On generalized Hermite constants.

Renaud Coulangeon (France)

We give a brief survey on some generalizations of the Hermite constant of quadratic forms. Our main concern is the theory of Humbert forms and quaternionic Humbert forms. In that cases, we proved an analogue of Voronoï’s theorem about extreme forms. We also give a brief account of Watanabe’s theory of generalized Hermite constants.

Introduction

The density of a regular sphere packing, the centers of which are the points of a Euclidean lattice Λ, is proportional to the Hermite invariant γ(Λ) of the underlying lattice. The lattices achieving a local maximum of this density function, the so-called extreme lattices, are characterized by a celebrated theorem of Voronoï, in terms of perfection and eutaxy.

Over the recent years, this theory has received several generalizations, due to the work of C. Bavard [2], A.-M. Bergé, J. Martinet [3], J. Opgenorth [14], and the author ([5,6]) among others .

Another active direction of research is the adelic geometry of numbers initiated by T. Watanabe [19], in which a very general notion of Hermite constant over the group of adèles G(𝔸) of an algebraic group G is defined. The classical Hermite constant corresponds to the general linear group GL_{n} over ℚ. The theory of Humbert forms studied in [6] might be viewed as a particular case of Watanabe’s theory, corresponding to the general linear group GL_{n} over a number field k. In a work in progress with T.Watanabe [7], we investigate in detail the case of GL_{n} over a quaternion algebra.

In this paper, we give a brief survey of these subjects, focusing on Humbert forms and quaternionic Humbert forms.

1. Hermite constant

To start with, we recall some classical definitions and results about the Hermite invariant of quadratic forms. Let S_{n}(ℝ) be the space of nxn real symmetric matrices, or equivalently, the space of n-ary real quadratic forms. Inside this space, we consider the cone P_{n} := S_{n}(ℝ)_{>0} of positive definite quadratic forms. If A is in P_{n}, the set {A[X], X ∈ ℤ^{n}\{0}} admits a minimum (as usual, the notation A[X] stands for XAX′), so one can set

(1)  m(A) = min_{X∈ℤ^{n}\{0}}A[X].

Renaud Coulangeon

The Hermite number of A is then the following homogenized version of m(A)

(2)  γ(A) = \frac{m(A)}{(det A)^{\frac{1}{n}}} .

Finally, we let S(A) be the (finite) set of minimal vectors of A i.e.

(3)  S(A) = {X ∈ ℤ^{n} : A[X] = m(A)}

The Hermite constant in dimension n is then defined as the supremum of γ(A) over all A ∈ P_{n}. Its actual value is known only in dimensions 1 to 8. Recall that the search for quadratic forms with high Hermite number is motivated by the following geometric interpretation : one writes A = P P′, with P ∈ GL_{n}(ℝ) and consider the lattice L = ℤ^{n}P in ℝ^{n}. Then it is readily checked that γ(A)^{n/2} is proportional to the density of the sphere packing associated to L.

In 1873, Korkine and Zolotareff initiated a method to compute γ_{n} by looking for the local maxima of the function A ↦ γ(A) ([11],[12]). A form achieving such a local maximum is extreme. Some thirty years later, Voronoï ([18)] completed their work by establishing the following fundamental theorem

Theorem 1 (Voronoï 1908). A is extreme if and only if it is both perfect and eutactic.

On generalized Hermite constants.

This is of course meaningless as long as we have not defined perfection and eutaxy. Since this will be done in full generality in the next section, we will content ourselves here with the definition of perfection: a positive definite quadratic form A is perfect if it is completly determined by the set of equations A[X] = m(A), X ∈ S(A). In other words if B[X] = m(A) for all X ∈ S(A), then B = A. As a noticeable consequence, one gets that γ_{n} = max_{A∈S_{n}(ℚ)_{>0}} γ(A), and that γ_{n}^{n} ∈ ℚ.

To conclude this introductory section, we give another geometric interpretation of Hermite constant in the case of binary forms, which will be of some help in the next section. In that case indeed, there is a well-known 1-1 correpondance between the Poincaré upper half-plane ℌ := {z ∈ ℂ : ℑ(z) > 0} and the cone P_{2}, up to scaling:

SL_{2}(ℤ)\ℌ  ⟷  ℝ_{>0}\P_{2}/SL_{2}(ℤ)
z = x + iy  ⟷ (\begin{matrix}1 & x \\ x & x^{2} + y^{2} \end{matrix})

Thus, a form in P_{2} with high γ on the right-hand side corresponds to a point in ℌ with small imaginary part. This is illustrated by the picture below:

2. Humbert forms.

There have been several attempts to develop a “geometry of numbers in a relative context”, i.e. replacing the field of rationals ℚ by an arbitrary number field (see for instance [16]). In the 40’s P. Humbert wrote two long papers about the reduction theory of quadratic forms over number fields ([8,9]). In fact, the objects under consideration in this reduction theory are not exactly quadratic forms, but rather tuples of such forms, which we will call Humbert forms in the sequel, according to the terminology introduced by Icaza ([10]). These are defined as follows: first, let k be a number field, with [k : ℚ] = d. We denote by {v_{1}, …, v_{r}} the set of real places, corresponding to the real embeddings of k, and by {v_{r+1}, …, v_{r+s}} the set of complex places, corresponding to the couples of pairwise conjugated complex embeddings of k, so that d = r + 2s. In keeping with the classical definitions recalled in the first section, we have to define a space of forms S_{n,k} and a cone P_{n,k} of positive elements inside it. This is done as follows:

S_{n,k} := S_{n}(ℝ)^{r} × H_{n}(ℂ)^{s}

∪

P_{n,k} := S_{n}(ℝ)_{>0}^{r} × H_{n}(ℂ)_{>0}^{s}.

Definition 1. A Humbert form of rank n over k is a (r + s)-tuple A = (A_{1}, …, A_{r}, A_{r+1}, …, A_{r+s}) in P_{n,k}.

The value of a Humbert form A at u ∈ k^{n} is defined as follows: to u ∈ k^{n} we associate its conjugates u^{(1)}, …, u^{(r+s)} and we set

A[u] := ∏_{i=1}^{r+s} A_{i}[u^{(i)}]^{d_{i}} with d_{i} = \begin{cases}1 if v_{i} real \\ 2 if v_{i} complex \end{cases}

= ∏_{i=1}^{r} A_{i}[u^{(i)}] ∏_{i=r+1}^{r+s} A_{i}[u^{(i)}]^{2}.

Let O_{k} be the ring of integers of k. From now on, we assume, for simplicity, that the class number h_{k} of k is 1, i.e. O_{k} is a PID. Restricting to vectors with entries in O_{k}, we set the following definition

Definition 2. The minimum of a Humbert form A ∈ P_{n,k} is defined as

m(A) = min{A[u] : u ∈ O_{k}^{n} \ {0}}.

The Hermite-Humbert number of A is

γ(A) = \frac{m(A)}{(det A)^{1/n}} ,

where det A = ∏_{i} det A_{i}^{d_{i}} = ∏_{i=1}^{r} det A_{i} ∏_{i=r+1}^{r+s}(det A_{i})^{2}

Remark 1. (1) If k = ℚ, then P_{n,k} = P_{n}, and the Hermite-Humbert func-

tion γ coincides with the classical Hermite function.

(2) When h_{k} > 1, one has to consider simultaneously h_{k} distinct Hermite numbers for a given Humbert form A, indexed by the ideal classes {[𝔞_{1}], ⋯, [𝔞_{h_{k}}]}, and defined by replacing the lattice O_{k}^{n} by the lattices O_{k}^{n−1} ⊕ 𝔞_{i}, i = 1, ⋯, h_{k} in the above definition.

Vectors in O_{k}^{n} achieving the minimum are called minimal. There might be infinitely many, since if u is minimal, then ξu is minimal for any ξ in U_{k}, the group of units of O_{k}. It turns out that modulo multiplication by units, the set of minimal vectors is indeed finite. We set

S(A) := {u ∈ O_{k}^{n} minimal for A}/U_{k}

In ([10]), Icaza established the following theorem

Theorem 2 (Icaza, 1997). γ_{n,k} := sup_{A∈P_{n,k}}{γ(A)} < ∞ and is attained.

Following Korkine and Zolotareff’s terminology, Humbert forms at which γ achieves a local maximum are called extreme. These extreme forms can be characterised by a Voronoï type theorem. This requires a few more definitions. To a vector u ∈ O_{k}^{n} we associate the tuple

u′u_{A} := (d_{1} \frac{\overline{u^{(1)}}′u^{(1)}}{A_{1}[u^{(1)}]} , …, d_{r+s} \frac{\overline{u^{(r+s)}}′u^{(r+s)}}{A_{r+s}[u^{(r+s)}]} ) ∈ S_{n,k}.

Definition 3. A Humbert form A = (A_{1}, …, A_{r+s}) is perfect if

dim_{ℝ} ∑_{u∈S(A)} ℝu′u_{A} = r \frac{n(n + 1)}{2} + sn^{2} − (r + s − 1).

Definition 4. A Humbert form A = (A_{1}, …, A_{r+s}) is eutactic if its adjoint form \overline{A^{−1}} = (\overline{A_{1}^{−1}}, …, \overline{A_{r+s}^{−1}}) lies in the open convex hull of the (r + s)-tuples u′u_{A}, u ∈ S(A). In other words, A is eutactic if there exist ρ_{u} > 0, u ∈ S(A) such that:

\overline{A_{i}^{−1}} = ∑_{u∈S(A)} ρ_{u}d_{i} \frac{\overline{u^{(i)}}′u^{(i)}}{A_{i}[u^{(i)}]} for i = 1, …, r + s.

Of course, when k = ℚ, these definitions coincide with the classical notion of perfect and eutactic form mentionned in the previous section. In [6], we obtained a characterization of extreme Humbert forms:

Theorem 3 (C, 2001). A Humbert form A is extreme if and only if it is both eutactic and perfect.

Although the statement of this theorem is quite similar to Voronoï’s theorem, its proof is slightly more complicated. A general tool to get this kind of results, is Bavard’s condition (C) (see [2]).

As a consequence, we get the following properties:

Theorem 4 (C, 2001). (1) (Algebraicity) Any perfect Humbert form A of dimension n is equivalent, modulo rescalation, to a Humbert form B with entries in a finite extension L of k.

(2) (Finiteness) The set of perfect n-ary Humbert forms over a number field k, up to scaling and equivalence under GL_{n}(O_{k}), is finite.

Unfortunately, there is no general Voronoï’s algorithm to determine all the perfect forms in a given dimension. Nevertheless, in dimension 2, in a joint work with M.I. Icaza, R. Baeza and M. O’Ryan (University of Talca, Chile), we were able to compute some Hermite-Humbert constants over real quadratic fields with small discriminant:

Theorem 5 (B-C-I-O, 2001). The value of γ_{2,k} for the real quadratic fields k = ℚ(\sqrt{d}), d = 2, 3, 5 are as follows:

\begin{matrix}d & 2 & 3 & 5 \\ γ_{2,ℚ(\sqrt{d})} & \frac{4}{2\sqrt{6}−3} & 4 & 4/\sqrt{5} \end{matrix}

The previous theorem may be interpretated in terms of Hilbert modular surfaces (see [1] for details). Indeed, if k is a totally real number field with [k : ℚ] = r, there is a natural action of SL_{2}(O_{k}) on ℌ^{r}, induced by the diagonal embedding of SL_{2}(O_{k}) in SL_{2}(ℝ)^{r} and the standard action of SL_{2}(ℝ) on ℌ. The quotient SL_{2}(O_{k})\ℌ^{r} is by definition a Hilbert modular surface. As in the first section, we get a 1-1 correspondance

SL_{2}(O_{k})\ℌ^{r} ⟷ ℝ_{>0}^{r}\P_{2,k}/SL_{2}(O_{k})

If we assume again, for simplicity, that h_{k} = 1, then the Hermite-Humbert number of a form A ∈ P_{2,k} can be interpreted as the distance to infinity of the corresponding point in SL_{2}(O_{k})\ℌ^{r}:

“distance to ∞” ⟷ “γ(A)”
“local lowest point” ⟷ “extreme Humbert form”

The problem of finding lowest points on Hilbert modular surface had been investigated by H. Cohn in the 60’s (see [4]). For the real quadratic fields ℚ(\sqrt{d}), d = 2, 3, 5, he had made some conjectures which, once translated in the appropriate language, appear as consequences of our theorem.

Remark 2. If h_{k} > 1, then there are h_{k} “cusps” and h_{k} Humbert invariants, corresponding to the distance to these various cusps.

3. Generalized Hermite constants (after T. Watanabe).

The Hermite type invariants mentionned so far, may be viewed as particular instances of a general theory developed recently by Watanabe (see [19],[13]). The general setting is as follows: we take k a number field, and G a connected reductive algebraic group over k. We let 𝔙 = 𝔙_{f} ∪ 𝔙_{∞} be the set of places of k, and 𝔸 the adele ring of k . One has to fix ρ : G ⟶ GL(V) a strongly k-rational absolutely irreducible representation of G on a k-vector space V (see [19],[13]for details). Let D be the highest weight space of ρ, with stabilizer P (parabolic subgroup). Then X = G/P is a smooth projective variety embedded in ℙ(V) via ρ. On each localisation V_{v} = V ⊗_{k} k_{v}, v ∈ 𝔙, we fix a norm || · ||_{v}. Then, one has to choose a suitable maximal compact subgroup K in G(𝔸). For x ∈ GL(V (𝔸))V (k), we set ||x||_{𝔸} := ∏_{v∈𝔙} ||x_{v}||_{v}. One of the conditions that K must satisfy is that ||.||_{𝔸} is K-invariant. Finally, ||.||_{𝔸} is normalized by the condition ||x_{0}||_{𝔸} = 1 for x ∈ D(k) \ {0} (this might be achieved by a suitable choice of the local norms || · ||_{v}).

For each g ∈ G(𝔸)^{1} := {g ∈ G(𝔸) : ∀χ ∈ X_{k}(G) |χ(g)|_{𝔸} = 1}, define H_{g}(x) := ||ρ(gγ)x_{0}||_{𝔸}^{1/[k:ℚ]}, where x = ρ(γ)x_{0}. Then

Theorem 6 (Watanabe, 2000).

K \ G(𝔸)^{1}/G(k) ⟶ ℝ_{+}
g ↦ min_{x∈X(k)} H_{g}(x)

is a bounded continuous function. The generalized Hermite constant associated to (ρ, ||.||_{𝔸}) is

μ(ρ, ||.||_{𝔸}) := max_{g∈G(𝔸)^{1}} min_{x∈X(k)} H_{g}(x)^{2}.

Example 1. G = GL_{n}

(1) k = ℚ,

(a) if ρ is the natural representation in ℚ^{n}, then μ(ρ, ||.||_{𝔸}) = γ_{n}, the classical Hermite constant.

(b) if ρ_{d} is the natural exterior representation in ⋀^{d} ℚ^{n}, then μ(ρ, ||.||_{𝔸}) =

γ_{n,d} the Rankin constant (see [15],[5]).

(2) k = number field,

(a) if ρ is the natural representation in k^{n}, then μ(ρ, ||.||_{𝔸}) = γ_{n,k} the Hermite-Humbert constant.

(b) if ρ_{d} is the natural exterior representation in ⋀^{d} k^{n}, then μ(ρ, ||.||_{𝔸}) = γ_{n,d} the Rankin-Thunder constant (see [17]).

An opened question is wether there exists a general Voronoï type theorem to characterize the local maxima of μ(ρ, ||.||_{𝔸}). In the next section, we present a new example where this holds true.

4. Quaternionic Humbert forms (work in progress with T. Watanabe).

Let k be a number field, and D a quaternion field over k. Let 𝔙 = 𝔙_{f} ∪ 𝔙_{∞} be the set of places of k. where 𝔙_{f} and 𝔙_{∞} stand for the set of finite and infinite places respectively. The set of infinite places splits into real and complex places, denoted 𝔙_{∞,1} and 𝔙_{∞,2} respectively, and 𝔙_{∞,1} splits into ramified (𝔙′_{∞,1}) and split (𝔙′′_{∞,1}) places. The cardinality of 𝔙_{∞,1} and 𝔙_{∞,2} are denoted r and s respectively. In other words, setting D_{v} = D ⊗ k_{v} for

v ∈ 𝔙, one has D_{v} = \begin{cases}ℍ if v ∈ 𝔙′_{∞,1} \\ M_{2}(ℝ) if v ∈ 𝔙′′_{∞,1} \\ M_{2}(ℂ) if v ∈ 𝔙_{∞,2} \end{cases}

We apply the previous theory to the affine algebraic k-group G defined by G(R) = GL_{n}(D⊗_{k} R) for any k-algebra R, and ρ the natural representation in D^{n}. In order to get a Voronoï type theorem for the corresponding μ(ρ, ||.||_{𝔸}), we first translate the definitions from the adelic to global setting. This is done via the introduction of quaternionic Humbert forms

Definition 5. A n-ary quaternionic Humbert form over D is a (r + s)-tuple S = (S_{v})_{v∈𝔙_{∞}}, where:

- if v ∈ 𝔙′_{∞,1}, S_{v} is an n-ary positive definite hermitian form on ℍ^{n}.

- if v ∈ 𝔙′′_{∞,1} (resp. v ∈ 𝔙_{∞,2}), S_{v} is a 2n-ary positive definite symmetric (resp. hermitian) form on ℝ^{2n} (resp. ℂ^{2n}).

The set of such n-ary quaternionic Humbert forms is denoted P_{n,D}. To a given S ∈ P_{n,D}, one can assign a value S[u] at u ∈ D^{n} (the precise definition is not straightforward, so we refer to [7] for the details).

Let 𝔒 be a maximal order of D. Assume, to simplify, that h_{D} = 1, and set

m(S) = min{S[u] : u ∈ 𝔒^{n} \ {0}}
γ(S) = \frac{m(S)}{(det S)^{1/n}}
and γ_{n,D} := sup_{S∈P_{n,D}}{γ(S)}.

Then

Theorem 7 (C-W, 2003). μ(ρ, ||.||_{𝔸}) = γ_{n,D}.

Moreover, we get a Voronoï type theorem, with suitable notions of perfection and eutaxy that are too complicated to be detailed here (see [7]).

Example 2. We consider the three euclidean quaternion fields over ℚ, namely, with the standard notation,

D_{2} = (−1, −1)_{ℚ}, D_{3} = (−1, −3)_{ℚ} and D_{5} = (−2, −5)_{ℚ}

(recall that (a, b)_{k} stands for the quaternion algebra over k generated by i and j with i^{2} = a, j^{2} = b and ij = −ji). Then

Proposition 1 (C-W, 2003). For m = 2, 3, 5, one has

(i) γ_{n}(D_{m}) ≤ m^{n(n−1)/2}, (ii) γ_{2}(D_{m}) = m.

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Université Bordeaux I - Institut de Mathématiques 351 cours de la Libération 33405 Talence FRANCE

e-mail: renaud.coulangeon@math.u-bordeaux.fr