数理解析研究所講究録 1200 巻 2001 年 65-70

65

A SURVEY ON GENERALIZED HERMITE CONSTANTS
TAKAO WATANABE (渡部 隆夫) 阪大理

This is an expository note on Hermite’s constant. We give an account of arecent
development of some generalizations of Hermite’s constant.
be the set of all lattices of rank in the
1. Hermite ankin’s constant. Let$d(L)$,
stands for the volume of the fundamental. For
Euclidean spaceparallelepiped of . It was proved by Hermite that$n$$\mathcal{L}^{n}$$L\in \mathcal{L}^{n}$$\mathrm{R}^{n}$$L$$\min_{0\neq x\in L}t_{\mathrm{I}\mathrm{I}}\leq(\frac{2}{\sqrt{3}})^{n-1}d(L)^{2/n}$holds for all
maximum$L\in \mathcal{L}^{n}$. Thus$\min_{0\neq x\in L}{}^{t}xx/d(L)^{2/n}$

is bounded and there exists the$\gamma_{n}=\max\min_{0L\in \mathcal{L}^{n}\neq x\in L}\frac{t_{XX}}{d(L)^{2/n}}$.

is called Hermite’s constant. Awell-known example of its appearance
The constantis the lattice sphere packing problem, namely the density of the densest lattice packing
equalsof spheres in$\gamma_{n}$$\mathbb{R}^{n}$$\delta_{n}=\gamma^{n/2}\frac{V(n)}{2^{n}}$,, i.e., $V(n)=\pi^{n/2}/\Gamma(1+$where $V(n)$ denotes the volume of the unit ball in$n/2)$ . Originally,arose from the reduction theory of positive definite quadraticisforms initiated by Lagrange, Seeber and Gauss. In terms of quadratic forms,
represented as$\mathbb{R}^{n}$$\gamma_{n}$$\gamma_{n}$(1)The exact value of$\gamma_{n}=\max\min_{0g\in GL_{n}(\mathrm{R})\neq x\in \mathrm{Z}^{n}}\frac{\iota_{x^{t}ggx}}{(\det g)^{2/n}}$$\gamma_{n}$.

, i.e.,
is known only for n. One has the estimate$\leq 8$$\gamma_{2}=2/\sqrt{3},\gamma_{3}=\sqrt[3]{2},\gamma_{4}=$$\sqrt{2},\gamma_{5}=\sqrt[6]{8},\gamma_{6}=\sqrt[6]{64}/3,\gamma_{7}=\sqrt[\tau]{64},\gamma_{8}=2$(2)$( \frac{2\zeta(n)}{V(n)})^{2/n}\leq\gamma_{n}\leq 4(\frac{1}{V(n)})^{2/n}$This upper bound was given by Minkowski and follows fromwas first stated by Minkowski and was proved by Hlawka$\delta_{n}\leq 1$. The lower bound66
The next step of Hermite’s constant is the following extension due to Rankin. For
every $1\leq d\leq n-1$ , define(3)$\gamma_{n,d}=\max L\in \mathcal{L}^{n}$Obviously,$\gamma_{n,1}$equals$\gamma_{n}$$x_{1} \Lambda\cdots\wedge x_{d}\neq 0x_{1},\cdots,x_{d}\in L\mathrm{m}\dot{\mathrm{m}}’\frac{\det(^{t}x_{}x_{\dot{f}})_{1\leq\cdot,j\leq d}}{d(L)^{2d/n}}.$. Rankin ([R]) proved.

satisfies the inequality$\gamma_{n,d}$$\gamma_{n,d}\leq\gamma_{m,d}(\gamma_{n,m})^{d/m}$

for $1\leq d<m\leq n-1$ , and he showed $\gamma_{4,2}=3/2$ . Rankin’s inequalty and the duality
yield Mordell’s inequalty.
$\gamma_{n}^{n-2}\leq\gamma_{n-1}^{n-1}$$\gamma_{n,d}=7\mathrm{n},\mathrm{n}-\mathrm{d}$2. Icaza-Thunder’s generalization. As ageneralzation of Hermite’s ankin constant, Thunder defined the constantfor any algebraic number field of finite
degree overin 1997. At first, we recall adefinition of twisted heights. Let,,
be astandard basis of . For any extension fieldover , $W_{n,d}(L)$stands for the -th exterior product of . Abasis of $W_{n,d}(k)$ is formed by the elements:with $I=\{1\leq i_{1}<i_{2}<\cdots<i_{d}\leq n\}$ . For each place of, let
be the completion of at andthe usual normalzed absolute value of. We define the local height onby$\gamma_{n,d}(k)$$r$$\mathrm{e}_{1}$$\cdots$$k$$\mathbb{Q}$$k^{n}$$\mathrm{e}_{n}$

$d$

$L$

$k$

$L^{n}$$\mathrm{e}_{I}=\mathrm{e}:_{1}\wedge\cdots\wedge \mathrm{e}$

$k$

$v$

$k$

$k_{v}$$k_{v}$$v$$|\cdot|_{v}$$W_{n,d}(k_{v})$$H_{v}( \sum_{I}a_{I}\mathrm{e}_{I})=\{\begin{array}{l}(\sum_{t}|a_{t}|_{v}^{[\mathbb{C}k_{v}])}1/([\mathbb{C}..k_{V}]r)(\sup_{t}|a_{I}|_{v})^{1/r}\end{array}$$(\mathrm{i}\mathrm{f}v\mathrm{i}s\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e})(\mathrm{i}\mathrm{f}v\mathrm{i}s\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e})$Then the global height $H$ on $W_{n,d}(k)$ is defined to be the product of$(x\in W_{n,d}(k))$$H(x)= \prod_{v}H_{v}(x)$$H_{v}$:.

Let Abe the adele ring of andthe idele norm on. Since $H(\alpha x)=$
, $H$ defines aheight on the projective space
for
.
$H$
By the Pliicker embedding,is regarded as aheight on the Grassmanianof all -dimensional subspaces of . For, $H(X)$ is precisely given by, where ,,is an arbitrary -basis of $X$ . More generally, foreach $g=(g_{v})$ in, the twisted heighton
is defined as$k$$|\alpha|_{\mathrm{A}}^{1/r}H(x)=H(x)$$|\cdot$$\mathrm{A}^{\mathrm{x}}$$|_{\mathrm{A}}$$\alpha\in k^{\mathrm{x}}$$\mathrm{P}W_{n,d}(k)$$\mathrm{G}\mathrm{r}_{n,d}(k)$$d$$k^{n}$$H(x_{1}\Lambda\cdots\wedge x_{d})$$x_{1}$$\cdots$$X\in \mathrm{G}\mathrm{r}_{n,d}(k)$$k$$x_{d}$$GL_{n}(\mathrm{A})$$H_{g}$$\mathrm{G}\mathrm{r}_{n,d}(k)$$H_{g}(X)= \prod_{v}H_{v}(g_{v}x_{1}\wedge\cdots\Lambda g_{v}x_{d})$Now the constant(4)

In the case of$\gamma_{n,d}(k)$$\gamma_{n,d}(k)$.

is defined to be$= \max\min_{Xg\in GL_{n}(\mathrm{A})\in \mathrm{G}\mathrm{r}_{n,d}(k)}\frac{H_{g}(X)^{2}}{|\det g|_{\mathrm{A}}^{2d/(nr)}}$.

, this definition is identical with (1) and (3), so that one has. As generalizations of Minkowski –Hlawka boimd and Rankin’s
inequality, Thunder showed$k$$\gamma_{n},a(\mathbb{Q})=\gamma_{n,d}$$=\mathbb{Q}$67
Theorem. ([T]) One has(5)$(_{\prod_{j=2}^{Z_{k}(j)}} \frac{n|D_{k}|^{d(n-d)/2}}{{\rm Res}_{s=1}\zeta_{k}(s)})j=n-d+1\leq\gamma nd’ d(k)\leq\prod^{n}Z_{k}(j)2/(nr)(\frac{2^{r_{1}+r_{2}}|D_{k}|^{1/2}}{V(n)^{r_{1}/n}V(2n)^{r_{2}/n}})^{2d/r}$

and$\gamma_{n},a(k)\leq\gamma_{m,d}(k)(\gamma_{n,m}(k))^{d/m}$$(1 \leq d<m\leq n-1)$.

Here $Z_{k}(s)=(\pi^{-s/2}\Gamma(s/2))^{r_{1}}((2\pi)^{1-s}\Gamma(s))^{r_{2}}\zeta_{k}(s)$ denotes the zeta function of ,the discriminant of and(resp.
) the number of real (resp. imaginary)places
$k$

$k$

of .

$r_{1}$$D_{k}$$r_{2}$$k$

We particularly writefor. Newmanand Icaza ([I]) also
consideredbased on Humbert’s reduction theory. Newman gave exact values offor some Eucledean imaginary quadratic fields. To be precise, one has
,
,
,
and
. As forof real quadratic fields, some numerical examples and conjectures were given by Cohn [C]. Recently, Coulangeon proved
apart of Cohn’s conjecture, i.e.,and, by using the Voronoi reduction. In ageneral , Ohno andthe author obtained an upper bound of
better than (5).
$\gamma_{n}(k)$$\gamma_{n,1}(k)$$([\mathrm{N}, \mathrm{X}\mathrm{I}])$$\gamma_{n}(k)$$\gamma_{2}(k)$$\gamma_{2}(\mathbb{Q}(\sqrt{-1}))=\sqrt{2}$$\gamma_{2}(\mathbb{Q}(\sqrt{-2}))=2$$\gamma_{2}(\mathbb{Q}(\sqrt{-11}))=\sqrt{22}/2$$\gamma_{2}(\mathbb{Q}(\sqrt{-3}))=\sqrt{6}/2$$\gamma_{2}(\mathbb{Q}(\sqrt{-7}))=\sqrt{21}/3$$\gamma_{2}(k)$$\gamma_{2}(\mathbb{Q}(\sqrt{2})=2/\sqrt{2\sqrt{6}-3}, \gamma_{2}(\mathbb{Q}(\sqrt{3}))=4$$\gamma_{2}(\mathbb{Q}(\sqrt{5}))=2/\sqrt[4]{5}$$k$$\gamma_{n}(k)$Theorem. ([T]) One has$\gamma_{n}(k)\leq|D_{k}|^{1/r}\frac{\gamma_{nr}(\mathbb{Q})}{r}$.

Combining this with (5), one obtains(6)$\frac{r}{\pi}\{\frac{nw_{k}\Gamma(n/2)^{r_{1}}\Gamma(n)^{r_{2}}\zeta_{k}(n)}{2^{r_{1}+nr_{2}}h_{k}R_{k}}\}^{2/(nr)}\leq\gamma_{nr}(\mathbb{Q})$anddenote the classfor any algebraic number field k of degree r. Here ,number of k, the regulator of k and the number of the roots of unity in k, respectively.
$h_{k}$$R_{k}$$w_{k}$

If asmall n is fixed, there are some numerical examples that (6) for asuitable k is
better than the Minkowski-Hlawka bound of.
$\gamma_{nr}(\mathbb{Q})$68
3. Generalized Hermite constants of flag varieties. Thunder’s definition of
Hermite’s constant can be extended to flag varieties. In order to do this, we use atheory of linear algebraic groups. Let
be aconnected reductive linear algebraic
group defined over and $\pi:Garrow GL(V_{\pi})$ a -rational absolutely irreducible representation. We denote by
the highest weight line inwith respect to afixed
Borel subgroup of . The stabilzer
of
in is aparabolic subgroup of . The
representation is said to be strongly -rational if
is defined over . Then theflag variety
is defined over and is embedded in the projective space. Let
be the adele group of andthe group consisting ofsuch thatfor any -rational character of . For each, atwistedheighton
is defined similarly to \S 2. Then we can prove that the following
exists for any strongly -rational ( , Proposition 2]):
$G$

$k$

$\mathrm{f}\mathrm{c}$$D_{\pi}$$V_{\pi}$$G$$Q_{\pi}$$\pi$$G/Q_{\pi}$

$G$

$k$

$Q_{\pi}$

$k$

$G$

$G(\mathrm{A})$$\mathrm{P}V_{\pi}$$G(\mathrm{A})^{1}$$k$$|\chi(g)|_{\mathrm{A}}=1$$H_{g}$$G$$D_{\pi}$$\mathrm{f}\mathrm{c}$$g\in G(\mathrm{A})$$G$$\chi$$g\in GL(V_{\pi}(\mathrm{A}))$$\mathrm{P}V_{\pi}(k)$$k$$\ovalbox{\tt\small REJECT} \mathrm{u}\mathrm{m}$$\pi$$[\mathrm{W}$,$\gamma_{\pi}^{G}=\mathrm{n}1\alpha \mathrm{m}\dot{\mathrm{m}}H_{\pi(g\gamma)}(D_{\pi})^{2}g\in G(\mathrm{A})^{1}\gamma\in G(k)$where we regard
as a -rational point in. If $G=GL_{n}$ and is a -th exterior
representation of , then one sees
. Amean value argument used toprove Minkowski-Hlawka bound works well in this general setting (cf. [M-W, \S 3.3]).
$k$$D_{\pi}$$\mathrm{P}V_{\pi}$$G$$\pi a$

$\pi$

$d$

$\gamma_{\pi_{d}}^{GL_{n}}=\gamma_{n,d}(k)$Theorem. ([W]) If $Q=Q_{\pi}$ is a maximal parabolic subgroup of , we have a lower
estimate of the form
$G$

(7)

$( \frac{C_{Q}d_{G}e_{Q}\tau(G)}{C_{G}d_{Q}\tau(Q)})^{2e_{\pi}/(e_{Q}r)}\leq\gamma_{\pi}^{G}$

.

Here
and
denote the Tamagawa numbers $ofG$ and , respectively, do,,$eQ$ andare some elementary positive rational numbers depending on $G,Q$ and ,and furthermore
andare the volumes of some maimal compact subgroups of, respectively.
and$\tau(G)$$\tau(Q)$$Q$$d_{Q}$$e_{\pi}$$\pi$$C_{G}$$G(\mathrm{A})$$C_{Q}$$Q(\mathrm{A})$If is split over , both constants
andare described by special values of
the Dedekind zeta function. Particularly, the estimate (7) in the case of $G=GL_{n}$and $\pi=\pi d$ coincides with the lower bound of (5). An upper bound of
is not yet
known in general.
$G$

$k$

$C_{G}$$C_{Q}$$\gamma_{\pi}^{G}$

4. Some examples. We show two examples. First, let $F:k^{n}\cross k^{n}arrow k$ be anondegenerate symmetric bilinear form of Witt index $q\geq 1$ and $G=SO_{F}$ be thespecial orthogonal group of $F$ . For $1\leq d\leq q$ , the -th exterior representation: $G(k)arrow GL(W_{n,d}(k))$ yields astrongly -rational representation of G. (The case$q=n/2=d$ is exceptional since
is not irreducible.) We write
for the generalized Hermite constant. As an analogue of (4),has the following geometricalrepresentation:
$d$

$k$

$\pi_{d}$$\gamma_{d}^{F}$$\pi_{q}$$\gamma_{\pi_{d}}^{G}$$\gamma_{d}^{F}$$\gamma_{d}^{F}=\mathrm{n}1m\dot{\mathrm{m}}\mathrm{n}H_{g}(X)^{2}g\in G(\mathrm{A})X\in \mathrm{G}\mathrm{r}_{n.d}(k,F)$

’

69
where
denotes asubset of
consisting of -dimensional totally
isotropic subspaces ofwith respect to F. In particular, vj is related to an existence of anontrivial small integral solution of the homogeneous quadratic equation
Q. If 2q$F(x,$r)n, (7) givesn or$\ovalbox{\tt\small REJECT}_{\mathrm{n},d}(k_{\mathrm{t}}F)$$d$$\mathrm{G}\mathrm{r}_{n,\ovalbox{\tt\small REJECT} 1}(\mathrm{A})$$k^{n}$$\ovalbox{\tt\small REJECT}$$2q+1\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$(2q+1=n)(2q=n)$$\gamma_{1}^{F}\geq\{\begin{array}{l}(\frac{|D_{k}|^{q-1}(2q-2)}{{\rm Res}_{\epsilon=1}\zeta_{k}(s)}\frac{Z_{k}(2(q-1))Z_{k}(q)}{Z_{k}(q-1)})^{1/((q-1)r)}(\frac{|D_{k}|^{q-1/2}(2q-1)}{{\rm Res}_{\epsilon=1}\zeta_{k}(s)}Z_{k}(2q))^{2/((2q-1)r)}\end{array}$Moreover, we can show the following estimate and an analogue of Rankin’s inequality.

Theorem. ([0-W],[W2]) For any nondegenerate $F$ , one has$(1 \leq d\leq q)$$\gamma_{d}^{F}\leq\gamma_{n-d}(k)^{n-d}(2H(F))^{n-d}$$(1\leq d<m\leq q)$$\gamma_{d}^{F}\leq\gamma_{m,d}(k)(\gamma_{m}^{F})^{d/m}$.

Here $H(F)$ denotes a height of the symmetric matrix corresponding to $F$ .
Second, let $D$ be acentral simple division algebra of dimension

overand $G$ be
an inner -form of $GL_{qn}$ whose group of -rational points equals $GLn(V)$ . If acyclic
extension of degree over contained in $D$ is fixed, then $GLn(V)$ is realized as a
subgroup of $GL_{qn}(L)$ . Since the $qd$-th exterior representation of $GL_{qn}(L)$ gives riseto afundamental -rational representationof $G$ for $1<d\leq n-1$ , one has thegeneralized Hermite constant. We write. Geometrically,forhas the following representation similar to (4):
$k$

$k$

$k$

$L$

$k$

$q$

$k$

$\pi_{d}$$\gamma_{\pi_{d}}^{G}$$\gamma_{n,d}(D)$where$q^{2}$$=$$\gamma_{n,d}(D)$$\gamma_{n,d}(D)$$\gamma_{\pi_{d}}^{\overline{G}}$$\max$$g \in G(\mathrm{A}_{k})X\in \mathrm{B}\mathrm{S}_{n}\min_{d(D)},\frac{H_{g}(X)^{2}}{|\mathrm{N}\mathrm{r}(g)|^{2d/(nr)}}$,denotes the set-dimensional $D$-subspace inand Nr the reduced$M_{n}(D)$norm on. The setis called the generalized Brauer-Severi variety
and is realized as asubset of the Grassmanian. The twisted heighton
is defined as the restriction of that on. By using this expression,
we can prove the following.
$D^{n}$$\mathrm{o}\mathrm{f}d$$\mathrm{B}\mathrm{S}_{n,d}(D)$$\mathrm{B}\mathrm{S}_{n,d}(D)$$\mathrm{G}\mathrm{r}_{qn,qd}(L)$$\mathrm{B}\mathrm{S}_{n,d}(D)$$H_{g}$$\mathrm{G}\mathrm{r}_{qn,qd}(L)$Theorem. ([W3]) One has$\gamma_{n,d}(D)\leq\epsilon_{\mathcal{D}}(\frac{2^{r_{1}(L)+r_{2}(L)}|D_{L}|^{1/2}}{V(qn)^{r_{1}(L)/(qn)}V(2qn)^{r_{2}(L)/(qn)}})^{2d/r}$

and$\gamma_{n,d}(D)\leq\gamma_{m,d}(D)(\gamma_{n,m}(D))^{d/m}$$(1 \leq d<m\leq n-1)$.

70(resp.
denotes the discriminant of L andcg(resp. imaginary)places of L. The constantis given by

Here$D_{L}$$\epsilon_{D}=(\prod_{w}\backslash \mathrm{m}$$r_{\mathrm{i}}(L)$へ。,$|a|_{w}$) the number of real

( $w$ runs over all places of )) $)^{2(q-1)n/(qr)}$

if we realize D as a cyclic algebra$L/k$ and an element a.

$r_{2}^{\ovalbox{\tt\small REJECT}}(L)$$L$a] by a generator$[L/k, \sigma,$$\sigma$of the Galois group of$\in k^{\mathrm{x}}$REFERENCES[C]

H. Cohn, A numerical survey of the floors of various HUbert fundamental domains, Math.
Comp. 21 (1965), 594 -605.

[C-S]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer Verlag,
1999.

[G-L]

P. M. Gruber and C. G. Lekkerkerker, Geometry of Numbers, North-Holland, 1987.

[I]

M. I. Icaza, Hermite constant and extreme forms for algebraic number fields, J. London
Math. Soc. 55 (1997), 11 -22.

[M-W] M. Morishita and T. Watanabe, Adele geometry of numbers, Class Field Theory-Its Centenary and Prospect, Advanced Studies in Pure Math, (to appear).
M. Newman, Integral Matrices, Academic Press, 1972.

[N][O-W] S. Ohno and T. Watanabe, Estimates of Hermite constants for algebraic number fields,(preprint).
[R]

R. A. Rankin, On positive definite quadratic forms, J. London Math. Soc. 28 (1953), 309314.

[T]

J. L. Thunder, Higher-dimensional analogs of Hermite(1998), 301 -314.

[W]

T. Watanabe, On an analog of Hermite’s constant, J. Lie Theory 10 (2000), 33- 52.

[W2]

T. Watanabe, Upper bounds of Hermite constants for orthogonal groups, Comm. Math. Univ.
Sancti Pauli 48 (1999), 25- 33.

[W3]

T. Watanabe, Hermite constants of division algebras, preprint.

constant, Michigan Math. J. 45

DEPARTMENT OF MATHEMATICS, GRADUATE School OF SCIENCE,ONAKA, OSAKA, 560-0043JAPAN-mail address: watanabe&nath.wani.

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