Small generators for S-unit groups of division algebras

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New York Journal of Mathematics

New York J. Math. 20(2014) 1175–1202.

Small generators for S-unit groups of division algebras

Ted Chinburg and Matthew Stover

To H. W. Lenstra, Jr.

Abstract. Letkbe a number field, suppose thatBis a central simple division algebra over k, and choose any maximal order D of B. The object of this paper is to show that the group D^∗S of S-units of B is generated by elements of small height onceScontains an explicit finite set of places of k. This generalizes a theorem of H. W. Lenstra, Jr., who proved such a result whenB=k. Our height bound is an explicit function of the number field and the discriminant of a maximal order in B used to define itsS-units.

Contents

1. Introduction 1176

2. Notation and definitions 1179

3. Absolute values and heights 1180

4. The main result 1185

5. Explicit bounds 1192

5.1. A maximal orderDand an archimedean setX 1192

5.2. The constantmX 1193

5.3. Choices forT1,T5,T3 and T₃⁰ 1195 5.4. Topological generators and the constantsT₂ and T₄ 1196

5.5. An upper bound onT6 1197

5.6. The explicit bound 1198

6. An explicit example 1199

6.1. S={∞} 1199

6.2. S={∞} ∪ {`_i}^h_i=1 for a finite set{`_i}^h_i=1 of odd primes 1200

Received November 7, 2014.

2010Mathematics Subject Classification. 17A35, 20H10, 22E40, 11F06, 16H10, 16U60, 20F05, 11H06.

Key words and phrases. Division algebras,S-unit groups,S-arithmetic lattices, heights on algebras, generators forS-unit groups, geometry of numbers.

Chinburg partially supported by NSF Grant DMS 1100355.

Stover partially supported by NSF RTG grant DMS 0602191 and NSF grant DMS 1361000.

ISSN 1076-9803/2014

1175

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References 1201

1. Introduction

Letkbe a number field, suppose thatB is a central simple division algebra overk, and choose any maximal order Dof B. The object of this paper is to show that the groupD^∗_S ofS-units ofk is generated by elements of small height once S contains an explicit finite set of places of k.

A result of this kind was shown by Lenstra in [7] when B is k itself. In this case, G is the multiplicative group Gm and the notion of height is the classical one. Lenstra showed that once S is sufficiently large, the group Gm(O_k,S) = O_k,S^∗ is generated by elements whose log heights are bounded by

1

2log|d_k/Q|+ logm_S+r₂(k) log(2/π),

whered_k is the discriminant ofk,m_S is the maximal norm of a finite place in S, and r2(k) is the number of complex places of k. One version of our results is the following.

Theorem 1.1. Suppose B is a central simple division algebra of dimension d² over a number field k, n = [k : Q], and s is the number of real places of k over which B ramifies. Then there is a maximal order D of B with discriminant d_D and functions f1(n, d) and f2(n, d) of integer variables n and dfor which the following is true. Define

e= 2n d(2n−s).

Suppose that S is a finite set of places of k containing all the archimedean places and that S contains all finite places v such that

Norm(v)≤f1(n, d)d^e_D.

Let mSf be the maximum norm of a finite place in S. Then e≤1 and the group Γ_S of S-units in B with respect to the order D is generated by the finite set of elements of height bounded above by

f2(n, d)mSf d^e_D.

See the remarks at the end of§5.6for explicit expressions forf1(n, d) and f2(n, d). In many cases (e.g., when B is a number field) we have

(1) f₁(n, d) =d^nd 2

π _n−s/2^ndr²

2√ 2 π

!_2n−s^nds

(2) f₂(n, d) =d^nd+n+s((d−1)!)^n−s2^s^d

2−2d−4 2

2 π

_n−s/2^ndr² 2√

2 π

!_2n−s^nds

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SMALL GENERATORS FOR 1177

wherer₂ is the number of complex places ofkand all other notation is from the statement of Theorem1.1. Thus, whenB is a number field this exactly reproduces Lenstra’s bound.

WhenSis empty, even for a number fieldkone does not expect to be able to generate the unit groupO^∗_kby elements whose log heights are bounded by a polynomial in log|d_k/Q|. For example, the Brauer–Siegel Theorem implies that ifkis real quadratic of class number 1, then the log height of a generator of O^∗_k is greater than cd^1/2−_k/

Q for all >0, where c >0 depends only on . However, to our knowledge there is no unconditional proof, even in the case of real quadratic fields, that there cannot be an upper bound on the log heights of generators forO^∗_k that is polynomial in log|d_k/_Q|.

To develop our counterpart of Lenstra’s results, we must first define an intrinsic notion of height for elements of B^∗. The role of |d_k/_Q| is played by the discriminant d_D of an O_k-order D in B that is used to define the S-integrality of points of G over k. The height bound we produce applies to all choices of DonceS is sufficiently large. It implies, in particular, that there is a maximal order Din B such that when S is sufficiently large (in an explicitly defined sense), one can generate the points of Gover Ok,S by elements whose log heights are bounded by ¹₂log|∆_k/Q|+ logm_S+µwhere µ depends only only the degree of B overQ. See [8], [9], [3] and references therein for work on heights on quaternion algebras.

To show that our bounds remain quite effective outside the number field setting, in§6we apply our results to Hamilton’s quaternion algebra overQ.

The main result is the following.

Theorem 1.2. Let B be Hamilton’s quaternion algebra over Q, that is, the rational quaternion algebra with basis {1, I, J, IJ} such that I² =J² =−1 and IJ=−J I. LetD be the maximal order

Z

1, I, J,1 +I +J+IJ 2

and S={∞, `₁, . . . , `h} be a set of places containing the archimedean place

∞ and any set {`_i}^h_i=1 of distinct odd primes. Then the unit group D^∗_S is generated by the finite set of elements with reduced norm contained in {1, `₁, . . . , `_h}.

As in Lenstra’s case, Theorem 1.1leads to an algorithm for finding generators forO^∗_k,S. After embedding B into a real vector space, the algorithm is reduced to the classical problem of enumerating lattice points of bounded norm. That an algorithm exists to generateS-integral points of an algebraic group over a number field, with no assumptions onS, was known by work of Grunewald–Segal [5], [6]. Unlike their algorithm, ours is primitive recursive, which answers a question raised in [6].

Lenstra went on to show that his algorithm, together with linear algebra, can be used to give a deterministic algorithm for finding generators for

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the unit group O^∗_k with running-time O(|d_k/Q|^3/4+). The output of the algorithm consists of the digits of a set of generators. Answering a question raised by Lenstra, Schoof recently announced an improvement of this run-time to O(|d_k/_Q|^1/2+). By the discussion of the Brauer–Siegel theorem above, one expects the length of the output may be on the order of

|d_k/_Q|^1/2− in some cases, but the input to the problem, namely enough in- formation to specify k, will in general be much shorter. For instance, a real quadratic field k can be specified by giving its discriminant d_k/_Q, and the number of bits necessary to specify d_k/_Q is proportional to log|d_k/_Q|.

Lenstra’s algorithm for finding generators forO_k^∗makes essential use of the fact thatO_k,S^∗ is abelian, and this is not the case whenB is noncommutative and D^∗_S is infinite. Consequently, we do not know a counterpart of this algorithm for producing generators in the general case.

We also note that there is a spectral approach to finding small generators for groups acting on symmetric spaces. For example, if V is a compact quotient of a rank one symmetric space other than hyperbolic 2- or 3-space, Burger and Schroeder [1] showed that one can bound the diameter of V from above in terms of its volume andλ₁(V)diam(V) in terms of log vol(V).

One can use this to bound the length (in the Riemannian metric on V) of a generating set. As noted in [1], these results fail for hyperbolic 2- and 3- space, though Peter Shalen informed us that one could prove an analogous theorem for arithmetic Fuchsian and Kleinian groups assuming Lehmer’s conjecture. However, the problem we solve in this paper is of a different kind, even where a result like that of [1] holds. Our methods find explicit matrix generators with small entries, and it is not at all clear that generators that are short in an associated Riemannian metric have representatives in GL_nwith small entries. It would be interesting to see if spectral methods for S-arithmetic subgroups of reductive groups could produce generators that have small height.

The main tool for producing small generators forO_k,S^∗ onceSis sufficiently large is Minkowski’s lattice point theorem. This determines elements of B^∗ which can be shown to beS-integral by careful consideration of the constants required to apply Minkowski’s theorem. In particular, the assumption that B is a division algebra is crucial, and it would be interesting to extend our results to unit groups of arbitrary central simple algebras. The reason we cannot work with a general central simple algebra is because Minkowski’s lattice point theorem returns a nonzero element of the algebraB. We prove that it is S-integral. However, it might not be invertible in B ifB is not a division algebra. Given that SL_n(Z) is generated by elementary matrices, which certainly have small height, we expect such a result to hold. It appears to us that it is a deep problem to extend such height results to generators for the S-integral points of more general linear algebraic groups over number fields.

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2. Notation and definitions

Let k be a number field with ring of integers O_k and [k : Q] = n. We denote byV∞(resp.Vf) the set of archimedean (resp. finite) places ofk. Let B be a division algebra over kwith degreedand letD⊂B be anO_k-order in B. The multiplicative group of units in a ring R will be denoted R^∗. For each place v ofk and anyk-algebra or O_k-moduleA, let A_v denote the associated completion at v.

Define a norm on B_v^∗ by

(3) Normv(xv) = Norm_k_v_/_Q_p(v)(det(xv yBv)),

where p(v) is the place of Q under v and x_v y B_v is the k_v-linear endo- morphism of Bv induced by left xv-multiplication. If detv :Bv → kv is the reduced norm, then

(4) det(x_vyB_v) = det_v(x_v)^d.

The idele group J(B) is the restricted direct product Q0

vB_v^∗ of the B_v^∗ with respect to the groupsD^∗v. For x=Q

vxv∈J(B), define

(5) Norm∞(x) = Y

v∈V∞

Normv(xv)

(6) Normf(x) = Y

v∈V_f

Normv(xv).

for all x∈B^∗.

LetS be a finite set of places of kcontainingV∞. SetSf =SrV∞, and consider the groups

B_R^∗ = Y

v∈V∞

B^∗_v

B_S^∗_f = Y

v∈S_f

B_v^∗ ⊂ B_f^∗= Y

v∈V_f 0B_v^∗

and the product

(8) B_S^∗ =Y

v∈S

B^∗_v =B_R^∗ ×B^∗_S_f ⊂J(B) =B_R^∗ ×B^∗_f.

LetG_S be the subgroup ofB_S^∗ that satisfies the product formula, so

(9) GS=

(x, β)∈B_S^∗ : |Norm_∞(x)|=|Norm_f(β)|⁻¹ .

IfOk,S denotes theS-integers ofk, i.e., those elements ofkwhich lie inOk,v

for all v /∈S, then the S-order of B associated withD is

(10) DS=O_k,S⊗_O_kD.

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The group of invertible elements ofDS will be denoted Γ_S and is called the group ofS-units of D.

We define a topology onG_S by its natural embedding intoB_S^∗. The image of Γ_S inG_S under the diagonal embedding is a discrete subgroup. We have diagonal embeddings ΓS → B^∗_S and ΓS → Q

v6∈SD^∗v, and the product of these embeddings is the natural diagonal embedding of ΓS intoJ(B).

For any element

α= Y

v∈V_f

αv ∈B_f^∗,

there is a right-D-module

(11) αD=B∩



 Y

v∈V_f

αvDv



,

where B is diagonally embedded in B_f. For α ∈ D, the index of αD inD equals |Norm_f(α)|⁻¹. We also have the left-D-module

(12) Dα⁻¹={x∈B : x(αD)⊆D}.

3. Absolute values and heights

In this section we define absolute values on the completions Bv of B.

These will be used to define our notion of height for elements of B^∗. We point the reader to [8], [9], [3] and references therein for earlier work on heights for quaternion algebras over number fields.

For each place v of k there is a division algebra Av over kv such that B_v =k_v⊗_kBis isomorphic to a matrix algebra M_m(v)(A_v). The dimension of Av overkvisd(v)²for some integerd(v) such thatd(v)m(v) =d=√

dim_kB.

Note thatA_v and B_v have center isomorphic tok_v.

For finite v let Ov be the ring of integers of kv. We fix isomorphisms ρ_v :B_v →M_m(v)(A_v) such that for almost all finitev,A_v =k_v andρ_v(Dv) = M_m(v)(Ov). Let Nv :Av→kv be the reduced norm. Then Nv(r) =r^d(v) for any r∈k_v ⊆A_v.

For all places v of k, let | |_v be the usual normalized absolute value on kv. We extend | |_v to an absolute value on Av by |α|_v = |N_v(α)|^1/d(v)_v for α ∈ A_v. This absolute value is clearly multiplicative and restricts to the usual absolute value on the centerkv ofAv.

Suppose v is nonarchimedean. There is a unique maximal order U_v in Av, namely the set of α ∈ Av such that |α|_v ≤ 1. When Av 6= kv, Uv

is a noncommutative local ring, and it is O_v when A_v = k_v. The unique maximal two-sided ideal of U_v is the set P_v of α ∈ A_v for which |α|_v <1.

There is an element λv of Pv such that Pv = Uvλv = λvUv; such λv are called prime elements by Weil in [12, Def. 3, Chap. I.4]. By [12, Prop. 5, Chap. I.4], N_v(λ_v) is a uniformizer in k_v. Thus |λ_v|_v = |N_v(λ_v)|^1/d(v)_v = (#k(v))^−1/d(v) wherek(v) is the residue field ofvand so the range of | |_v on

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A^∗_v is (#k(v))^1/d(v))^Z. The set of α ∈A^∗_v such that |α|_v ≤(#k(v))^−t/d(v) is exactlyP_v^t. This implies that|α+β|_v ≤max(|α|_v,|β|_v) for everyα, β∈Av.

We now prove a simple lemma.

Lemma 3.1. With notation as above, suppose that v is archimedean. For allm-element subsets{α_i}^m_i=1⊂Av,

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m

X

i=1

αi

v

≤m^[k^v^:^R^]−1

m

X

i=1

|α_i|_v.

Proof. If v is complex, then Av = kv ∼= C and | |_v is the square of the usual Euclidean absolute value. The result reduces to the Cauchy–Schwarz inequality. Ifv is real andA_v =k_v ∼=R, the lemma is again clear.

The final case is when v is real and Av is Hamilton’s quaternions H = R+RI+RJ+RIJ whereI²=J² =−1 andIJ =−J I. Here,

|a+bI+cJ +dIJ|_v = (a²+b²+c²+d²)^1/2.

We can view this as the Euclidean length in R⁴ of the vector (a, b, c, d). It is clear from the triangle inequality that the optimal constant in this case is

again 1. This proves the lemma.

Recall that for each placev of kwe fixed an isomorphism ρ_v :B_v →M_m(v)(A_v).

For each v and each γ ∈ B_v^∗, let γ^i,j(v) denote the (i, j)-component of the m(v)×m(v) matrix ρv(γ) ∈ GL_m(v)(Av). Define |γ|_v = maxi,j|γ^i,j(v)|_v. EmbedB intoB_v =k_v⊗_kB in the natural way. Then theheight ofγ ∈B^∗ is defined by

(14) H(γ) = Y

v∈V

max{1,|γ|^d(v)_v },

where the product is over the set V =V∞∪V_f of all places of k. From the definition of| |_v we see that

(15) H(γ) = Y

v∈V

max

1, max

i,j |N_v(γ^i,j(v))|_v

.

For anyS and any positive real numberx, the set (16) BH_S(x) ={γ ∈Γ_S : H(γ)≤x}

of S-units of D with height bounded by x is finite. Indeed, bounding the nonarchimedean height bounds the denominator of each matrix entry under the image of everyρv, so the set ofγ ∈ΓS with bounded height is contained in a lattice in B_R. Bounding the archimedean height immediately implies finiteness.

We end this section by proving some inequalities we will need later con- cerning the behavior of absolute values on taking products and inverses of elements of Bv.

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Lemma 3.2. Suppose v is a finite place of k and fix an isomorphism ρv :Bv →M_m(v)(Av).

Let det_v :B_v →k_v be the reduced norm. For y, y⁰ ∈B_v: 1. |yy⁰|_v ≤ |y|_v|y⁰|_v.

2. If y is invertible, then |det_v(y)|_v|y⁻¹|^d(v)_v ≤ |y|d(v)(m(v)−1)

v .

Proof. Recall that

|y|_v = max

i,j {|y^i,j(v)|_v}, where

ρ_v(y) = (y^i,j(v))_i,j ∈M_m(v)(A_v).

Here|q|_v =|N_v(q)|^1/d(v)_v whenq ∈A_v, N_v :A_v →k_vis the reduced norm and dimkv(Av) = d(v)². We noted earlier that|qq⁰|_v = |q|_v|q⁰|_v and |q+q⁰|_v ≤ max{|q|_v,|q⁰|_v} for q, q ∈ A_v, so statement 1. of the lemma is clear by the usual matrix multiplication formula.

Let λv be a prime element of the unique maximal Ov-order Uv in Av, so that λ_vU_v = U_vλ_v = P_v is the maximal two-sided proper ideal of U_v. For 0 6= q ∈ Av there is an integer ` such that qUv = λ^`_vUv, and |q|_v =

|N_v(λ^`_v)|^1/d(v)_v = (#k(v))^−`/d(v). This interpretation of|q|_v implies that|a|= max_i,j|a_i,j|_v is unchanged if we multiply a matrixa = (a_i,j) ∈ M_m(v)(A_v) on the left or right by a permutation matrix, by a matrix which multiplies a single row or column by an element of U_v^∗, or by an elementary matrix associated with some element of Uv. Thus to prove inequality (2) of Lem- ma 3.2, we can use these operations to reduce to the case where y is a diagonal matrix with entriesλ^z_v¹, . . . , λ^z_v^m(v) for some integers z₁, . . . , z_m(v).

When y has this form,

|y|^d(v)_v = max{|N_v(λ^z_vⁱ)|_v: 1≤i≤m(v)}= (#k(v))⁻^minⁱ^{zⁱ^},

|det_v(y)|_v =|N_v(λ)^z|_v = (#k(v))^−z where z=

m(v)

X

i=1

zi,

and y⁻¹ is the diagonal matrix with entriesλ^−z_v ¹, . . . , λ^−zv ^m(v). Therefore,

|y⁻¹|^d(v)_v = (#k(v))⁻^minⁱ^{−zⁱ^}= (#k(v))^maxⁱ^{zⁱ^}.

The inequality in statement 2. of the lemma is therefore equivalent to maxi {z_i} −z≤ −(m(v)−1) min

i {z_i}.

This is the same as

z−max

i {z_i} ≥(m(v)−1) min

i {z_i},

which is certainly true.

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Lemma 3.3. Suppose vis an infinite place, so that there is an isomorphism ρv :Bv → M_m(v)(Av) with Av = kv if v is complex and either Av =kv or Av =Hifv is real. Definedet⁰_v :Bv →kv by det⁰_v(q) =|det_v(q)|^1/d(v)_v where det_v :B_v →k_v is the reduced norm. Then, there are minimal real constants δ1(Av, m(v))and δ2(Av, m(v)) such that for all y, y⁰ ∈Bv:

1. |yy⁰|_v ≤δ1(Av, m(v))|y|_v|y⁰|_v.

2. |yy⁰|_v =|y|_v|y⁰|_v if eithery or y⁰ is a scalar matrix or a permutation matrix.

3. |det⁰_v(y)y⁻¹|_v ≤δ₂(A_v, m(v))|y|^m(v)−1_v for all y∈B_v^∗. We also have the bounds

(17) 1≤δ₁(A_v, m(v))≤m(v)^[k^v^:R]

(18) 1≤δ₂(A_v, m(v))≤2^[k^v:R]m(v)(m(v)−1). Furthermore, ifA_v =k_v then

(19) 1≤δ₂(A_v, m(v))≤((m(v)−1)!)^[k^v^:^R^]. Proof. As before,

|y|_v = max

i,j {|y^i,j(v)|_v} where

ρ_v(y) = (y^i,j(v))_i,j ∈M_m(v)(A_v)

and |q|_v =|N_v(q)|^1/d(v)_v forq ∈Av, and where Nv :Av → kv is the reduced norm and dim_k_v(Av) =d(v)². We noted earlier that |qq⁰|_v =|q|_v|q⁰|_v for all q, q ∈A_v, and by Lemma 3.1,

m(v)

X

i=1

q_i v

≤m(v)^[k^v^:^R^]−1

m(v)

X

i=1

|q_i|_v ≤m(v)^[k^v^:^R^]max_i{|q_i|_v}

for{q_i}_i ⊂Av. By writing the matrix entries of yy⁰ as sums of products of the entries ofyandy⁰ this leads to (1) in Lemma3.3and the stated bounds on δ₁(A_v, m(v)). The bound (2) in Lemma3.3is clear.

Now suppose that y ∈ B_v^∗. We can find permutation matrices r and r⁰ such that the entry q of ryr⁰ for which | |_v is maximal lies in the upper left corner. We then perform Gauss–Jordan elimination on the rows and columns ofryr⁰ to produce matrices eand e⁰ in M_m(v)(A_v) such thateand e⁰ are products of elementary matrices and the nonzero off-diagonal entry of each elementary matrix for e or e⁰ has the form −τ /q for some entry τ of ryr⁰, where | −τ /q|_v = |τ|_v/|q|_v ≤ 1. The matrix y₁ = eryr⁰e⁰ has the same entryq asy in the upper left corner, and all of the other entries in the first row and the first column are 0. Finally, the other entries ofy₁ have the form α−(τ /q)β where α, β, andτ are entries of ryr⁰. Since|τ|_v ≤ |q|_v we see from Lemma 3.1that

|α−(τ /q)β|_v ≤2^[k^v^:^R^]−1(|α|_v+|β|_v)≤2·2^[k^v^:^R^]−1|q|_v = 2^[k^v^:^R^]|y|_v.

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Since q is an entry of y₁, we deduce that

|y|_v ≤ |y₁|_v≤2^[k^v^:^R^]|y|_v and det(y1) =±det(y).

We now continue withy1 and construct matricess, s⁰ ∈M_m(v)(Av) such that sands⁰ are products of elementary matrices and permutation matrices, and the off-diagonal entries of the elementary matrices involved in each product have absolute value with respect to | |_v bounded above by 1. The matrix y⁰=sys⁰ is diagonal and

|y|_v ≤ |y⁰|_v ≤2^[k^v^:^R^](m(v)−1)|y|_v (20)

det(y⁰) =±det(y).

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Then (y⁰)⁻¹ = (s⁰)⁻¹y⁻¹s⁻¹ and s⁰(y⁰)⁻¹s = y⁻¹, where s, s⁰,(s⁰)⁻¹, and s⁻¹ are products of elementary matrices and permutation matrices such that the off diagonal entries in each elementary matrix has absolute value with respect to | |_v bounded by 1. This leads by the above reasoning to the bounds

|(y⁰)⁻¹|_v ≤2^[k^v^:^R^](m(v)−1)|y⁻¹|_v (22)

|y⁻¹|_v ≤2^[k^v^:^R^](m(v)−1)|(y⁰)⁻¹|_v. (23)

Write y⁰ = diag(c1, . . . , c_m(v)) for some ci ∈ Av. Define ri = |c_i|_v =

|N_v(c_i)|^1/d(v)_v . Then

|y⁰|_v = max_i{r_i}, det⁰(y⁰) =Y

i

r_i =r,

det⁰(y⁰)(y⁰)⁻¹ = diag(rc⁻¹₁ , . . . , rc⁻¹_m(v)).

We deduce from this that

|det⁰(y⁰)(y⁰)⁻¹)|_v = max

i {|rc⁻¹_i |_v}

= rmax

i {|c⁻¹_i |_v}

= rmax

i {r_i⁻¹} (24)

≤ (max

i {r_i})^m(v)−1 (25)

= |y⁰|^m(v)−1_v . (26)

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Combining this with (20) and (22) gives

|det⁰(y)(y)⁻¹)|_v = |det⁰(y)|_v|y⁻¹|_v

= |det⁰(y⁰)|_v|y⁻¹|_v

≤ 2^[k^v^:^R^](m(v)−1)|det⁰(y⁰)|_v|(y⁰)⁻¹|_v

≤ 2^[k^v^:^R^](m(v)−1)|det⁰(y⁰)(y⁰)⁻¹|_v

≤ 2^[k^v^:^R^](m(v)−1)|y⁰|^m(v)−1_v

≤ 2^[k^v^:^R^](m(v)−1)

2^[k^v^:^R^](m(v)−1)|y|_vm(v)−1

= 2^[k^v:R]m(v)(m(v)−1)|y|^m(v)−1_v . (27)

This gives (2) in Lemma3.3 and the bound (18) onδ₂(A_v, m(v)).

Now, suppose that Av = kv. We can improve the above bound on δ₂(A_v, m(v)) using the fact that det(y)y⁻¹ is the transpose of the cofactor matrix ofy. Using the formula for the determinant as a sum over permuta- tions, every entry det(y)y⁻¹is the sum of (m(v)−1)! terms, each of which is

±1 times a product ofm(v)−1 entries of the matrixy. The absolute value with respect to| |_v of each entry of yis bounded by|y|_v, which implies that

|det(y)y⁻¹|_v ≤((m(v)−1)!)^[k^v^:^R^]−1(m(v)−1)!|y|^m(v)−1_v = ((m(v)−1)!)^[k^v^:R]|y|^m(v)−1_v .

This is the bound in (19).

4. The main result

We retain all notation and definitions from§§2-3. Let{ω_i}^nd_i=1² be aZ-basis forD. The discriminantd_D of Dis defined to be

d_D= det(M),

whereM is the matrix (T(ω_iω_j))_1≤i,j≤nd2 and T :B_R→R is the trace. As a real vector space, B_R∼=R^nd

2. The additive Tamagawa measure Vol onB described in [2,§X.3] is defined in such a way that

(28) d_D= Vol(B_R/D) =|d_D|^1/2.

Consider a compact convex symmetric subset X of B_R. By Minkowski’s lattice point theorem, if

(29) Vol(X)≥2^dim^Q^B d_D,

then X contains a nonzero element of D. Since X is bounded, there is a constant mX such that |Norm_v(y)|_v is bounded by m^[k_X^v^:^R^]/n for every y∈B_v∩X and v∈V∞. Then the set

(30) F_X ={(x, β)∈G_S : x∈X, βD⊆D, [D:βD]≤m_X} is a compact subset ofG_S.

(12)

Proposition 4.1. With notation as above, suppose thatS contains all finite placesv ofk such that|Norm_k/_Q(v)|^d≤mX. ThenFX is a fundamental set for the action of Γ_S onG_S in the sense that Γ_SF_X =G_S.

Proof. Given (x, β)∈G_S, we must show that there existsc∈Γ_S such that (cx, cβ)∈FX. This happens if and only if

cβD ⊆ D (31)

[D:cβD] ≤ mX, and (32)

cx ∈ X.

(33)

By definition, (31) means thatc∈Dβ⁻¹. Ifcx∈X, then [D:cβD] = |Norm_f(cβ)|⁻¹

= |Norm_f(c)|⁻¹|Norm_f(β)|⁻¹

= |Norm_∞(c)| |Norm_∞(x)|

(34)

= |Norm∞(cx)|

≤ Y

v∈V∞

m^[k_X^v^:^Q^]/n=mX

by (9) and the definitions of GS and mX. Therefore (33) implies (32).

Combining these facts, it suffices to show that Dβ⁻¹ ∩Xx⁻¹ contains an element of ΓS.

Since Xx⁻¹ is convex and symmetric with volume Vol(X)|Norm_∞(x)|⁻¹ and the latticeDβ⁻¹ inR^nd

2 has covolume

Covol(Dβ⁻¹) =d_D|Norm_f(β⁻¹)|⁻¹=d_D|Norm_∞(x)|⁻¹, this implies that

Vol(Xx⁻¹)≥2^dim^Q^BCovol(Dβ⁻¹)

if and only if Vol(X) ≥ 2^dim^Q^Bd_D. Since this holds by definition of X, it follows that Xx⁻¹∩Dβ⁻¹ contains a nonzero element c of Dβ⁻¹. By construction,c is an element ofB^∗ such that (cx, cβ)∈F_X. We claim that c∈ΓS.

Since cβ ∈D, it follows from (4) that |Norm_v((cβ)_v)|⁻¹ is a nonnegative integral power of Norm_k/_Q(v)^d for each v∈V_f. We know that

Y

v6∈V∞

|Norm_k/

Q(v)|^d≤m_X, which implies that v∈S_f. It follows that

|Norm_v((cβ)v)|= 1

(13)

for allv∈V_frS_f. Thus (cβ)_vDv =Dv for thesev, since cβ∈D. However, βv = 1 if v /∈S, so

cDv =c_vDv = (cβ)_vDv =Dv

for allv∈V_f rS_f. This implies that c∈D^∗v for allv /∈S_f, so c∈ΓS. This

proves the proposition.

We now describe howFX determines generators for ΓS. A subsetP ofGS

will be called a set oftopological generators forG_S if for any open subsetO of GS, the group generated byO and P is all of GS. The following lemma should be compared with [7, Lemma 6.3].

Lemma 4.2. Let P be a set of topological generators for G_S that contains the identity, and let FX be as in Proposition 4.1. Then ΓS is generated by its intersection withF_XP F_X⁻¹.

Proof. We have an equality of sets

FX(P ∪P⁻¹)F_X⁻¹ = (FXP F_X⁻¹)∪(FXP F_X⁻¹)⁻¹.

Therefore we can replace P by P∪P⁻¹ for the remainder of the proof and assume that P is symmetric, i.e., that P = P⁻¹. We emphasize that this does not mean we must assume P is symmetric in the statement of the lemma.

Consider the subset

O = (GSrΓS)∪(ΓS∩FXF_X⁻¹)

of GS. This is an open neighborhood of FXF_X⁻¹ in GS because ΓS is a discrete subgroup ofG_S. SinceF_X is a fundamental set for the action of Γ_S on GS, we can find a subsetF ⊂FX such that ΓS×F →GS is a bijection.

We claim that there is a small open neighborhood U of the identity in G_S such thatF U F⁻¹⊂O.

It will be enough to find a U such that Γ_S∩(F U F⁻¹) ⊂ F_XF_X⁻¹. Let T be the set of γ ∈ Γ_S such that γF ∩F U 6=∅. We want to show that if γ ∈T, then γFX ∩FX 6=∅. SinceF is a bounded fundamental domain for the action of Γ_S on G_S and Γ_S is discrete inG_S, the setT is finite when U is bounded. We can then shrink U further and assume that if γ ∈T, then γF⁰∩F⁰ 6=∅ for each open neighborhoodF⁰ of the closure of F inG_S. If γFX ∩FX = ∅ for some γ ∈ T, then since FX is compact there will be an open neighborhoodF⁰ ofF_X for whichγF⁰∩F⁰ =∅. This is a contradiction, since the closure of F is contained inF_X, which proves the claim.

LetP⁰ =P∪U, sohP⁰i=GS, and let ∆<ΓS be the subgroup generated by Γ_S∩F P⁰F⁻¹. We claim that ∆ = Γ_S. Indeed, if xp∈F P⁰, there exist y∈F and γ ∈ΓS such thatxp=γy. Then

γ =xpy⁻¹ ∈F P⁰F⁻¹,

soγ ∈∆. This implies thatF P⁰⊆∆F, so ∆F P⁰⊆∆F. Therefore, ∆F is right P⁰-invariant, but hP⁰i =GS, so ∆F =GS. Since ΓS×F → GS is a bijection, it follows that ∆ = Γ_S.

(14)

This proves that Γ_S is generated by

ΓS∩F P⁰F⁻¹ ⊆(ΓS∩F P F⁻¹)∪(ΓS∩F U F⁻¹).

However, F U F⁻¹ ⊂O, and Γ_S∩O ⊂F_XF_X⁻¹ by definition, so (35) ΓS∩F P⁰F⁻¹ ⊆(ΓS∩FXP F_X⁻¹)∪(ΓS∩FXF_X⁻¹).

Since P contains the identity, the right side of (35) equals Γ_S∩F_XP F_X⁻¹.

This proves the lemma.

We now define several constants that we need to state our main result.

(1) For X,FX, and `as above, let T1 be the supremum of 1 and n

|x_v|^d(v)/[k_v ^v^:R])o over all

x= Y

v∈V∞

xv ∈B_R

for which (x, β)∈FX for someβ.

(2) Let P be a finite set of topological generators for G_S which contains the identity element (see §5.4 for an example of such a set). We assume that every element ofP has the form (z, ζ) withz=Q

v∈S∞zv ∈ B^∗

Randζ =Q

v∈S_fζv ∈B_S^∗

f, wherezv is a real scalar and eachζv lies in the local maximal order M_m(v)(U_v) ofB_v = M_m(v)(A_v) (cf. §5.4).

Let T2 be the supremum of 1 and n

|z_v|^d(v)/[k_v ^v^:^R^]o over all z=Q

vz_v ∈P and all v∈V∞. (3) Let T3 be

Y

v∈Sf

maxn

1,|α_v|^d(v)_v o where as (x, α) ranges overF_X and α=Q

v∈Sf α_v ∈B_S^∗

f. Note that for suchαandαv we have thatαvDv ⊆Dv. Suchαv are contained in Dv, so this constant is finite. Similarly, define T₃⁰ to be the maximum of

Y

v∈S_f

max n

1,|α_v|d(v)(m(v)−1) v

o ,

where α and theαv range as above.

(4) Let T₄ be the smallest number such that Y

v∈S_f

max n

1,|g_v|^d(v)_v o

≤T4

for all g=Q

vgv ∈P.

(15)

(5) Let T₅ be the supremum of 1 and

n|det_v(a_v)|^1/[k_v ^v^:^R^] : a∈F_X and v∈V∞

o .

where we write a ∈ F_X as a = (a_v)_v with a_v ∈ B_v. (Recall that detv :Bv →kv is the reduced norm.)

(6) Let T₆ be the maximum over all subsetsW of V∞ of T₁^a(W⁾T₂^b(W⁾T₅^b(V^∞^r^W)

where

(36) a(W) = X

v∈W

[kv :R]m(v) and b(W) = X

v∈W

[kv :R].

Now we are ready to state and prove our main result.

Theorem 4.3. Let k be an algebraic number field of degree n over Q and B a central simple k-division algebra of degree d. Let S be a finite set of places of k containing all the archimedean places V∞ and let D⊂B be an O_k-order. We suppose that the k_v isomorphisms ρ_v :B_v → M_m(v)(A_v) are chosen such that ρ_v(Dv) ⊆ M_m(v)(U_v) for v 6∈ S, where U_v is the unique maximal Ov-order in the kv-division algebra Av. Suppose s is the number of (real) places v at which A_v is isomorphic to H. Let G_S be the topological group defined in (9), P be a topological generating set for GS satisfying the above conditions, and let Γ_S be the group ofS-units associated with D.

Suppose thatX is a convex symmetric subset of B_R such that (29) holds, and let m_X be the smallest real number such that|Norm_v(y)|_v is bounded by m^[k_X^v^:R]/n for everyy ∈B_v∩X and v∈V∞. Suppose that S contains every finite place v of k such that Norm(v) ≤m^1/d_X . Finally, let T₁, . . . , T₆ be the constants defined immediately above.

Then the set Γ_S∩F_XP F_X⁻¹ from Lemma 4.2is contained in (37) G_S,X = BHS ((d−1)!d)ⁿ 2^(d/2)(d−2)d

4(d−1)!

!s

T6T3T₃⁰T4

! .

Consequently, G_S,X is a finite generating set for ΓS.

Proof. Suppose thatγ ∈ΓS∩FXP F_X⁻¹. Then, there exist elements (z, ζ)∈ P and

(x, α),(y, β)∈F_X, x, y∈X, α= Y

v∈S_f

α_v, β= Y

v∈S_f

β_v

so that (γ, γ) = (x, α)(z, ζ)(y⁻¹, β⁻¹). That is,x_vz_vy_v⁻¹ =γ for eachv ∈V∞

and αvζvβ_v⁻¹ =γ for each v∈Sf.

LetW(γ) =W∞(γ)∪W_f(γ) be the set of placesv ofkat which|γ|_v >1.

By assumption, if v 6∈ S, then v is finite and ρ_v(Dv) ⊆ M_m(v)(U_v). Thus γ ∈ΓS implies that |γ|_v ≤ 1 ifv 6∈S. Thus W(γ) ⊆S,W∞(γ) ⊂V∞ and W_f(γ)⊆S_f.

(16)

By definition ofH(γ) we have H(γ) = Y

v∈W∞(γ)

|x_vz_vy_v⁻¹|^d(v)_v × Y

v∈W_f(γ)

|α_vζ_vβ⁻¹_v |^d(v)_v ,

Recall that det_v : B_v → k_v is the reduced norm and that d(v)² is the dimension ofAv overkv. Ifv is archimedean, we defined det⁰_v :Bv→kv by det⁰_v(q) =|det_v(q)|^1/d(v)_v forq ∈Bv.

We have |cα|_v = |c|_v|α|_v for c∈ k_v and α ∈B_v, where |c|_v here denotes the absolute value ofc∈kv with respect to| |_v :kv →R. Therefore we can rewrite the above expression as

H(γ) = Y

v∈W∞(γ)

|det⁰_v(y_v)x_vz_vy⁻¹_v |^d(v)_v (38)

× Y

v∈W_f(γ)

|det_v(βv)|_v|α_vζvβ_v⁻¹|^d(v)_v (39)

× Y

v∈W∞( γ)

|det_v(yv)|⁻¹_v (40)

× Y

v∈W_f(γ)

|det_v(β_v)|⁻¹_v , (41)

where the last two products are computed using the absolute values on the completionsk_v. We now proceed to bound each of these terms.

Sublemma 1. One has that

(42) Y

v∈W∞(γ)

|det⁰_v(y_v)(x_vz_vy_v⁻¹)|^d(v)_v ≤µ₁ T₁^a(W^∞^(γ)) T₂^b(W^∞^(γ)) where a(W∞(γ))and b(W∞(γ)) are as in (36) and

(43) µ₁ = Y

v∈W∞(γ)

δ₁(A_v, m(v))^d(v)δ₂(A_v, m(v))^d(v). Furthermore,

(44) µ1 ≤((d−1)!d)^[k:^Q^] 2^(d/2)(d−2)d 4(d−1)!

!s

.

if Av =H at exactly s real places of k.

Proof. By assumption each zv is a real scalar. Therefore, Lemma3.3gives

|det⁰_v(yv)(xvzvy_v⁻¹)|^d(v)_v

≤δ₁(A_v, m(v))^d(v)|x_v|^d(v)_v |z_v|^d(v)_v |det⁰_v(y_v)y_v⁻¹|^d(v)_v (45)

≤δ1(Av, m(v))^d(v)|x_v|^d(v)_v |z_v|^d(v)_v δ2(Av, m(v))^d(v)|y_v|d(v)(m(v)−1) v

≤δ₁(A_v, m(v))^d(v)δ₂(A_v, m(v))^d(v)T₁^[k^v^:^R^]T₂^[k^v^:^R^]T₁^[k^v^:^R^](m(v)−1)