On Kronecker’s Theorem - Universiteit Leidenalso use geometry of numbers to prove a quantitative version of Theorem 1 for the case n= 1, published in [9]. Their theorem gives an

Thijs Vorselen

On Kronecker’s Theoremover the adeles

Master’s thesis, defended on April 27, 2010

Thesis advisor: Jan-Hendrik Evertse

Mathematisch Instituut

Universiteit Leiden

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Kronecker’s Theorem 11.1 An introduction to the geometry of numbers . . . . . . . . . . . . . . 11.2 Kronecker’s approximation theorem . . . . . . . . . . . . . . . . . . . 41.3 Systems of linear equations over principal ideal domains . . . . . . . . 71.4 A more general Kronecker’s Theorem . . . . . . . . . . . . . . . . . . 8

2 Valuations 122.1 Some algebraic preliminaries . . . . . . . . . . . . . . . . . . . . . . . 122.2 An introduction to absolute values . . . . . . . . . . . . . . . . . . . 152.3 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Absolute values on number fields . . . . . . . . . . . . . . . . . . . . 19

3 An effective Kronecker’s Theorem 243.1 Uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 An effective Kronecker’s Theorem . . . . . . . . . . . . . . . . . . . . 283.3 Subspace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Geometry of numbers over the adeles 344.1 Adeles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Strong Approximation Theorem . . . . . . . . . . . . . . . . . . . . . 374.3 Fundamental domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.4 The Haar measure on the adele space . . . . . . . . . . . . . . . . . . 414.5 Minkowski’s Theorem for adele spaces . . . . . . . . . . . . . . . . . . 43

5 Kronecker’s Theorem over the adeles 455.1 Kronecker’s theorem for adele spaces . . . . . . . . . . . . . . . . . . 455.2 An effective adelic Kronecker’s Theorem . . . . . . . . . . . . . . . . 535.3 A more general adelic Kronecker’s Theorem . . . . . . . . . . . . . . 55

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

iii

Introduction

In this thesis we state and prove “effective” versions and adelic generalizationsof Kronecker’s Theorem. Kronecker’s Theorem takes an important place in the fieldof mathematics called Diophantine approximation. This field of mathematics is con-cerned with approximating real numbers by rational numbers. Kronecker’s Theoremdeals with inhomogeneous Diophantine inequalities and is published in 1884 by Kro-necker in a paper [10] called “Naherungsweise ganzzahlige Auflosung linearer Glei-chungen”.

Let Li(q) = αi1q1 + · · · + αinqn (i = 1, . . . ,m) be m linear forms with realcoefficients αij and let A be the m × n matrix with elements αij. The followingtheorem is a special case of Kronecker’s Theorem.

Theorem 1. (Kronecker) Assume that

{z ∈ Qm : ATz ∈ 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 theoremfrom Diophantine approximation is Dirichlet’s Theorem. This theorem is proven byDirichlet in 1840 and deals with homogeneous Diophantine inequalities. If we compareKronecker’s Theorem with Dirichlet’s Theorem, then we come across an interestingdifference.

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, ‖q‖ ≤ ε−m/n.

In contrast to Kronecker’s Theorem this theorem gives an upper bound for ‖q‖that is easy to calculate in terms of ε. We call such an upper bound an effectiveupper bound. Motivated by this observation we ask ourselves if there also exists aneffective upper bound in the case of Kronecker’s Theorem. In this thesis we answerthis 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 fromthe geometry of numbers. Geometry of numbers is concerned with convex bodies andinteger vectors in n-dimensional Euclidean space. In 1988 R. Kannan and L. Lovasz

iv

also use geometry of numbers to prove a quantitative version of Theorem 1 for thecase n = 1, published in [9]. Their theorem gives an ineffective upper bound for ‖q‖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 generalizethe proof of R. Kannan and L. Lovasz to arbitrary m. We also give a more generalquantitative version of Kronecker’s Theorem.

In the second chapter we recall some results from algebraic number theory andmore specifically the theory of valuations. These results are used in Chapter 3 toprove an effective version of Kronecker’s Theorem in the case that the elements of Aare algebraic.

In Chapter 4 we introduce the adele space. This space was introduced by Chevalleyin 1940 and is useful to solve number theoretic problems. Many of the concepts andtheorems in geometry of numbers have been generalized over the adeles. An importantresult is the adelic version of the Second Theorem of Minkowki. This theorem ispublished by R. McFeat [16] in 1971. Unaware of McFeat’s result E. Bombieri andJ. Vaaler [3] proved the same theorem in 1983. Motivated by the results in geometryof numbers over the adeles we wondered if it would be possible to generalize a versionof Kronecker’s Theorem over the adeles.

In Chapter 5 we prove some adelic versions of Kronecker’s Theorem. Again thearticle by R. Kannan and L. Lovasz forms the starting point of our proof. The adelicSecond Theorem of Minkowski also plays a crucial role.

I would like to thank Jan-Hendrik Evertse for the idea of this thesis and for manysuggestions and corrections.

v

Chapter 1

Kronecker’s Theorem

In this chapter we give an introduction to the geometry of numbers and prove someversions 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-dimensionalcolumn 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 + tylies 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 isLebesgue measurable, so it has a finite volume, which we denote by V (C). For λ ∈ Rwe define

λC = {x ∈ Rn : x = λy, y ∈ C}.Let 〈 , 〉 denote the standard inner product on Rn, given by

〈x,y〉 =n∑i=1

xiyi for x,y ∈ Rn.

The polar of C, denoted by C∗, is given by

C∗ := {x ∈ Rn : 〈x,y〉 ≤ 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 thedeterminant of L and is denoted by detL. Define the n× n matrix

A := [x1, . . . ,xn],

where x1, . . . ,xn are the columns of A. This matrix A is invertible, because x1, . . . ,xnare linearly independent. We have

L = {Ax : x ∈ Zn}.

1

The dual of L is given by

L∗ = {x ∈ Rn : 〈x,y〉 ∈ Z for all y ∈ L}.

It is a corollary of the next lemma that L∗ is again a lattice in Rn. We use the followingnotation. 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 inRn. Then its dual is given by L∗ = {(A−1)Tx : x ∈ Zn}.

Proof. We have

〈(A−1)Tx, Ay〉 = xTA−1Ay = xTy = 〈x,y〉.

Hence, 〈(A−1)Tx, Ay〉 ∈ Z if and only if 〈x,y〉 ∈ Z. Further, it is easy to see that

{x ∈ Rn : 〈x,y〉 ∈ Z for all y ∈ Zn } = Zn.

We conclude thatL∗ = {(A−1)Tx : x ∈ Zn}.

This proves the lemma.

We define the n successive minima λ1(C,L), . . . , λn(C,L) of C with respect to Lby

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 tosee that 0 < λ1 ≤ · · · ≤ λn < ∞. We defined successive minima as infima, but theyare in fact minima, as suggested by their name.

Theorem 1.2. (Second Minkowski’s Convex Body Theorem) Let C be abounded symmetric convex body and L a lattice in Rn. Then the successive minimaλ1, . . . , λn of C with respect to L satisfy

2n

n!≤ λ1 . . . λn

V (C)detL

≤ 2n.

Proof. See Minkowski [17], Kapitel V, for the original proof. We refer to Gruberand Lekkerkerker [7], Chapter 2, Section 9, Theorem 1 for a shorter proof by Ester-mann.

2

Let C be a bounded symmetric convex body and L a lattice in Rn. For any u ∈ Lwe define the set

λC + u = {x + u : x ∈ λC}.

Then µ(C,L), defined by

µ(C,L) = inf{λ ∈ R>0 :⋃u∈L

(λC + u) = Rn},

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∗) ≤ 12n2.

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 andcovering radius µ satisfy

12λn ≤ µ ≤ 1

2(λ1 + · · ·+ λn).

Proof. First we prove the lower bound by contradiction. Let t be the number oflinearly independent vectors in (µ+ 1

2λn)C ∩L. Suppose that µ+ 1

2λn < λn, implying

that t < n. Let x1, . . . ,xt ∈ (µ + 12λn)C ∩ L be linearly independent. Choose x ∈

λnC ∩ L such that x 6∈ Span{x1, . . . ,xt}. There exists u ∈ L such that 12x− u ∈ µC

by the definition of the covering radius. By symmetry and convexity of C we findthat both u = u − 1

2x + 1

2x and x − u = 1

2x − u + 1

2x are in (µ + 1

2λn)x. Hence,

u,x−u ∈ Span{x1, . . . ,xt}, which contradicts x 6∈ Span{x1, . . . ,xt}. This proves thelower bound.

Now, we prove the upper bound. Choose linearly independent x1, . . . ,xn ∈ Lsuch that xi ∈ λiC for i = 1, . . . , n. Let us take an arbitrary x ∈ Rn. There existα1, . . . , αn ∈ R such that x = α1x1 + · · ·+αnxn, because x1, . . . ,xn span Rn. Choosea1, . . . , an ∈ Z such that |αi−ai| ≤ 1

2for i = 1, . . . , n. Then x− (a1x1 + · · ·+anxn) ∈

12(λ1 + · · ·+ λn)C. This proves the theorem.

3

1.2 Kronecker’s approximation theorem

The aim of this section is to obtain a quantitative version of Kronecker’s approx-imation theorem for linear forms. The proof of this version is based on geometry ofnumbers. This idea comes originally from K. Mahler. He published a paper [15] in1966 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 andthe ring of m×n matrices over R by Rm,n. We define the following two norms on Rn:

1. ‖x‖1 :=∑n

i=1 |xi| for all x ∈ Rn, called the sum norm, and2. ‖x‖∞ := max{|x1|, . . . , |xn|} for all x ∈ Rn called the maximum norm.

Let Li(q) = αi1q1 + · · ·+αinqn (i = 1, . . . ,m) be m linear forms with real coefficientsand define

A :=

α11 · · · α1n...

. . ....

αm1 · · · αmn

.

The following theorem is a special case of Kronecker’s approximation theorem forlinear forms.

Theorem 1.5. (Kronecker) Let A ∈ Rm,n and assume that

{z ∈ Qm : ATz ∈ 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 z′ ∈ Qm such thatATz′ ∈ Qn and z′ 6= 0. Let z be the vector obtained by multiplying z′ with the lowestcommon multiple of the denominators of the coordinates in z′ and ATz′. Note that

4

z ∈ Zm and ATz ∈ Zn. There exists b ∈ Rm with 〈z,b〉 6∈ Z since z 6= 0. For allvectors p ∈ Zm, q ∈ Zn we have

|〈z, Aq− p− b〉| = |(Aq)Tz− 〈z,p〉 − 〈z,b〉|= |qTATz− 〈z,p〉 − 〈z,b〉|.

Note that qTATz − 〈z,p〉 ∈ Z for all p ∈ Zm, q ∈ Zn and 〈z,b〉 6∈ Z. Hence,|〈z, Aq−p−b〉| ≥ d〈z,b〉c > 0 independent of p and q. This proves the lemma.

Theorem 1.5 does not give an upper bound for ‖q‖∞. The following theorem byDirichlet 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, ‖q‖∞ ≤ 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 ‖q‖∞ in thecase of a system of inhomogenous linear forms. The following theorem implies a non-effective upper bound. The inequalities in (1.2) can be stated in the following moreefficient way

‖Aq− p− b‖∞ ≤ 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 allzero,

Q

n∑j=1

da1αj1 + · · ·+ amαjmc+ ε

m∑i=1

|ai| ≥ 12(m+ n)2. (1.3)

Then for all b ∈ Rn there exist p ∈ Zm, q ∈ Zn such that

‖Aq− p− b‖∞ ≤ ε, ‖q‖∞ ≤ Q.

Kannan and Lovasz 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 : ‖x‖∞ ≤ 1} be the unit cube and let L be the latticegenerated by the columns of the matrix

B =

(Im −A0 ε

QIn

),

where In denotes the n × n identity matrix. It is easy to see that C∗ = {x ∈ Rm+n :‖x‖1 ≤ 1} and that the dual lattice L∗ is generated by the columns of the inversetranspose of B

(B−1)T =

(Im 0QεAT Q

εIn

).

That is, lattice points in L∗ are of the form(a

Qε

(ATa + c)

)with a ∈ Zm, c ∈ Zn. Condition (1.3) in Theorem 1.8 implies that for all (a, c) 6= (0,0)one has ∥∥∥∥( a

Qε

(ATa + c)

)∥∥∥∥1

≥ 12(m+ n)2/ε.

Hence, by definition of λ1(C∗,L∗) we have

λ1(C∗,L∗) ≥ 12(m+ n)2/ε.

For all b ∈ Rm there exist vectors p ∈ Zm, q ∈ Zn such that ‖B(pq

)+(b0

)‖∞ ≤ µ(C,L)

by definition of µ(C,L). Since µ(C,L) ≤ ε by Lemma 1.3 we have

‖p− Aq + b‖∞ ≤ ε

‖q‖∞ ≤ Q.

This proves the theorem.

Lemma 1.9. Theorem 1.8 implies Theorem 1.5.

Proof. Choose an arbitrary ε > 0. For every a = (a1, . . . , am) ∈ Zm with a 6= 0 wefind

∑nj=1da1α1j + · · ·+ amαmjc > 0, because of condition (1.1). Now take

Q =1

2(m+ n)2

min‖a‖1≤ 1

2(m+n)2/ε

a6=0

n∑j=1

da1α1j + · · ·+ anαmjc

−1

.

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 equationsover principal ideal domains. Recall that a principal ideal domain is a domain, of whichall ideals are principal. This result is used for the proof of a more general Kronecker’sTheorem in the next section.

A diagonal matrix is a square matrix in which all elements outside the maindiagonal are zero. Let R be a principal ideal domain. For a, b ∈ R, a 6= 0 we denotea|b if a divides b.

Theorem 1.10. Let A ∈ Rm,n. Then there exist V1 ∈ GLm(R), V2 ∈ GLn(R) suchthat A′ given by A′ := V1AV2 is a diagonal matrix

A′ =

δ1

. . .

δt0

. . .

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 Propo-sition 4. For a proof in the special case that R = Z, see Smith [21].

The matrix A′ is called the Smith normal form of A. Bachem and Kannan, [1],published a polynomial time algorithm to calculate the Smith normal form in thespecial case that R = Z.

Theorem 1.11. Let R be a principal ideal domain with field of fractions K and letA ∈ Rm,n, b ∈ Rm. Then the following two assertions are equivalent.

(i) There exists x ∈ Rn such that Ax = b.

(ii) {z ∈ Km : ATz ∈ Rn} ⊆ {z ∈ Km : bTz ∈ R}.

Proof. First we prove (i) ⇒ (ii). Let z ∈ Km, such that ATz ∈ Rn. Then

bTz = xTATz ∈ xTRn ⊂ R.

This proves (i) ⇒ (ii).

7

Now, we prove (ii) ⇒ (i). Let A ∈ Rm,n and b ∈ Rm satisfy assertion (ii). Let A′

be the Smith normal form of A and let V1, V2 be the matrices such that A′ = V1AV2

as in Theorem 1.10.If A′Tz ∈ Rn then we have V T

2 ATV T

1 z ∈ Rn and as V2 ∈ GLn(R) we get ATV T1 z ∈

Rn. By assertion (ii) we see that (V1b)Tz = bTV T1 z ∈ R. We conclude that A′ and

b′ := (b′1, . . . , b′m)T = V1b also satisfy assertion (ii).

For every z = (z1, . . . , zm) ∈ Km we have

A′Tz =

δ1

. . .

δt0

. . .

0

z = (δ1z1, . . . , δtzt, 0, . . . , 0)T .

It is easy to see that δ1|b′1, . . . , δt|b′t and that b′t+1 = 0, . . . , b′m = 0. Define

x′ := (b′1/δ1, . . . , b′t/δt, 0, . . . , 0)

and note that x′ ∈ Rn and A′x′ = b′. Define x := V −12 x′, 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 thissection we use that result to prove the following more general form of Kronecker’stheorem. We use the following notation in this proof. Let again A ∈ Rr,s be the r× smatrix

A :=

α11 · · · α1s...

. . ....

αr1 · · · αrs

.

Define a norm ‖·‖ on Rr,s by

‖A‖ := max1≤i≤r

s∑j=1

|αij|.

Note that ‖Ax‖∞ ≤ ‖A‖‖x‖∞ for all A ∈ Rr,s, x ∈ Rs.

8

Theorem 1.12. Let A ∈ Rr,s and b ∈ Rr. Then the following two assertions areequivalent.

(i) For every ε > 0 there exists x ∈ Zs such that ‖Ax− b‖∞ ≤ ε.

(ii) For all z ∈ Rr with ATz ∈ Zs we have bTz ∈ Z.

In the proof of this theorem we need some lemmas, in which we refer repeatedly toassertions (i) and (ii).

Lemma 1.13. Let U ∈ GLr(R), V ∈ GLs(Z) and put A := UAV , b := Ub. Then:(i) is equivalent to the assertion that for every ε > 0 there is x ∈ Zs such that‖Ax− b‖∞ ≤ ε,(ii) is equivalent to the assertion that

{z ∈ Rr : ATz ∈ Zs} ⊆ {z ∈ Rr : bTz ∈ Z}.

Proof. Suppose assertion (i) holds. Then for every ε > 0 there exists x ∈ Zs suchthat

‖Ax− b‖∞ ≤ε

‖U‖.

Define x := V −1x. Then

‖Ax− b‖∞ = ‖UAx− Ub‖∞ ≤ ε.

The proof of the reverse implication is entirely similar.Suppose assertion (ii) holds. If there exists z ∈ Rr, w ∈ Z such that V TAT (UTz) =

w then AT (UTz) = (V T )−1w ∈ Z. By assertion (ii) we have bTUTz ∈ Z. We concludethat

{z ∈ Rr : ATz ∈ Zs} ⊆ {z ∈ Rr, bTz ∈ 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) suchthat

UAV =

It 0 −A1

0 Im−t −A2

0 0 0

, Ub =

b1

b2

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 mrows of A are linearly independent. Then by elementary linear algebra, there existsU1 ∈ GLr(R) such that

U1A =

(Im −A′0 0

)with A′ ∈ Rm,s−m.

By Lemma 1.13, the validity of (ii) is unaffected if we replace A by U1A. That is, (ii)is equivalent to the assertion that

{z ∈ Rr :

(Im −A′0 0

)z ∈ Zs} ⊆ {z ∈ Rr : bTUTz ∈ Z}. (1.4)

For this to hold, we must have U1b = (b′,0)T with b′ ∈ Rm. Thus, (1.4) becomes

{z′ ∈ Zm : A′Tz′ ∈ Zs−m} ⊆ {z′ ∈ Zm : b′Tz′ ∈ Z}. (1.5)

The left-hand side is a sub-Z-module M of Zs−m, hence free of rank t ≤ m. If t = 0we are done. Suppose t > 0. Then by Theorem 1.11 there is a basis {d1, . . . ,dm} ofZm and there are positive integers δ1, . . . , δt, such that {δ1d1, . . . , δtdt} is a Z-basisof M .

Now, let D := [d1, . . . ,dm] be the matrix with columns d1, . . . ,dm. Then D ∈GLm(Z). Define A := DTA′, b := DTb′. Then(

DT 00 Ir−t

)(Im −A′0 0

)((DT )−1 0

0 Is−m

)=

(Im −A0 0

). (1.6)

The right-hand side is clearly of the shape UAV with U ∈ GLr(R), V ∈ GLs(Z).Writing z := D−1z′ we see that (1.5) is equivalent to

{z ∈ Zm : AT z ∈ Zs−m} ⊆ {z ∈ Zm : bT z ∈ Z}. (1.7)

Let A1 consist of the first t rows of A and A2 of the last m−t rows. Further, let b1 andb2 consist of respectively the first t and the last m− t coordinates of b. Notice thatthe left-hand side of (1.7) consists of all vectors of the shape (δ1x1, . . . , δrxt, 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


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 lossof generality, that

A =

It 0 −A1

0 Im−t −A2

0 0 0

, b =

b1

b2

0

,

with A1, A2,b1,b2 as in Lemma 1.14. Writing xT = (qT ,pT1 ,pT2 ), We can rewrite (i)

as

‖A1q− p1 − b1‖∞ ≤ ε, ‖A2q− p2 − b2‖∞ ≤ ε, (1.8)

to be solved in q ∈ Zs−m,p1 ∈ Zt,p2 ∈ Zm−t. By Theorem 1.11 there exist q′ ∈Zs−m,p′1 ∈ Zt such that

A1q′ − p′1 = b1.

Recall that A1 ∈ Qt,s−m. Let d be a positive integer, such that dA1 ∈ Zt,s−m. ByTheorem 1.8, there exist q′′ ∈ Zs−m, p′2 ∈ Zt, such that

‖A2q′′ − p′2 −

1

d(b2 − A2q

′)‖∞ ≤ε

d. (1.9)

Hence,

A1(q′ + dq′′)− (p′1 + dA1q′′)− b1 = 0,

‖A2(q′ + dq′′)− dp′2 − b2‖∞< ε,

which implies that (1.8) is satisfied with q = q′ + dq′′, p1 = p′1 + dA1q′′, p2 = dp′2.


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 morespecifically, the theory of valuations.

2.1 Some algebraic preliminaries

We refer to “Algebraic Number Theory” by J. Neukirch [18] for algebraic numbertheory. 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 aDedekind domain, which means that it is a noetherian, integrally closed, integraldomain in which every non-zero prime ideal is a maximal ideal. Ideals in Dedekinddomains 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 idealof 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 · · ·Peg

g

for certain prime ideals P1, . . . ,Pg of OL and positive integers e1, . . . , eg. We call eithe ramification index of Pi over p and fi := [OL/Pi : OK/p] the residue class degreeof Pi over p.

Theorem 2.1. (Fundamental identity) Let K ⊂ L be an extension of numberfields of degree d. Then one has

g∑i=1

eifi = d.

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 · · · pkn

n ,

where p1, . . . , pn are prime ideals and k1, . . . , kn ∈ Z. For a non-zero prime ideal p ofOK we define ordpi

(a) := ki for i = 0, . . . , n and ordp(a) := 0 if p 6∈ {p1, . . . , pn}. Wehave ordp(a) 6= 0 for only finitely many prime ideals p. For an element x ∈ K, wedefine 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 fractionalideals generated by one element of K. The factor group IK/PK is the ideal class groupof 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 Kis trivial.

Proposition 2.3. Let K be a number field of degree d = [K : Q] over Q and let a bea 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 7−→∑σ

σ(x)

is called the trace and the map

NL/K : L −→ K

x 7−→∏σ

σ(x)

is called the norm. Here the sum and product range over all K-isomorphic embeddingsσ of L in C.

13

We can also define a norm on the group of fractional ideals. Any fractional ideala can be expressed uniquely as a = bc−1, where b, c are integral ideals such thatb + 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 discriminantof a basis {x1, . . . , xd} of L over K by

DL/K(x1, · · · , xd) =(

det(x(j)i )di,j=1

)2

,

where x(1)i , . . . , x

(d)i are the d images of xi under the K-isomorphic embeddings of L

in C. By some elementary calculations, we can rewrite this as

DL/K(x1, · · · , xd) = det(

TrK/Q(xixj))di,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, thenwe 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 Proposition2.4, 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(ωiωj))di,j=1.

As ωiωj ∈ OK for i, j = 1, · · · , d, we find that TrK/Q(ωiωj) ∈ Z. Now, it follows thatDK ∈ Z.

14

Proposition 2.5. Let a be a fractional ideal of OK. Then

DK/Q(a) = N(a)2DK .

Proof. This follows directly by expressing a as a product of two integral idealsa = bc−1 and applying Lang [13], Chapter I, Section 2, Proposition 12 to these ide-als.

2.2 An introduction to absolute values

Definition 2.6. An absolute value on a field K is a function | · |v : K → R≥0 suchthat

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 thata 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 absolutevalue.

Example 2.8. The most trivial example of an absolute value, which can be defined onany field K, is the absolute value that sends all x ∈ K with x 6= 0 to 1. This absolutevalue is called trivial. Henceforth, all absolute values we consider are assumed to benon-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 = bc

with b, c ∈ Z. If wesubsequently extract the highest possible power of p from b and c, we get

x = pmb

c, gcd(bc, p) = 1.

The p-adic absolute value |x|p of x is then defined by |x|p := 1pm . Further, we put

|0|p := 0.

Definition 2.11. Let K be a field. Two absolute values | · |v and | · |w on K arecalled 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. Wedenote 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 thefollowing 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 itsatisfies 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 isarchimedean. The following lemma shows that we can define this property for places.

Lemma 2.14. Equivalent absolute values are either both archimedean or both non-archimedean.

Proof. Let | · |v and | · |w be equivalent absolute values on a field K. Hence, thereexists 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 otherimplication.

For a field K, denote by MK the set of places of K, by M∞K the set of archimedean

places of K, and by MfinK 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 equivalenteither 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 primenumber. Thus, we may write MQ = {∞} ∪ {primes}.

Theorem 2.16. (Product formula) We have∏p∈MQ

|x|p = 1 for all x ∈ Q∗.

Proof. Take an arbitrary x ∈ Q∗. The theorem follows directly from writing

x = ±pk11 p

k22 · · · pkt

t

and calculating∏

p∈MQ|x|p.

2.3 Completions

We are already familiar with the construction of the real numbers as completionof 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 anyabsolute value | · |v on this field K.

A field K is called complete with respect to an absolute value | · |v, if every Cauchysequence in K converges in K with respect to | · |v. Let R be the ring of Cauchysequences in K with respect to | · |v. This is a ring under pointwise addition andmultiplication of the sequences. The subset of R consisting of all sequences in Rconverging 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 easyto 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

k→∞xk = x. We define | · |v

on Kv by |x|v := limk→∞|xk|v. The field Kv is complete with respect to this valuation.

The completion Kv of a field K with respect to an archimedean absolute value isequal to R or C. This follows directly from the following result regarding fields thatare complete with respect to an archimedean absolute value.

Theorem 2.17. (Ostrowski) Every field that is complete with respect to an archi-medean 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 veryimportant 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 absolutevalues 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. LetK ⊂ L be a field extension and let v, w be places on respectivelyK and L. A place w extends v, denoted w|v, if the restriction of a representative ofw 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 andlet K ⊂ L be a field extension of finite degree d. Then there exists an unique extensionof | · |v to L given by

|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 extensionof | · |v from Kv to Kv also denoted by | · |v, given by

|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

|x|v1 := d

√|NL1/Kv(x)|v for x ∈ L1

|x|v2 := d

√|NL2/Kv(x)|v for x ∈ L2

on respectively L1 and L2. These absolute values agree on L1 ∩ L2, because the ex-tension of | · |v to L1 ∩ L2 is unique by Theorem 2.20. We conclude that there existsan 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 morespecific case of number fields. Henceforth, let K be a number field and let K denotethe algebraic closure of K.

Let σ1, . . . , σr be the real embeddings of K, and let σr+1, . . . , σr+2s be the complexembeddings 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 | · |σassociated to an embedding σ by

|x|σ := |σ(x)| for all x ∈ K if σ is real,

|x|σ := |σ(x)|2 for all x ∈ K if σ is complex.

Let σ be a complex embedding. We denote its complex conjugate by σ. Note that wehave |σ(x)| = |σ(x)| for all x ∈ K. So, conjugate embeddings give rise to the sameabsolute value. We call a place associated to a real embedding a real place and a placeassociated to a complex embedding a complex place. There are r real places and scomplex 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 withrespect 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 eitherR or C. Every archimedean absolute value on R or C is equivalent to the standardabsolute value. Hence, | · |v is equivalent to | · |σ. This proves the lemma.

Henceforth, we identify the embeddings {σ1, . . . , σr+s} with the correspondingarchimedean 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 ∈MfinK }.

Proof. First we prove that OK ⊂ {x ∈ K : |x|v ≤ 1 for all v ∈ MfinK }. Take an

arbitrary x ∈ OK and v ∈ MfinK . 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 =Xn + an−1X

n−1 + · · ·+ a0 ∈ Z[X] such that

xn + an−1xn−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−1xn−1 + · · ·+ a1x+ a0|v ≤ max

0≤i≤n−1|aixi|.

We get a contradiction if |x|v > 1. We conclude that |x|v ≤ 1 for all v ∈MfinK .

We still have to show that {x ∈ K : |x|v ≤ 1 for all v ∈ MfinK } ⊂ OK . Choose an

x ∈ K such that |x|v ≤ 1 for all v ∈MfinK . Let f be the monic minimal polynomial of

x over Q. It is given by

f(X) = Xn + an−1Xn−1 + · · ·+ a1X + a0

with a0, . . . , an−1 ∈ Q. We have to prove that a0, . . . , an−1 ∈ Z. Let α1, . . . , αn be theroots of f in Kv. Sending x to one of the values α1, . . . , αn induces an embedding ofK(x) in Kv. Let σ1, . . . , σn be those embeddings. Extend | · |v to Kv and define anabsolute 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 areup to sign sums of products of subsets of {α1, . . . , αn}. We conclude that |a1|v ≤1, . . . , |an−1|v ≤ 1 for all v ∈ Mfin

K . Hence, |ai|p ≤ 1 for all p ∈ MfinQ 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 | · |pby 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 non-archimedean 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 idealsgive representatives for different places by Proposition 2.12. This proves the injectiv-ity 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 fgives back an absolute value equivalent to | · |v by Proposition 2.12.

Henceforth, for an algebraic number field K, we identify every prime ideal of OKwith 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 primefactorization

(x) = pk11 . . . pkg

g .

Hence, |x|v ≤ 1 for all v ∈MfinK 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|x|w = |NLw/Kv(x)|[Lw:Kv ]

v for all x ∈ L.

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 . . .Peg

g .

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)

−eiordp(x)

= (#OK/p)−eifiordp(x) = |x|eifip .

By Neukirch [18], Chapter II, Proposition 6.8 (with Remark) and the explanationunder Proposition 8.5 we have [Lw : Kv] = eifi. 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:∏

w|v

|x|w = |NL/K(x)|v for all x ∈ L.

Proof. Let v ∈MK . We have∏w|v

|x|w =∏w|v

|NLw/Kv(x)|v

and by Neukirch [18], Chapter II, Corollary 8.4∏w|v

|NLw/Kv(x)|v = |NL/K(x)|v.


Theorem 2.28. (Product formula) For a number field K we have the followingproduct formula: ∏

v∈MK

|x|v = 1 for all x ∈ K∗.

22

Proof. Using Theorem 2.27 for a field extension Q ⊂ K, we get∏v∈MK

|x|v =∏p∈MQ

∏v|p

|x|v =∏p∈MQ

|NK/Q(x)|p = 1 for all x ∈ K∗.


Now, we are able to define the height for x ∈ Qn.

Definition 2.29. Let x = (x1, . . . , xn) ∈ Kn with x 6= 0, then

HK(x) =∏v∈MK

max(|x1|v, . . . , |xn|v)

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 ∈ Kn. Then we have

HL(x) =∏w∈ML

max(|x1|w, . . . , |xn|w)

=∏v∈MK

∏w|v

max(|x1|w, . . . , |xn|w)

=∏v∈MK

∏w|v

max(|x1|v, . . . , |xn|v)[Lw:Kv ]

=∏v∈MK

max(|x1|v, . . . , |xn|v)[L:K]

= HK(x)d.

Now, we define the absolute height for x ∈ Qnby choosing a number field K such

that x ∈ Kn, and putting H(x) = HK(x)1/[K:Q]. By what we just observed, this isindependent 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) ofKronecker’s Theorem. Then by Kronecker’s Theorem there exist p ∈ Zm, q ∈ Zn

such that

‖Aq− p− b‖∞ ≤ ε. (3.1)

In Theorem 1.8 we proved a non-effective upper bound for ‖q‖∞. In the next sectionswe also prove an effective upper bound and in this section we will use the theory ofuniform distribution to get some heuristics on the order of the upper bound for ‖q‖∞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−bxkc ∈ [a, b). Thesequence (xk)

∞k=1 is said to be uniformly distributed (modulo 1) if

limK→∞

T ([a, b);K)

K= b− a

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

limK→∞

1

K

K∑k=1

e2πihxk = 0.

Proof. See Kuipers and Niederreiter [11], Chapter 1, Theorem 2.1. See Weyl [22] forWeyl’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∑k=0

xk =1− xK+1

1− x.

Since α 6∈ Q, we have e2πihα 6= 1 for all h 6= 0. Hence,

limK→∞

1

K

K∑k=1

e2πihαk = limK→∞

1

K

(1− (e2πiα)K

1− e2πiα− 1

)= 0,

since the numerator is bounded. Corollary 3.3 follows directly from Weyl’s criterion,Theorem 3.2.

Corollary 3.4. Let (qk)∞k=1 be a sequence in Rn, ordered such that if s, t are any

positive integers with ‖qs‖∞ < ‖qt‖∞ then s < t. If α = (α1, . . . , αn) 6∈ Qn then thesequence (xk)

∞k=1 defined by xk = 〈qk,α〉 is uniformly distributed.

Proof. By Weyl’s criterion, it suffices to prove

limK→∞

1

K

K∑k=1

e2πihxk = 0 for all integers h 6= 0.

Choose an integer h 6= 0. For every K ∈ N there is a non-negative integer L such that(2L+ 1)n ≤ K < (2L+ 3)n. We split 1

K

∑(2L+1)n

k=1 e2πihxk into S1 + S2, where

S1 :=1

K

(2L+1)n∑k=1

e2πihxk and

S2 :=1

K

K∑k=(2L+1)n+1

e2πihxk .

25

Without loss of generality, assume α1 6∈ Q. Then

S1 ≤1

(2L+ 1)n

∣∣∣∣∣L∑

l1=−L

· · ·L∑

ln=−L

e2πih(α1l1+···+αnln)

∣∣∣∣∣≤ 1

(2L+ 1)n

∣∣∣∣∣L∑

l1=−L

e2πihα1l1

(L∑

l2=−L

· · ·L∑

ln=−L

e2πih(α2l2+···+αnln)

)∣∣∣∣∣=

1

(2L+ 1)n(2L+ 1)n−1

∣∣∣∣∣L∑

l1=−L

e2πihα1l1

∣∣∣∣∣=

1

2L+ 1

∣∣∣∣∣L∑

l1=−L

e2πihα1l1

∣∣∣∣∣ .Further,

S2 ≤(2L+ 3)n − (2L+ 1)n

(2L+ 1)n

=

(1 +

2

2L+ 1

)n− 1.

Note that S1 → 0 as K →∞ and S2 → 0 as K →∞. We conclude

limK→∞

1

K

K∑k=1

e2πihxk = 0.

Corollary 3.4 follows.

The theory of uniform distribution can be extended to higher dimensions. Butbefore stating the definition of uniform distribution in n-dimensional space, we needsome new notation. Let a = (a1, . . . , an), b = (b1, . . . , bn) be two real vectors in Rn

with ai < bi for i = 1, . . . , n. We denote the n-dimensional interval∏n

i=1[ai, bi) by[a,b). For x ∈ Rn we denote the vector (x1 − bx1c, · · · , xn − bxnc) by x − bxc. By0 and 1 we denote the two n-dimensional vectors defined by 0 = (0, · · · , 0) and1 = (1, · · · , 1).

Definition 3.5. Let (xk)∞k=1 be a sequence in Rn. 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−bxkc ∈[a,b). A sequence (xk)

∞k=1 is said to be uniformly distributed (modulo 1) in Rn if

limK→∞

T ([a,b);K)

K=

n∏j=1

(bj − aj)

26

for all intervals [a,b) ⊂ [0,1).

Theorem 3.6. (Weyl’s criterion) A sequence (xk)∞k=1 in Rn is uniformly distributed

if and only if for every h ∈ Zn, h 6= 0,

limK→∞

1

K

K∑k=1

e2πi〈h,xk〉 = 0.


Corollary 3.7. A sequence (xk)∞k=1 in Rn is uniformly distributed if and only if

for every h ∈ Zn, h 6= 0, the sequence of real numbers (〈h,xk〉)∞k=1, is uniformlydistributed.


Corollary 3.8. Define α = (α1, . . . , αm) with α1, . . . , αm Q-linearly independent.The sequence (x)∞k=1 defined by xk = kα is uniformly distributed in Rm.

Proof. For every h ∈ Zn with h 6= 0, we have 〈h,α〉 6∈ Q by the Q-linear indepen-dence of α1, . . . , αm. The result follows from Corollary 3.3.

Lemma 3.9. Let (qk)∞k=1 be a sequence in Zn, ordered such that if s, t are any positive

integers with ‖qs‖∞ < ‖qt‖∞ then s < t. Further, let A be an m× n-matrix with realentries, that fulfills the condition of Kronecker’s theorem,

{z ∈ Qm : ATz ∈ Qn} = {0}. (3.2)

Finally, let xk = Aqk for k = 1, . . . , n. Then the sequence (xk)∞k=1 is uniformly

distributed in Rm.

Proof. For any h ∈ Zn we have

〈h,xk〉 = 〈h, Aqk〉= 〈ATh,qk〉.

27

As A satisfies equation (3.2) one has ATh 6∈ Qn for all h 6= 0. The sequence(〈ATh,qk〉)∞k=1 is a sequence as in Corollary 3.4. This implies that it is uniformlydistributed. Now Theorem 3.6 gives the required result.

For x ∈ Rn define dxc = max(dx1c, . . . , dxnc). Let A be a matrix as in Lemma3.9. This lemma implies that for all b ∈ Rm and 0 < ε < 1

2we have the limit

limQ→∞

#{q ∈ Zn : ‖q‖∞ ≤ Q, dAq− bc ≤ ε}Qn

= (2ε)m.

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 existq ∈ Zn and p ∈ Zm such that ‖Aq − p − b‖ ≤ ε. This also follows from the limitjust stated. For an effective Kronecker’s theorem we want to give an upper bound Qfor ‖q‖ in terms of ε. The limit suggests that for “most” A and b there should be asolution q ∈ Zn of dAq− bc ≤ ε with ‖q‖∞ 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 versionof 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

|q0 + q1α1 + · · ·+ qnαn| ≥ ‖q‖1−d∞ (n+ 1)1−dH(1, α1, . . . , αn)−d

for every q = (q0, . . . , qn) ∈ Zn+1 with q0 + q1α1 + · · ·+ qnαn 6= 0.

Proof. By assumption K ⊂ R. Denote by v the place represented by | · |. As q0 +q1α1 + · · ·+ qnαn 6= 0 we may rewrite the product formula, Theorem 2.28, as

|q0 + q1α1 + · · ·+ qnαn| =∏

w∈MKw 6=v

|q0 + q1α1 + · · ·+ qnαn|−1w

≥ ‖q‖1−d∞ (n+ 1)1−d

∏w∈MKw 6=v

max{1, |α1|w, . . . , |αn|w}−1

≥ ‖q‖1−d∞ (n+ 1)1−dH(1, α1, . . . , αn)−d.

28

This proves the proposition.

With this proposition we can make Theorem 1.8 effective for the case that A isan m× n matrix with real algebraic elements satisfying condition (1.1). Let

A =

α11 . . . α1n...

. . ....

αm1 . . . αmn

with real algebraic elements αij. The extensions Q(α1j, . . . , αmj) over Q are finite forj = 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 maxj=1,...,n

H(1, α1j, . . . , αmj)d.

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 suchthat

{z ∈ Qm : ATz ∈ Qn} = {0}. (3.4)

Let d be given by (3.3) and define for ε > 0

Q(ε) := d2d−1 (m+ n)2dH∗(A)

(1

ε

)d−1

.

Then for every ε > 0, b ∈ Rm, there exist p ∈ Zm, q ∈ Zn with

‖Aq− p− b‖∞ ≤ ε, ‖q‖∞ ≤ Q(ε).

Proof. Let a ∈ Zm with a 6= 0. Condition (3.4) implies that there is a j ∈ {1, . . . ,m},such that α1ja1 + · · ·+ αmjam 6= 0. Proposition 3.10 gives

|α1ja1 + · · ·+ αmjam| ≥ H∗(A)−1‖a‖1−dj∞ .

Let ε > 0. If we choose Q such that

Qn∑j=1

da1α1j + · · ·+ amαmjc+ εm∑i=1

|ai| ≥ 12(m+ n)2,

29

then we can apply Theorem 1.8. In view of Proposition 3.10 this inequality is satisfiedif

QH∗(A)−1‖a‖1−d∞ + ε‖a‖∞ ≥ 1

2(m+ n)2.

By elementary calculus the minimum of the function f defined by

f(x) = QH∗(A)−1x1−d + εx

is assumed at

xmin =

(QH∗(A)−1(d− 1)

εn

) 1d

.

Hence, we must choose Q such that

QH∗(A)−1x1−dmin + εxmin ≥ 1

2(m+ n)2.

An easy calculation shows that this inequality is satisfied for

Q = d2d−1 (m+ n)2dH∗(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 Kro-necker’s Theorem should hold with ‖q‖∞ � ε−m. We only get this expected bestresult 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 > 0an upper bound for ‖q‖∞ of order (ε−1)m+δ. This upper bound is independent of thedegree 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 generalform 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 extensionof 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 forall z ∈ Rr with ATz ∈ Zs we have bTz ∈ Z. Then there exists C ∈ R>0, effectivelycomputable and depending on A and b, such that for all ε > 0, there exists x ∈ Zs

with

‖Ax− b‖∞ ≤ ε, ‖x‖∞ ≤ C

(1

ε

)c(A)−1

.

30

Proof. For the proof of this theorem we follow the steps of Theorem 1.12. By Lemma1.13 and Lemma 1.14 we can find U ∈ GLr(R) such that UA is in the shape asdemanded at the start of the proof of Theorem 1.12. Let A1, A2, b1, and b2 be definedas in Lemma 1.14. These matrices and vectors are effectively computable, as thereexists a polynomial time algorithm to calculate the Smith normal form. See Bachemand Kannan, [1], for this. A basis for the module M in the proof of Theorem 1.12 isalso effectively computable by expressing the elements of A as Q-linear combinationsof a chosen Q-basis of K, and then using linear algebra over Z. By Theorem 1.11there exists q′ ∈ Zs−m,p′1 ∈ Zt such that

A1q′ − p′1 = b1.

By Lemma 1.13 and inequality (1.9) of Theorem 1.12 we have

‖A2q′′ − p′2 −

1

d(b2 − A2q

′)‖∞ ≤ε

d‖U‖.

We want to give an upper bound for ‖q′′‖ in this inequality. By Theorem 3.11 thereexist q′′ ∈ Zn, p′2 ∈ Zm−t such that

‖A′2q′′ − p′2 −1

d(b2 + A2q

′)‖∞ ≤ ‖U‖ε

d,

‖q′′‖∞ ≤c(A)

2c(A)+1(m+ n)2c(A)H∗(A2)

(d

ε

)c(A)−1

.

The proof of this theorem follows if we use this result and continue with the proof ofTheorem 1.12.

3.3 Subspace Theorem

In this section we use the Subspace Theorem, see below, to prove an effectiveversion 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 = αi1x1 + · · ·+ αinxn, i = 1, . . . , n,

are linearly independent if

det(L1, . . . , Ln) =

∣∣∣∣∣∣∣α11 · · · α1n

.... . .

...αn1 · · · αnn

∣∣∣∣∣∣∣ 6= 0.

31

Theorem 3.13. (Subspace Theorem) Let L1, . . . , Ln be n linearly independentforms with algebraic coefficients in C and let δ > 0. Then the set of solutions of theequation

0 ≤ |L1(x) · · ·Ln(x)| ≤ ‖x‖−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 Qand let δ > 0. Then there exist only finitely many x ∈ Zn with |α1x1 + · · ·+ αnxn| ≤‖x‖1−n−δ

∞ .

Proof. This proof is by induction. If α1 6= 0 then there exist only finitely many x ∈ Zsuch that |α1x| ≤ |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 inde-pendent over Q. Take L1(x) = α1x1 + · · · + αnxn 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|L1(x)| ≤ ‖x‖1−n−δ

∞ is contained in the set of solutions of |L1(x) · · ·Ln(x)| ≤ ‖x‖−δ∞ .By the Subspace theorem we find that the solutions with x ∈ Zn are contained inthe union of finitely many proper linear subspaces of Qn of dimension n − 1. Thesesubspaces are of the form c1x1 + · · ·+ cnxn = 0 with (c1, . . . , cn) ∈ Qn\{0}. For sucha subspace the solutions of |L1(x)| ≤ ‖x‖1−n−δ

∞ are also solutions of |(α1 − c1cnαn)x1 +

· · · + (αn−1 − cn−1

cnαn)xn−1| ≤ ‖x‖1−n−δ

∞ . Note that α1 − c1cnαn, · · · , αn−1 − cn−1

cnαn are

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 existsa constant C ∈ R>0 such that all x ∈ Zn satisfy the inequality

dα1x1 + · · ·+ αnxnc ≥ C‖x‖−n−δ∞ .

Hence, we proved a lower bound for the sum dα1x1 + · · · + αnxnc as in Proposition3.10. In contrast to Proposition 3.10 we do not have an effective lower bound and wehave the stronger condition that 1, α1, . . . , αn must be linearly independent. But forthis result the lower bound is in many cases much stronger than the one depending onthe degree of the field extension. As in Section 3.2 we would now like to formulate aversion of Kronecker’s Theorem using this result. This can not be done for Kronecker’sTheorem for linear forms, because of the stronger condition that 1, α1, . . . , αn mustbe linearly independent. This is why we only consider the problem of inhomogeneoussimultaneous approximation of real numbers by rational numbers.

32

Theorem 3.15. Let 1, α1, . . . , αm ∈ R be algebraic numbers that are linearly inde-pendent over Q and let δ > 0. Then there exist a constant D > 0 depending onα1, . . . , 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, q ≤ D

(1

ε

)m+δ

.

Proof. Let δ > 0. According to Corollary (3.14) there exists a constant C such that

da1α1 + · · ·+ amαmc ≥ C‖a‖−m−δ∞ for all a ∈ Zm.

With the same calculation as in the proof of Theorem 3.11 we conclude that thereexists a constant D such that with

Q(ε) = Dε−m−δ

we satisfyQ(ε)C‖a‖−m−δ∞ + ε‖a‖∞ ≥ 1

2(m+ 1)2.

for all ε > 0. We conclude that

Q(ε)da1α1 + · · ·+ amαmc+ εm∑i=1

|ai| ≥ 12(m+ 1)2.

We can apply Theorem 1.8 with n = 1. We conclude that there exist p ∈ Zm, q ∈ Zwith

|qαi − pi − bi| ≤ ε for all 0 < i ≤ m, q ≤ Dε−m−δ.


Based on the heuristic argument in Section 3.1 the best possible result would beproving that Q grows as ε−m. We proved that Q grows as ε−m−δ, so we came prettyclose.

33

Chapter 4

Geometry of numbers over the adeles

4.1 Adeles

Let K be a number field. For any finite set P of places on K containing M∞K ,

defineAK(P ) =

∏v∈P

Kv ×∏v 6∈P

Ov,

where Ov, defined byOv := {x ∈ Kv : |x|v ≤ 1},

is the maximal compact subring 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 adele ringover K. Elements of this ring are called adeles and denoted by (av), with av ∈ Kv

for all v ∈ MK and av ∈ Ov for almost all1 v ∈ MfinK . We call (av)v∈M∞K the infinite

part and (av)v∈MfinK

the finite part of the adele (av). On every completion Kv of K we

have a topology induced by the absolute value | · |v on Kv. These topologies induce atopology on AK , with a basis of the following shape∏

v∈MK

Uv,

where Uv is a non-empty open subset of Kv for all v ∈ MK , and Uv = Ov for almostall v ∈Mfin

K . With this toplogy the adele ring AK is the restricted topological productof Kv with respect to Ov. We can say a bit more about the topological structure ofAK , 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 alsoan abelian group, such that the opposite map

G −→ G

x 7−→ −x

and the group operation

G×G −→ G

(x, y) 7−→ x+ y

1With 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 locallycompact topological group.

Proof. See Cassels and Frohlich [6], Chapter 2, Section 14.

The n-dimensional Cartesian product of AK is called the n-dimensional adele spacedenoted An

K . Elements of this space are denoted by (av) with av ∈ Knv for all v ∈MK

and av ∈ Onv for almost all v ∈MfinK . We call (av)v∈M∞K and (av)v∈Mfin

Krespectively the

infinite part and finite part of (av). We endow AnK with the n-fold product topology

of AK .For an adele (av) ∈ An

K and λ ∈ R>0 we define scalar multiplication by λ(av) =(λvav) with

λv = 1 for v ∈MfinK ;

λv = λ for v ∈M∞K .

In the forthcoming sections we will state and prove results in the geometry ofnumbers in the adele space. For this it is essential to generalize the notion of a convexbody to the adele space. In order to do this we first need a definition of a convex bodyin Cn. We define a convex body in Cn by identifying Cn with R2n. A body in Cn isa non-empty connected subset of Cn, which has the same closure as its interior. It isconvex under the same condition as in Rn. A body C in Cn is called C-symmetric ifit 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:

C =∏

v∈M∞K

Cv ×∏

v∈MfinK

Mv,

where Cv is a convex body in Knv for all v ∈M∞

K and Mv is a free Ov-module of rank nfor all v ∈ Mfin

K , with Mv = Onv for almost all v ∈ MfinK . This convex body is called

symmetric if Cv is symmetric for all real v ∈ M∞K and C-symmetric for all complex

v ∈M∞K . It is called bounded if Cv is bounded for all v ∈M∞

K .

For every v ∈MK there is a natural embedding of K in Kv. Define φ := Kn ↪→ AnK

as the map that is the natural embedding on each coordinate. Let k ∈ Kn 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 ‖k‖v ≤ 1 for almost all v ∈MK . This follows directly from Lemma 2.25.The set φ(Kn) may be viewed as a lattice in An

K . We will take a closer look atthis embeddding in the one-dimensional case. First we will look specifically at theembedding restricted to

∏v∈M∞K

Kv.

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+ifor i = 1, . . . , s. Define x(i) = σi(x) for i = 1, . . . , d. Define

φ∞ : K ↪→ Rr × Cs =∏

σi∈M∞K

Kσi

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

C =∏

v∈M∞K

Cv ×∏

v∈MfinK

Mv,

with Cv and Mv as in Definition 4.2. Note that the set defined by {x ∈ K : x ∈Mv for v ∈Mfin

K } 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

of determinant detL = 2−s√|DK/Q(a)|.

Proof. Choose a Z-basis {α1, . . . , αd} of a. We define a d× d matrix A by

A = [a1, . . . , ad] = (α(k)l )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 = 12(ak + ak+s) and b2k−r =

−i(ak−b2k−r−1) for k = r+ 1, . . . , r+ s. With elementary linear algebra we calculate| detB| = 2s| detA| 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 byProposition 2.4. Hence, L is a lattice in Rd. Now, the result follows from detL =| det(φ∞(α1), . . . , φ∞(αd))|.

Corollary 4.4. A bounded symmetric convex body C in AnK contains only finitely

many points of φ(Kn).

Proof. We prove this in the case n = 1. For arbitrary n the argument must beapplied to each of the coordinates. Let again C in AK be given by

C =∏

v∈M∞K

Cv ×∏

v∈MfinK

Mv.

Definea := {x ∈ K : x ∈Mv for all v ∈Mfin

K }.Note that a is a fractional ideal of OK . The set

{φ∞(x) : x ∈ a}

is a lattice of∏

v∈M∞KKv by Lemma 4.3. We conclude that the intersection of φ(K)

and C is finite, because C is bounded.

Analogously to the classical case, we can define successive minima of convex bodiesin An

K . 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(Kn)) ≥ 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 ∈ K∗v , |iv|v = 1 for almost all v ∈MfinK }.

37

If we endow IK with componentwise multiplication, it becomes a group. We call thisset the idele group of K. Let ‖(iv)‖ be defined by

‖(iv)‖ =∏v∈MK

|iv|v.

Further, define

c(v) = 0 for v ∈MfinK

= 1 for real v ∈M∞K

= 2 for complex v ∈M∞K .

Theorem 4.5. (Strong Approximation Theorem) Let (iv) ∈ IK and let v0 ∈MK. Then for every adele (av) ∈ AK there exists an x ∈ K such that

|x− av|v ≤ |iv|v for all v ∈MK\ {v0}.

Proof. See Cassels and Frohlich [6], Chapter 2, Section 15.

Theorem 4.6. (Effective Strong Approximation Theorem) Denote again by rthe number of real places of K and by s the number of complex places of K. Choosean idele (iv) ∈ IK such that

‖(iv)‖ ≥(d

2

(2

π

)s√|DK |

)d. (4.1)

Then for every adele (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 ∈MfinK }.

Let L = φ∞(a) be a lattice in Rd as in Lemma 4.3. Define A1, . . . , Ar+s ∈ R>0 byAj = |iσj

|σjfor j = 1, . . . , r + s. Define

C :=

(y1, . . . , yr, yr+1, zr+1, . . . , yr+s, zr+s) ∈ Rd with

|yi| ≤ Ai for i = 1, . . . , ry2i + z2

i ≤ Ai for i = r + 1, . . . , r + s

.

38

Note that C is a convex body in Rd. By the Strong Approximation Theorem, Theorem4.5, there exists b ∈ K such that |b− av|v ≤ |iv|v for all v ∈Mfin

K . By definition of thecovering radius µ = µ(C,L) there exists an u ∈ L such that φ∞(b)− (av)v∈M∞K + 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 λdusing Minkowski’s convex body theorem and then apply Theorem 1.4 for the requiredupper bound for µ. In order to use Minkowski’s Convex Body Theorem, we have tocalculate detL and V (C).

We will start by calculating detL. We already know that

detL = 2−s√|DK/Q(a)| = 2−sN(a)

√|DK | (4.2)

by Lemma 4.3 and Proposition 2.5. So, we need to take a closer look at N(a). Forv ∈Mfin

K , 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 ∈MfinK }

we finda =

∏v∈Mfin

K

pordpv (iv)v .

Hence,

N(a) =∏

v∈MfinK

N(pv)ordpv (iv) =

∏v∈Mfin

K

|iv|−1v .

Putting together this result with equation (4.2), we get

detL = 2−s√|DK |

∏v∈Mfin

K

|iv|−1v .

Calculating the volume of the convex body C is actually quite easy:

V (C) = 2rπs∏

v∈M∞K

|iv|.

Minkowski’s Convex Body Theorem states that

λ1 · · ·λd ≤ 2ddetLV (C)

.

39

If we have a lower bound for λ1, . . . , λd−1, then we get an upper bound for λd. Thereexists an u ∈ Rd with u 6= 0 and u ∈ λ1C ∩ L and there exists a κ ∈ a such thatφ∞(κ) = u. We have |κ|v ≤ |iv|v for all v ∈Mfin

K and |κ|v ≤ λc(v)1 |iv|v for all v ∈M∞

K .Hence, we get

1 =∏v∈MK

|κ|v ≤ λd1‖(iv)‖.

We can use this to find the following upper bound

λd ≤(

2π

)s√|DK |‖(iv)‖−1

λd−11

=

(2

π

)s√|DK |‖(iv)‖−1/d.

Using Theorem 1.4, we have

µ ≤ d

2λd ≤

d

2

(2

π

)s√|DK |‖(iv)‖−1/d.

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 forevery (av) ∈ AK there exists precisely one (a′v) ∈ F , such that (av)− (a′v) ∈ φ(K).

Let φ∞ be the canonical embedding of K in∏

v∈M∞KKv as defined in Section 4.1.

Let U be the set given by

U = {ξ1φ∞(ω1) + · · ·+ ξnφ

∞(ωn) : ξi ∈ R,−12≤ ξi <

12

(i = 1, . . . , n)},

where {ω1, · · · , ωn} is a Z-basis of OK . We define the set F by

F = U ×∏

v∈MfinK

Ov. (4.3)

Theorem 4.8. The set F is a fundamental domain for AK/φ(K).

40

Proof. Let (av) ∈ AK be an arbitrary adele. We have to show that there existsprecisely one (a′v) ∈ F such that (av)− (a′v) ∈ φ(K). This is equivalent with provingthat there exists precisely one x ∈ K with (av)−φ(x) ∈ F . We first show the existenceand then the uniqueness.

There exists an x′ ∈ K, such that |x′ − av|v ≤ 1 for all v ∈ MfinK by the Strong

Approximation Theorem, Theorem 4.5. The set {φ∞(ω1), . . . , φ∞(ωn)} is an R-basisby Lemma 4.3. Hence, we have

φ∞(x′) =n∑i=1

αiφ∞(ωi),

where αi ∈ R for i = 1, . . . , n. We can choose z1, . . . , zn ∈ Z such that

−12≤ αi − zi < 1

2.

Now, x = x′ − (z1ω1 + · · ·+ znωn) 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 allv ∈ Mfin

K . Hence x − y ∈ OK by Lemma 2.23. Expressing φ∞(x) and φ∞(y) on thebasis {φ∞(ω1), . . . , φ∞(ωn)} of φ∞(K), we get

φ∞(x) =n∑i=1

xiφ∞(ωi),

φ∞(y) =n∑i=1

yiφ∞(ωi),

with xi, yi ∈ R and xi− yi in Z for i = 1, . . . , n As we have both −12≤ xi− (avi

) < 12

and −12≤ yi − (avi

) < 12

we find −1 < xi − yi < 1 for i = 1, . . . , n. Hence, xi = yi fori = 1, . . . , n and x = y. This proves the theorem.

4.4 The Haar measure on the adele space

For an introduction to measure theory we refer to Bartle [2]. The definitions of aσ-algebra, Borel set and measure can be found in this book. Let X be a topologicalspace. The Borel σ-algebra on X, denoted B(X), is the σ-algebra generated by theopen subsets of X. The sets in this σ-algebra are called Borel sets. This σ-algebraalso 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 calleda Borel measure if all Borel sets are µ-measurable, and µ(S) < ∞ for all compactsubsets 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 Lebesguemeasure 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 Borelmeasure 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 adelespace An

K .We will define a Haar measure on the adele space AK as the product of Haar

measures on the spaces Kv with v ∈ MK . Let v ∈ MfinK . 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 thatβv(Ov) = |DK |1/2dv . With this information it is possible to calculate the volume of freeOv-modules. A free Ov-module is of the form xOv for some x ∈ Kv. The volume ofxOv is equal to [Ov : xOv]−1|DK |1/2dv = |NKv/Qp(x)|v|DK |1/2dv .

As the adele space is a locally compact group, we can define a Haar measure onit. We will do this in the following way:

1. If v ∈MfinK , let βv denote the Haar measure on Kv as explained above.

42

2. If v ∈M∞K with Kv = R, let βv denote the ordinary Lebesgue measure on R.

3. If v ∈ M∞K with Kv = C, let βv denote the ordinary Lebesgue measure on C

multiplied by 2.

For every set of finite places P the product measure β =∏

v βv is a Haar measure onAK(P ). The Haar measure on AK is the measure β that agrees with all these measureson all subsets AK(P ). The n-fold product measure of β gives a Haar measure on An

K .Denote the volume of a convex set C in An

K induced by this measure by V (C).The choice of the scaling βv(Ov) = |DK |1/2dv 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 andlet r and s be respectively the number of real and complex embeddings. Let F be thefundamental domain AK/φ(K) as defined in equation (4.3). Then we have V (F) = 1.

Proof. Let φ∞ be the natural embedding of K in∏

v∈M∞KKv as defined in Section

4.1. Using Lemma 4.3 and the product formula, Theorem 2.28, we find that

V (F) = det(φ∞(ω1), . . . , φ∞(ωn)) · 2s ·∏

v∈MfinK

βv(Ov) =

√|DK |

∏v∈M∞K

|DK |−1/2dv = 1.


4.5 Minkowski’s Theorem for adele spaces

In this section we state an adelic version of the Minkowski’s Theorem. This the-orem will be of crucial importance for our proof of an adelic version of Kronecker’sTheorem. It was first proven in 1971 by R. McFeat [16] in a dissertation called “Ge-ometry of numbers in adele spaces”. Unaware of this, E. Bombieri and J. Vaaler [3]published the same result in 1983.

Theorem 4.11. (Second Theorem of Minkowski for adele spaces) Let K be anumber field of degree d = [K : Q] over Q and let r and s be respectively the numberof real and complex embeddings. Let C be a bounded symmetric convex body in theadele space An

K. Then its successive minima λ1, . . . , λn satisfy

2dnπsn

(n!)r(2n!)s|DK |12n≤ (λ1 · · ·λn)dV (C) ≤ 2dn.

43

Proof. See E. Bombieri and J. Vaaler [3], Theorem 3 and Theorem 6 for respectivelythe upper and the lower bound.

We have the following relationship between the adelic and classical version ofMinkowski’s Theorem, i.e., Theorem 1.2.

Corollary 4.12. The adelic 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 thelattice Zn. To prove this result for other lattices we have to apply an invertible lineartransformation. Define a convex body in An

Q by:

C ′ = C ×∏

v∈MfinQ

Ov.

Note that the number of real embeddings r = 1, the number of complex embeddingss = 0, the degree d = 1 and the discriminant DQ = 1. After filling in these constants,Minkowski’s Theorem for adele spaces states that

2n

n!λ1(C ′) · · ·λn(C ′)V (C ′) ≤ 2n.

As DQ = 1, we have V (C ′) = V (C) by definition. Hence, the corollary is proven if weshow that λi(C ′) = λi(C,Zn).

Let us recall the exact definitions of λi(C ′) and λi(C,Zn):

λi(C ′) = inf{λ ∈ R>0 : dim(λC ′ ∩ φ(Qn)) ≥ i},λi(C,Z) = inf{λ ∈ R>0 : dim(λC ∩ Zn) ≥ i}.

Note that λi(C ′) = λi(C,Zn) follows once we have proved that x ∈ λC∩Zn if and onlyif φ(x) ∈ λC ∩φ(Qn). Let x ∈ Qn with φ(x) ∈ λC ′. We have x ∈ λC and |xi|p ≤ 1 fori = 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 ′.This proves the corollary.

44

Chapter 5

Kronecker’s Theorem over the adeles

5.1 Kronecker’s theorem for adele spaces

In this section we prove an adelic version of Kronecker’s Theorem. We need someadditional definitions and results first.

We already defined successive minima for convex bodies in the adele space. Thereexists an adelic analog of the covering radius too. The covering radius µ(C) of a convexbody C in An

K is defined by

µ(C) = inf{λ ∈ R>0 :⋃

x∈Kn

(λC + φ(x)) = AnK}.

O’Leary and Vaaler proved a lower and upper bound for the covering radius µof convex body in terms of its successive minima. To this purpose they introduced aconstant ν(K) depending on the field K. For every number field K we can define aconvex body SK by

SK =∏

v∈M∞K

Cv ×∏

v∈MfinK

Ov,

where Cv = {x ∈ Kv : |x|v ≤ 1} for all v ∈M∞K . 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

ν(K) ≤ d

2

(2

π

)s√|DK |.

Proof. Define (iv) ∈ IK by iv = 1 for all v ∈ MfinK and iv = d

2( 2π)s√|DK | for all

v ∈M∞K . Then

‖(iv)‖ =

(d

2

(2

π

)s√|DK |

)d.

Now choose an arbitrary adele (av) ∈ AK . By the effective Strong ApproximationTheorem, Theorem 4.6, there exists an x ∈ K such that |x − av|v ≤ |iv|v for allv ∈MK . Hence, we have

(av)− φ(x) ∈(d

2

(2

π

)s√|DK |

)SK .

45

Hence, for λ =(d2

(2π

)s√|DK |)

we have⋃x∈K

(λSK + φ(x)) = AK

and

ν(K) = µ(SK) ≤ d

2

(2

π

)s√|DK |.


Theorem 5.2. Let C be a convex body in AnK. Its successive minima λ1, . . . , λn and

covering radius µ satisfy

12λn ≤ µ ≤ ν(K)(λ1 + · · ·+ λn).

For the original proof by O’Leary and Vaaler, see [19], Theorem 5.

Proof. LetC =

∏v∈M∞K

Cv ×∏

v∈MfinK

Mv

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 linearlyindependent vectors in (µ + 1

2λn)C ∩ φ(K). Suppose that µ + 1

2λn < λn, implying

that t < n. Choose linearly independent x1, . . . ,xt ∈ Kn such that φ(x1), . . . , φ(xt) ∈(µ + 1

2λn)C. Choose x ∈ Kn such that φ(x) ∈ λnC and x 6∈ Span{x1, . . . ,xt}. There

exists u ∈ Kn such that 12φ(x)− φ(u) ∈ µC by the definition of the covering radius.

Recall that 12φ(x) = (xv) with xv = 1

2x for v ∈ M∞

K , xv = x for v ∈ MfinK . By

symmetry and convexity of Cv we find that u = u− 12x+ 1

2x and x−u = 1

2x−u+ 1

2x

are both in (µ + 12λn)Cv for all v ∈ M∞

K . Further, x − u,x ∈ Mv for all v ∈ MfinK .

Hence, because Mv is a module, u,x−u ∈Mv for all v ∈MfinK . We conclude that both

φ(u) ∈ (µ + 12λn)C and φ(x− u) ∈ (µ + 1

2λn)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 ∈ Kn

such that φ(xi) ∈ λiC. Let us take an arbitrary (av) ∈ AnK . For every v ∈ MK there

exist αv,1, . . . , αv,n ∈ R such that av = αv,1x1 + · · ·+αv,nxn. Note that (αv,i) ∈ AnK for

i = 1, . . . , n. By definition of ν(K) there exist κ1, . . . , κn ∈ K such that ‖αv,i− κi‖ ≤

46

ν(K) for all v ∈ M∞K and ‖αv,i − κi‖v ≤ 1 for all v ∈ Mfin

K . Recall that xi ∈ λiCv forall v ∈Mfin

K . Hence, for all v ∈MK we find

av −n∑i=1

κixi =n∑i=1

(αv,i − κi)xi ∈ ν(K)(λ1 + · · ·+ λn)Cv.


For v ∈MK we define the maximum norm ‖·‖v with respect to v by

‖x‖v := max{|x1|v, . . . , |xn|v} for x = (x1, . . . , xn) ∈ Knv .

Let A ∈ Km,nv be a matrix given by

A =

α11 · · · α1n...

. . ....

αm1 · · · αmn

.

Define the norm ‖·‖v on Km,nv analogously to the norm on matrices in Section 1.4 by

‖A‖v = max1≤i≤m1≤j≤n

|αij|v for v ∈MfinK ,

‖A‖v = max1≤i≤m

n∑j=1

|αij|v for real v ∈M∞K .

‖A‖v = max1≤i≤m

(n∑j=1

|αij|12v

)2

for complex v ∈M∞K .

Note that we have ‖Ax‖v ≤ ‖A‖v‖x‖v for all v ∈MK , A ∈ Km,nv , and x ∈ Kn

v .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 MfinK and then

detA ∈ IK , ‖detA‖ =∏v∈MK

| detAv|v.

Let A ∈ GLn(AnK) and define

Π := {(av) ∈ AnK : ‖Avav‖v ≤ 1 for all v ∈MK}.

47

This set may be viewed as the adelic analog of a parallelepiped. The set given by

Π∗ = {(av) ∈ AnK : ‖(A−1

v )Tav‖v ≤ 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 ≤ 2dnV (Π)−1 = | detA|−1, (5.1)

and

(λ∗1 . . . λ∗n)d ≤ 2dnV (Π∗)−1 = | detA|. (5.2)

In the following theorem we will prove an upper bound for λ∗iλn+1−i.

Theorem 5.3. The successive minima of Π and Π∗ satisfy

n−1 ≤ λ∗iλn+1−i ≤ nn−1 for i = 1, . . . , n.

Proof. First we prove the lower bound. Choose linearly independent x1, . . . ,xn ∈ Kn

such that φ(xi) ∈ λiΠ for i = 1, . . . , n and linearly independent x∗1, . . . ,x∗n ∈ Kn such

that φ(x∗i ) ∈ λ∗iΠ∗ for i = 1, . . . , n. Note that

‖Avxi‖v ≤ λc(v)i , ‖(A−1

v )Tyj‖v ≤ (λ∗j)c(v) for all v ∈MK .

We have for all v ∈MfinK

|xTi x∗j |v = |xTi ATv (ATv )−1x∗j |v= |(Avxi)T (ATv )−1x∗j |v≤ ‖Avxi‖v‖(ATv )−1x∗j‖v≤ 1.

Hence, xTi x∗j ∈ OK and NK/Q(xTi x∗j) ∈ Z. By Theorem 2.27 we have∏v∈M∞K

|xTi x∗j |v = |NK/Q(xTi x∗j)| ∈ Z

and so we get xTi x∗j = 0 or |xTi x∗j |v ≥ 1 for at least one v ∈M∞K .

48

In a similar way we derive for all v ∈M∞K that

|xTi x∗j |v = |xTi ATv (ATv )−1x∗j |v= |(Avxi)T (ATv )−1x∗j |v≤ nc(v)‖Avxi‖v‖(ATv )−1x∗j‖v≤ (nλiλ

∗j)c(v).

The set {x1, . . . ,xi,x∗1, . . . ,x

∗n+1−i} contains n+1 vectors inKn. 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 nK-linearly independent vectors in Kn. We conclude that

1 ≤ |xTk x∗l |v ≤ (nλkλ∗l )c(v) ≤ (nλiλ

∗n+1−i)

c(v)

for at least one v ∈M∞K . This proves the lower bound

λiλ∗n+1−i ≥

1

n.

The upper bound is a direct consequence of the lower bound and inequalities (5.1)and (5.2).

(λiλ∗n+1−i)

d =(λ1 . . . λn)d(λ∗1 . . . λ

∗n)d

n∏k=1k 6=i

(λkλ∗n+1−k)

d

≤ n(n−1)d.


Corollary 5.4. The first successive minimum of Π and the covering radius of Π∗

satisfyµ(Π)λ1(Π∗) ≤ ν(K)nn.

Proof. This follows directly from Theorem 5.2 and Theorem 5.3.

Using this corollary we derive an adelic version of Kronecker’s Theorem in a wayvery similar to the proof of the effective version of the classical form of Kronecker’sTheorem (Theorem 1.8). First we introduce some new notation to state the theoremmore efficiently.

Let P be a finite set of places in MK . We denote AnK,P =

∏v∈P K

nv . A matrix

A ∈ Am,nK,P may be represented as a tuple (Av)v∈P with Av ∈ Km,n

v .

49

Theorem 5.5. Let P be a finite set of places of a number field K and let A ∈ Am,nK,P .

Suppose

{z ∈ Km : ∃w ∈ Kn such that ATv z = w for all v ∈ P} = {0}. (5.3)

Then for every ε > 0, (bv) ∈ AnK,P there exist vectors x ∈ Km, y ∈ Kn such that

‖Avx− y − bv‖v ≤ ε for all v ∈ P ,

‖x‖v ≤ 1, ‖y‖v ≤ 1 for all v ∈MK\P .

Proof. Let ε > 0, (bv) ∈ AnK . Define

τ := ν(K)(m+ n)m+n.

The set

V :=

{k ∈ Km+n :

‖k‖v ≤ τ c(v) 1ε

+ ‖Av‖vτ c(v) 1ε

for all v ∈ P ,‖k‖v ≤ 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∈PVv 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 for v ∈ P with Vv empty,

|Q|v > mink∈Vv

τ c(v)

‖(ATv In) k‖vfor v ∈ P with Vv non-empty .

Define B ∈ GLm+n(AK) by

Bv =

(κ−1Im −κ−1Av

0 Q−1In

)for all v ∈ P ,

Bv = I for all v ∈MK\P .

We have

(B−1v )T =

(κIm 0QATv QIn

)for all v ∈ P .

DefineΠ := {(av) ∈ Am+n

K : ‖Bvav‖v ≤ 1 for all v ∈MK}.

50

Its reciprocal Π∗ is given by

Π∗ = {(av) ∈ Am+nK : ‖(B−1

v )Tav‖v ≤ 1 for all v ∈MK}.

Suppose there exists a k ∈ Km+n with k ∈ τΠ, then k ∈ V . Hence, k = 0 or

‖BTv k‖ ≥ ‖Q(ATv In) k‖v ≥ 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 (b′v) ∈ AnK there exist

x ∈ Km, y ∈ Kn such that∥∥∥∥Bv

(y

x

)+

(b′v0

)∥∥∥∥v

≤ 1 for all v ∈ P,∥∥∥∥Iv(y

x

)∥∥∥∥v

≤ 1 for all v ∈MK\P.

Take (b′v) = (κ−1bv). We conclude that there exist x ∈ Km, y ∈ Kn such that

‖Avx− y + bv‖v ≤ ε for all v ∈ P,‖x‖v ≤ 1, ‖y‖v ≤ 1 for all v ∈MK\P.


In this Kronecker’s Theorem for adele spaces we demanded that

{z ∈ Km : ∃w ∈ Kn such that ATv z = w for all v ∈ P} = {0}.

In the next lemma and theorem we prove that this condition is necessary for theadelic 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 ∈ Km : ∃w ∈ Kn such that ATv z = w for all v ∈ P} = {0} (5.4)

is equivalent to

{z ∈ OmK : ∃w ∈ OnK such that ATv z = w for all v ∈ P} = {0}. (5.5)

51

Proof. Suppose there exist z ∈ Km, w ∈ Kn with z 6= 0 and ATv z = w for allv ∈ P . There exists κ ∈ K∗ such that κ ≤ max{‖z‖v, ‖w‖v}−1 for all v ∈Mfin

K by theStrong Approximation Theorem 4.5. Note that κz ∈ OmK , κw ∈ OnK with z 6= 0 andATv κz = κw.

Further,

{z ∈ OmK : ∃w ∈ OnK such that ATv z = w for all v ∈ P} ⊆{z ∈ Km : ∃w ∈ Kn 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) ∈ An

K,P there exist vectors x ∈ Km, y ∈ Kn such that


‖x‖v ≤ 1, ‖y‖v ≤ 1 for all v ∈MK\P ,

then condition (5.5) is satisfied.

Proof. This proof is by contradiction. Suppose there exist z ∈ OmK , w ∈ OnK suchthat z 6= 0 and ATv z = w for all v ∈ P . Then

zT (Avx− y − bv) = zTAvx− zTy − zTbv = wTx− zTy − zTbv.

Choose (bv) ∈ AnK,P such that 0 < |zTbv|v < (1

2)d+1 for all v ∈ P and choose ε > 0

such that ε < m−c(v)‖z‖−1v |zTbv|v for all v ∈ P . There exist x ∈ Km, y ∈ Kn such

that


‖x‖v ≤ 1, ‖y‖v ≤ 1 for all v ∈MK/P .

Then

|xTw − yTz− zTbv|v ≤ εmc(v)‖z‖v < |zTbv|v for all v ∈ P . (5.6)

Hence,

|xTw − yTz|v ≤ |zTbv|v + εmc(v)‖z‖v < (12)d for all v ∈ P ,

|xTw − yTz|v ≤ 2c(v) for all v ∈MK/P .

52

Here we used that ‖x‖v ≤ 1, ‖y‖v ≤ 1 for v ∈ MK/P and that z ∈ OmK , w ∈ OnK .We conclude that ∏

v∈MK

|xTw − yTz|v < 1.

Since xTw−yTz ∈ K we have xTw−yTz = 0 by the product formula. We concludethat

|xTw − zTy − zTbv|v = |zTbv|v for all v ∈ P .

This contradicts with inequality (5.6), which proves the theorem.

5.2 An effective adelic Kronecker’s Theorem

In this section we prove an effective version of the adelic 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 bealgebraic over K, H the absolute height. Define L := K(α1, . . . , αn) and d := [L : Q].Then we have

|q0 + q1α1 + · · ·+ qnαn|v ≥ H(1, α1, . . . , αn)−d(n+ 1)−dH(q)−d‖q‖v

for every q ∈ Kn+1 for which q0 + q1α1 + · · ·+ qnαn 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

|q0 + q1α1 + · · ·+ qnαn|v =∏w∈MLw 6=v

|q0 + q1α1 + · · ·+ qnαn|−1w

≥∏w∈MLw 6=v

(n+ 1)−1 max{1, |α1|w, . . . , |αn|w}−1‖q‖−1w

≥ H(1, α1, . . . , αn)−d(n+ 1)−dH(q)−d‖q‖v.


53

Let K be a number field. Let A ∈ Km,nv be a matrix given by

A =

α11 · · · α1n...

. . ....

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

H∗K(A) = (n+ 1)d maxj=1,...,n

H(1, α1j, . . . , αmj)d.

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 ∈ Km : ∃w ∈ Kn such that ATv z = w for all v ∈ P} = {0}. (5.7)

For every ε > 0 and for every v ∈ P define

Qv(ε) := τ dH∗K(Av)

(∏w∈P

‖BTw‖w

)−d(Av)

‖Bv‖v(

1

ε

)t·d(Av)−1

.

Then for every ε > 0, (bv) ∈ AnK, there exist vectors x ∈ Kn, y ∈ Km, such that

‖Avx− y − bv‖v ≤ ε, ‖x‖v ≤ Qv(ε) for all v ∈ P ,

‖x‖v ≤ 1, ‖y‖v ≤ 1 for all v ∈MK\P .

Proof. Let ε > 0. Denote Qv(ε) by Qv. Define

|εv|v = maxκ∈Kv :|κ|v≤ε

|κ|v for all v ∈ P .

Let B ∈ GL(m+ n,AK) be the linear transformation of Am+nK given by

Bv =

(ε−1v Im −ε−1

v Av0 Q−1

v In

)for all v ∈ P ,

Bv = I for all v ∈MK\P .

54

We define a parallelepiped Π given by

Π = {(av) ∈ Am+nK : ‖Bvav‖v ≤ 1 for all v ∈MK}.

Its reciprocal Π∗ is given by

Π∗ = {(av) ∈ Am+nK : ‖(B−1

v )Tav‖v ≤ 1 for all v ∈MK}.

Note that (B−1v )T is given by

(B−1v )T =

(εvIm 0QvA

Tv QvIn

)for all v ∈ P ,

(B−1v )T = I for all v ∈MK\P .

Define again τ := ν(K)(m + n)m+n. Let k ∈ Km+n\{0}. Suppose that φ(k) ∈τΠ∗, then we have ‖(BT

v )−1k‖v ≤ τ c(v) for every v ∈ MK . We can rewrite this as‖k‖v ≤ ‖(BT

v )‖vτ c(v). Condition (5.7) gives that (ATv In)k 6= 0 for at least one v ∈ P .By proposition 5.8 we get

‖(ATv In)k‖v ≥ H∗K(Av)−1

( ∏w∈MK

‖k‖w

)−d(Av)

‖k‖v

≥ H∗K(Av)−1τ−d

(∏w∈P

‖BTw‖w

)−d(Av)

‖Bv‖vτ c(v)ε1−t·d(Av).

Hence,‖(BT

v )−1k‖v ≥ ‖Qv(ATv In)k‖v ≥ τ c(v).

It follows that λ1(Π∗) ≥ τ . We conclude in the same way as in the proof of thenon-effective version of this theorem that µ(Π) ≤ 1. This proves the theorem.

5.3 A more general adelic Kronecker’s Theorem

In this section we prove a more general form of an adelic Kronecker’s Theoremas in Section 1.4. First we introduce some new notation. Let A = (Av) ∈ Ar,s

K,P ,

U = (Uv) ∈ Aq,rK,P and V ∈ Ks,t. The product A = UA is given by Av = UvAv and

the product A = AV is given by A = AvV .Let P be a finite set of places of a number field K containing M∞

K . 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 M∞K

such that OP is a principal ideal domain, let A = (Av) ∈ Ar,sK,P , m ∈ Z≥0 such that

rank(Av) = m for all v ∈ P , and let (bv) ∈ ArK,P . Then the following two assertions

are equivalent.(i) For every ε > 0 there exists an x ∈ OsP such that

‖Avx− bv‖v ≤ ε for all v ∈ P .

(ii)

{(zv) ∈ ArK,P : ∃ w ∈ OsP 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 A = (Av) :=UAV and define (bv) by bv := Uvbv for all v ∈ P . Then (i) is equivalent to theassertion that for every ε > 0 there exists x ∈ OsP such that


Proof. Suppose A and (bv) satisfy assertion (i). For every ε > 0 there exists x ∈ OsPsuch that


Define x := V −1x. Using that ‖UvAvV x− Uvbv‖v = ‖Uv(Avx− bv)‖v we get

‖UvAvx− Uvbv‖v ≤ ‖Uv‖vε for all v ∈ P ,‖x‖v = ‖V −1x‖v ≤ ‖V −1‖v‖x‖v ≤ 1 for all v ∈MK\P .


Lemma 5.12. Let U = (Uv) ∈ GLr(AK,P ), V = (Vv) ∈ GLs(OP ). Put A = (Av) :=UAV , and define (bv) by bv := Uvbv for all v ∈ P . Then (ii) is equivalent to theassertion that



56

Proof. Suppose there exist (zv) ∈ ArK , w ∈ OsP such that V TATv (UT

v zv) = w for allv ∈ P then we have ATv (UT

v zv) = (V T )−1w ∈ OsP for all v ∈ P . By assertion(ii) thereexists an ω ∈ OP such that bTv U

Tv zv = ω for all v ∈ P . We conclude that




Lemma 5.13. There exist U = (Uv) ∈ GLr(AK,P ), V = (Vv) ∈ GLs(OP ) such that

UvAvVv =

(Im −A′v0 0

)for all v ∈ P , (5.8)

where A′ = (A′v) ∈ Am,s−mK,P .

Proof. This proof is by induction on k. Let 0 ≤ k ≤ m. Our induction hypothesis isthat there exist U ∈ GLr(AK,P ), V ∈ GLs(OK) such that

UvAvVv =

(Ik A′v0 A′′v

)with A′v ∈ Kk,s−k

v , A′′v ∈ Ks−k,s−kv for all v ∈ P .

This is trivial for k = 0. Assume that this induction hypothesis is true for some k with0 ≤ k < m. Note that A′′v 6= 0 for all v ∈ P . It is easy to find v = (v1, . . . , vs−k) ∈ Os−kK ,such that v1 = 1 and A′′vv 6= 0 for all v ∈ P . Let e1, . . . , es−k be the unit vectors inKs−k. Define V ′k = [v, e2 . . . , es−k]. Note that V ′k ∈ Os−kK . Define

Vk :=

(Ik 00 V ′k

).

Note that Vk ∈ GL(s,K) and that ‖Vk‖v ≤ 1 for all v ∈ MK\P . Now, define A =(Av) = UAV Vk. We have

Av =

(Ik A′v0 A′′v

)for all v ∈ P ,

where the first column of A′′v, which is A′′vv1, is not equal to 0 for all v ∈ P . UsingGaussian elimination we can find an Uk ∈ GLr(AK,P ) with UkA = UkUAV Vk in thedesired 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

UAV =

It 0 −A1

0 Im−t −A2

0 0 0

, Ubv =

b1

b(v)2

0

for all v ∈ P ,

where 0 ≤ t ≤ m ≤ r, A1 ∈ Kt,s−m, A2 = (A(v)2 ) ∈ Am−t,s−m

K , b1 ∈ Kt, (b(v)2 ) ∈ Am−t

K ,and

{z1 ∈ OtP : AT1 z1 ∈ Os−mP } ⊆ {z1 ∈ OtP : bT1 z1 ∈ OP},{(zv) ∈ Om−tP : ∃ w ∈ Os−mP such that A

(v)T2 zv = w for all v ∈ P} = {0}.

Proof. By Lemma 5.13 there exist U1 = U(v)1 ∈ GLr(AK), V1 ∈ GLs(OP ) such that

U1AV1 =

(Im −A′0 0

)with A′ ∈ Am,s−m

K .

By Lemmas 5.11 and 5.12, the validity of (ii) is unaffected if we replace A by U1AV1.

Define (b′v) by U(1)v bv = (b′v,0)T for all v ∈ P . Thus, assertion (ii) becomes

{(z′v) ∈ OmP : ∃ w ∈ Os−mP such that A′Tv z′v = w for all v ∈ P} ⊆ (5.9)

{(z′v) ∈ OmP : ∃ ω ∈ OP such that b′Tv z′v = ω for all v ∈ P}

in the same way as in the proof of Theorem 1.12. The left-hand side is a sub-OP -module M of Os−mP , 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 OmP and there are δ1, . . . , δt ∈OP , such that {δ1d1, . . . , δtdt} is a OP -basis of M .

Now, let D := [d1, . . . ,dm] be the matrix with columns d1, . . . ,dm. Then D ∈GLm(OP ). Define A := DTA′, b := DTb′. Then(

DT 00 Ir−t

)(Im −A′0 0

)((DT )−1 0

0 Is−m

)=

(Im −A0 0

). (5.10)

The right-hand side is clearly of the shape UAV with U ∈ GLr(AK), V ∈ GLs(OP ).Writing zv := D−1z′v for all v ∈ P we see that (5.9) is equivalent to

{(zv) ∈ OmP : ∃ w ∈ Os−mP such that ATv zv = w for all v ∈ P} ⊆ (5.11)

{(zv) ∈ OmP : ∃ ω ∈ OP such that bTv zv = ω for all v ∈ P}.

58

Let A1 = (A(v)1 ) ∈ At,n

K,P and A2 = (A(v)2 )Am−t,n

K,P be defined by A(v)1 and A

(v)2 , which

consist respectively of the first t rows and the last m − t rows of Av for all v ∈ P .Let b

(v)1 = (b

(v)1 , . . . , b

(v)t ), b

(v)2 = (b

(v)t+1, . . . , b

(v)m ) be defined by b

(v)1 and b

(v)2 , which

consist of respectively the first t and the last m− t coordinates of b. Notice that theleft-hand side of (5.11) consists of all vectors of the shape (δ1z1, . . . , δrzt, 0, . . . , 0)T

with z1, . . . , zt ∈ OP for all v ∈ P . We have Av(δ1z1, . . . , δrzt, 0, . . . , 0) = w ∈ Os−mP

and b(v)T1 (δ1z1, . . . , δrzt, 0, . . . , 0) = ω ∈ OP for all v ∈ P and all z1, . . . , zt ∈ OP .

Hence, A1 ∈ Kt,s−m, b1 = (b(v)1 ) ∈ Kt, and

{z1 ∈ OtP : AT1 z1 ∈ Os−mP } ⊆ {z1 ∈ OtP : bT1 z1 ∈ OP}.

Further, by applying (5.11) with vectors (0, . . . , 0, zt+1, . . . , zm)T , we see that

{(zv) ∈ Om−tP : ∃ w ∈ Os−mP such that A(v)T2 zv = w for all v ∈ P} = {0}.


Proof of Theorem 5.10. First we prove (i) ⇒ (ii). This proof is by contradiction.Assume (ii) does not hold. Suppose there exist (zv) ∈ Ar

K , w ∈ OsP 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. DefineC :=

∏v∈MK

Cv,

where

Cv := {x ∈ Kv : |x|v ≤ |bTv zv|v + εrc(v)} for all v ∈ P ,

Cv := {x ∈ Kv : |x|v ≤ rc(v)‖w‖v} 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) for all v ∈ P ,

|κ|v ≤ rc(v)‖w‖v 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 doesnot exist a ω ∈ OP such bTv zv = ω for all v ∈ P . Hence, we can find an ε > 0 suchthat there does not exist κ satisfying these inequalities.

Hence, there does not exist x ∈ OsP , such that

|xTw − bTv zv|v ≤ εrc(v) for all v ∈ P ,

|xTw|v ≤ rc(v)‖w‖v for all v ∈MK\P .

59

Using that (Avx− bv)Tzv = xTw− bTv zv, we conclude that there is no x ∈ OsP , such

that

‖Avx− bv‖v ≤ ε for all v ∈ P ,

‖x‖v ≤ 1 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 withoutloss of generality, that

A = (Av) =

It 0 −A1

0 Im−t −A2

0 0 0

, bv =

b1

b(v)2

0

for all v ∈ P ,

with A1, A2,b1, (b(v)2 ) as in Lemma 5.14. Writing xT = (qT ,pT1 ,p

T2 ), We can rewrite

(i) as

‖A1q− p1 − b1‖v ≤ ε for all v ∈ P , (5.12)

‖A2q− p2 − b2‖v ≤ ε for all v ∈ P ,

to be solved in q ∈ Os−mP ,p1 ∈ OtP ,p2 ∈ Om−tP . By Theorem 1.11 there exist q ∈Os−mP ,p′1 ∈ OtP such that

A1q′ − p′1 = b1.

Recall that A1 ∈ Kt,s−m. By the Strong Approximation Theorem, i.e., Theorem 4.5there exists non-zero d ∈ OP such that dAT1 ∈ O

m,mP . By Theorem 1.8, there exist

q′′ ∈ Os−mP , p′2 ∈ OtP , such that

‖A2q′′ − p′2 −

1

d(b2 − A2q

′)‖v ≤ε

max(1, |d|v)for all v ∈ P .

Hence,

A1(q′ + dq′′)− (p′1 + dA1q′′)− b1 = 0,

‖A2(q′ + dq′′)− dp′2 − b2‖v< ε for all v ∈ P ,

which implies that (5.12) is satisfied with q = q′ + dq′′, p1 = p′1 + dA1q′′, p2 = dp′2.


60

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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, wedenote adeles by (av) and ideles by (iv).

C convex body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

C∗ polar of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1V (C) volume of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1L lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1L∗ dual of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2detL determinant of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1λi(C,L) ith successive minima of C with respect to L . . . . . . . . . . . . . . . . . . . . . . . . . 2µ(C,L) covering radius of C with respect to L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

F fundamental domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40e1, . . . , en the n unit vectors of Rn

Z ring of rational integersQ field of rational numbersR field of real numbersC field of complex numbers

K fieldR ring

Rn n-dimensional column vectors over RRm,n m× n matrices over RGLn(R) group of invertible n× n matrices over RQp field of p-adic rationalsKv completion of K with respect to absolute value | · |v . . . . . . . . . . . . . . . . 17

K algebraic closure of KAK adele ring over K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34IK idele group over K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37φ natural diagonal embedding of Kn in An

K . . . . . . . . . . . . . . . . . . . . . . . . . . . 35ordp valuation induced by prime ideal p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

63

OK ring of integers of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Ov maximal compact subring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34OP ring of P -integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55IK group of fractional ideals of OK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13PK group of principal fractional ideals of OK . . . . . . . . . . . . . . . . . . . . . . . . . . . 13DK discriminant of a number field K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14TrL/K trace of L over K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13NL/K norm of L over K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

(xk)∞k=1 sequence of reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

(xk)∞k=1 sequence of real vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

0 zero vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 (1, . . . , 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26

| · |v absolute value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15| · |∞ standard absolute value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15| · |p p-adic absolute value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16| · |p absolute value induced by a prime ideal p . . . . . . . . . . . . . . . . . . . . . . . . . . 21| · |σ absolute value induced by an embedding σ . . . . . . . . . . . . . . . . . . . . . . . . . 19MQ set of places of Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17MK set of places of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Mfin

K set of non-archimedean places of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17M∞

K set of archimedean places of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17H(x) height of x ∈ Qn

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

# cardinality of a set〈 , 〉 standard inner product on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1[ : ] degree of a field extensionbxc floor function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24dxc distance from x to the nearest integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5dxc distance from x to nearest integer vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 28‖x‖∞ maximum norm of a real vector x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4‖x‖1 sum norm of x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4‖x‖v maximum norm of x with respect to an absolute value | · |v . . . . . . . . . 47‖A‖ norm of matrix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8‖A‖v norm of matrix A with respect to | · |v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

64

Index

absolute value, 15archimedean –, 16non-archimedean –, 16equivalent –s, 16extension of an –, 18non-trivial –, 15p-adic –, 16trivial –, 15

adele ring, 34adeles, 34adelic convex body, 35adelic covering radius, 45adelic successive minima, 37archimedean absolute value, 16

completion, 18complex place, 19convex body, 1

polar –, 1symmetric –, 1volume of a –, 1adelic –, 35

covering radius, 3adelic –, 45

diagonal matrix, 7discriminant, 14dual lattice, 2

equivalent absolute value, 16extension of a place, 18extension of an absolute value, 18

fractional ideal, 13fundamental domain, 40

height, 23

idele group, 37ideal class group, 13infinite prime, 15inhomogeneous minimum, 3

lattice, 1dual –, 2

locally compact abelian group, 34

maximal compact subring, 34maximum norm, 4

non-archimedean absolute value, 16non-trivial absolute value, 15norm, 4, 13

maximum –, 4sum –, 4

number field, 12

p-adic absolute value, 16P -integers

ring of –, 55place, 16

complex –, 19extension of an –, 18real –, 19

polar convex body, 1principal ideal domain, 7product formula, 23

ramification index, 12real place, 19residue class degree, 12ring of integers, 12

65

ring of P -integers, 55

Smith normal form, 7successive minima, 3

adelic –, 37sum norm, 4symmetric convex body, 1

trace, 13trivial absolute value, 15

uniform distribution, 24

valuation, 15volume of a convex body, 1

66

On Kronecker’s Theorem - Universiteit Leidenalso use geometry of numbers to prove a quantitative version of Theorem 1 for the case n= 1, published in [9]. Their theorem gives an

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