Discrete subgroups of real vector spaces and lattices. Definitions: A ... Suppose
that Λ be a discrete subgroup of the real vector space Rn, i.e., Λ is a subgroup of
...
DISCRETE SUBGROUPS, LATTICES, AND UNITS. IAN KIMING
1. Discrete subgroups of real vector spaces and lattices. Definitions: A lattice in a real vector space V of dimension d is a subgroup of form: Zv1 + . . . + Zvd where v1 , . . . , vd is a basis of V . A subset B of the real vector space Rn is called discrete if the intersection between B and any bounded subset of Rn is finite (here we envision Rn as being equipped with the usual metric coming from the usual measure kvk of length of vectors v ∈ Rn ). Proposition 1. (i). If Λ is a lattice in Rn then Λ is discrete in Rn . (ii). Suppose that Λ be a discrete subgroup of the real vector space Rn , i.e., Λ is a subgroup of Rn that is discrete as a subset of Rn . Then Λ is a lattice in the subspace V of Rn that it generates as vector space. In particular, Λ is a free abelian group of rank ≤ n. Proof. (i). Write Λ = Zv1 + . . . Zvn with v1 , . . . , vn a basis of Rn . The matrix M having the vi ’s as columns is then invertible. If x is a lattice point we can write: ! a1 . x=M· .. an
with integers ai and where we view x as a column vector. We obtain: ! a1 .. = M −1 · x . an
and consequently (with just a rough estimate), |ai | ≤ c · nkxk where c is the maximum of the numerical values of entries of M −1 . Thus, if we require a lattice point x to be in some bounded subset of Rn there are only finitely many possibilities for the ai , — and hence only finitely many such lattice points. (ii). Let V be the subspace of Rn that is generated by Λ (i.e., V is the subspace consisting of all R-linear combinations of elements of Λ). Let d := dim V so that d ≤ n. Now, Λ must contain a basis of V as vector space; suppose that v1 , . . . , vd ∈ Λ is a basis of V . It is easily seen that Λ is discrete as a subset of V . 1
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IAN KIMING
Consider then the subgroup Λ0 := Zv1 + . . . + Zvd of Λ. Then Λ0 ∼ = Zd as the vi are R-linearly independent, and hence in particular Z-linearly independent. Since V is generated over R by the vi we see that any element v ∈ V can be written: (∗)
with ai ∈ R , 0 ≤ ai < 1 , x0 ∈ Λ0 .
v = a1 v1 + . . . ad vd + x0 ,
As Λ ⊆ V we see in particular that any element of Λ can be written in the form (∗). We deduce that any coset of Λ/Λ0 has a representative x of form (])
x = a1 v1 + . . . ad vd ,
with ai ∈ R , 0 ≤ ai < 1 .
But if x ∈ Λ has shape (]) then kxk ≤
X
|ai |kvi k ≤
i
X
kvi k .
i
But then x is in the intersection Λ ∩ {v ∈ V | kvk ≤
X
kvi k}
i
which is a finite set because Λ is discrete in V . We conclude that Λ/Λ0 is finite. If we put a := [Λ : Λ0 ] we have then that a · Λ ⊆ Λ0 . By exercise 24, page 44 of [1], we have that a · Λ is a free abelian group of rank ≤ d. Since x 7→ ax is clearly an isomorphism of Λ onto a · Λ, we have that Λ is a free abelian group of rank ≤ d. But since Λ contains Λ0 which is free of rank d, we conclude that Λ is free of rank d. If now x1 , . . . , xd is a Z-basis of Λ then the vi are Z-linear combinations of the xi . We conclude that the xi generate V as a real vector space, and hence that Λ is a lattice in V . 2. The logarithmic map on units in algebraic number fields. In this section we work with the following setup: K is an algebraic number field of degree n over Q. Denote by UK the abelian group of units, that is, multiplicatively invertible elements of the ring OK of algebraic integers in K. We denote by r and s the number of real embeddings K ,→ R, and the number of pairs of complex conjugate embeddings K ,→ C, respectively. Thus, r + 2s = n. Let σ1 , . . . , σr denote the r real embeddings K ,→ R, and let (τ1 , τ¯1 ), . . . , (τs , τ¯s ) denote the s pairs of complex conjugate embeddings K ,→ C. We then define the ‘logarithmic map’ ` : UK → Rr+s as follows: `(u) := (log |σ1 (u)|, . . . , log |σr (u)|, log |τ1 (u)|2 , . . . , log |τs (u)|2 ) . Notice that this definition is independent of whether we use τj or τ¯j from a pair (τj , τ¯j ) of complex conjugate embeddings in the definition. Proposition 2. (i). The logarithmic map ` is a homomorphism of abelian groups with image `(UK ) contained in the hyperplane H := {(x1 , . . . , xr+s ) | x1 + . . . + xr+s = 0} . (ii). If B is a bounded subset of Rr+s then `−1 (B) is a finite subset of UK .
DISCRETE SUBGROUPS, LATTICES, AND UNITS.
3
Proof. (i). That ` is a homomorphism, i.e., that `(u1 u2 ) = `(u1 ) + `(u2 ) is immediately clear. Let u ∈ UK . We know that then NK/Q (u) = ±1; but this means: 1 = | NK/Q (u)| = =
r Y i=1 r Y
|σi (u)| · |σi (u)| ·
i=1
s Y j=1 s Y
|τj (u)||¯ τj (u)| |τj (u)|2
j=1
as |τj (u)| = |¯ τj (u)| for j = 1, . . . , s. So, 0=
r X
log |σi (u)| +
i=1
s X
log |τj (u)|2
j=1
which just means that `(u) ∈ H. (ii). Let u ∈ UK . Observe that any bound on k`(u)k implies a bound on the numerical values of the coordinates of `(u) and hence a bound on the numerical value of all conjugates σi (u) and τj (u), τ¯j (u) of u. This again implies a bound on the numerical values of the coordinates of φ(u) where φ : K → Rn is the map that we introduced earlier (cf. [1], p. 133): φ(u) := (σ1 (u), . . . , σr (u), Re(τ1 (u)), Im(τ1 (u)), . . . , Re(τs (u)), Im(τs (u))) . We have shown that if `(u) lies in a given bounded subset B of Rr+s then φ(u) lies in a certain bounded subset of Rn (that could be specified explicitly in terms of B). But since we already know, cf. the theorem 36, p. 134, of [1], that φ(OK ) is a lattice in Rn , and since UK ⊆ OK , we conclude by Proposition 1 (i) that there are only finitely many possibilities for u ∈ UK if `(u) is required to lie in a given bounded subset B. Corollary 1. The logarithmic map ` maps UK to a lattice in some subspace of the hyperplane H := {(x1 , . . . , xr+s ) | x1 + . . . + xr+s = 0} . The kernel of ` consists of the roots of unity in K. Proof. Proposition 2 implies that `(UK ) is a discrete subgroup of H. The first claim then follows from Proposition 1 (ii). Let u ∈ UK . If u is a root of unity then so is every conjugate of u and so every conjugate of u has numerical value 1. We see then that `(u) = 0. On the other hand, consider the set: W := {u ∈ UK | `(u) = 0} . We certainly have that W is contained in `−1 of the unit ball of Rr+s ; proposition 2 (ii) then implies that W is finite. But since ` is a homomorphism UK → Rr+s of abelian groups we have ui ∈ W for all i ∈ Z if u ∈ W . So if u ∈ W there are necessarily 2 distinct integers i and j such that ui = uj . Then ui−j = 1 and u is a root of unity.
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IAN KIMING
Notice that the previous proof gives in particular a new proof of the fact that there are only finitely many roots of unity in K. Corollary 2. For the abelian group UK we have: UK ∼ =W ×F where W is the finite subgroup of roots of unity in K and F is a free abelian group of rank ≤ r + s − 1. Proof. The image `(UK ) under the logarithmic map is a lattice in a subspace of Rr+s of dimension r+s−1; in particular this image is free of some rank d ≤ r+s−1. Choose units u1 , . . . , ud ∈ UK such that `(u1 ), . . . , `(ud ) is a Z-basis of the free abelian group `(UK ). Since the kernel of ` is W it is now clear that UK is generated as abelian group by W and the ui . Letting F denote the subgroup of UK generated by the ui , we have that F is free of rank d with Z-basis u1 , . . . , ud : For any relation ua1 1 · · · uadd = 1 with ai ∈ Z implies a1 `(u1 ) + . . . + ad `(ud ) = 0 and thus a1 = . . . ad = 0. In particular, the only element of finite order in F is 1 so we have W ∩ F = {1}. But then UK ∼ =W ×F follows.
The following lemma is an application of Minkowski’s lemma on convex bodies. We will need it for the proof of Dirichlet’s unit theorem. Lemma 1. Fix any integer k with 1 ≤ k ≤ r + s. Then given any nonzero element α ∈ OK there is a nonzero β ∈ OK such that s p 2 | NK/Q (β)| ≤ |disc(K)| , π and such that: |σi (β)| < |σi (α)|
if
i 6= k ,
and: |τj (β)| < |τj (α)|
if
r + j 6= k .
Proof. Since α 6= 0 we can, and will, choose positive real numbers ci , i = 1, . . . , r +s such that: ci < |σi (α)| for i = 1, . . . , r , i 6= k , cr+j < |τj (α)|2
for j = 1, . . . , s , r + j 6= k ,
and furthermore:
s p 2 c1 · · · cr+s = |disc(K)| . π Consider now the subset E of Rn defined by the inequalities: |x1 | ≤ c1 , . . . , |xr | ≤ cr , x2r+1 + x2r+2 ≤ cr+1 , . . . , x2n−1 + x2n ≤ cr+s .
DISCRETE SUBGROUPS, LATTICES, AND UNITS.
5
Then E is compact and easily seen to be centrally symmetric and convex with volume: vol(E)
=
(2c1 ) · · · (2cr ) · (πcr+1 ) · · · (πcr+s ) Y 2r π s ci
=
2r+s
= =
2r+2s vol(Rn /φ(OK )) 2n vol(Rn /φ(OK ))
=
i
p |disc(K)|
where again φ is the embedding K ,→ Rn discussed earlier, and where we used theorem 36 of [1]. Now Minkowski’s lemma implies the existence of a nonzero point in the intersection between E and the lattice φ(OK ). Writing this lattice point as φ(β) with β ∈ OK , we have β 6= 0. Since β ∈ E we find: s Y p 2 |disc(K)| , | NK/Q (β)| ≤ ci = π i and also that β has the remaining desired properties: |σi (β)| ≤ ci < |σi (α)| for i = 1, . . . , r , i 6= k , √ |τj (β)| ≤ cr+j < |τj (α)| for j = 1, . . . , s , r + j 6= k . Lemma 2. Fix any integer k with 1 ≤ k ≤ r + s. Then there is u ∈ UK such that if `(u) = (y1 , . . . , yr+s ) then yi < 0 for i 6= k. Proof. Choosing any nonzero element α1 ∈ OK and applying Lemma 1 repeatedly we find a sequence α1 , α2 , . . . of nonzero elements of OK such that, for all m ∈ N, |σi (αm+1 )| < |σi (αm )| if i 6= k , and: |τj (αm+1 )| < |τj (αm )| if r + j 6= k , and such that all numbers | NK/Q (αm )| are bounded by a certain constant. Since k(αm )k = | NK/Q (αm )|, a previous argument shows that there are only finitely many distinct ideals among the principal ideals (αm ) (consider the prime factorizations of the ideals). Consequently, there exist m, t ∈ N with m < t and such that (αm ) = (αt ). But then αt = u · αm with a unit u, and we find (using t > m): log |σi (u)| = log |σi (αt )| − log |σi (αm )| < 0 log |τj (u)|2 = 2 log |τj (αt )| − 2 log |τj (αm )| < 0
for i = 1, . . . , r , i 6= k , for
j = 1, . . . , s , r + j 6= k .
As `(u) = (log |σ1 (u)|, . . . , log |σr (u)|, log |τ1 (u)|2 , . . . , log |τs (u)|2 ) , the unit u has the desired properties.
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References [1] D. A. Marcus: ‘Number fields’. Springer-Verlag 1995 (Corr. 3rd printing). Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK2100 Copenhagen Ø, Denmark. E-mail address:
[email protected]
