paper [15] in
1966 in which he applies geometry of numbers to prove Kronecker’s Theorem.
Let R be a ring. We denote the space of n-dimensional column vectors by Rn and
the ring of m × n matrices over R by Rm,n . We define the following two norms on Rn :
P
1. kxk1 := ni=1 |xi | for all x ∈ Rn , called the sum norm, and
2. kxk∞ := max{|x1 |, . . . , |xn |} for all x ∈ Rn called the maximum norm.
Let Li (q) = αi1 q1 + · · · + αin qn (i = 1, . . . , m) be m linear forms with real coefficients
and define


α11 · · · α1n

..  .
..
A :=  ...
.
. 
αm1 · · · αmn
The following theorem is a special case of Kronecker’s approximation theorem for
linear forms.
Theorem 1.5. (Kronecker) Let A ∈ Rm,n and assume that
{z ∈ Qm : AT z ∈ Qn } = {0}.
(1.1)
Then for every ε > 0, b1 , . . . , bm ∈ R, there exist p1 , . . . , pm ∈ Z, q ∈ Zn such that
|Li (q) − pi − bi | ≤ ε
for i = 1, . . . , m.
(1.2)
Proof. A proof is given in this section. See [10] for Kronecker’s original paper.
The next lemma states that condition (1.1) is necessary for this theorem.
Lemma 1.6. If for every ε > 0, b = (b1 , . . . , bm ) ∈ Rm there exist p = (p1 , . . . , pm ) ∈
Zm , q ∈ Zn , such that |Li (q) − pi − bi | < ε then condition (1.1) is satisfied.
Proof. This is a proof by contradiction. Suppose there exists z0 ∈ Qm such that
AT z0 ∈ Qn and z0 6= 0. Let z be the vector obtained by multiplying z0 with the lowest
common multiple of the denominators of the coordinates in z0 and AT z0 . Note that
4
z ∈ Zm and AT z ∈ Zn . There exists b ∈ Rm with hz, bi 6∈ Z since z 6= 0. For all
vectors p ∈ Zm , q ∈ Zn we have
|hz, Aq − p − bi| = |(Aq)T z − hz, pi − hz, bi|
= |qT AT z − hz, pi − hz, bi|.
Note that qT AT z − hz, pi ∈ Z for all p ∈ Zm , q ∈ Zn and hz, bi 6∈ Z. Hence,
|hz, Aq − p − bi| ≥ dhz, bic > 0 independent of p and q. This proves the lemma.
Theorem 1.5 does not give an upper bound for kqk∞ . The following theorem by
Dirichlet proves an effective upper bound for homogeneous linear forms.
Theorem 1.7. (Dirichlet) For every ε with 0 < ε < 1, there exist p ∈ Zm , q ∈ Zn
with q 6= 0 such that
|Li (q) − pi | ≤ ε for i = 1, . . . , m,
kqk∞ ≤ ε−m/n .
Proof. Dirichlet proved this in 1842. See Cassels [5] for a proof.
We wonder if we can calculate such an effective upper bound for kqk∞ in the
case of a system of inhomogenous linear forms. The following theorem implies a noneffective upper bound. The inequalities in (1.2) can be stated in the following more
efficient way
kAq − p − bk∞ ≤ ε.
We denote the distance from a real number x to the nearest integer by dxc.
Theorem 1.8. Let Q, ε be positive reals such that for all integers a1 , . . . , am , not all
zero,
n
m
X
X
Q
da1 αj1 + · · · + am αjm c + ε
|ai | ≥ 12 (m + n)2 .
j=1
(1.3)
i=1
Then for all b ∈ Rn there exist p ∈ Zm , q ∈ Zn such that
kAq − p − bk∞ ≤ ε,
kqk∞ ≤ Q.
Kannan and Lovász proved this theorem for n = 1. The following proof expands theirs
([9], Theorem 5.5) to arbitrary n.
5
Proof. Let C = {x ∈ Rm+n : kxk∞ ≤ 1} be the unit cube and let L be the lattice
generated by the columns of the matrix
Im −A
B=
,
0 Qε In
where In denotes the n × n identity matrix. It is easy to see that C ∗ = {x ∈ Rm+n :
kxk1 ≤ 1} and that the dual lattice L∗ is generated by the columns of the inverse
transpose of B
Im
0
−1 T
(B ) =
.
Q T
A Qε In
ε
That is, lattice points in L∗ are of the form
a
Q
(AT a + c)
ε
with a ∈ Zm , c ∈ Zn . Condition (1.3) in Theorem 1.8 implies that for all (a, c) 6= (0, 0)
one has
a
Q
≥ 1 (m + n)2 /ε.
2
T
(A
a
+
c)
ε
1
Hence, by definition of λ1 (C ∗ , L∗ ) we have
λ1 (C ∗ , L∗ ) ≥ 21 (m + n)2 /ε.
For all b ∈ Rm there exist vectors p ∈ Zm , q ∈ Zn such that kB pq + b0 k∞ ≤ µ(C, L)
by definition of µ(C, L). Since µ(C, L) ≤ ε by Lemma 1.3 we have
kp − Aq + bk∞ ≤ ε
kqk∞ ≤ Q.
This proves the theorem.
Lemma 1.9. Theorem 1.8 implies Theorem 1.5.
m
Proof.
Pn Choose an arbitrary ε > 0. For every a = (a1 , . . . , am ) ∈ Z with a 6= 0 we
find j=1 da1 α1j + · · · + am αmj c > 0, because of condition (1.1). Now take

−1
n
X
1


Q = (m + n)2 
min
da1 α1j + · · · + an αmj c .
2
kak1 ≤ 12 (m+n)2 /ε j=1
a6=0
With this ε and Q, condition (1.3) is satisfied. The lemma follows.
6
1.3
Systems of linear equations over principal ideal domains
In this section we prove a criterion for the solvability of systems of linear equations
over principal ideal domains. Recall that a principal ideal domain is a domain, of which
all ideals are principal. This result is used for the proof of a more general Kronecker’s
Theorem in the next section.
A diagonal matrix is a square matrix in which all elements outside the main
diagonal are zero. Let R be a principal ideal domain. For a, b ∈ R, a 6= 0 we denote
a|b if a divides b.
Theorem 1.10. Let A ∈ Rm,n . Then there exist V1 ∈ GLm (R), V2 ∈ GLn (R) such
that A0 given by A0 := V1 AV2 is a diagonal matrix


δ1
..


.




δ


t
A0 = 
,
0




.
..


0
where δ1 , . . . , δt ∈ R with δ1 | . . . |δt and t = rank(A).
Proof. See Bourbaki [4], Livre II, Chapitre VII, Section 4.5, Corollaire 1 of Proposition 4. For a proof in the special case that R = Z, see Smith [21].
The matrix A0 is called the Smith normal form of A. Bachem and Kannan, [1],
published a polynomial time algorithm to calculate the Smith normal form in the
special case that R = Z.
Theorem 1.11. Let R be a principal ideal domain with field of fractions K and let
A ∈ Rm,n , b ∈ Rm . Then the following two assertions are equivalent.
(i) There exists x ∈ Rn such that Ax = b.
(ii) {z ∈ K m : AT z ∈ Rn } ⊆ {z ∈ K m : bT z ∈ R}.
Proof. First we prove (i) ⇒ (ii). Let z ∈ K m , such that AT z ∈ Rn . Then
bT z = xT AT z ∈ xT Rn ⊂ R.
This proves (i) ⇒ (ii).
7
Now, we prove (ii) ⇒ (i). Let A ∈ Rm,n and b ∈ Rm satisfy assertion (ii). Let A0
be the Smith normal form of A and let V1 , V2 be the matrices such that A0 = V1 AV2
as in Theorem 1.10.
If A0 T z ∈ Rn then we have V2T AT V1T z ∈ Rn and as V2 ∈ GLn (R) we get AT V1T z ∈
Rn . By assertion (ii) we see that (V1 b)T z = bT V1T z ∈ R. We conclude that A0 and
b0 := (b01 , . . . , b0m )T = V1 b also satisfy assertion (ii).
For every z = (z1 , . . . , zm ) ∈ K m we have


δ1
..


.




δt


0T
A z=
 z = (δ1 z1 , . . . , δt zt , 0, . . . , 0)T .
0




..


.
0
It is easy to see that δ1 |b01 , . . . , δt |b0t and that b0t+1 = 0, . . . , b0m = 0. Define
x0 := (b01 /δ1 , . . . , b0t /δt , 0, . . . , 0)
and note that x0 ∈ Rn and A0 x0 = b0 . Define x := V2−1 x0 , then we have x ∈ Rm and
Ax = b. This proves the theorem.
1.4
A more general Kronecker’s Theorem
We already proved a special case of Kronecker’s Theorem for linear forms. In this
section we use that result to prove the following more general form of Kronecker’s
theorem. We use the following notation in this proof. Let again A ∈ Rr,s be the r × s
matrix


α11 · · · α1s

..  .
..
A :=  ...
.
. 
αr1 · · · αrs
Define a norm k·k on Rr,s by
kAk := max
1≤i≤r
s
X
|αij |.
j=1
Note that kAxk∞ ≤ kAkkxk∞ for all A ∈ Rr,s , x ∈ Rs .
8
Theorem 1.12. Let A ∈ Rr,s and b ∈ Rr . Then the following two assertions are
equivalent.
(i) For every ε > 0 there exists x ∈ Zs such that kAx − bk∞ ≤ ε.
(ii) For all z ∈ Rr with AT z ∈ Zs we have bT z ∈ Z.
In the proof of this theorem we need some lemmas, in which we refer repeatedly to
assertions (i) and (ii).
Lemma 1.13. Let U ∈ GLr (R), V ∈ GLs (Z) and put à := U AV , b̃ := U b. Then:
(i) is equivalent to the assertion that for every ε > 0 there is x ∈ Zs such that
kÃx − b̃k∞ ≤ ε,
(ii) is equivalent to the assertion that
{z ∈ Rr : ÃT z ∈ Zs } ⊆ {z ∈ Rr : b̃T z ∈ Z}.
Proof. Suppose assertion (i) holds. Then for every ε > 0 there exists x ∈ Zs such
that
ε
.
kAx − bk∞ ≤
kU k
Define x̃ := V −1 x. Then
kÃx̃ − b̃k∞ = kU Ax − U bk∞ ≤ ε.
The proof of the reverse implication is entirely similar.
Suppose assertion (ii) holds. If there exists z ∈ Rr , w ∈ Z such that V T AT (U T z) =
w then AT (U T z) = (V T )−1 w ∈ Z. By assertion (ii) we have bT U T z ∈ Z. We conclude
that
{z ∈ Rr : ÃT z ∈ Zs } ⊆ {z ∈ Rr , b̃T z ∈ Z}.
Again, the reverse implication is proved in the same manner.
Lemma 1.14. Assume (ii) holds. Then there exist U ∈ GLr (R), V ∈ GLs (Z) such
that




It
0
−A1
b1
U AV =  0 Im−t −A2  , U b =  b2 
0
0
0
0
where 0 ≤ t ≤ m ≤ r, A1 ∈ Qt,s−m , A2 ∈ Rm−t,s−m , b1 ∈ Qt , b2 ∈ Rm−t , and
{z1 ∈ Zt : AT1 z1 ∈ Zs−m } ⊆ {z1 ∈ Zt : bT1 z1 ∈ Z},
{z2 ∈ Zm−t : AT2 z2 ∈ Zs−m } = {0}.
9
Proof. Let m := rank(A) and suppose without loss of generality that the first m
rows of A are linearly independent. Then by elementary linear algebra, there exists
U1 ∈ GLr (R) such that
Im −A0
U1 A =
with A0 ∈ Rm,s−m .
0
0
By Lemma 1.13, the validity of (ii) is unaffected if we replace A by U1 A. That is, (ii)
is equivalent to the assertion that
Im −A0
r
z ∈ Zs } ⊆ {z ∈ Rr : bT U T z ∈ Z}.
(1.4)
{z ∈ R :
0
0
For this to hold, we must have U1 b = (b0 , 0)T with b0 ∈ Rm . Thus, (1.4) becomes
{z0 ∈ Zm : A0T z0 ∈ Zs−m } ⊆ {z0 ∈ Zm : b0T z0 ∈ Z}.
(1.5)
The left-hand side is a sub-Z-module M of Zs−m , hence free of rank t ≤ m. If t = 0
we are done. Suppose t > 0. Then by Theorem 1.11 there is a basis {d1 , . . . , dm } of
Zm and there are positive integers δ1 , . . . , δt , such that {δ1 d1 , . . . , δt dt } is a Z-basis
of M .
Now, let D := [d1 , . . . , dm ] be the matrix with columns d1 , . . . , dm . Then D ∈
GLm (Z). Define  := DT A0 , b̂ := DT b0 . Then
T
(DT )−1
0
D
0
Im −A0
Im −Â
=
.
(1.6)
0
0
0
Is−m
0 Ir−t
0
0
The right-hand side is clearly of the shape U AV with U ∈ GLr (R), V ∈ GLs (Z).
Writing ẑ := D−1 z0 we see that (1.5) is equivalent to
{ẑ ∈ Zm : ÂT ẑ ∈ Zs−m } ⊆ {ẑ ∈ Zm : b̂T ẑ ∈ Z}.
(1.7)
Let A1 consist of the first t rows of  and A2 of the last m−t rows. Further, let b1 and
b2 consist of respectively the first t and the last m − t coordinates of b̂. Notice that
the left-hand side of (1.7) consists of all vectors of the shape (δ1 x1 , . . . , δr xt , 0, . . . , 0)T
with x1 , . . . , xt ∈ Z. Hence, A1 ∈ Qt,s−m , b1 ∈ Qt and
{z1 ∈ Zt : AT1 z1 ∈ Zs−m } ⊆ {z1 ∈ Zt : bT1 z1 ∈ Z}.
Further, by applying (1.7) with vectors (0, . . . , 0, zt+1 , . . . , zm )T , we see that
{z2 ∈ Zt : AT2 z2 ∈ Zs−m } = {0}.
10
This proves the lemma.
Proof of Theorem 1.12. The proof of (i) ⇒ (ii) is very similar to that of (i) ⇒
(ii) of Theorem 1.11, and is therefore left to the reader.
Now, we prove (ii) ⇒ (i). By Lemmas 1.13 and 1.14, we may assume without loss
of generality, that




It
0
−A1
b1
A =  0 Im−t −A2  , b =  b2  ,
0
0
0
0
with A1 , A2 , b1 , b2 as in Lemma 1.14. Writing xT = (qT , pT1 , pT2 ), We can rewrite (i)
as
kA1 q − p1 − b1 k∞ ≤ ε,
kA2 q − p2 − b2 k∞ ≤ ε,
(1.8)
to be solved in q ∈ Zs−m , p1 ∈ Zt , p2 ∈ Zm−t . By Theorem 1.11 there exist q0 ∈
Zs−m , p01 ∈ Zt such that
A1 q0 − p01 = b1 .
Recall that A1 ∈ Qt,s−m . Let d be a positive integer, such that dA1 ∈ Zt,s−m . By
Theorem 1.8, there exist q00 ∈ Zs−m , p02 ∈ Zt , such that
1
ε
kA2 q00 − p02 − (b2 − A2 q0 )k∞ ≤ .
d
d
(1.9)
Hence,
A1 (q0 + dq00 ) − (p01 + dA1 q00 ) − b1 = 0,
kA2 (q0 + dq00 ) − dp02 − b2 k∞ < ε,
which implies that (1.8) is satisfied with q = q0 + dq00 , p1 = p01 + dA1 q00 , p2 = dp02 .
This proves the theorem.
11
Chapter 2
Valuations
In Chapter 3 we give an effective version of Kronecker’s Theorem for a matrix A ∈
Rm,n with algebraic entries. For this we need some algebraic number theory and more
specifically, the theory of valuations.
2.1
Some algebraic preliminaries
We refer to “Algebraic Number Theory” by J. Neukirch [18] for algebraic number
theory. In this section we have stated some definitions and results.
A number field K is a finite extension of Q. Define
OK = {x ∈ K : ∃ monic f ∈ Z[X] such that f (x) = 0}
This set is a subring of K and is called the ring of integers of K. This ring is a
Dedekind domain, which means that it is a noetherian, integrally closed, integral
domain in which every non-zero prime ideal is a maximal ideal. Ideals in Dedekind
domains can be factorized into prime ideals in a unique way.
Let K ⊂ L be an extension of number fields and let p be a non-zero prime ideal
of OK . Then the set
pOL = {xy : x ∈ p, y ∈ OL }
is an ideal of OL . Using the unique ideal factorization in OL we get
pOL = Pe11 · · · Pegg
for certain prime ideals P1 , . . . , Pg of OL and positive integers e1 , . . . , eg . We call ei
the ramification index of Pi over p and fi := [OL /Pi : OK /p] the residue class degree
of Pi over p.
Theorem 2.1. (Fundamental identity) Let K ⊂ L be an extension of number
fields of degree d. Then one has
g
X
ei fi = d.
i=1
12
Proof. See Neukirch [18], Chapter I, Proposition 8.2.
A fractional ideal of OK is a finitely generated non-zero sub-OK -module a 6= (0)
of K. Every fractional ideal a of OK can be expressed uniquely as a product
a = pk11 · · · pknn ,
where p1 , . . . , pn are prime ideals and k1 , . . . , kn ∈ Z. For a non-zero prime ideal p of
OK we define ordpi (a) := ki for i = 0, . . . , n and ordp (a) := 0 if p 6∈ {p1 , . . . , pn }. We
have ordp (a) 6= 0 for only finitely many prime ideals p. For an element x ∈ K, we
define ordp (x) := ordp (b), where b = (x) is the fractional OK -ideal generated by x.
The set of fractional ideals of OK is a group under multiplication, denoted by IK .
Let PK denote the group of principal fractional ideals of OK , i.e., the set of fractional
ideals generated by one element of K. The factor group IK /PK is the ideal class group
of K and its order is called the class number of K.
Theorem 2.2. The ideal class group IK /PK is finite.
Proof. See Janusz [8], Chapter I, Theorem 11.10.
Note that OK is a principal ideal domain if and only if the ideal class group of K
is trivial.
Proposition 2.3. Let K be a number field of degree d = [K : Q] over Q and let a be
a fractional ideal of OK . Then as a Z-module, a is free of rank d.
Proof. This follows directly from Neukirch [18], Chapter I, Proposition 2.10.
Let K ⊂ L be an extension of number fields. The map
TrL/K : L −→ K
X
x 7−→
σ(x)
σ
is called the trace and the map
NL/K : L −→ K
Y
x 7−→
σ(x)
σ
is called the norm. Here the sum and product range over all K-isomorphic embeddings
σ of L in C.
13
We can also define a norm on the group of fractional ideals. Any fractional ideal
a can be expressed uniquely as a = bc−1 , where b, c are integral ideals such that
b + c = (1). We then define N (a) by
N (a) := (#OK /b) · (#OK /c)−1 .
Let K ⊂ L an extension of number fields of degree d. We define the discriminant
of a basis {x1 , . . . , xd } of L over K by
2
(j) d
DL/K (x1 , · · · , xd ) = det(xi )i,j=1 ,
(1)
(d)
where xi , . . . , xi are the d images of xi under the K-isomorphic embeddings of L
in C. By some elementary calculations, we can rewrite this as
d
DL/K (x1 , · · · , xd ) = det TrK/Q (xi xj )
.
i,j=1
We have the following important proposition regarding the discriminant.
Proposition 2.4. Let {x1 , . . . , xd } be a Q-basis of K. Then
DK/Q (x1 , . . . , xd ) 6= 0.
If {x1 , . . . , xd } and {y1 , . . . , yd } are Z-bases of the same fractional ideal of OK , then
we have
DK/Q (x1 , . . . , xd ) = DK/Q (y1 , . . . , yd ).
Proof. See Lang [13], Chapter I, Section 2, Proposition 8 and Proposition 12.
Let again K be a number field and a a fractional ideal of OK . By Proposition
2.4, we can define the discriminant DK/Q (a) of a fractional ideal a by DK/Q (a) =
DK/Q (x1 , . . . , xd ), where {x1 , . . . , xd } is a Z-basis of a as a Z-module. In particular,
we define the discriminant DK of a number field K by
DK = DK/Q (OK ).
Hence, if we choose a basis {ω1 , . . . , ωd } of OK , then we get
DK = det(TrK/Q (ωi ωj ))di,j=1 .
As ωi ωj ∈ OK for i, j = 1, · · · , d, we find that TrK/Q (ωi ωj ) ∈ Z. Now, it follows that
DK ∈ Z.
14
Proposition 2.5. Let a be a fractional ideal of OK . Then
DK/Q (a) = N (a)2 DK .
Proof. This follows directly by expressing a as a product of two integral ideals
a = bc−1 and applying Lang [13], Chapter I, Section 2, Proposition 12 to these ideals.
2.2
An introduction to absolute values
Definition 2.6. An absolute value on a field K is a function | · |v : K → R≥0 such
that
1. |x|v = 0 if and only if x = 0 for all x ∈ K;
2. |xy|v = |x|v |y|v for all x, y ∈ K;
3. there exists C ∈ R>0 , such that |x + y|v ≤ C max{|x|v , |y|v } for all x, y ∈ K.
Definition 2.7. A valuation on a field K is a function v : K → R ∪ {∞} such that
1. v(x) = +∞ if and only if x = 0 for all x ∈ K;
2. v(xy) = v(x) + v(y) for all x, y ∈ K;
3. v(x + y) ≥ min {v(x), v(y)} for all x, y ∈ K.
An example of a valuation on a number field is given by ordp . It is easy to see that
a valuation v gives rise to an absolute value | · |v : K → R≥0 defined by |x|v = c−v(x)
with c ∈ R>1 . In other literature, sometimes the term valuation is used for absolute
value.
Example 2.8. The most trivial example of an absolute value, which can be defined on
any field K, is the absolute value that sends all x ∈ K with x 6= 0 to 1. This absolute
value is called trivial. Henceforth, all absolute values we consider are assumed to be
non-trivial.
Example 2.9. The standard absolute value on Q, | · |∞ , is defined by
| · |∞ : Q −→ R≥0
x 7−→ |x|.
We call ∞ the infinite prime.
15
Example 2.10. Let x ∈ Q\{0}. We can write x as x = cb with b, c ∈ Z. If we
subsequently extract the highest possible power of p from b and c, we get
b
x = pm ,
c
gcd(bc, p) = 1.
The p-adic absolute value |x|p of x is then defined by |x|p :=
|0|p := 0.
1
.
pm
Further, we put
Definition 2.11. Let K be a field. Two absolute values | · |v and | · |w on K are
called equivalent if and only if there exists a real number c > 0 such that
|x|v = |x|cw
for all x ∈ K. An equivalence class of absolute values on K is called a place of K. We
denote the collection of all places of K by MK .
Proposition 2.12. Let | · |v and | · |w be two absolute values on a field K. Then the
following two assertions are equivalent:
(i) | · |v and | · |w are equivalent,
(ii) |x|v < 1 ⇔ |x|w < 1 for all x ∈ K.
Proof. See Neukirch [18], Chapter II, Proposition 3.3.
Definition 2.13. An absolute value | · |v on a field K is called non-archimedean if it
satisfies the ultrametric inequality
|x + y|v ≤ max{|x|v , |y|v } for all x, y ∈ K.
Otherwise it is called archimedean .
The p-adic absolute values are non-archimedean and the standard absolute value is
archimedean. The following lemma shows that we can define this property for places.
Lemma 2.14. Equivalent absolute values are either both archimedean or both nonarchimedean.
Proof. Let | · |v and | · |w be equivalent absolute values on a field K. Hence, there
exists an c > 0 such that |x|v = |x|cw for all x ∈ K. Suppose | · |v is non-archimedean,
then
|x + y|w = |x + y|cv ≤ max{|x|v , |y|v }c = max{|x|cv , |y|cv } = max{|x|w , |y|w }
16
for all x, y ∈ K. Hence, | · |w is archimedean too. By symmetry we get the other
implication.
For a field K, denote by MK the set of places of K, by MK∞ the set of archimedean
places of K, and by MKfin the set of non-archimedean places of K. By the following
theorem we can identify these places in the case that K = Q.
Theorem 2.15. (Ostrowski) Every non-trivial absolute value on Q is equivalent
either to | · |∞ or to | · |p for some prime number p.
Proof. For a proof see Neukirch [18], Chapter II, Proposition 3.7.
Hence, we may identify a non-archimedean place of Q with its corresponding prime
number. Thus, we may write MQ = {∞} ∪ {primes}.
Theorem 2.16. (Product formula) We have
Y
|x|p = 1 for all x ∈ Q∗ .
p∈MQ
Proof. Take an arbitrary x ∈ Q∗ . The theorem follows directly from writing
x = ±pk11 pk22 · · · pkt t
and calculating
2.3
Q
p∈MQ
|x|p .
Completions
We are already familiar with the construction of the real numbers as completion
of the rational numbers. This is done with respect to the standard absolute value
| · |∞ . We can adjust the same construction to arbitrary fields K with respect to any
absolute value | · |v on this field K.
A field K is called complete with respect to an absolute value | · |v , if every Cauchy
sequence in K converges in K with respect to | · |v . Let R be the ring of Cauchy
sequences in K with respect to | · |v . This is a ring under pointwise addition and
multiplication of the sequences. The subset of R consisting of all sequences in R
converging to 0 with respect to | · |v form a maximal ideal of R which we denote by m.
The field Kv := R/m is called the completion of K with respect to | · |v . It is easy
to see that Kv contains K. We can extend | · |v to Kv in the following way. For every
17
x ∈ Kv there exists a Cauchy sequence (xk )∞
k=1 in K with lim xk = x. We define | · |v
k→∞
on Kv by |x|v := lim |xk |v . The field Kv is complete with respect to this valuation.
k→∞
The completion Kv of a field K with respect to an archimedean absolute value is
equal to R or C. This follows directly from the following result regarding fields that
are complete with respect to an archimedean absolute value.
Theorem 2.17. (Ostrowski) Every field that is complete with respect to an archimedean absolute value is isomorphic to either R or C.
Proof. See Neukirch [18], Chapter II, Theorem 4.2.
The next definition is that of an extension of an absolute value. This is very
important in the next section, where we look at absolute values on number fields.
Definition 2.18. Let K ⊂ L be a field extension and let | · |v and | · |w be absolute
values on K and L respectively. We say that | · |w extends | · |v if the restriction of
| · |w to K is equal to | · |v .
Now, it is natural to speak of an extension of a place as in the following definition.
Definition 2.19. Let K ⊂ L be a field extension and let v, w be places on respectively
K and L. A place w extends v, denoted w|v, if the restriction of a representative of
w to K is equivalent to a representative of v. We say that w lies above v.
We have the following important theorem regarding extensions of absolute values.
Theorem 2.20. Let K be a complete field with respect to an absolute value | · |v and
let K ⊂ L be a field extension of finite degree d. Then there exists an unique extension
of | · |v to L given by
q
|x|w :=
d
|NL/K (x)|v
for x ∈ L
and L is complete with respect to | · |w .
Proof. See Neukirch [18], Chapter II, Theorem 4.8.
Theorem 2.21. Let | · |v be an absolute value on K. There exists a unique extension
of | · |v from Kv to Kv also denoted by | · |v , given by
q
|x|v := d |NKv (x)/Kv (x)|v for x ∈ Kv .
18
Proof. Let x ∈ Kv . Let L1 and L2 be any two finite field extions of Kv containing x.
There exist two unique absolute values by Theorem 2.20, on respectively L1 and L2 ,
given by
q
|x|v1 := d |NL1 /Kv (x)|v
for x ∈ L1
q
|x|v2 := d |NL2 /Kv (x)|v
for x ∈ L2
on respectively L1 and L2 . These absolute values agree on L1 ∩ L2 , because the extension of | · |v to L1 ∩ L2 is unique by Theorem 2.20. We conclude that there exists
an unique extension of | · |v to Kv .
2.4
Absolute values on number fields
In this section we state and prove some results of absolute values in the more
specific case of number fields. Henceforth, let K be a number field and let K denote
the algebraic closure of K.
Let σ1 , . . . , σr be the real embeddings of K, and let σr+1 , . . . , σr+2s be the complex
embeddings of K ordered such that σr+s+i = σr+i for i = 1, . . . , s. Define d := [K : Q].
Note that r + 2s = d.
Let σ : K ,→ C be an embedding of K in C. We define an absolute value | · |σ
associated to an embedding σ by
|x|σ := |σ(x)|
|x|σ := |σ(x)|2
for all x ∈ K if σ is real,
for all x ∈ K if σ is complex.
Let σ be a complex embedding. We denote its complex conjugate by σ. Note that we
have |σ(x)| = |σ(x)| for all x ∈ K. So, conjugate embeddings give rise to the same
absolute value. We call a place associated to a real embedding a real place and a place
associated to a complex embedding a complex place. There are r real places and s
complex places of K.
Lemma 2.22. Every archimedean place on K is induced by an embedding
σ : K ,→ C.
Proof. Let |·|v be an archimedean abolute value on K. The completion Kv of K with
respect to | · |v is either isomorphic to R or C as stated in Theorem 2.17. Hence, the
19
natural embedding composed with this isomorphism is an embedding of K in either
R or C. Every archimedean absolute value on R or C is equivalent to the standard
absolute value. Hence, | · |v is equivalent to | · |σ . This proves the lemma.
Henceforth, we identify the embeddings {σ1 , . . . , σr+s } with the corresponding
archimedean places and we choose | · |σi as standard representative for the place σi .
This is possible by Lemma 2.22.
Lemma 2.23. The ring of integers OK of K is equal to
{x ∈ K : |x|v ≤ 1 for all v ∈ MKfin }.
Proof. First we prove that OK ⊂ {x ∈ K : |x|v ≤ 1 for all v ∈ MKfin }. Take an
arbitrary x ∈ OK and v ∈ MKfin . Note that v|p for some prime number p, because
v is non-archimedean. As x is integral over Z, we can find a monic polynomial f =
X n + an−1 X n−1 + · · · + a0 ∈ Z[X] such that
xn + an−1 xn−1 + · · · + a0 = 0.
There is a c ∈ R>0 such that | · |v = | · |cp on Q. Hence, |ai |v ≤ 1 for i = 0, . . . , n − 1.
If we take the absolute value and use the ultrametric inequality we get
|xn |v = |an−1 xn−1 + · · · + a1 x + a0 |v ≤ max |ai xi |.
0≤i≤n−1
We get a contradiction if |x|v > 1. We conclude that |x|v ≤ 1 for all v ∈ MKfin .
We still have to show that {x ∈ K : |x|v ≤ 1 for all v ∈ MKfin } ⊂ OK . Choose an
x ∈ K such that |x|v ≤ 1 for all v ∈ MKfin . Let f be the monic minimal polynomial of
x over Q. It is given by
f (X) = X n + an−1 X n−1 + · · · + a1 X + a0
with a0 , . . . , an−1 ∈ Q. We have to prove that a0 , . . . , an−1 ∈ Z. Let α1 , . . . , αn be the
roots of f in Kv . Sending x to one of the values α1 , . . . , αn induces an embedding of
K(x) in Kv . Let σ1 , . . . , σn be those embeddings. Extend | · |v to Kv and define an
absolute value | · |w by
|y|w := |σi (y)|v for y ∈ K(x).
The restriction to K of this absolute value is a non-archimedean, hence |x|w ≤ 1.
It follows that |αi |v ≤ 1 for i = 1, . . . , n. The coefficients a0 , . . . , an−1 of f are
up to sign sums of products of subsets of {α1 , . . . , αn }. We conclude that |a1 |v ≤
1, . . . , |an−1 |v ≤ 1 for all v ∈ MKfin . Hence, |ai |p ≤ 1 for all p ∈ MQfin and so ai ∈ Z
20
for i = 0, . . . , n − 1. This proves the lemma.
Every prime ideal p 6= 0 of OK gives rise to a non-archimedean absolute value | · |p
by defining
|x|p = N (p)−ordp (x) for x ∈ K ∗ , |0|p = 0.
Lemma 2.24. The map f from the set of prime ideals of OK to the set of nonarchimedean places of K given by
p 7→ place of | · |p
is a bijection.
Proof. Note that the set {x ∈ OK : |x|p < 1} is equal to p. So, different prime ideals
give representatives for different places by Proposition 2.12. This proves the injectivity of f . To prove surjectivity of f take an arbitrary non-archimedean absolute value
| · |v on K. The set {x ∈ OK : |x|v < 1} is an ideal. By Lemma 2.23 and the property
|xy|v = |x|v |y|v we get primality of this ideal. The image of this prime ideal under f
gives back an absolute value equivalent to | · |v by Proposition 2.12.
Henceforth, for an algebraic number field K, we identify every prime ideal of OK
with the corresponding non-archimedean place. Thus,
MK = {prime ideals of OK } ∪ {σ1 , . . . , σr+s }.
Lemma 2.25. If x ∈ K then |x|v ≤ 1 for almost all v ∈ MK .
Proof. The fractional ideal (x) of OK generated by x ∈ K has a unique prime
factorization
(x) = pk11 . . . pkg g .
Hence, |x|v ≤ 1 for all v ∈ MKfin with v 6= p1 , . . . , v 6= pn .
Lemma 2.26. Let K ⊂ L be an extension of number fields and let v ∈ MK , w ∈ ML
with w|v. Then
w :Kv ]
|x|w = |NLw /Kv (x)|[L
for all x ∈ L.
v
Proof. Note that there exists a c ∈ R>0 such that
|x|w = |NLw /Kv (x)|cv for all x ∈ L
21
by Theorem 2.20.
Now, we calculate c for v, w non-archimedean. We have v = p for a prime ideal p
in OK . The unique prime factorization in OL gives
pOL = Pe11 . . . Pegg .
As {x ∈ K : |x|w < 1} = p we know that w = Pi for some 1 ≤ i ≤ g. Expressing
|x|Pi in terms of |x|p for x ∈ K, we get
|x|Pi
= (#OL /Pi )−ordPi (x) = (#OL /Pi )−ei ordp (x)
= (#OK /p)−ei fi ordp (x) = |x|pei fi .
By Neukirch [18], Chapter II, Proposition 6.8 (with Remark) and the explanation
under Proposition 8.5 we have [Lw : Kv ] = ei fi . We conclude that
|x|w = |NLw /Kv (x)|v for all x ∈ L.
It is easy to derive the same relation for v, w archimedean.
Theorem 2.27. Let K ⊂ L be an extension of number fields of degree d. Let v ∈ MK .
For the places w ∈ ML extending v we have the following formula:
Y
|x|w = |NL/K (x)|v for all x ∈ L.
w|v
Proof. Let v ∈ MK . We have
Y
w|v
|x|w =
Y
|NLw /Kv (x)|v
w|v
and by Neukirch [18], Chapter II, Corollary 8.4
Y
|NLw /Kv (x)|v = |NL/K (x)|v .
w|v
This proves the theorem.
Theorem 2.28. (Product formula) For a number field K we have the following
product formula:
Y
|x|v = 1 for all x ∈ K ∗ .
v∈MK
22
Proof. Using Theorem 2.27 for a field extension Q ⊂ K, we get
Y
Y Y
Y
|x|v =
|x|v =
|NK/Q (x)|p = 1 for all x ∈ K ∗ .
v∈MK
p∈MQ v|p
p∈MQ
This proves the theorem.
n
Now, we are able to define the height for x ∈ Q .
Definition 2.29. Let x = (x1 , . . . , xn ) ∈ K n with x 6= 0, then
Y
HK (x) =
max(|x1 |v , . . . , |xn |v )
v∈MK
is called the height of x with respect to K and denoted by HK (x).
Let L be a field extension of K and let x ∈ K n . Then we have
Y
max(|x1 |w , . . . , |xn |w )
HL (x) =
w∈ML
=
Y Y
max(|x1 |w , . . . , |xn |w )
v∈MK w|v
=
Y Y
max(|x1 |v , . . . , |xn |v )[Lw :Kv ]
v∈MK w|v
=
Y
max(|x1 |v , . . . , |xn |v )[L:K]
v∈MK
= HK (x)d .
n
Now, we define the absolute height for x ∈ Q by choosing a number field K such
that x ∈ K n , and putting H(x) = HK (x)1/[K:Q] . By what we just observed, this is
independent of the choice of K.
23
Chapter 3
An effective Kronecker’s Theorem
3.1
Uniform distribution
Let A be m × n matrix with real coefficients satisfying the condition (1.1) of
Kronecker’s Theorem. Then by Kronecker’s Theorem there exist p ∈ Zm , q ∈ Zn
such that
kAq − p − bk∞ ≤ ε.
(3.1)
In Theorem 1.8 we proved a non-effective upper bound for kqk∞ . In the next sections
we also prove an effective upper bound and in this section we will use the theory of
uniform distribution to get some heuristics on the order of the upper bound for kqk∞
in terms of ε.
Define bxc as the largest integer not greater than x.
Definition 3.1. Let (xk )∞
k=1 be a sequence of reals. For a, b ∈ R with 0 ≤ a < b < 1
let T ([a, b); K) denote the number of xk with k ≤ K such that xk − bxk c ∈ [a, b). The
sequence (xk )∞
k=1 is said to be uniformly distributed (modulo 1) if
T ([a, b); K)
=b−a
K→∞
K
lim
for all intervals [a, b) in [0, 1).
Theorem 3.2. (Weyl’s criterion) A sequence (xk )∞
k=1 is uniformly distributed if
and only if for all integers h 6= 0
K
1 X 2πihxk
e
= 0.
lim
K→∞ K
k=1
Proof. See Kuipers and Niederreiter [11], Chapter 1, Theorem 2.1. See Weyl [22] for
Weyl’s original proof.
Corollary 3.3. Let α ∈ R\Q. Then the sequence (xk )∞
k=1 defined by xk = kα is
uniformly distributed.
24
Proof. If x 6= 1, we have the following identity
K
X
xk =
k=0
1 − xK+1
.
1−x
Since α 6∈ Q, we have e2πihα 6= 1 for all h 6= 0. Hence,
K
1 X 2πihαk
1 1 − (e2πiα )K
lim
e
= lim
− 1 = 0,
K→∞ K
K→∞ K
1 − e2πiα
k=1
since the numerator is bounded. Corollary 3.3 follows directly from Weyl’s criterion,
Theorem 3.2.
n
Corollary 3.4. Let (qk )∞
k=1 be a sequence in R , ordered such that if s, t are any
positive integers with kqs k∞ < kqt k∞ then s < t. If α = (α1 , . . . , αn ) 6∈ Qn then the
sequence (xk )∞
k=1 defined by xk = hqk , αi is uniformly distributed.
Proof. By Weyl’s criterion, it suffices to prove
K
1 X 2πihxk
e
= 0 for all integers h 6= 0.
lim
K→∞ K
k=1
Choose an integer h 6= 0. For every K ∈ N
is a non-negative integer L such that
Pthere
(2L+1)n 2πihxk
(2L + 1)n ≤ K < (2L + 3)n . We split K1 k=1
e
into S1 + S2 , where
n
S1
(2L+1)
1 X 2πihxk
:=
e
K k=1
S2
1
:=
K
K
X
k=(2L+1)n +1
25
and
e2πihxk .
Without loss of generality, assume α1 6∈ Q. Then
L
L
X
X
1
2πih(α1 l1 +···+αn ln ) S1 ≤
···
e
(2L + 1)n l =−L
ln =−L
1
!
L
L
L
X
X
X
1
2πihα1 l1
2πih(α2 l2 +···+αn ln ) e
·
·
·
≤
e
(2L + 1)n l =−L
l2 =−L
ln =−L
1
L
X
1
n−1 2πihα1 l1 (2L + 1)
e
=
(2L + 1)n
l1 =−L
L
1 X 2πihα1 l1 =
e
.
2L + 1 l1 =−L
Further,
(2L + 3)n − (2L + 1)n
(2L + 1)n
n
2
=
1+
− 1.
2L + 1
S2 ≤
Note that S1 → 0 as K → ∞ and S2 → 0 as K → ∞. We conclude
K
1 X 2πihxk
e
= 0.
K→∞ K
k=1
lim
Corollary 3.4 follows.
The theory of uniform distribution can be extended to higher dimensions. But
before stating the definition of uniform distribution in n-dimensional space, we need
some new notation. Let a = (a1 , . . . , an ), b = (b1 , . . . , bn ) be two realQ
vectors in Rn
with ai < bi for i = 1, . . . , n. We denote the n-dimensional interval ni=1 [ai , bi ) by
[a, b). For x ∈ Rn we denote the vector (x1 − bx1 c, · · · , xn − bxn c) by x − bxc. By
0 and 1 we denote the two n-dimensional vectors defined by 0 = (0, · · · , 0) and
1 = (1, · · · , 1).
n
Definition 3.5. Let (xk )∞
k=1 be a sequence in R . Let [a, b) be an n-dimensional
interval and let T ([a, b); K) denote the number of xk with k ≤ K such that xk −bxk c ∈
n
[a, b). A sequence (xk )∞
k=1 is said to be uniformly distributed (modulo 1) in R if
n
T ([a, b); K) Y
lim
=
(bj − aj )
K→∞
K
j=1
26
for all intervals [a, b) ⊂ [0, 1).
n
Theorem 3.6. (Weyl’s criterion) A sequence (xk )∞
k=1 in R is uniformly distributed
n
if and only if for every h ∈ Z , h 6= 0,
K
1 X 2πihh,xk i
lim
e
= 0.
K→∞ K
k=1
Proof. See Kuipers and Niederreiter [11], Chapter 1, Theorem 6.2. See Weyl [22] for
Weyl’s original proof.
n
Corollary 3.7. A sequence (xk )∞
k=1 in R is uniformly distributed if and only if
for every h ∈ Zn , h 6= 0, the sequence of real numbers (hh, xk i)∞
k=1 , is uniformly
distributed.
Proof. See Kuipers and Niederreiter [11], Chapter 1, Theorem 6.3. See Weyl [22] for
Weyl’s original proof.
Corollary 3.8. Define α = (α1 , . . . , αm ) with α1 , . . . , αm Q-linearly independent.
m
The sequence (x)∞
k=1 defined by xk = kα is uniformly distributed in R .
Proof. For every h ∈ Zn with h 6= 0, we have hh, αi 6∈ Q by the Q-linear independence of α1 , . . . , αm . The result follows from Corollary 3.3.
n
Lemma 3.9. Let (qk )∞
k=1 be a sequence in Z , ordered such that if s, t are any positive
integers with kqs k∞ < kqt k∞ then s < t. Further, let A be an m × n-matrix with real
entries, that fulfills the condition of Kronecker’s theorem,
{z ∈ Qm : AT z ∈ Qn } = {0}.
(3.2)
Finally, let xk = Aqk for k = 1, . . . , n. Then the sequence (xk )∞
k=1 is uniformly
m
distributed in R .
Proof. For any h ∈ Zn we have
hh, xk i = hh, Aqk i
= hAT h, qk i.
27
As A satisfies equation (3.2) one has AT h 6∈ Qn for all h 6= 0. The sequence
(hAT h, qk i)∞
k=1 is a sequence as in Corollary 3.4. This implies that it is uniformly
distributed. Now Theorem 3.6 gives the required result.
For x ∈ Rn define dxc = max(dx1 c, . . . , dxn c). Let A be a matrix as in Lemma
3.9. This lemma implies that for all b ∈ Rm and 0 < ε < 21 we have the limit
#{q ∈ Zn : kqk∞ ≤ Q, dAq − bc ≤ ε}
= (2ε)m .
Q→∞
Qn
lim
Note that dAq − bc < ε if and only if Aq − b − bAq − bc ∈ [0, ε1) ∪ (1 − ε1, 1).
This explains the factor 2m . Kronecker’s theorem states that for every ε there exist
q ∈ Zn and p ∈ Zm such that kAq − p − bk ≤ ε. This also follows from the limit
just stated. For an effective Kronecker’s theorem we want to give an upper bound Q
for kqk in terms of ε. The limit suggests that for “most” A and b there should be a
solution q ∈ Zn of dAq − bc ≤ ε with kqk∞ of the order ε−m/n .
3.2
An effective Kronecker’s Theorem
At this point we know enough algebraic number theory to prove an effective version
of Kronecker’s approximation theorem for linear forms with algebraic coefficients.
Proposition 3.10. Let K = Q(α1 , . . . , αn ) ⊂ R be a number field and let d = [K : Q].
Then we have
1−d
|q0 + q1 α1 + · · · + qn αn | ≥ kqk∞
(n + 1)1−d H(1, α1 , . . . , αn )−d
for every q = (q0 , . . . , qn ) ∈ Zn+1 with q0 + q1 α1 + · · · + qn αn 6= 0.
Proof. By assumption K ⊂ R. Denote by v the place represented by | · |. As q0 +
q1 α1 + · · · + qn αn 6= 0 we may rewrite the product formula, Theorem 2.28, as
Y
|q0 + q1 α1 + · · · + qn αn | =
|q0 + q1 α1 + · · · + qn αn |−1
w
w∈MK
w6=v
1−d
≥ kqk∞
(n + 1)1−d
Y
max{1, |α1 |w , . . . , |αn |w }−1
w∈MK
w6=v
1−d
≥ kqk∞
(n + 1)1−d H(1, α1 , . . . , αn )−d .
28
This proves the proposition.
With this proposition we can make Theorem 1.8 effective for the case that A is
an m × n matrix with real algebraic elements satisfying condition (1.1). Let


α11 . . . α1n

.. 
...
A =  ...
. 
αm1 . . .
αmn
with real algebraic elements αij . The extensions Q(α1j , . . . , αmj ) over Q are finite for
j = 1, . . . , n, because α1j , . . . , αmj are algebraic over Q. Define
dj := [Q(α1j , . . . , αmj ) : Q] (j = 1, . . . , n) d := max(d1 , . . . , dn ).
(3.3)
We define the height of A as
H ∗ (A) = (n + 1)d−1 max H(1, α1j , . . . , αmj )d .
j=1,...,n
This notation is not standard, but it turns out to be very useful in the next theorem.
Theorem 3.11. Let A = (αij ) be an m × n matrix with real algebraic elements such
that
{z ∈ Qm : AT z ∈ Qn } = {0}.
(3.4)
Let d be given by (3.3) and define for ε > 0
d−1
1
.
Q(ε) := 2d−1 (m + n) H (A)
ε
d
2d
∗
Then for every ε > 0, b ∈ Rm , there exist p ∈ Zm , q ∈ Zn with
kAq − p − bk∞ ≤ ε,
kqk∞ ≤ Q(ε).
Proof. Let a ∈ Zm with a 6= 0. Condition (3.4) implies that there is a j ∈ {1, . . . , m},
such that α1j a1 + · · · + αmj am 6= 0. Proposition 3.10 gives
j
|α1j a1 + · · · + αmj am | ≥ H ∗ (A)−1 kak1−d
∞ .
Let ε > 0. If we choose Q such that
n
m
X
X
Q
da1 α1j + · · · + am αmj c + ε
|ai | ≥ 21 (m + n)2 ,
j=1
i=1
29
then we can apply Theorem 1.8. In view of Proposition 3.10 this inequality is satisfied
if
1−d
+ εkak∞ ≥ 12 (m + n)2 .
QH ∗ (A)−1 kak∞
By elementary calculus the minimum of the function f defined by
f (x) = QH ∗ (A)−1 x1−d + εx
is assumed at
xmin =
QH ∗ (A)−1 (d − 1)
εn
d1
.
Hence, we must choose Q such that
1−d
QH ∗ (A)−1 xmin
+ εxmin ≥
1
(m
2
+ n)2 .
An easy calculation shows that this inequality is satisfied for
Q=
d
(m
2d−1
+ n)2d H ∗ (A)ε1−d .
Theorem 3.11 now follows by applying Theorem 1.8.
Recall that the heuristic argument in Section 3.1 suggests that in general Kronecker’s Theorem should hold with kqk∞ ε−m . We only get this expected best
result if A is a m × 1 matrix and the extension Q ⊂ Q(α11 , . . . , αm1 ) is of degree m.
In Section 3.3 below we derive in the case that A is a m × 1 matrix and any δ > 0
an upper bound for kqk∞ of order (ε−1 )m+δ . This upper bound is independent of the
degree of the extension, but is not effectively computable by the method of proof.
It is also possible to apply this result to Kronecker’s theorem in the more general
form as in Section 1.4. Before stating this theorem we need the following definitions.
Let A be a r×s matrix with real algebraic elements αij . Define K as the field extension
of Q generated by the elements of A. We define c(A) := [K : Q].
Theorem 3.12. Let A ∈ Rr,s , b ∈ Rr both with algebraic elements, such that for
all z ∈ Rr with AT z ∈ Zs we have bT z ∈ Z. Then there exists C ∈ R>0 , effectively
computable and depending on A and b, such that for all ε > 0, there exists x ∈ Zs
with
c(A)−1
1
.
kAx − bk∞ ≤ ε, kxk∞ ≤ C
ε
30
Proof. For the proof of this theorem we follow the steps of Theorem 1.12. By Lemma
1.13 and Lemma 1.14 we can find U ∈ GLr (R) such that U A is in the shape as
demanded at the start of the proof of Theorem 1.12. Let A1 , A2 , b1 , and b2 be defined
as in Lemma 1.14. These matrices and vectors are effectively computable, as there
exists a polynomial time algorithm to calculate the Smith normal form. See Bachem
and Kannan, [1], for this. A basis for the module M in the proof of Theorem 1.12 is
also effectively computable by expressing the elements of A as Q-linear combinations
of a chosen Q-basis of K, and then using linear algebra over Z. By Theorem 1.11
there exists q0 ∈ Zs−m , p01 ∈ Zt such that
A1 q0 − p01 = b1 .
By Lemma 1.13 and inequality (1.9) of Theorem 1.12 we have
1
ε
kA2 q00 − p02 − (b2 − A2 q0 )k∞ ≤
.
d
dkU k
We want to give an upper bound for kq00 k in this inequality. By Theorem 3.11 there
exist q00 ∈ Zn , p02 ∈ Zm−t such that
ε
1
kÂ02 q00 − p02 − (b2 + A2 q0 )k∞ ≤ kU k ,
d
d
c(A)−1
d
c(A)
kq00 k∞ ≤ c(A)+1 (m + n)2c(A) H ∗ (A2 )
.
2
ε
The proof of this theorem follows if we use this result and continue with the proof of
Theorem 1.12.
3.3
Subspace Theorem
In this section we use the Subspace Theorem, see below, to prove an effective
version of Kronecker’s Theorem for matrix A ∈ Rm,1 with algebraic entries.
First we need the following definition. We say that n linear forms
Li = αi1 x1 + · · · + αin xn ,
i = 1, . . . , n,
are linearly independent if
α11 · · · α1n
..
...
det(L1 , . . . , Ln ) = ...
.
αn1 · · · αnn
31
6= 0.
Theorem 3.13. (Subspace Theorem) Let L1 , . . . , Ln be n linearly independent
forms with algebraic coefficients in C and let δ > 0. Then the set of solutions of the
equation
0 ≤ |L1 (x) · · · Ln (x)| ≤ kxk−δ
∞
with x ∈ Zn is contained in the union of finitely many proper linear subspaces of Qn .
Proof. See Schmidt [20].
Corollary 3.14. Let α1 , . . . , αn ∈ C be algebraic and linearly independent over Q
and let δ > 0. Then there exist only finitely many x ∈ Zn with |α1 x1 + · · · + αn xn | ≤
.
kxk1−n−δ
∞
Proof. This proof is by induction. If α1 6= 0 then there exist only finitely many x ∈ Z
such that |α1 x| ≤ |x|−δ . This proves the theorem for the case n = 1.
Assume that the theorem is true for n − 1. Let α1 , . . . , αn ∈ C be linearly independent over Q. Take L1 (x) = α1 x1 + · · · + αn xn and Li (x) = xi for 2 ≤ i ≤ n.
Note that L1 , . . . , Ln are chosen such that det(L1 , . . . , Ln ) 6= 0. The set of solutions of
is contained in the set of solutions of |L1 (x) · · · Ln (x)| ≤ kxk−δ
|L1 (x)| ≤ kxk1−n−δ
∞.
∞
By the Subspace theorem we find that the solutions with x ∈ Zn are contained in
the union of finitely many proper linear subspaces of Qn of dimension n − 1. These
subspaces are of the form c1 x1 + · · · + cn xn = 0 with (c1 , . . . , cn ) ∈ Qn \{0}. For such
1−n−δ
are also solutions of |(α1 − ccn1 αn )x1 +
a subspace the solutions of |L1 (x)| ≤ kxk∞
1−n−δ
αn )xn−1 | ≤ kxk∞
. Note that α1 − ccn1 αn , · · · , αn−1 − cn−1
αn are
· · · + (αn−1 − cn−1
cn
cn
linearly independent over Q. Hence, these equations have only finitely many solutions
(x1 , . . . , xn−1 ) ∈ Zn−1 by the induction hypothesis. This proves the corollary.
So, if 1, α1 , . . . , αn are linearly independent with α1 , . . . , αn ∈ C, then there exists
a constant C ∈ R>0 such that all x ∈ Zn satisfy the inequality
dα1 x1 + · · · + αn xn c ≥ Ckxk−n−δ
.
∞
Hence, we proved a lower bound for the sum dα1 x1 + · · · + αn xn c as in Proposition
3.10. In contrast to Proposition 3.10 we do not have an effective lower bound and we
have the stronger condition that 1, α1 , . . . , αn must be linearly independent. But for
this result the lower bound is in many cases much stronger than the one depending on
the degree of the field extension. As in Section 3.2 we would now like to formulate a
version of Kronecker’s Theorem using this result. This can not be done for Kronecker’s
Theorem for linear forms, because of the stronger condition that 1, α1 , . . . , αn must
be linearly independent. This is why we only consider the problem of inhomogeneous
simultaneous approximation of real numbers by rational numbers.
32
Theorem 3.15. Let 1, α1 , . . . , αm ∈ R be algebraic numbers that are linearly independent over Q and let δ > 0. Then there exist a constant D > 0 depending on
α1 , . . . , αm , δ such that for all ε > 0 and b ∈ Rm there exist p ∈ Zm , q ∈ Z with
|qαi − pi − bi | ≤ ε
for i = 1, . . . , m,
m+δ
1
q≤D
.
ε
Proof. Let δ > 0. According to Corollary (3.14) there exists a constant C such that
da1 α1 + · · · + am αm c ≥ Ckak−m−δ
∞
for all a ∈ Zm .
With the same calculation as in the proof of Theorem 3.11 we conclude that there
exists a constant D such that with
Q(ε) = Dε−m−δ
we satisfy
Q(ε)Ckak−m−δ
+ εkak∞ ≥ 12 (m + 1)2 .
∞
for all ε > 0. We conclude that
Q(ε)da1 α1 + · · · + am αm c + ε
m
X
|ai | ≥ 12 (m + 1)2 .
i=1
We can apply Theorem 1.8 with n = 1. We conclude that there exist p ∈ Zm , q ∈ Z
with
|qαi − pi − bi | ≤ ε for all 0 < i ≤ m, q ≤ Dε−m−δ .
This proves the theorem.
Based on the heuristic argument in Section 3.1 the best possible result would be
proving that Q grows as ε−m . We proved that Q grows as ε−m−δ , so we came pretty
close.
33
Chapter 4
Geometry of numbers over the adèles
4.1
Adèles
Let K be a number field. For any finite set P of places on K containing MK∞ ,
define
Y
Y
AK (P ) =
Kv ×
Ov ,
v∈P
v6∈P
where Ov , defined by
Ov := {x ∈ Kv : |x|v ≤ 1},
is the maximal compact subring
S of Kv with respect to the topology induced by | · |v .
Let AK be the union AK = P AK (P ) over all such P ⊂ MK . This set forms a
ring under componentwise addition and multiplication and is called the adèle ring
over K. Elements of this ring are called adèles and denoted by (av ), with av ∈ Kv
for all v ∈ MK and av ∈ Ov for almost all1 v ∈ MKfin . We call (av )v∈MK∞ the infinite
part and (av )v∈MKfin the finite part of the adèle (av ). On every completion Kv of K we
have a topology induced by the absolute value | · |v on Kv . These topologies induce a
topology on AK , with a basis of the following shape
Y
Uv ,
v∈MK
where Uv is a non-empty open subset of Kv for all v ∈ MK , and Uv = Ov for almost
all v ∈ MKfin . With this toplogy the adèle ring AK is the restricted topological product
of Kv with respect to Ov . We can say a bit more about the topological structure of
AK , but first we need to define the notion of a locally compact abelian group.
A locally compact abelian group is a locally compact Hausdorff space that is also
an abelian group, such that the opposite map
G −→ G
x 7−→ −x
and the group operation
G × G −→ G
(x, y) 7−→ x + y
1
With almost all places we mean all but finitely many places.
34
with the product topology on G × G are continuous.
Lemma 4.1. With the topology defined above the additive group of AK is a locally
compact topological group.
Proof. See Cassels and Fröhlich [6], Chapter 2, Section 14.
The n-dimensional Cartesian product of AK is called the n-dimensional adèle space
denoted AnK . Elements of this space are denoted by (av ) with av ∈ Kvn for all v ∈ MK
and av ∈ Ovn for almost all v ∈ MKfin . We call (av )v∈MK∞ and (av )v∈MKfin respectively the
infinite part and finite part of (av ). We endow AnK with the n-fold product topology
of AK .
For an adèle (av ) ∈ AnK and λ ∈ R>0 we define scalar multiplication by λ(av ) =
(λv av ) with
λv = 1 for v ∈ MKfin ;
λv = λ for v ∈ MK∞ .
In the forthcoming sections we will state and prove results in the geometry of
numbers in the adèle space. For this it is essential to generalize the notion of a convex
body to the adèle space. In order to do this we first need a definition of a convex body
in Cn . We define a convex body in Cn by identifying Cn with R2n . A body in Cn is
a non-empty connected subset of Cn , which has the same closure as its interior. It is
convex under the same condition as in Rn . A body C in Cn is called C-symmetric if
it satisfies the following condition
C = αC for all α ∈ C with |α| = 1.
Definition 4.2. We define a convex body in AnK to be a Cartesian product of the
following shape:
Y
Y
C=
Cv ×
Mv ,
∞
v∈MK
fin
v∈MK
where Cv is a convex body in Kvn for all v ∈ MK∞ and Mv is a free Ov -module of rank n
for all v ∈ MKfin , with Mv = Ovn for almost all v ∈ MKfin . This convex body is called
symmetric if Cv is symmetric for all real v ∈ MK∞ and C-symmetric for all complex
v ∈ MK∞ . It is called bounded if Cv is bounded for all v ∈ MK∞ .
For every v ∈ MK there is a natural embedding of K in Kv . Define φ := K n ,→ AnK
as the map that is the natural embedding on each coordinate. Let k ∈ K n and let
35
(av ) = φ(k) then we have av = k for each v ∈ MK . To prove that (av ) ∈ AnK we have
to show that kkkv ≤ 1 for almost all v ∈ MK . This follows directly from Lemma 2.25.
The set φ(K n ) may be viewed as a lattice in AnK . We will take a closer look at
this embeddding in the Q
one-dimensional case. First we will look specifically at the
embedding restricted to v∈M ∞ Kv .
K
Let K be a number field of degree d over Q. Let σ1 , . . . , σr be the real embeddings,
and let σr+1 , . . . , σr+2s be the complex embeddings ordered such that σr+s+i = σr+i
for i = 1, . . . , s. Define x(i) = σi (x) for i = 1, . . . , d. Define
Y
φ∞ : K ,→ Rr × Cs =
Kσ i
∞
σi ∈MK
x 7−→ (x(1) , · · · , x(r) , x(r+1) , · · · , x(r+s) ).
We may view φ∞ as a map from K to Rd
φ∞ : K −→ Rd
x 7−→ (x(1) , . . . , x(r) , Re(x(r+1) ), Im(x(r+1) ), . . . , Re(x(r+s) ), Im(x(r+s) )).
Let C be a bounded symmetric convex body in AK . We have
Y
Y
C=
Cv ×
Mv ,
∞
v∈MK
fin
v∈MK
with Cv and Mv as in Definition 4.2. Note that the set defined by {x ∈ K : x ∈
Mv for v ∈ MKfin } is a fractional ideal of OK . The following lemma explains why φ(K)
may be viewed as a lattice.
Lemma 4.3. Let a be a fractional
ideal of OK . The set L = φ∞ (a) is a lattice of Rd
p
−s
|DK/Q (a)|.
of determinant det L = 2
Proof. Choose a Z-basis {α1 , . . . , αd } of a. We define a d × d matrix A by
(k)
A = [a1 , . . . , ad ] = (αl )dk,l=1
and a d × d matrix B by
B = [b1 , . . . , bd ] = [φ∞ (α1 ), . . . , φ∞ (αd )].
Note that bk = ak for k = 1, . . . , r and that b2k−r−1 = 21 (ak + ak+s ) and b2k−r =
−i(ak − b2k−r−1 ) for k = r + 1, . . . , r + s. With elementary linear algebra we calculate
| det B| = 2s | det A| and now we have
|DK/Q (a)| = | det(A)2 | = 22s (det(φ∞ (α1 ), . . . , φ∞ (αd ))2 .
36
We conclude that {φ∞ (α1 ), . . . , φ∞ (αd )} is an R-basis of Rd as DK/Q (a) 6= 0 by
Proposition 2.4. Hence, L is a lattice in Rd . Now, the result follows from det L =
| det(φ∞ (α1 ), . . . , φ∞ (αd ))|.
Corollary 4.4. A bounded symmetric convex body C in AnK contains only finitely
many points of φ(K n ).
Proof. We prove this in the case n = 1. For arbitrary n the argument must be
applied to each of the coordinates. Let again C in AK be given by
Y
Y
C=
Cv ×
Mv .
∞
v∈MK
fin
v∈MK
Define
a := {x ∈ K : x ∈ Mv for all v ∈ MKfin }.
Note that a is a fractional ideal of OK . The set
{φ∞ (x) : x ∈ a}
Q
is a lattice of v∈M ∞ Kv by Lemma 4.3. We conclude that the intersection of φ(K)
K
and C is finite, because C is bounded.
Analogously to the classical case, we can define successive minima of convex bodies
in AnK . For C a convex body in AnK and λ ∈ R>0 let λC be the convex body given by
λC := {λ(av ) : (av ) ∈ C}. We define the n successive minima λ1 (C), . . . , λn (C) of C by
λi (C) := inf{λ ∈ R>0 : dim(λC ∩ φ(K n )) ≥ i}.
They satisfy 0 < λ1 ≤ · · · ≤ λn < ∞. As in the classical case successive minima,
defined as infima, are minima.
4.2
Strong Approximation Theorem
In this section we prove an effective version of the Strong Approximation Theorem.
In the proof we use a non-effective version of the Strong Approximation Theorem,
which is in fact a strong version of the Chinese Remainder Theorem.
Let IK ⊂ AK be the set defined by
IK = {(iv ) : iv ∈ Kv∗ , |iv |v = 1 for almost all v ∈ MKfin }.
37
If we endow IK with componentwise multiplication, it becomes a group. We call this
set the idèle group of K. Let k(iv )k be defined by
Y
k(iv )k =
|iv |v .
v∈MK
Further, define
c(v) = 0 for v ∈ MKfin
= 1 for real v ∈ MK∞
= 2 for complex v ∈ MK∞ .
Theorem 4.5. (Strong Approximation Theorem) Let (iv ) ∈ IK and let v0 ∈
MK . Then for every adèle (av ) ∈ AK there exists an x ∈ K such that
|x − av |v ≤ |iv |v
for all v ∈ MK \ {v0 }.
Proof. See Cassels and Fröhlich [6], Chapter 2, Section 15.
Theorem 4.6. (Effective Strong Approximation Theorem) Denote again by r
the number of real places of K and by s the number of complex places of K. Choose
an idèle (iv ) ∈ IK such that
d
s
d 2 p
|DK | .
k(iv )k ≥
2 π
(4.1)
Then for every adèle (av ) ∈ AK there exists x ∈ K, such that
|x − av |v ≤ |iv |v
for all v ∈ MK .
Proof. Let a be the fractional ideal of OK given by
a = {x ∈ K : |x|v ≤ |(iv )|v for all v ∈ MKfin }.
Let L = φ∞ (a) be a lattice in Rd as in Lemma 4.3. Define A1 , . . . , Ar+s
Aj = |iσj |σj for j = 1, . . . , r + s. Define

 (y1 , . . . , yr , yr+1 , zr+1 , . . . , yr+s , zr+s ) ∈ Rd with
|yi | ≤ Ai
for i = 1, . . . , r
C :=

2
2
yi + zi ≤ Ai
for i = r + 1, . . . , r + s
38
∈ R>0 by



.
Note that C is a convex body in Rd . By the Strong Approximation Theorem, Theorem
4.5, there exists b ∈ K such that |b − av |v ≤ |iv |v for all v ∈ MKfin . By definition of the
covering radius µ = µ(C, L) there exists an u ∈ L such that φ∞ (b) − (av )v∈MK∞ + u ∈
µC. There exists a κ ∈ a such that φ∞ (κ) = u. Take x = b − κ, then we get
|x − av |v ≤ µc(v) |iv |v for all v ∈ MK .
Now, we are ready if we prove that µ ≤ 1. We will first give an upper bound for λd
using Minkowski’s convex body theorem and then apply Theorem 1.4 for the required
upper bound for µ. In order to use Minkowski’s Convex Body Theorem, we have to
calculate det L and V (C).
We will start by calculating det L. We already know that
q
p
−s
|DK/Q (a)| = 2−s N (a) |DK |
(4.2)
det L = 2
by Lemma 4.3 and Proposition 2.5. So, we need to take a closer look at N (a). For
v ∈ MKfin , let pv be the prime ideal given by {x ∈ OK : |x|v < 1}. Using that
a = {x ∈ K : ordpv (x) ≤ ordpv (a) for all v ∈ MKfin }
we find
a=
Y
pv (iv )
pord
.
v
fin
v∈MK
Hence,
N (a) =
Y
Y
N (pv )ordpv (iv ) =
|iv |−1
v .
fin
v∈MK
fin
v∈MK
Putting together this result with equation (4.2), we get
Y
p
|iv |−1
det L = 2−s |DK |
v .
fin
v∈MK
Calculating the volume of the convex body C is actually quite easy:
Y
V (C) = 2r π s
|iv |.
∞
v∈MK
Minkowski’s Convex Body Theorem states that
λ1 · · · λd ≤ 2d
39
det L
.
V (C)
If we have a lower bound for λ1 , . . . , λd−1 , then we get an upper bound for λd . There
exists an u ∈ Rd with u 6= 0 and u ∈ λ1 C ∩ L and there exists a κ ∈ a such that
c(v)
φ∞ (κ) = u. We have |κ|v ≤ |iv |v for all v ∈ MKfin and |κ|v ≤ λ1 |iv |v for all v ∈ MK∞ .
Hence, we get
Y
1=
|κ|v ≤ λd1 k(iv )k.
v∈MK
We can use this to find the following upper bound
p
s
2 s
|DK |k(iv )k−1
2 p
π
=
λd ≤
|DK |k(iv )k−1/d .
d−1
π
λ1
Using Theorem 1.4, we have
d
d
µ ≤ λd ≤
2
2
s
2 p
|DK |k(iv )k−1/d .
π
We conclude that µ ≤ 1 by inequality (4.1). This proves the theorem.
4.3
Fundamental domain
Let again K be an algebraic number field.
Definition 4.7. A set F ⊂ AK is called a fundamental domain for AK /φ(K) if for
every (av ) ∈ AK there exists precisely one (a0v ) ∈ F, such that (av ) − (a0v ) ∈ φ(K).
Q
Let φ∞ be the canonical embedding of K in v∈M ∞ Kv as defined in Section 4.1.
K
Let U be the set given by
U = {ξ1 φ∞ (ω1 ) + · · · + ξn φ∞ (ωn ) : ξi ∈ R, − 21 ≤ ξi <
1
2
(i = 1, . . . , n)},
where {ω1 , · · · , ωn } is a Z-basis of OK . We define the set F by
Y
F = U×
Ov .
fin
v∈MK
Theorem 4.8. The set F is a fundamental domain for AK /φ(K).
40
(4.3)
Proof. Let (av ) ∈ AK be an arbitrary adèle. We have to show that there exists
precisely one (a0v ) ∈ F such that (av ) − (a0v ) ∈ φ(K). This is equivalent with proving
that there exists precisely one x ∈ K with (av )−φ(x) ∈ F. We first show the existence
and then the uniqueness.
There exists an x0 ∈ K, such that |x0 − av |v ≤ 1 for all v ∈ MKfin by the Strong
Approximation Theorem, Theorem 4.5. The set {φ∞ (ω1 ), . . . , φ∞ (ωn )} is an R-basis
by Lemma 4.3. Hence, we have
∞
0
φ (x ) =
n
X
αi φ∞ (ωi ),
i=1
where αi ∈ R for i = 1, . . . , n. We can choose z1 , . . . , zn ∈ Z such that
− 12 ≤ αi − zi < 12 .
Now, x = x0 − (z1 ω1 + · · · + zn ωn ) satisfies.
We still have to prove that there exists only one x ∈ K with (av ) − φ(x) ∈ F. Let
x, y ∈ K with (av ) − φ(x) ∈ F and (av ) − φ(y) ∈ F. We have |x − y|v ≤ 1 for all
v ∈ MKfin . Hence x − y ∈ OK by Lemma 2.23. Expressing φ∞ (x) and φ∞ (y) on the
basis {φ∞ (ω1 ), . . . , φ∞ (ωn )} of φ∞ (K), we get
φ∞ (x) =
n
X
xi φ∞ (ωi ),
i=1
φ∞ (y) =
n
X
yi φ∞ (ωi ),
i=1
with xi , yi ∈ R and xi − yi in Z for i = 1, . . . , n As we have both − 12 ≤ xi − (avi ) < 12
and − 21 ≤ yi − (avi ) < 12 we find −1 < xi − yi < 1 for i = 1, . . . , n. Hence, xi = yi for
i = 1, . . . , n and x = y. This proves the theorem.
4.4
The Haar measure on the adèle space
For an introduction to measure theory we refer to Bartle [2]. The definitions of a
σ-algebra, Borel set and measure can be found in this book. Let X be a topological
space. The Borel σ-algebra on X, denoted B(X), is the σ-algebra generated by the
open subsets of X. The sets in this σ-algebra are called Borel sets. This σ-algebra
also contains all closed sets of X. From now on let X be Hausdorff. Every compact
41
set in a Hausdorff space is closed and hence a Borel set. A measure µ on X is called
a Borel measure if all Borel sets are µ-measurable, and µ(S) < ∞ for all compact
subsets S ⊂ X. A Borel measure is called
1. inner regular if
µ(E) = sup{µ(S) : S ⊂ E, S compact} for all E ∈ B(X).
2. outer regular if
µ(E) = inf{µ(U ) : U ⊃ E, U open} for all E ∈ B(X).
3. regular if it is both inner and outer regular.
On the Euclidean space Rn there is Borel measure, which agrees with the Lebesgue
measure on Borel sets. This measure is unique up to a constant factor.
Let G be a locally compact abelian group. A Haar measure on G is a regular Borel
measure µ on G such that the following two properties hold:
1. µ(E) < ∞ if E is compact;
2. µ(E + x) = µ(E) for all measurable subsets E ⊂ G and all x ∈ G.
We have the following important theorem concerning the Haar measure.
Theorem 4.9. On every locally compact abelian group G there exists a Haar measure,
which is unique up to a multiplicative constant.
Proof. See Loomis [14], Section 29, Chapter VI.
Hence, by Theorem 4.9 and Lemma 4.1 there exists a Haar measure on the adèle
space AnK .
We will define a Haar measure on the adèle space AK as the product of Haar
measures on the spaces Kv with v ∈ MK . Let v ∈ MKfin . Let p be the place on Q
with v|p. From Theorem 4.9, we know there exists a Haar measure on Kv , since Kv
is a locally compact group. Let βv denote the Haar measure on Kv scaled such that
1/2d
βv (Ov ) = |DK |v . With this information it is possible to calculate the volume of free
Ov -modules. A free Ov -module is of the form xOv for some x ∈ Kv . The volume of
1/2d
1/2d
xOv is equal to [Ov : xOv ]−1 |DK |v = |NKv /Qp (x)|v |DK |v .
As the adèle space is a locally compact group, we can define a Haar measure on
it. We will do this in the following way:
1. If v ∈ MKfin , let βv denote the Haar measure on Kv as explained above.
42
2. If v ∈ MK∞ with Kv = R, let βv denote the ordinary Lebesgue measure on R.
3. If v ∈ MK∞ with Kv = C, let βv denote the ordinary Lebesgue measure on C
multiplied by 2.
Q
For every set of finite places P the product measure β = v βv is a Haar measure on
AK (P ). The Haar measure on AK is the measure β that agrees with all these measures
on all subsets AK (P ). The n-fold product measure of β gives a Haar measure on AnK .
Denote the volume of a convex set C in AnK induced by this measure by V (C).
1/2d
The choice of the scaling βv (Ov ) = |DK |v probably looks a bit arbitrary. This
is not the case as shown in the following lemma.
Proposition 4.10. Let K be a number field of degree d = [K : Q] over Q and
let r and s be respectively the number of real and complex embeddings. Let F be the
fundamental domain AK /φ(K) as defined in equation (4.3). Then we have V (F) = 1.
Q
Proof. Let φ∞ be the natural embedding of K in v∈M ∞ Kv as defined in Section
K
4.1. Using Lemma 4.3 and the product formula, Theorem 2.28, we find that
Y
V (F) = det(φ∞ (ω1 ), . . . , φ∞ (ωn )) · 2s ·
βv (Ov ) =
fin
v∈MK
Y
p
|DK |
|DK |−1/2d
= 1.
v
∞
v∈MK
This proves the proposition.
4.5
Minkowski’s Theorem for adèle spaces
In this section we state an adèlic version of the Minkowski’s Theorem. This theorem will be of crucial importance for our proof of an adèlic version of Kronecker’s
Theorem. It was first proven in 1971 by R. McFeat [16] in a dissertation called “Geometry of numbers in adèle spaces”. Unaware of this, E. Bombieri and J. Vaaler [3]
published the same result in 1983.
Theorem 4.11. (Second Theorem of Minkowski for adèle spaces) Let K be a
number field of degree d = [K : Q] over Q and let r and s be respectively the number
of real and complex embeddings. Let C be a bounded symmetric convex body in the
adèle space AnK . Then its successive minima λ1 , . . . , λn satisfy
2dn π sn
1
(n!)r (2n!)s |DK | 2 n
≤ (λ1 · · · λn )d V (C) ≤ 2dn .
43
Proof. See E. Bombieri and J. Vaaler [3], Theorem 3 and Theorem 6 for respectively
the upper and the lower bound.
We have the following relationship between the adèlic and classical version of
Minkowski’s Theorem, i.e., Theorem 1.2.
Corollary 4.12. The adèlic version of Minkowski’s Theorem implies Theorem 1.2.
Proof. Let C be a symmetric convex body in Rn . We only prove the corollary for the
lattice Zn . To prove this result for other lattices we have to apply an invertible linear
transformation. Define a convex body in AnQ by:
Y
C0 = C ×
Ov .
v∈MQfin
Note that the number of real embeddings r = 1, the number of complex embeddings
s = 0, the degree d = 1 and the discriminant DQ = 1. After filling in these constants,
Minkowski’s Theorem for adèle spaces states that
2n
λ1 (C 0 ) · · · λn (C 0 )V (C 0 ) ≤ 2n .
n!
As DQ = 1, we have V (C 0 ) = V (C) by definition. Hence, the corollary is proven if we
show that λi (C 0 ) = λi (C, Zn ).
Let us recall the exact definitions of λi (C 0 ) and λi (C, Zn ):
λi (C 0 ) = inf{λ ∈ R>0 : dim(λC 0 ∩ φ(Qn )) ≥ i},
λi (C, Z) = inf{λ ∈ R>0 : dim(λC ∩ Zn ) ≥ i}.
Note that λi (C 0 ) = λi (C, Zn ) follows once we have proved that x ∈ λC ∩Zn if and only
if φ(x) ∈ λC ∩ φ(Qn ). Let x ∈ Qn with φ(x) ∈ λC 0 . We have x ∈ λC and |xi |p ≤ 1 for
i = 1, . . . , n and all primes primes p. Hence, x ∈ λC ∩ Zn . Conversely, if x ∈ λC ∩ Zn
then |xi |p ≤ 1 for all p, implying φ(x) ∈ λC 0 .
This proves the corollary.
44
Chapter 5
Kronecker’s Theorem over the adèles
5.1
Kronecker’s theorem for adèle spaces
In this section we prove an adelic version of Kronecker’s Theorem. We need some
additional definitions and results first.
We already defined successive minima for convex bodies in the adèle space. There
exists an adèlic analog of the covering radius too. The covering radius µ(C) of a convex
body C in AnK is defined by
[
µ(C) = inf{λ ∈ R>0 :
(λC + φ(x)) = AnK }.
x∈K n
O’Leary and Vaaler proved a lower and upper bound for the covering radius µ
of convex body in terms of its successive minima. To this purpose they introduced a
constant ν(K) depending on the field K. For every number field K we can define a
convex body SK by
Y
Y
SK =
Cv ×
Ov ,
∞
v∈MK
fin
v∈MK
where Cv = {x ∈ Kv : |x|v ≤ 1} for all v ∈ MK∞ . Define ν(K) := µ(SK ). The following
lemma gives an upper bound for ν(K).
Lemma 5.1. The constant ν(K) is bounded by the following upper bound
s
d 2 p
|DK |.
ν(K) ≤
2 π
p
Proof. Define (iv ) ∈ IK by iv = 1 for all v ∈ MKfin and iv = d2 ( π2 )s |DK | for all
v ∈ MK∞ . Then
d
s
d 2 p
|DK | .
k(iv )k =
2 π
Now choose an arbitrary adèle (av ) ∈ AK . By the effective Strong Approximation
Theorem, Theorem 4.6, there exists an x ∈ K such that |x − av |v ≤ |iv |v for all
v ∈ MK . Hence, we have
s
d 2 p
(av ) − φ(x) ∈
|DK | SK .
2 π
45
Hence, for λ =
d
2
2 s
π
p
|DK | we have
[
(λSK + φ(x)) = AK
x∈K
and
d
ν(K) = µ(SK ) ≤
2
s
2 p
|DK |.
π
This proves the lemma.
Theorem 5.2. Let C be a convex body in AnK . Its successive minima λ1 , . . . , λn and
covering radius µ satisfy
1
λ
2 n
≤ µ ≤ ν(K)(λ1 + · · · + λn ).
For the original proof by O’Leary and Vaaler, see [19], Theorem 5.
Proof. Let
C=
Y
Cv ×
∞
v∈MK
Y
Mv
fin
v∈MK
be a convex body in AnK as in Definition 4.2.
First we prove the lower bound by contradiction. Let t be the number of linearly
independent vectors in (µ + 21 λn )C ∩ φ(K). Suppose that µ + 12 λn < λn , implying
that t < n. Choose linearly independent x1 , . . . , xt ∈ K n such that φ(x1 ), . . . , φ(xt ) ∈
(µ + 21 λn )C. Choose x ∈ K n such that φ(x) ∈ λn C and x 6∈ Span{x1 , . . . , xt }. There
exists u ∈ K n such that 21 φ(x) − φ(u) ∈ µC by the definition of the covering radius.
Recall that 12 φ(x) = (xv ) with xv = 21 x for v ∈ MK∞ , xv = x for v ∈ MKfin . By
symmetry and convexity of Cv we find that u = u − 12 x + 12 x and x − u = 12 x − u + 12 x
are both in (µ + 12 λn )Cv for all v ∈ MK∞ . Further, x − u, x ∈ Mv for all v ∈ MKfin .
Hence, because Mv is a module, u, x−u ∈ Mv for all v ∈ MKfin . We conclude that both
φ(u) ∈ (µ + 12 λn )C and φ(x − u) ∈ (µ + 12 λn )C. Hence, u, x − u ∈ Span{x1 , . . . , xt },
which contradicts with x 6∈ Span{x1 , . . . , xt }. This proves the lower bound.
Now we prove the upper bound. Choose linearly independent x1 , . . . , xn ∈ K n
such that φ(xi ) ∈ λi C. Let us take an arbitrary (av ) ∈ AnK . For every v ∈ MK there
exist αv,1 , . . . , αv,n ∈ R such that av = αv,1 x1 + · · · + αv,n xn . Note that (αv,i ) ∈ AnK for
i = 1, . . . , n. By definition of ν(K) there exist κ1 , . . . , κn ∈ K such that kαv,i − κi k ≤
46
ν(K) for all v ∈ MK∞ and kαv,i − κi kv ≤ 1 for all v ∈ MKfin . Recall that xi ∈ λi Cv for
all v ∈ MKfin . Hence, for all v ∈ MK we find
av −
n
X
n
X
κi xi =
(αv,i − κi )xi ∈ ν(K)(λ1 + · · · + λn )Cv .
i=1
i=1
This proves the theorem.
For v ∈ MK we define the maximum norm k·kv with respect to v by
kxkv := max{|x1 |v , . . . , |xn |v }
for x = (x1 , . . . , xn ) ∈ Kvn .
Let A ∈ Kvm,n be a matrix given by


α11 · · · α1n

..  .
..
A =  ...
.
. 
αm1 · · · αmn
Define the norm k·kv on Kvm,n analogously to the norm on matrices in Section 1.4 by
kAkv = max |αij |v
1≤i≤m
1≤j≤n
n
X
kAkv = max
1≤i≤m
kAkv = max
1≤i≤m
|αij |v
for v ∈ MKfin ,
for real v ∈ MK∞ .
j=1
n
X
1
2
!2
for complex v ∈ MK∞ .
|αij |v
j=1
Note that we have kAxkv ≤ kAkv kxkv for all v ∈ MK , A ∈ Kvm,n , and x ∈ Kvn .
Let GLn (AK ) be the group of invertible matrices in An,n
K . Each element A ∈
GLn (AK ) may be represented by a tuple (Av ) with Av ∈ GLn (Kv ) for all v ∈ MK
and Av ∈ GLn (Ov ) for almost all v in MKfin and then
Y
det A ∈ IK , kdet Ak =
| det Av |v .
v∈MK
Let A ∈ GLn (AnK ) and define
Π := {(av ) ∈ AnK : kAv av kv ≤ 1 for all v ∈ MK }.
47
This set may be viewed as the adèlic analog of a parallelepiped. The set given by
T
Π∗ = {(av ) ∈ AnK : k(A−1
v ) av kv ≤ 1 for all v ∈ MK }
may be viewed as the reciprocal of Π. Both Π and Π∗ are convex bodies in AnK . Let
λ1 , . . . , λn be the successive minima of Π and λ∗1 , . . . , λ∗n the successive minima of Π∗ .
By Theorem 4.11 we have
(λ1 . . . λn )d ≤ 2dn V (Π)−1 = | det A|−1 ,
(5.1)
(λ∗1 . . . λ∗n )d ≤ 2dn V (Π∗ )−1 = | det A|.
(5.2)
and
In the following theorem we will prove an upper bound for λ∗i λn+1−i .
Theorem 5.3. The successive minima of Π and Π∗ satisfy
n−1 ≤ λ∗i λn+1−i ≤ nn−1
for i = 1, . . . , n.
Proof. First we prove the lower bound. Choose linearly independent x1 , . . . , xn ∈ K n
such that φ(xi ) ∈ λi Π for i = 1, . . . , n and linearly independent x∗1 , . . . , x∗n ∈ K n such
that φ(x∗i ) ∈ λ∗i Π∗ for i = 1, . . . , n. Note that
c(v)
kAv xi kv ≤ λi
T
∗ c(v)
, k(A−1
v ) yj kv ≤ (λj )
for all v ∈ MK .
We have for all v ∈ MKfin
|xTi x∗j |v = |xTi ATv (ATv )−1 x∗j |v
= |(Av xi )T (ATv )−1 x∗j |v
≤ kAv xi kv k(ATv )−1 x∗j kv
≤ 1.
Hence, xTi x∗j ∈ OK and NK/Q (xTi x∗j ) ∈ Z. By Theorem 2.27 we have
Y
|xTi x∗j |v = |NK/Q (xTi x∗j )| ∈ Z
∞
v∈MK
and so we get xTi x∗j = 0 or |xTi x∗j |v ≥ 1 for at least one v ∈ MK∞ .
48
In a similar way we derive for all v ∈ MK∞ that
|xTi x∗j |v = |xTi ATv (ATv )−1 x∗j |v
= |(Av xi )T (ATv )−1 x∗j |v
≤ nc(v) kAv xi kv k(ATv )−1 x∗j kv
≤ (nλi λ∗j )c(v) .
The set {x1 , . . . , xi , x∗1 , . . . , x∗n+1−i } contains n+1 vectors in K n . We have xTk x∗l 6= 0
for a k and l with 1 ≤ k ≤ i and 1 ≤ l ≤ n + 1 − i, because there are maximal n
K-linearly independent vectors in K n . We conclude that
1 ≤ |xTk x∗l |v ≤ (nλk λ∗l )c(v) ≤ (nλi λ∗n+1−i )c(v)
for at least one v ∈ MK∞ . This proves the lower bound
λi λ∗n+1−i ≥
1
.
n
The upper bound is a direct consequence of the lower bound and inequalities (5.1)
and (5.2).
(λi λ∗n+1−i )d =
(λ1 . . . λn )d (λ∗1 . . . λ∗n )d
n
Y
(λk λ∗n+1−k )d
k=1
k6=i
≤ n(n−1)d .
This proves the lemma.
Corollary 5.4. The first successive minimum of Π and the covering radius of Π∗
satisfy
µ(Π)λ1 (Π∗ ) ≤ ν(K)nn .
Proof. This follows directly from Theorem 5.2 and Theorem 5.3.
Using this corollary we derive an adèlic version of Kronecker’s Theorem in a way
very similar to the proof of the effective version of the classical form of Kronecker’s
Theorem (Theorem 1.8). First we introduce some new notation to state the theorem
more efficiently.
Q
Let P be a finite set of places in MK . We denote AnK,P = v∈P Kvn . A matrix
m,n
A ∈ Am,n
.
K,P may be represented as a tuple (Av )v∈P with Av ∈ Kv
49
Theorem 5.5. Let P be a finite set of places of a number field K and let A ∈ Am,n
K,P .
Suppose
{z ∈ K m : ∃ w ∈ K n such that ATv z = w for all v ∈ P } = {0}.
(5.3)
Then for every ε > 0, (bv ) ∈ AnK,P there exist vectors x ∈ K m , y ∈ K n such that
kAv x − y − bv kv ≤ ε
kxkv ≤ 1, kykv ≤ 1
for all v ∈ P ,
for all v ∈ MK \ P .
Proof. Let ε > 0, (bv ) ∈ AnK . Define
τ := ν(K)(m + n)m+n .
The set
for all v ∈ P ,
kkkv ≤ τ c(v) 1ε + kAv kv τ c(v) 1ε
m+n
V := k ∈ K
:
kkkv ≤ 1
for all v ∈ MK \ P
is finite by Lemma 4.4. Define for every v ∈ P the set
Vv := {k ∈ V : (ATv In )k 6= 0}.
Note that V \{0} = ∪v∈P Vv by (5.3).
By the Strong Approximation Theorem there exist κ ∈ K such that |κ|v ≤ ε for
all v ∈ P and Q ∈ K such that
|Q|v > 1
τ c(v)
|Q|v > min
k∈Vv k(AT
v In ) kkv
for v ∈ P with Vv empty,
for v ∈ P with Vv non-empty .
Define B ∈ GLm+n (AK ) by
−1
κ Im −κ−1 Av
Bv =
for all v ∈ P ,
0
Q−1 In
Bv = I for all v ∈ MK \ P .
We have
(Bv−1 )T
=
κIm
0
T
QAv QIn
for all v ∈ P .
Define
m+n
Π := {(av ) ∈ AK
: kBv av kv ≤ 1 for all v ∈ MK }.
50
Its reciprocal Π∗ is given by
Π∗ = {(av ) ∈ Am+n
: k(Bv−1 )T av kv ≤ 1 for all v ∈ MK }.
K
Suppose there exists a k ∈ K m+n with k ∈ τ Π, then k ∈ V . Hence, k = 0 or
kBvT kk ≥ kQ(ATv In ) kkv ≥ τ c(v)
for at least one v ∈ P . We conclude that
λ1 (Π∗ ) > τ = ν(K)(m + n)m+n .
So, by Corollary 5.4 we get µ(Π) ≤ 1. This implies that for all (b0v ) ∈ AnK there exist
x ∈ K m , y ∈ K n such that
0 Bv y + bv ≤ 1
for all v ∈ P,
0 v
x
Iv y ≤ 1
for all v ∈ MK \ P.
x v
Take (b0v ) = (κ−1 bv ). We conclude that there exist x ∈ K m , y ∈ K n such that
kAv x − y + bv kv ≤ ε
kxkv ≤ 1, kykv ≤ 1
for all v ∈ P,
for all v ∈ MK \ P.
This proves the theorem.
In this Kronecker’s Theorem for adèle spaces we demanded that
{z ∈ K m : ∃ w ∈ K n such that ATv z = w for all v ∈ P } = {0}.
In the next lemma and theorem we prove that this condition is necessary for the
adelic Kronecker’s Theorem. We need the following auxiliary result.
Lemma 5.6. Let P be a finite set of places of K. The condition
{z ∈ K m : ∃ w ∈ K n such that ATv z = w for all v ∈ P } = {0}
(5.4)
is equivalent to
m
n
: ∃ w ∈ OK
such that ATv z = w for all v ∈ P } = {0}.
{z ∈ OK
51
(5.5)
Proof. Suppose there exist z ∈ K m , w ∈ K n with z 6= 0 and ATv z = w for all
v ∈ P . There exists κ ∈ K ∗ such that κ ≤ max{kzkv , kwkv }−1 for all v ∈ MKfin by the
m
n
Strong Approximation Theorem 4.5. Note that κz ∈ OK
, κw ∈ OK
with z 6= 0 and
T
Av κz = κw.
Further,
m
n
{z ∈ OK
: ∃ w ∈ OK
such that ATv z = w for all v ∈ P } ⊆
{z ∈ K m : ∃ w ∈ K n such that ATv z = w for all v ∈ P } = {0}.
Hence, (5.4) implies (5.5). The converse can be proved in a similar manner.
Theorem 5.7. Let K be a number field and let d = [K : Q]. If for every ε > 0,
(bv ) ∈ AnK,P there exist vectors x ∈ K m , y ∈ K n such that
kAv x − y − bv kv ≤ ε
kxkv ≤ 1, kykv ≤ 1
for all v ∈ P ,
for all v ∈ MK \ P ,
then condition (5.5) is satisfied.
n
m
such
, w ∈ OK
Proof. This proof is by contradiction. Suppose there exist z ∈ OK
T
that z 6= 0 and Av z = w for all v ∈ P . Then
zT (Av x − y − bv ) = zT Av x − zT y − zT bv = wT x − zT y − zT bv .
Choose (bv ) ∈ AnK,P such that 0 < |zT bv |v < ( 12 )d+1 for all v ∈ P and choose ε > 0
T
m
n
such that ε < m−c(v) kzk−1
v |z bv |v for all v ∈ P . There exist x ∈ K , y ∈ K such
that
kAv x − y − bv kv ≤ ε
kxkv ≤ 1, kykv ≤ 1
for all v ∈ P ,
for all v ∈ MK / P .
Then
|xT w − yT z − zT bv |v ≤ εmc(v) kzkv < |zT bv |v
for all v ∈ P .
Hence,
|xT w − yT z|v ≤ |zT bv |v + εmc(v) kzkv < ( 21 )d
|xT w − yT z|v ≤ 2c(v)
52
for all v ∈ P ,
for all v ∈ MK / P .
(5.6)
m
n
Here we used that kxkv ≤ 1, kykv ≤ 1 for v ∈ MK / P and that z ∈ OK
, w ∈ OK
.
We conclude that
Y
|xT w − yT z|v < 1.
v∈MK
Since xT w − yT z ∈ K we have xT w − yT z = 0 by the product formula. We conclude
that
|xT w − zT y − zT bv |v = |zT bv |v for all v ∈ P .
This contradicts with inequality (5.6), which proves the theorem.
5.2
An effective adèlic Kronecker’s Theorem
In this section we prove an effective version of the adèlic Kronecker’s Theorem.
We need the following proposition to calculate a lower bound.
Proposition 5.8. Let K be a number field, let v ∈ MK , and let α1 , . . . , αn ∈ Kv be
algebraic over K, H the absolute height. Define L := K(α1 , . . . , αn ) and d := [L : Q].
Then we have
|q0 + q1 α1 + · · · + qn αn |v ≥ H(1, α1 , . . . , αn )−d (n + 1)−d H(q)−d kqkv
for every q ∈ K n+1 for which q0 + q1 α1 + · · · + qn αn 6= 0.
The place v of K gives rise to a place of L as L ⊂ Kv , which we denote by v again.
The standard representatives for these places are equal on K as Kv = Lv .
Proof. Using the product formula, Theorem 2.28, we find that
Y
|q0 + q1 α1 + · · · + qn αn |v =
|q0 + q1 α1 + · · · + qn αn |−1
w
w∈ML
w6=v
≥
Y
(n + 1)−1 max{1, |α1 |w , . . . , |αn |w }−1 kqk−1
w
w∈ML
w6=v
≥ H(1, α1 , . . . , αn )−d (n + 1)−d H(q)−d kqkv .
This proves the proposition.
53
Let K be a number field. Let A ∈ Kvm,n be a matrix given by


α11 · · · α1n

.. 
..
A =  ...
.
. 
αm1 · · · αmn
with elements αij algebraic over K. Define
dj := [K(α1j , . . . , α1n ) : Q] (j = 1, . . . , n) d := d(A) := max(d1 , . . . , dn ).
We define the height of A with respect to K by
∗
(A) = (n + 1)d max H(1, α1j , . . . , αmj )d .
HK
j=1,...,n
This notation is not standard.
Theorem 5.9. Let K be a number field, P a finite set of places of K and put d :=
[K : Q], t := #P . Further, let A ∈ Am,n
K,P . Suppose
{z ∈ K m : ∃ w ∈ K n such that ATv z = w for all v ∈ P } = {0}.
(5.7)
For every ε > 0 and for every v ∈ P define
!−d(Av )
Qv (ε) := τ
d
∗
HK
(Av )
Y
kBwT kw
w∈P
t·d(Av )−1
1
kBv kv
.
ε
Then for every ε > 0, (bv ) ∈ AnK , there exist vectors x ∈ K n , y ∈ K m , such that
kAv x − y − bv kv ≤ ε, kxkv ≤ Qv (ε)
kxkv ≤ 1, kykv ≤ 1
for all v ∈ P ,
for all v ∈ MK \ P .
Proof. Let ε > 0. Denote Qv (ε) by Qv . Define
|εv |v =
max
κ∈Kv :|κ|v ≤ε
|κ|v
for all v ∈ P .
Let B ∈ GL(m + n, AK ) be the linear transformation of Am+n
given by
K
−1
εv Im −ε−1
v Av
Bv =
for all v ∈ P ,
0
Q−1
v In
Bv = I for all v ∈ MK \ P .
54
We define a parallelepiped Π given by
Π = {(av ) ∈ Am+n
: kBv av kv ≤ 1 for all v ∈ MK }.
K
Its reciprocal Π∗ is given by
Π∗ = {(av ) ∈ Am+n
: k(Bv−1 )T av kv ≤ 1 for all v ∈ MK }.
K
Note that (Bv−1 )T is given by
(Bv−1 )T
=
(Bv−1 )T = I
εv Im
0
Qv ATv Qv In
for all v ∈ P ,
for all v ∈ MK \ P .
Define again τ := ν(K)(m + n)m+n . Let k ∈ K m+n \{0}. Suppose that φ(k) ∈
τ Π∗ , then we have k(BvT )−1 kkv ≤ τ c(v) for every v ∈ MK . We can rewrite this as
kkkv ≤ k(BvT )kv τ c(v) . Condition (5.7) gives that (ATv In )k 6= 0 for at least one v ∈ P .
By proposition 5.8 we get
!−d(Av )
Y
∗
k(ATv In )kkv ≥ HK
(Av )−1
kkkw
kkkv
w∈MK
!−d(Av )
∗
≥ HK
(Av )−1 τ −d
Y
kBwT kw
kBv kv τ c(v) ε1−t·d(Av ) .
w∈P
Hence,
k(BvT )−1 kkv ≥ kQv (ATv In )kkv ≥ τ c(v) .
It follows that λ1 (Π∗ ) ≥ τ . We conclude in the same way as in the proof of the
non-effective version of this theorem that µ(Π) ≤ 1. This proves the theorem.
5.3
A more general adèlic Kronecker’s Theorem
In this section we prove a more general form of an adèlic Kronecker’s Theorem
as in Section 1.4. First we introduce some new notation. Let A = (Av ) ∈ Ar,s
K,P ,
q,r
s,t
U = (Uv ) ∈ AK,P and V ∈ K . The product à = U A is given by Ãv = Uv Av and
the product  = AV is given by  = Av V .
Let P be a finite set of places of a number field K containing MK∞ . Define
OP = {x ∈ K : |x|v ≤ 1 for all v ∈ MK \ P }.
This is a ring called the ring of P -integers of K. We need the following theorem.
55
Theorem 5.10. Let P be a finite set of places of a number field K containing MK∞
such that OP is a principal ideal domain, let A = (Av ) ∈ Ar,s
K,P , m ∈ Z≥0 such that
r
rank(Av ) = m for all v ∈ P , and let (bv ) ∈ AK,P . Then the following two assertions
are equivalent.
(i) For every ε > 0 there exists an x ∈ OPs such that
kAv x − bv kv ≤ ε
for all v ∈ P .
(ii)
{(zv ) ∈ ArK,P : ∃ w ∈ OPs such that ATv zv = w for all v ∈ P } ⊆
{(zv ) ∈ ArK,P : ∃ ω ∈ OP such that bTv zv = ω for all v ∈ P }.
In the proof of this theorem we need some lemmas, in which we refer repeatedly to
(i) and (ii). We follow the steps of the proof in Section 1.4 as much as possible.
Lemma 5.11. Let U = (Uv ) ∈ GLr (AK,P ), V = (Vv ) ∈ GLs (OP ). Put à = (Ãv ) :=
U AV and define (b̃v ) by b̃v := Uv bv for all v ∈ P . Then (i) is equivalent to the
assertion that for every ε > 0 there exists x ∈ OPs such that
kÃv x − b̃v kv ≤ ε for all v ∈ P .
Proof. Suppose A and (bv ) satisfy assertion (i). For every ε > 0 there exists x ∈ OPs
such that
kAv x − bv kv ≤ ε
for all v ∈ P .
Define x̃ := V −1 x. Using that kUv Av V x̃ − Uv bv kv = kUv (Av x − bv )kv we get
kUv Av x̃ − Uv bv kv ≤ kUv kv ε
kx̃kv = kV −1 xkv ≤ kV −1 kv kxkv ≤ 1
for all v ∈ P ,
for all v ∈ MK \ P .
This proves the lemma.
Lemma 5.12. Let U = (Uv ) ∈ GLr (AK,P ), V = (Vv ) ∈ GLs (OP ). Put à = (Ãv ) :=
U AV , and define (b̃v ) by b̃v := Uv bv for all v ∈ P . Then (ii) is equivalent to the
assertion that
{(zv ) ∈ ArK,P : ∃ w ∈ OPs such that ÃTv zv = w for all v ∈ P } ⊆
{(zv ) ∈ ArK,P : ∃ ω ∈ OP such that b̃Tv zv = ω for all v ∈ P }.
56
Proof. Suppose there exist (zv ) ∈ ArK , w ∈ OPs such that V T ATv (UvT zv ) = w for all
v ∈ P then we have ATv (UvT zv ) = (V T )−1 w ∈ OPs for all v ∈ P . By assertion(ii) there
exists an ω ∈ OP such that bTv UvT zv = ω for all v ∈ P . We conclude that
{(zv ) ∈ ArK,P : ∃ w ∈ OPs such that ÃTv zv = w for all v ∈ P } ⊆
{(zv ) ∈ ArK,P : ∃ ω ∈ OP such that b̃Tv zv = ω for all v ∈ P }.
This proves the lemma.
Lemma 5.13. There exist U = (Uv ) ∈ GLr (AK,P ), V = (Vv ) ∈ GLs (OP ) such that
Im −A0v
Uv Av Vv =
for all v ∈ P ,
(5.8)
0
0
where A0 = (A0v ) ∈ Am,s−m
.
K,P
Proof. This proof is by induction on k. Let 0 ≤ k ≤ m. Our induction hypothesis is
that there exist U ∈ GLr (AK,P ), V ∈ GLs (OK ) such that
Ik A0v
with A0v ∈ Kvk,s−k , A00v ∈ Kvs−k,s−k for all v ∈ P .
Uv Av Vv =
0 A00v
This is trivial for k = 0. Assume that this induction hypothesis is true for some k with
s−k
0 ≤ k < m. Note that A00v 6= 0 for all v ∈ P . It is easy to find v = (v1 , . . . , vs−k ) ∈ OK
,
such that v1 = 1 and A00v v 6= 0 for all v ∈ P . Let e1 , . . . , es−k be the unit vectors in
s−k
K s−k . Define Vk0 = [v, e2 . . . , es−k ]. Note that Vk0 ∈ OK
. Define
Ik 0
Vk :=
.
0 Vk0
Note that Vk ∈ GL(s, K) and that kVk kv ≤ 1 for all v ∈ MK \ P . Now, define  =
(Âv ) = U AV Vk . We have
Ik Â0v
Âv =
for all v ∈ P ,
0 Â00v
where the first column of Â00v , which is Ã00v v1 , is not equal to 0 for all v ∈ P . Using
Gaussian elimination we can find an Uk ∈ GLr (AK,P ) with Uk  = Uk U AV Vk in the
desired shape for k + 1. This proves the proposition.
57
Lemma 5.14. Assume (ii) holds. Then there exist U ∈ GLr (AK,P ), V ∈ GLs (OP )
such that




b1
It
0
−A1
 for all v ∈ P ,
U AV =  0 Im−t −A2  , U bv =  b(v)
2
0
0
0
0
(v)
(v)
where 0 ≤ t ≤ m ≤ r, A1 ∈ K t,s−m , A2 = (A2 ) ∈ Am−t,s−m
, b1 ∈ K t , (b2 ) ∈ Am−t
K
K ,
and
{z1 ∈ OPt : AT1 z1 ∈ OPs−m } ⊆ {z1 ∈ OPt : bT1 z1 ∈ OP },
(v)T
{(zv ) ∈ OPm−t : ∃ w ∈ OPs−m such that A2
zv = w for all v ∈ P } = {0}.
(v)
Proof. By Lemma 5.13 there exist U1 = U1 ∈ GLr (AK ), V1 ∈ GLs (OP ) such that
Im −A0
U1 AV1 =
with A0 ∈ Am,s−m
.
K
0
0
By Lemmas 5.11 and 5.12, the validity of (ii) is unaffected if we replace A by U1 AV1 .
(1)
Define (b0v ) by Uv bv = (b0v , 0)T for all v ∈ P . Thus, assertion (ii) becomes
0
{(z0v ) ∈ OPm : ∃ w ∈ OPs−m such that A0T
v zv = w for all v ∈ P } ⊆
0
{(z0v ) ∈ OPm : ∃ ω ∈ OP such that b0T
v zv = ω for all v ∈ P }
(5.9)
in the same way as in the proof of Theorem 1.12. The left-hand side is a sub-OP module M of OPs−m , hence free of rank t ≤ m. If t = 0 we are done. Suppose t > 0.
Then by Theorem 1.11 there is a basis {d1 , . . . , dm } of OPm and there are δ1 , . . . , δt ∈
OP , such that {δ1 d1 , . . . , δt dt } is a OP -basis of M .
Now, let D := [d1 , . . . , dm ] be the matrix with columns d1 , . . . , dm . Then D ∈
GLm (OP ). Define  := DT A0 , b̂ := DT b0 . Then
T
D
0
Im −A0
(DT )−1
0
Im −Â
=
.
(5.10)
0 Ir−t
0
0
0
Is−m
0
0
The right-hand side is clearly of the shape U AV with U ∈ GLr (AK ), V ∈ GLs (OP ).
Writing ẑv := D−1 z0v for all v ∈ P we see that (5.9) is equivalent to
{(ẑv ) ∈ OPm : ∃ w ∈ OPs−m such that ÂTv ẑv = w for all v ∈ P } ⊆
{(ẑv ) ∈
OPm
: ∃ ω ∈ OP such that
58
b̂Tv ẑv
= ω for all v ∈ P }.
(5.11)
(v)
(v)
(v)
(v)
m−t,n
Let A1 = (A1 ) ∈ At,n
be defined by A1 and A2 , which
K,P and A2 = (A2 )AK,P
consist respectively of the first t rows and the last m − t rows of Âv for all v ∈ P .
(v)
(v)
(v)
(v)
(v)
(v)
(v)
(v)
Let b1 = (b1 , . . . , bt ), b2 = (bt+1 , . . . , bm ) be defined by b1 and b2 , which
consist of respectively the first t and the last m − t coordinates of b̂. Notice that the
left-hand side of (5.11) consists of all vectors of the shape (δ1 z1 , . . . , δr zt , 0, . . . , 0)T
with z1 , . . . , zt ∈ OP for all v ∈ P . We have Av (δ1 z1 , . . . , δr zt , 0, . . . , 0) = w ∈ OPs−m
(v)T
and b1 (δ1 z1 , . . . , δr zt , 0, . . . , 0) = ω ∈ OP for all v ∈ P and all z1 , . . . , zt ∈ OP .
(v)
Hence, A1 ∈ K t,s−m , b1 = (b1 ) ∈ K t , and
{z1 ∈ OPt : AT1 z1 ∈ OPs−m } ⊆ {z1 ∈ OPt : bT1 z1 ∈ OP }.
Further, by applying (5.11) with vectors (0, . . . , 0, zt+1 , . . . , zm )T , we see that
(v)T
{(zv ) ∈ OPm−t : ∃ w ∈ OPs−m such that A2
zv = w for all v ∈ P } = {0}.
This proves the lemma.
Proof of Theorem 5.10. First we prove (i) ⇒ (ii). This proof is by contradiction.
Assume (ii) does not hold. Suppose there exist (zv ) ∈ ArK , w ∈ OPs such that ATv zv =
w for all v ∈ P and there does not exist ω ∈ OP such that bTv zv = ω for all v ∈ P .
Let ε > 0. Define
Y
C :=
Cv ,
v∈MK
where
Cv := {x ∈ Kv : |x|v ≤ |bTv zv |v + εrc(v) }
Cv := {x ∈ Kv : |x|v ≤ rc(v) kwkv }
for all v ∈ P ,
for all v ∈ MK \ P .
This convex set contains only finitely many points of φ(K) by Corollary 4.4. Hence,
there are only finitely many κ ∈ OP such that
|κ − bTv zv |v ≤ εrc(v)
|κ|v ≤ rc(v) kwkv
for all v ∈ P ,
for all v ∈ MK \ P .
For each of these κ we have κ − bTv zv 6= 0 for at least one v ∈ P , because there does
not exist a ω ∈ OP such bTv zv = ω for all v ∈ P . Hence, we can find an ε > 0 such
that there does not exist κ satisfying these inequalities.
Hence, there does not exist x ∈ OPs , such that
|xT w − bTv zv |v ≤ εrc(v)
|xT w|v ≤ rc(v) kwkv
59
for all v ∈ P ,
for all v ∈ MK \ P .
Using that (Av x − bv )T zv = xT w − bTv zv , we conclude that there is no x ∈ OPs , such
that
kAv x − bv kv ≤ ε
kxkv ≤ 1
for all v ∈ P ,
for all v ∈ MK \ P .
Hence (i) does not hold.
Now we prove (ii) ⇒ (i). By Lemmas 5.11, 5.12, and 5.14 we may assume without
loss of generality, that




b1
It
0
−A1
 for all v ∈ P ,
A = (Av ) =  0 Im−t −A2  , bv =  b(v)
2
0
0
0
0
(v)
with A1 , A2 , b1 , (b2 ) as in Lemma 5.14. Writing xT = (qT , pT1 , pT2 ), We can rewrite
(i) as
kA1 q − p1 − b1 kv ≤ ε
kA2 q − p2 − b2 kv ≤ ε
for all v ∈ P ,
for all v ∈ P ,
(5.12)
to be solved in q ∈ OPs−m , p1 ∈ OPt , p2 ∈ OPm−t . By Theorem 1.11 there exist q ∈
OPs−m , p01 ∈ OPt such that
A1 q0 − p01 = b1 .
Recall that A1 ∈ K t,s−m . By the Strong Approximation Theorem, i.e., Theorem 4.5
there exists non-zero d ∈ OP such that dAT1 ∈ OPm,m . By Theorem 1.8, there exist
q00 ∈ OPs−m , p02 ∈ OPt , such that
ε
1
kA2 q00 − p02 − (b2 − A2 q0 )kv ≤
d
max(1, |d|v )
for all v ∈ P .
Hence,
A1 (q0 + dq00 ) − (p01 + dA1 q00 ) − b1 = 0,
kA2 (q0 + dq00 ) − dp02 − b2 kv < ε for all v ∈ P ,
which implies that (5.12) is satisfied with q = q0 + dq00 , p1 = p01 + dA1 q00 , p2 = dp02 .
This proves the theorem.
60
Bibliography
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61
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62
List of symbols
We denote column vectors with boldface, x, y, z, p, q, a, b, prime ideals by p and P,
fractional ideals by a, fields by K and L and valuations by | · |v and | · |w . Further, we
denote adèles by (av ) and idèles by (iv ).
C
C∗
V (C)
L
L∗
det L
λi (C, L)
µ(C, L)
F
e1 , . . . , en
convex body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
polar of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
volume of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
dual of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
determinant of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
ith successive minima of C with respect to L . . . . . . . . . . . . . . . . . . . . . . . . . 2
covering radius of C with respect to L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
fundamental domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
the n unit vectors of Rn
Z
Q
R
C
K
R
Rn
Rm,n
GLn (R)
Qp
Kv
K
AK
IK
φ
ordp
ring of rational integers
field of rational numbers
field of real numbers
field of complex numbers
field
ring
n-dimensional column vectors over R
m × n matrices over R
group of invertible n × n matrices over R
field of p-adic rationals
completion of K with respect to absolute value | · |v . . . . . . . . . . . . . . . . 17
algebraic closure of K
adèle ring over K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
idèle group over K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
natural diagonal embedding of K n in AnK . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
valuation induced by prime ideal p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
63
OK
Ov
OP
IK
PK
DK
TrL/K
NL/K
ring of integers of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
maximal compact subring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
ring of P -integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
group of fractional ideals of OK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
group of principal fractional ideals of OK . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
discriminant of a number field K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
trace of L over K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
norm of L over K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
(xk )∞
k=1
(xk )∞
k=1
0
1
sequence of reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
sequence of real vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
zero vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
(1, . . . , 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
| · |v
| · |∞
| · |p
| · |p
| · |σ
MQ
MK
MKfin
MK∞
H(x)
absolute value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
standard absolute value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
p-adic absolute value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
absolute value induced by a prime ideal p . . . . . . . . . . . . . . . . . . . . . . . . . . 21
absolute value induced by an embedding σ . . . . . . . . . . . . . . . . . . . . . . . . . 19
set of places of Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
set of places of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
set of non-archimedean places of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
set of archimedean places of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
n
height of x ∈ Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
#
h, i
[:]
bxc
dxc
dxc
kxk∞
kxk1
kxkv
kAk
kAkv
cardinality of a set
standard inner product on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
degree of a field extension
floor function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
distance from x to the nearest integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
distance from x to nearest integer vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
maximum norm of a real vector x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
sum norm of x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
maximum norm of x with respect to an absolute value | · |v . . . . . . . . . 47
norm of matrix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
norm of matrix A with respect to | · |v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
64
Index
absolute value, 15
archimedean –, 16
non-archimedean –, 16
equivalent –s, 16
extension of an –, 18
non-trivial –, 15
p-adic –, 16
trivial –, 15
adèle ring, 34
adèles, 34
adèlic convex body, 35
adèlic covering radius, 45
adèlic successive minima, 37
archimedean absolute value, 16
completion, 18
complex place, 19
convex body, 1
polar –, 1
symmetric –, 1
volume of a –, 1
adèlic –, 35
covering radius, 3
adèlic –, 45
diagonal matrix, 7
discriminant, 14
dual lattice, 2
equivalent absolute value, 16
extension of a place, 18
extension of an absolute value, 18
fractional ideal, 13
fundamental domain, 40
height, 23
idèle group, 37
ideal class group, 13
infinite prime, 15
inhomogeneous minimum, 3
lattice, 1
dual –, 2
locally compact abelian group, 34
maximal compact subring, 34
maximum norm, 4
non-archimedean absolute value, 16
non-trivial absolute value, 15
norm, 4, 13
maximum –, 4
sum –, 4
number field, 12
p-adic absolute value, 16
P -integers
ring of –, 55
place, 16
complex –, 19
extension of an –, 18
real –, 19
polar convex body, 1
principal ideal domain, 7
product formula, 23
ramification index, 12
real place, 19
residue class degree, 12
ring of integers, 12
65
ring of P -integers, 55
Smith normal form, 7
successive minima, 3
adèlic –, 37
sum norm, 4
symmetric convex body, 1
trace, 13
trivial absolute value, 15
uniform distribution, 24
valuation, 15
volume of a convex body, 1
66