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\topmatter
\title
A survey and a complement of fundamental Hermite constants
\endtitle
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\author
Takao Watanabe 
\endauthor
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\address
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043 Japan
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\email
watanabe\@math.wani.osaka-u.ac.jp
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In this note, we give an account of further development of generalized Hermite constants after [W1]. In [W5], we introduced the fundamental Hermite constant $\gamma(G, Q, k)$ of a pair $(G, Q)$ of a connected reductive group $G$ and a maximal parabolic subgroup $Q$ of $G$ both defined over a global field $k$. Though we use adelic language, the definition of $\gamma(G, Q, k)$ is given as a natural generalization of the definition of the original Hermite constant $\gamma_n$. It was proved in [W5], among other things, that some properties of Hermite--Rankin's constant, e.g., Rankin's inequality,
 can be generalized to fundamental Hermite constants. We will give a survey of these results in the first two sectons of this note. In Section 1, we recall Hermite--Rankin's constant and its generalization due to Thunder [T2]. Section 2 is a summary of our papers [W5] and [W6], in which we define the fundamental Hermite constant $\gamma(G, Q, k)$ and state properties of $\gamma(G, Q, k)$. In Example 1, we show that Thunder's generalization is none other than the fundamental Hermite constant of $GL_n$ defined over an algebraic number field. Section 3 is a complement of properties of fundamental Hermite constants, in which we will study a behavior of fundamental Hermite constants under central $k$-isogenies. 

%
\vskip 8mm
\subhead 1. Hermite's constant and some generalizations \endsubhead 
Let $\cL^n$ be the set of all lattices of rank $n$ in the Euclidean space $\bR^n$. For $L \in \cL^n$, we denote by $d(L)$ the volume of the fundamental parallelepiped of $L$ and by $m_1(L)$ the square of the length of minimal vectors in $L$, i.e., $m_1(L) = \min_{0 \ne x \in L} ||x||^2$. Hermite proved that the inequality 
$$ m_1(L) \le \left ( \frac {2} {\sqrt 3} \right )^{n-1} d(L)^{2/n} $$
holds for all $L \in \cL^n$. This implies $m_1(L)/d(L)^{2/n}$ is bounded on $\cL^n$. As a consequence of the reduction theory, it is known that the function $L \mapsto m_1(L) /d(L)^{2/n}$ defined on $\cL^n$ has the maximum:
 $$ \gamma_n = \max_{L \in \cL^n} \frac {m_1(L)}{d(L)^{2/n}}, $$
which is called Hermite's constant. Hermite's constant is connected with the density $\delta_n$ of the densest lattice packing of spheres in $\bR^n$ as follows:$$ \delta_n = \gamma_n^{n/2} \frac {V(n)} {2^n} \,,$$
where $V(n)$ denotes the volume of the unit ball in $\bR^n$, i.e., $V(n) = \pi^{n/2}/\Gamma(1 + n/2)$. By $\delta_n \le 1$ and the mean value argument of geometry of numbers, one has an estimate of the form
$$ \left (\frac {2\zeta(n)} {V(n)} \right )^{2/n} \le \gamma_n \le 4\left (\frac {1} {V(n)} \right )^{2/n} \,. $$
This upper bound was given by Minkowski. The lower bound was first stated by Minkowski and was proved by Hlawka. Korkine and Zolotareff [K-Z, \S 5, $5^\circ$] proved that $\gamma_n^n$ is a rational number for each $n$.   
\par
The next step of Hermite's constant is the following extension due to Rankin. For $1 \le d \le n-1$, define the lattice invariant $m_d(L)$ by
$$ m_d(L) = \min_{\Sb x_1, \cdots, x_d \in L \\ x_1 \wedge \cdots \wedge x_d \ne 0 \endSb} \det ({}^tx_i x_j)_{1 \le i,j \le d} \,. $$
Then Rankin [R] defined the constant: 
$$ \gamma_{n,d} = \max_{L \in \cL^n} \frac {m_d(L)} {d(L)^{2d/n}} \,, $$
where the maximum of the right-hand side is attained. Obviously, $\gamma_{n,1}$ equals $\gamma_n$. Rankin proved that $\gamma_{n,d}$ satisfies the inequality
$$ \gamma_{n,d} \le \gamma_{m,d}(\gamma_{n,m})^{d/m} $$
for $1 \le d < m \le n-1$, and obtained $\gamma_{4,2} = 3/2$. Rankin's inequality and the duality relation $\gamma_{n, d} = \gamma_{n, n-d}$ yield Mordell's inequality $\gamma_n^{n-2} \le \gamma_{n-1}^{n-1}$ ([Mo]).

\par
As a generalization of Hermite--Rankin's constant, Thunder [T2] defined the constant $\gamma_{n,d}(k)$ for any algebraic number field $k$. We will recall Thunder's definition of $\gamma_{n,d}(k)$ in the next section (see Example 1) and express $\gamma_{n,d}(k)$ in terms of fundamental Hermite constants of $GL_n$ defined over $k$. Thunder proved the following results:
\roster
\item $\gamma_{n,d}(\bQ)$ is equal to Rankin's constant $\gamma_{n,d}$.
\item $\gamma_{n, d}(k) = \gamma_{n, n-d}(k)$ for $1 \le d \le n-1$.
\item $ \gamma_{n,d}(k) \le \gamma_{m,d}(k)(\gamma_{n,m}(k))^{d/m} $ for $1 \le d < m \le n-1$.
\item $\dsize{\left ( \frac {n |D_k|^{d(n-d)/2}} {\Res_{s = 1} \zeta_k(s)} \frac {\dsize\prod_{j=n-d+1}^n Z_k(j)} {\dsize\prod_{j=2}^{d} Z_k(j)} \right )^{2/(n[k:\bQ])} \hskip -3mm \le \gamma_{n,d}(k) \le \left ( \frac {2^{r_1+r_2}|D_k|^{1/2}} {V(n)^{r_1/n}V(2n)^{r_2/n}} \right )^{2d/[k:\bQ]}}.$
\endroster
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 $k$, $D_k$ the discriminant of $k$ and $r_1$ \rom(resp. $r_2$ \rom) the number of real \rom(resp. imaginary \rom) places of $k$. 
\par
We particularly write $\gamma_n(k)$ for $\gamma_{n,1}(k)$. Newman ([N, XI]) and Icaza ([I]) also considered $\gamma_n(k)$ based on Humbert's reduction theory. Tables below show the known explicit values of $\gamma_n(k)$ (cf. [BCIO], [G-L], [N]). 
$$
\gather
\matrix
n & \strut\vrule & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\noalign{\vskip -7mm}
\omit & \vrule height3pt & \omit & \omit & \omit & 
\omit & \omit & \omit & \omit \\
\multispan9\hrulefill \\
\noalign{\vskip -2mm}
\omit & \vrule height8pt & \omit & \omit & \omit & 
\omit & \omit & \omit & \omit \\
\noalign{\vskip -2mm}
\gamma_n & \strut\vrule & 2/\sqrt{3} & \root{3}\of{2} & \sqrt{2} & \root{5}\of{8} & \root{6}\of{64/3} & \root{7}\of{64} & 2 \endmatrix 
\\ \vspace{5mm}
\matrix
d & \strut\vrule & -1 & -2 & -3 & -7 & -11 & 2 & 3 & 5 \\
\noalign{\vskip -7mm}
\omit & \vrule height3pt & \omit & \omit & \omit & 
\omit & \omit & \omit & \omit & \omit \\
\multispan{10}\hrulefill \\
\noalign{\vskip -2mm}
\omit & \vrule height8pt & \omit & \omit & \omit & 
\omit & \omit & \omit & \omit & \omit \\
\noalign{\vskip -2mm}
\gamma_2(\bQ(\sqrt{d})) & \strut\vrule & \sqrt{2} & 2 & \sqrt{6}/2 & \sqrt{21}/3 & \sqrt{22}/2 & 2/\sqrt{2\sqrt{6} - 3} & 2 & 2/\root{4}\of{5} \endmatrix  
\endgather
$$
By using the Voronoi theory, Coulangeon [Co] proved that $\gamma_n(k)$ is an algebraic number for all $n$ if the class number of $k$ is equal to one. 


\vskip 8mm
\subhead 2. Fundamental Hermite constants \endsubhead
Thunder's definition shows that $\gamma_{n,d}(k)$ is a quantity attached to the Grassmann variety of $d$-dimensional subspaces in $k^n$. This suggests that there exists an analogue of Hermite's constant for any generalized flag variety $Q \backslash G$, where $G$ denotes a connected reductive algebraic group defined over $k$ and $Q$ a $k$-parabolic subgroup of $G$. 
We introduced such a constant in terms of a strongly $k$-rational representation $\pi$ of $G$ in [W1]. This constant, say $\gamma_\pi^G$, was named a generalized Hermite constant attached to $\pi$, because $\gamma_{n,d}(k)$ is equal to $\gamma_{\pi_d}^{GL_n}$ of the $d$-th exterior representation $\pi_d$ of $GL_n$. A strongly $k$-rational representation is used for embedding $k$-rationally $Q \backslash G$ into a projective space. We note that there are infinitely many strongly $k$-rational representations of $G$ if $G$ is isotropic. 
In a subsequent paper [W5], we gave a more natural definition of the generalized Hermite constant of $Q \backslash G$ provided that $Q$ is maxmal. This new definition depends only on $G, Q$ and does not need a strongly $k$-rational representaton $\pi$. We write $\gamma(G,Q,k)$, or simply $\gamma_Q$, for this new constant. Two constants $\gamma_\pi^G$ and $\gamma_Q$ have a relation of the form $\gamma_\pi^G = (\gamma_Q)^{c_\pi}$, where $c_\pi$ is a positive rational number depending on $\pi$. In other words, $\gamma_Q$ is considered as an essential part of $\gamma_\pi^G$ in the sense that it is independent of any embedding of $Q \backslash G$ into a projective space. In this section, we first recall the definition of $\gamma_Q$, and then we state some properties of $\gamma_Q$.   
\par
In the following, $k$ denotes a global field, i.e., an algebraic number field or a function field of one variable over a finite field. We fix a connected reductive algebraic group $G$ defined over $k$, a minimal $k$-parabolic subgroup $P$ of $G$ and a maximal standard $k$-parabolic subgroup $Q$ of $G$. By ``standard'', we means $Q$ contains $P$. To define notations, we take a connected $k$-subgroup $R$ of $G$. Let $R(k)$ denote the group of $k$-rational points of $R$, $R(\bA)$ the adele group of $R$ and $\bX^*_k(R)$ the module of $k$-rational characters of $R$. For $a \in R(\bA)$, define the homomorphism $\vartheta_R(a)$ from $\bX^*_k(R)$ into the group $\bR_+$ of positive real numbers by $\vartheta_R(a)(\chi) = |\chi(a)|_\bA$ for $\chi \in \bX^*_k(R)$, where $|\cdot|_\bA$ stands for the idele norm of the idele group of $k$. Then $\vartheta_R$ gives rise to a homomorphism from $R(\bA)$ into $\Hom(\bX^*_k(R), \bR_+)$. The kernel of $\vartheta_R$ is denoted by $R(\bA)^1$. 
If $R$ is a standard $k$-parabolic subgroup, $U_R$ and $M_R$ stand for the unipotent radical and a Levi subgroup of $R$, respectively. If $R$ is a minimal $k$-parabolic subgroup $P$, we can take $M_P$ as the centralizer of a maximal $k$-split torus $S$ of $G$. In general, we take $M_R$ such that $M_P \subset M_R$. The maximal central $k$-split torus of $M_R$ is denoted by $Z_R$. 
We fix a good maximal compact subgroup $K$ of $G(\bA)$. 
\par
We define the height function $H_Q$ on $G(\bA)$. 
Since $Q$ is maximal, $\bX^*_k(M_Q/Z_G)$ is of rank one and has a generator $\what\alpha_Q$ such that $\what\alpha_Q|_S$ is contained in the closed cone generated by the simple roots with respect to $(P,S)$ over $\bR$. 
Define the map $z_Q \: G(\bA) \longrightarrow Z_G(\bA)M_Q(\bA)^1 \backslash M_Q(\bA)$ by $z_Q(g) = Z_G(\bA)M_Q(\bA)^1m$ if $g = umh$, $u \in U_Q(\bA), m \in M_Q(\bA)$ and $h \in K$. This is well defined and a left $Z_G(\bA)Q(\bA)^1$-invariant.  Then the function $H_Q \: G(\bA) \longrightarrow \bR_+$ is defined by $H_Q(g) = |\what\alpha_Q(z_Q(g))|_\bA^{-1}$ for $g \in G(\bA)$.
\par
We set $Y_Q = Q(\bA)^1 \backslash G(\bA)^1$ and $X_Q = Q(k) \backslash G(k)$. Then $X_Q$ is regarded as a subset of $Y_Q$. Since $Z_G(\bA)^1 = Z_G(\bA) \cap G(\bA)^1 \subset M_Q(\bA)^1$, $z_Q$ maps $Y_Q = Q(\bA)^1 \backslash G(\bA)^1$ to $M_Q(\bA)^1 \backslash (M_Q(\bA) \cap G(\bA)^1)$. Namely, we have the following commutative diagram:
$$ \CD
Y_Q @>{z_Q}>> M_Q(\bA)^1 \backslash (M_Q(\bA) \cap G(\bA)^1) \\
@VVV @VVV \\
Z_G(\bA)Q(\bA)^1 \backslash G(\bA) @>{z_Q}>> Z_G(\bA)M_Q(\bA)^1 \backslash M_Q(\bA)
\endCD $$
Since both vertical arrows are injective, $H_Q$ is restricted to $Y_Q$. Let $B_T = \{ y \in Y_Q \: H_Q(y) \le T \}$ for $T > 0$. We can prove the following.
%
\medskip
\proclaim{Proposition} For $T > 0$ and any $g \in G(\bA)^1$, $B_T \cap X_Qg$ is a finite subset of $Y_Q$. Hence, one can define the function
$$ \Gamma_Q(g) = \min \{ T > 0 \: B_T \cap X_Qg \ne \emptyset \} = \min_{y \in X_Q g} H_Q(y) $$
on $G(\bA)^1$. Then the maximum
$$ \gamma(G, Q, k) = \max_{g \in G(\bA)^1} \Gamma_Q(g) $$
exists. 
\endproclaim
\medskip
%
The existence of the maximum is a result of the reduction theory due to Borel--Harich-Chandra and Harder. The constant $\gamma_Q = \gamma(G,Q,k)$ is called the fundamental Hermite constant of $(G, Q)$ over $k$. An interesting thing is a similarity between the definitions of $\gamma_n$ and $\gamma_Q$. Namely, $\gamma_n$ is represented as
$$ \gamma_n = \max_{\Sb g \in GL_n(\bR) \\ |\det g| = 1 \endSb} \min\{T > 0 \: B_T^n \cap g\bZ^n \ne \{0\} \} \,, $$
where $B_T^n$ denotes the ball of radius $T$ with center $0$ in $\bR^n$. On the other hand, by definition,
$$ \gamma_Q = \max_{g \in G(\bA)^1}\min \{ T > 0 \: B_T \cap X_Qg \ne \emptyset \} \,. $$
Thus $X_Q$ plays a role of the lattice $\bZ^n$ and $B_T$ is an analogue of the ball $B_T^n$. In some cases, it is more convenient to consider the constant 
$$ \wtilde\gamma(G, Q, k) = \max_{g \in G(\bA)} \min_{y \in X_Qg} H_Q(g) \,. $$
If $k$ is an algebraic number field, then $\wtilde\gamma(G, Q, k)$ is always equal to $\gamma(G, Q, k)$ as the natural map $Y_Q \longrightarrow Z_G(\bA)Q(\bA)^1 \backslash G(\bA)$ is bijective. The next example shows a relation between $\gamma(GL_n, Q, k)$ and $\gamma_{n,d}(k)$. 
%

\demo{Example 1} Let $V_{n,d}(k) = \bigwedge^d k^n$ be the $d$-th exerior product of $k^n$ and $V_{n,d}(\bA) = V_{n,d}(k) \otimes_k \bA$ the adele space of $V_{n,d}(k)$. We fix a $k$-basis $\be_1, \cdots, \be_n$ of $k^n$, and identify the group of linear automorphisms of $k^n$ with $GL_n(k)$. For $1 \le d \le n-1$, $Q_d(k)$ denotes the stabilizer of the subspace spanned by $\be_1, \cdots, \be_d$ in $GL_n(k)$. A $k$-basis of $V_{n,d}(k)$ is formed by the elements $\be_I = \be_{i_1} \wedge \cdots \wedge \be_{i_d}$ with $I = \{ 1 \le i_1 < i_2 < \cdots < i_d \le n \}$. Let $k_v$ be the completion field of $k$ at a place $v$ and $|\cdot|_v$ the usual normalized absolute value of $k_v$. The local height $H_v \: V_{n,d}(k) \otimes_k k_v \longrightarrow \bR_+$ is defined by
$$ H_v(\sum_{I} a_I \be_I) = \cases \dsize{\left(\sum_{I} |a_I|_v^{2/[k_v:\bR]} \right)^{[k_v:\bR]/2}} & \;\; (\text{$v$ is infinite}) \\ \dsize\sup_I (|a_I|_v)  & \;\; (\text{$v$ is finite}) 
\endcases $$
Note that $|a|_v = a \overline a$ if $k_v = \bC$. Then the global height $H_{n,d} \: V_{n,d}(k) \longrightarrow \bR_+$ is defined to be the product of all $H_v$, i.e.,
$$ H_{n,d}(x) = \prod_{v} H_{v}(x) $$ 
for $x \in V_{n,d}(k)$. This is immediately extended to the subset $GL(V_{n,d}(\bA))V_{n,d}(k)$ of the adele space $V_{n,d}(\bA)$ by 
$$ H_{n,d}(Ax) = \prod_{v} H_v(A_vx) $$
for $A = (A_v)_v \in GL(V_{n,d}(\bA))$ and $x \in V_{n,d}(k)$. Especially, for $g \in GL_n(\bA)$ and $x_1, \cdots, x_d \in k^n$, the height $H_{n,d}(gx_1 \wedge \cdots \wedge gx_d)$ of $gx_1 \wedge \cdots \wedge gx_d$ is defined. Then there exists the following maximum: 
$$ \what\gamma_{n,d}(k) = \max_{g \in GL_n(\bA)} \min_{\Sb x_1, \cdots, x_d \in k^n \\ x_1 \wedge \cdots \wedge x_d \ne 0 \endSb} \frac {H_{n,d}(gx_1 \wedge \cdots \wedge gx_d)} {|\det g|_\bA^{d/n}}. $$
In the case that $k$ is an algebraic number field, Thunder's $\gamma_{n,d}(k)$ is defined by
$$ \gamma_{n,d}(k) = \what\gamma_{n,d}(k)^{2/[k:\bQ]} $$
It is immediate to see that 
$$ \frac {H_{n,d}(g^{-1}\be_1 \wedge \cdots \wedge g^{-1}\be_d)} {|\det g^{-1}|_\bA^{d/n}} = H_{Q_d}(g)^{\gcd(d,n-d)/n} $$
for $g \in GL_n(\bA)$, and hence
$$ \what\gamma_{n,d}(k) = \widetilde \gamma(GL_n, Q_d, k)^{\gcd(d,n-d)/n} \,. $$
In general, $Z_{GL_n}(\bA)GL_n(\bA)^1$ is an index finite normal subgroup of $GL_n(\bA)$, but it is not necessarily equal to $GL_n(\bA)$ if $k$ is a function field. Let $\Xi$ be a complete set of representatives for the cosets of $Z_{GL_n}(\bA)GL_n(\bA)^1 \backslash GL_n(\bA)$. If we put 
$$ \align
\what\gamma_{n,j}(k)_\xi & = \max_{g \in Z_{GL_n}(\bA)GL_n(\bA)^1\xi} \min_{\Sb x_1, \cdots, x_d \in k^n \\ x_1 \wedge \cdots \wedge x_d \ne 0 \endSb} \frac {H_{n,d}(gx_1 \wedge \cdots \wedge gx_d)} {|\det g|_\bA^{d/n}} \\
& = \frac {1} {|\det \xi|_\bA^{d/n}} \max_{g \in GL_n(\bA)^1\xi} \min_{\Sb x_1, \cdots, x_d \in k^n \\ x_1 \wedge \cdots \wedge x_d \ne 0 \endSb} H_{n,d}(gx_1 \wedge \cdots \wedge gx_d) 
\endalign $$
for $\xi \in \Xi$, then 
$$ \what\gamma_{n,d}(k) = \max_{\xi \in \Xi} \gamma_{n,d}(k)_\xi , $$ 
and in particular, for the unit element $\xi = 1$,  
$$ \what\gamma_{n,d}(k)_1 = \gamma(GL_n, Q_d, k)^{\gcd(d,n-d)/n} . $$
If $k$ is a number field, $Z_{GL_n}(\bA)GL_n(\bA)^1 = GL_n(\bA)$ holds, and hence one has
$$ \wtilde\gamma(GL_n, Q_d, k) = \gamma(GL_n, Q_d, k) = \gamma_{n,d}(k)^{n[k:\bQ]/(2\gcd(d,n-d))}. $$
\enddemo 
%
We summarize the properties of $\gamma(G, Q, k)$. 
%
\medskip
\proclaim{Theorem 1} Assume the exact sequence  
$$ \CD 1 @>>> Z @>>> G @>{\beta}>> G^\prime @>>> 1 \endCD $$
of connected reductive groups defined over $k$ satisfies the following two conditions: 
\par
$\bullet$ $Z$ is central in $G$. 
\par
$\bullet$ $Z$ is isomorphic to a product of tori of the form $R_{k^\prime/k}(GL_1)$, where each $k^\prime/k$ is a finite separable extension and $R_{k^\prime/k}$ denotes the functor of restriction of scalars from $k^\prime$ to $k$. 
\par
Then $\gamma(G, Q, k)$ is equal to $\gamma(G^\prime, \beta(Q), k)$. 
\endproclaim
%
\medskip   
\proclaim{Theorem 2} If $k/\ell$ is a finite separable extension, then $\gamma(R_{k/\ell}(G), R_{k/\ell}(Q), \ell)$ is equal to $\gamma(G, Q, k)$. 
\endproclaim
%
\medskip
\proclaim{Theorem 3} 
Let $R$ and $Q$ be two different maximal standard $k$-parabolic subgroups of $G$, $Q^R = M_R \cap Q$ a maximal standard parabolic subgroup of $M_R$ and $M^R_Q = M_R \cap M_Q$ a Levi subgroup of $Q^R$. We write  $\what \alpha^R_Q$ for the $\bZ$-basis $\what \alpha_{Q^R}$ of $\bX^*_k(M^R_Q/Z_R)$. Then $\bQ$-vector space $\bX^*_k(M^R_Q/Z_G) \otimes_\bZ \bQ$ is spanned by $\what \alpha^R_Q$ and $\what \alpha_R|_{M^R_Q}$. If we take $\omega_1, \omega_2 \in \bQ$ such that 
$$ \what \alpha_Q|_{M^R_Q} = \omega_1 \what \alpha^R_Q + \omega_2 \what \alpha_R|_{M^R_Q} \,, $$
then one has an inequality of the form
$$ \gamma(G, Q, k) \le \widetilde \gamma(M_R, Q^R, k)^{\omega_1} \gamma(G, R, k)^{\omega_2} \,.$$
\endproclaim
%
\par
\demo{Example 2} We illustrate that Theorem 1 and Theorem 3 are generalizations of the duality relation (2) and Rankin's inequality (3) in \S 1, respectively. We use the same notations as in Example 1. First, we consider the automorphism $\beta \: GL_n \longrightarrow GL_n$ defined by $\beta(g) = w_0 ({}^tg^{-1})w_0^{-1}$, where
$$ w_0 = \pmatrix 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & \cdots & 1 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 1 & \cdots & 0 & 0 \\ 1 & 0 & \cdots & 0 & 0 \endpmatrix \in GL_n(k). $$
Since $\beta(Q_d) = Q_{n-d}$, Theorem 1 deduces
$$ \gamma(GL_n, Q_d, k) = \gamma(GL_n, Q_{n-d}, k). $$
If $k$ is a number field, this implies the duality relation (2). Next, for $i, j \in \bZ$ with $1 \le i < j \le n-1$, we take two maximal standard $k$-parabolic subgroups $R = Q_j$ and $Q = Q_i$ of $GL_n$. Then, $M_R = GL_j \times GL_{n-j}$, $M_Q = GL_i \times GL_{n-i}$ and $M^R_Q = GL_i \times GL_{j-i} \times GL_{n-j}$. It is easy to see 
$$ \omega_1 = \frac {n} {j} \frac {\gcd(i,j-i)} {\gcd(i,n-i)} \,, \qquad \omega_2 = \frac {i} {j} \frac {\gcd(j,n-j)} {\gcd(i, n-i)}\,. $$
Theorem 3 deduces
$$ \gamma(GL_n, Q_i, k) \le \widetilde\gamma(M_{Q_j}, Q_i^{Q_j}, k)^{\frac {n} {j} \frac {\gcd(i,j-i)} {\gcd(i,n-i)}} \gamma(GL_n, Q_j, k)^{\frac {i} {j} \frac {\gcd(j,n-j)} {\gcd(i, n-i)}} \,. $$
If $k$ is a number field, this and Example 1 imply Rankin's inequality (3).
\enddemo
%
Let $\tau(G)$ (resp. $\tau(Q)$) be the Tamagawa number of $G$ (resp. $Q$) and $\omega^G_\bA$ (resp. $\omega^{U_Q}_{\bA}$ and $\omega^{M_Q}_{\bA}$) the Tamagawa measure of $G(\bA)$ (resp. $U_Q(\bA)$ and $M_Q(\bA)$). The modular character $\delta_Q^{-1}$ of $Q(\bA)$ is defined by the relation $d\omega^{U_Q}_\bA(m^{-1}um) = \delta_Q(m)^{-1} d\omega^{U_Q}_\bA(u)$ for $u \in U_Q(\bA)$ and $m \in M_Q(\bA)$. We define constants $\what e_Q$ and $C_{G,Q}$ as follows:
\newline
$\bullet$ $\delta_Q(m)  = |\what\alpha_Q(m)|_\bA^{\what e_Q}$ for all $m \in M(\bA)$.
\newline
$\bullet$ $d\omega^G_\bA(g) = C_{G,Q}^{-1} \delta_Q(m)^{-1} d\omega^{U_Q}_{\bA}(u) d\omega^{M_Q}_{\bA}(m) d\nu_K(h) $ for all $g = umh$, $u \in U_Q(\bA)$, $m \in M_Q(\bA)$ and $h \in K$. 
\newline
Here $\nu_K$ denotes the Haar measure of $K$ normalized so that $\nu_K(K) = 1$.  By an argument of the mean value theorem, we can show the following theorem.
%
\medskip
\proclaim{Theorem 4}
One has an estimate of the form
$$ \left ( C_{G,Q} \cdot D_{G,Q} \cdot  E_Q \cdot \frac {\tau(G)} {\tau(Q)} \right )^{1/\what{e_Q}} \le \gamma(G, Q, k) \,, $$
where $D_{G,Q}$ and $E_Q$ are given as follows:
$$ \align
D_{G,Q} & = \cases \dsize\frac{[\bX^*_k(Z_G) : \bX^*_k(G)]} {[\bX^*_k(Z_Q) : \bX^*_k(M_Q)]} & \quad (\ch(k) = 0), \\ \dsize\frac{(\log q)^{\rank \bX^*_k(G)}[\Hom(\bX^*_k(G), q^\bZ) : \im\vartheta_G]} {(\log q)^{\rank \bX^*_k(M_Q)}[\Hom(\bX^*_k(M_Q), q^\bZ) : \im\vartheta_{M_Q}]} & \quad (\ch(k) > 0), \endcases \\
E_Q & = \cases \what e_Q [\bX^*_k(Z_Q/Z_G) : \bX^*_k(M_Q/Z_G)] & \quad (\ch(k) = 0), \\ (1 - q_0^{-\what e_Q}) & \quad (\ch(k) > 0). \endcases \\
\endalign $$
Here, if $\ch(k) > 0$, then $q$ denotes the cardinality of the constant field of $k$ and $q_0 > 1$ the generator of the subgroup $|\what\alpha_Q(M_Q(\bA) \cap G(\bA)^1)|_\bA$ of the cyclic group $q^\bZ$ generated by $q$. Moreover, this inequality is strict if $\ch(k) > 0$.  
\endproclaim
\medskip
%
We note that $\gamma(G, Q, k) \in q_0^\bZ$ if $\ch(k) > 0$.
%
\medskip
\demo{Example 3} 
Let $G = GL_n$ and $Q = Q_d$. If $\ch(k) = 0$, Theorem 4 is essentially the same as the lower bound of (4) in \S 1. If $\ch(k) > 0$, we obtain $q_0 = q^{n/\gcd(d,n-d)}$ and 
$$ \left ( \frac {q^{(g(k) - 1)(d(n-d) + 1)}(q - 1)(1 - q^{-n})} {h_k} \frac {\dsize\prod_{i=n-d+1}^n \zeta_k(i)} {\dsize\prod_{i=2}^d \zeta_k(i)} \right )^{1/\gcd(d,n-d)}  \hskip -5mm < \gamma(GL_n, Q_d, k) \,, 
$$
where $g(k)$ denotes the genus of $k$, $h_k$ the divisor class number of $k$ and $\zeta_k(s)$ the congruence zeta function of $k$. On the other hand, from the definition of $\what\gamma_{n,d}(k)$ and Thunder's theorem on an analogue of Minkowski's second convex bodies theorem ([T1]), it follows that 
$$ 1 \le \gamma(GL_n, Q_d, k) \le \wtilde\gamma(GL_n, Q_d, k) \le  q^{ndg(k)/\gcd(d,n-d)} = q_0^{dg(k)} \,. $$
If $g(k) = 0$, i.e., $k$ is a rational function field over $\bF_q$, this implies $\gamma(GL_n, Q_d, k) = \wtilde\gamma(GL_n, Q_d, k) = 1$. If $g(k) = 1$ and $d = 1$, the first inequality and the upper bound of the second inequality give
$$ q^{n-1} \cdot \frac{(q - 1)(q^{2n} + a_1q^n + q)} {(q + a_1 + 1)(q^{2n} - q^{n+1})} < \gamma(GL_n, Q_1, k) \le \wtilde\gamma(GL_n, Q_1, k) \le q^n \,,$$
where $h_k = a_1 + q + 1$. Combining this with the Hasse--Weil bound $|a_1| \le 2\sqrt{q}$, we have $\gamma(GL_n, Q_1, k) = \wtilde\gamma(GL_n, Q_1, k) = q^n$ provided that $h_k \le q - 1$.  
\enddemo
\medskip
%
Except for the case where $G$ is either an inner form of a general linear group or an orthogonal group defined over an algebraic number field ([W2], [W3]), we have no any result on an upper bound of $\gamma(G,Q,k)$. 
\par
Theorems 1 -- 4 and Example 3 were proved in [W5]. Furthermore, we can add a small result on $\wtilde\gamma(GL_n, Q_1, k)$.  
%
\medskip
\proclaim{Theorem 5}
We define the constant $\Delta_k$ as follows:
$$ \Delta_k = \cases |D_k| & \;\; (\text{$k$ is an algebraic number field of absolute discriminant $D_k$}). \\ q^{2g(k) - 2} & \;\; (\text{$k$ is a function field of genus $g(k)$ and constant field $\bF_q$}). \endcases $$
If $\ell$ is a separable extension of $k$ with degree $r$, then
$$ \frac {\wtilde\gamma(GL_n, Q_1, \ell)} {\Delta_\ell^{n/2}} \le r^{-nrs_k/2} \cdot \frac {\wtilde\gamma(GL_{nr}, Q_1, k)} {\Delta_k^{nr/2}} \,, $$
where $s_k$ denotes the number of infinite places of $k$. 
\endproclaim
\medskip
%
This theorem was first proved in [O-W] in the case of $k = \bQ$. See [W6] for a genral case. 
%
\medskip
\demo{Remark}
By definition, $\gamma(G, Q, k)$ measures the existence of rational points in $B_T$. Namely, if $T \ge \gamma(G, Q, k)$, then $B_T \cap X_Q g \ne \emptyset$ for any $g \in G(\bA)^1$, especially $g = e$. If $k$ is an algebraic number field, the cardinality of $B_T \cap X_Qg$ is increasing in proportion to the volume of $B_T$ as $T \to \infty$. More precisely, one has 
$$ \lim_{T \to \infty} \frac {\sharp(B_T \cap X_Qg)} {\omega_Y(B_T)} = \frac {\tau(Q)} {\tau(G)} $$
for all $g \in G(\bA)^1$, where $\omega_Y = \omega_{\bA}^Q \backslash \omega_{\bA}^G$ is the Tamagawa measure on $Y$. The volume $\omega_Y(B_T)$ is equal to $(C_{G,Q}D_{G,Q}E_Q)^{-1}T^{\what e_Q}$. See [W4] for details. 
\enddemo
%
%
\vskip 8mm
\subhead 3. Behavior of fundamental Hermite constants under isogenies \endsubhead
Theorem 1 asserts that the fundamental Hermite constants is invariant under some kind of central extensions. It is natural to ask how the fundamental Hermite constatns behaves under central isogenies. We show the following.
%
\medskip
\proclaim{Theorem 6}
Let
$$ \CD
1 @>>> F @>>> \what G @>{\beta}>> G @>>> 1
\endCD
$$
be a separable central $k$-isogeny of a connected reductive $k$-group $G$ and $Q$ a maximal $k$-parabolic subgroup of $G$. Then 
$$ \gamma(\what G, \what Q, k)^{d_\beta} \le \gamma(G, Q, k) \,, $$
where $\what Q = \beta^{-1}(Q)$ and $d_\beta = [\bX^*_k(M_{\what Q}/Z_{\what G}) : \bX^*_k(M_Q/Z_G)]$.
\endproclaim 
\smallskip
\demo{Proof}
We note that $X_{\what Q} = \what Q(k) \backslash \what G(k)$ is isomorphic with $X_Q = Q(k) \backslash G(k)$ as $F$ is central in $G$. Since $\bX^*_k(F)$ is a torsion group, $\beta$ gives rise to an isomorphism between $\Hom(\bX^*_k(\what G), \bR_+)$ and $\Hom(\bX^*_k(G), \bR_+)$. This implies that $\beta(\what G(\bA)^1)$ is contained in $G(\bA)^1$. By the similar reason, $\beta(\what Q(\bA)^1)$ is contained in $Q(\bA)^1$. Therefore, one has the following commutative diagram:
$$ \CD
X_{\what Q} @>{\beta}>{\cong}> X_Q \\
@VVV @VVV \\
Y_{\what Q} @>{\beta}>> Y_Q
\endCD
$$
The central isogeny
$$ \CD
1 @>>> F/F \cap Z_{\what G} @>>> M_{\what Q}/Z_{\what G} @>{\beta}>> M_Q/Z_G @>>>1
\endCD $$
yields the exact sequence
$$ \CD
1 @>>> \bX^*_k(M_Q/Z_G) @>{\beta^*}>> \bX^*_k(M_{\what Q}/Z_{\what G}) @>>> \bX^*_k(F/F\cap Z_{\what G}).
\endCD $$
Since $\bX^*_k(M_Q/Z_G) = \bZ \what\alpha_Q$ and $\bX^*_k(M_{\what Q}/Z_{\what G}) = \bZ \what \alpha_{\what Q}$ by definition, $\beta^*(\what \alpha_Q)$ is equal to $d_\beta \what\alpha_{\what Q}$. Moreover, by definition of the map $z_Q$, it is easy to see that the following diagram is commutative:
$$ \CD
\what G(\bA) @>{z_{\what Q}}>> Z_{\what G}(\bA)M_{\what Q}(\bA)^1 \backslash M_{\what Q}(\bA) \\
@V{\beta}VV @VV{\beta}V \\
G(\bA) @>{z_Q}>> Z_G(\bA)M_Q(\bA)^1 \backslash M_Q(\bA)
\endCD $$
Then, for any $g \in \what G(\bA)$, one has
$$ H_Q(\beta(g)) = |\what \alpha_Q(z_Q(\beta(g)))|_\bA^{-1} = |\what \alpha_Q(\beta(z_{\what Q}(g)))|_\bA^{-1} = |\what \alpha_{\what Q}(z_{\what Q}(g))|_\bA^{-d_\beta} = H_{\what Q}(g)^{d_\beta}. $$
From $\beta(\what G(\bA)^1) \subset G(\bA)^1$ and $X_{\what Q} \cong X_Q$, it follows that 
$$ \max_{g \in G(\bA)^1} \min_{x \in X_Q} H_Q(xg) 
\ge \max_{g \in \what G(\bA)^1} \min_{x \in X_{\what Q}} H_Q(\beta(xg)) 
= \max_{g \in \what G(\bA)^1} \min_{x \in X_{\what Q}} H_{\what Q}(xg)^{d_\beta}. $$
This leads us to $\gamma(G, Q, k) \ge \gamma(\what G, \what Q, k)^{d_\beta}$. \qed
\enddemo
\medskip
%
If $k$ is an algebraic number field, we have a more precise result. In the following, we assume $G$ is an almost simple isotropic group and 
$$ \CD
1 @>>> F @>>> \wtilde G @>{\beta}>> G @>>> 1
\endCD
$$
is the simply connected covering of $G$ defined over an algebraic number field $k$. We put
$$ G(\bA_\infty) = \prod_{\Sb w \\ \text{infinite} \endSb} G(k_v) \times \prod_{\Sb v \\ \text{finite} \endSb} K_v \,, $$
where $K_v$ denotes the $v$-component of the fixed good maximal compact subgroup $K$ of $G(\bA)$. It is known that $G(k)G(\bA_\infty)$ is a normal subgroup of $G(\bA)$ and $G(k)G(\bA_\infty) \backslash G(\bA) = G(k) \backslash G(\bA) / G(\bA_\infty)$ is a finite set ([P-R, Proposition 8.8]). Let $\Xi_G$ be a complete set of representatives of $G(k)G(\bA_\infty) \backslash G(\bA)$. For each $\xi \in \Xi_G$, we set
$$ \gamma(G, Q, k)_\xi = \max_{g \in G(k)G(\bA_\infty)\xi} \min_{x \in X_Qg} H_Q(x) \,, $$
and especially 
$$ \gamma(G, Q, k)_{\roman{pr}} = \max_{g \in G(k)G(\bA_\infty)} \min_{x \in X_Qg} H_Q(x) \,. $$
From $G(\bA) = G(\bA)^1$, it follows that 
$$ \gamma(G, Q, k) = \max_{\xi \in \Xi_G} \gamma(G, Q, k)_\xi \,. $$
%
\medskip
\proclaim{Theorem 7}
Being the notations and assumptions as above, we have
$$ \gamma(\wtilde G, \wtilde Q, k)^{d_\beta} = \gamma(G, Q, k)_{\roman{pr}} \,, $$
where $\wtilde Q = \beta^{-1}(Q)$ and $d_\beta = [\bX^*_k(M_{\wtilde Q}) : \bX^*_k(M_Q)]$.
\endproclaim
\smallskip
\demo{Proof}
Since $\Gamma_Q(g) = \min_{x \in X_Q} H_Q(xg)$ is left $G(k)$-invariant and right $K$-invariant, it is sufficient to prove that $G(k)\beta(\wtilde G(\bA))K = G(k)G(\bA_\infty)$. By the proof of [P-R, Proposition 8.8], one has $[G(\bA) : G(\bA)] \subset \beta(\wtilde G(\bA)) \subset G(k)G(\bA_\infty)$. This implies $G(k)\beta(\wtilde G(\bA))K \subset G(k)G(\bA_\infty)$. To prove the converse, we must show $G(k_w) \subset \beta(\wtilde G(k_w))K_w$ for any infinite place $w$. This is obvious if $k_w = \bC$. Thus we assume $K_w = \bR$. Since $\wtilde G(\bR)$ is connected as a real Lie group ([B-T, (4.7)]), $\beta(\wtilde G(\bR))$ coincides with the topologically connected component $G(\bR)^0$ of $G(\bR)$. Then $G(\bR) = G(\bR)^0K_w$ follows from the maximality of $K_w$ (cf. [P-R, Proposition 3.10]). \qed
\enddemo
\medskip
%
In general, if $\beta_1 \: G_1 \longrightarrow G$ is a central finite covering of $G$, we have $G(k)G(\bA_\infty) \subset G(k) \beta(G_1(\bA))K \subset G(\bA)$. Therefore, there exists a subset $\Xi_{G, G_1}$ of $\Xi_G$ such that 
$$ \gamma(G_1, Q_1, k)^{d_{\beta_1}} = \max_{\xi \in \Xi_{G, G_1}} \gamma(G, Q, k)_\xi \,, $$
where $Q_1 = \beta_1^{-1}(Q)$ and $d_{\beta_1} = [\bX^*_k(M_{Q_1}) : \bX^*_k(M_Q)]$. 
\par
As a corollary of Theorems 1 and 7, we obtain the following.
%
\medskip
\proclaim{Corollary}
If $k$ is an algebraic number field and $Q_d$ the maximal parabolic subgroup of $GL_n$ given as in Example 1, then
$$ \gamma(SL_n, Q_d \cap SL_n, k)^{n/\gcd(d, n-d)} =  \gamma(PGL_n, Z_{GL_n} \backslash Q_d, k)_{\roman{pr}} $$
In particular, if the ideal class group $I_k = k^\times \bA_\infty^\times \backslash \bA^\times$ of $k$ satisfies $I_k = I_k^n$, then
$$ \gamma(SL_n, Q_d \cap SL_n, k)^{n/\gcd(d, n-d)}  = \gamma(PGL_n, Z_{GL_n} \backslash Q_d, k)= \gamma(GL_n, Q_d, k). $$
\endproclaim


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\enddocument

\end