A SURVEY AND A COMPLEMENT OF

FUNDAMENTAL HERMITE CONSTANTS

Takao Watanabe

In this note, we give an account of further development of generalized Hermite constants after [W1]. In [W5], we introduced the fundamental Hermite constant γ(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 γ(G, Q, k) is given as a natural generalization of the definition of the original Hermite constant γ_{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 γ(G, Q, k) and state properties of γ(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.

1. Hermite’s constant and some generalizations. Let L^{n} be the set of all lattices of rank n in the Euclidean space ℝ^{n}. For L ∈ L^{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≠x∈L} ||x||^{2}. Hermite proved that the inequality

m_{1}(L) ≤ (\frac{2}{\sqrt{3}})^{n−1} d(L)^{2/n}

holds for all L ∈ L^{n}. This implies m_{1}(L)/d(L)^{2/n} is bounded on L^{n}. As a consequence of the reduction theory, it is known that the function L ↦ m_{1}(L)/d(L)^{2/n} defined on L^{n} has the maximum:

γ_{n} = max_{L∈L^{n}} \frac{m_{1}(L)}{d(L)^{2/n}} ,

which is called Hermite’s constant. Hermite’s constant is connected with the density δ_{n} of the densest lattice packing of spheres in ℝ^{n} as follows:

δ_{n} = γ_{n}^{n/2} \frac{V (n)}{2^{n}} ,

1 where V (n) denotes the volume of the unit ball in ℝ^{n}, i.e., V (n) = π^{n/2}/Γ(1 + n/2). By δ_{n} ≤ 1 and the mean value argument of geometry of numbers, one has an estimate of the form

(\frac{2ζ(n)}{V (n)})^{2/n} ≤ γ_{n} ≤ 4 (\frac{1}{V (n)})^{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, §5, 5^{∘}] proved that γ_{n}^{n} is a rational number for each n.

The next step of Hermite’s constant is the following extension due to Rankin. For 1 ≤ d ≤ n − 1, define the lattice invariant m_{d}(L) by

m_{d}(L) = min_{\begin{matrix}x_{1},··· ,x_{d}∈L \\ x_{1}∧···∧x_{d}≠0 \end{matrix}} det(^{t}x_{i}x_{j})_{1≤i,j≤d} .

Then Rankin [R] defined the constant:

γ_{n,d} = max_{L∈L^{n}} \frac{m_{d}(L)}{d(L)^{2d/n}} ,

where the maximum of the right-hand side is attained. Obviously, γ_{n,1} equals γ_{n}. Rankin proved that γ_{n,d} satisfies the inequality

γ_{n,d} ≤ γ_{m,d}(γ_{n,m})^{d/m}

for 1 ≤ d < m ≤ n − 1, and obtained γ_{4,2} = 3/2. Rankin’s inequality and the duality relation γ_{n,d} = γ_{n,n−d} yield Mordell’s inequality γ_{n}^{n−2} ≤ γ_{n−1}^{n−1} ([Mo]).

As a generalization of Hermite–Rankin’s constant, Thunder [T2] defined the constant γ_{n,d}(k) for any algebraic number field k. We will recall Thunder’s definition of γ_{n,d}(k) in the next section (see Example 1) and express γ_{n,d}(k) in terms of fundamental Hermite constants of GL_{n} defined over k. Thunder proved the following results:

(1) γ_{n,d}(ℚ) is equal to Rankin’s constant γ_{n,d}.

(2) γ_{n,d}(k) = γ_{n,n−d}(k) for 1 ≤ d ≤ n − 1.

(3) γ_{n,d}(k) ≤ γ_{m,d}(k)(γ_{n,m}(k))^{d/m} for 1 ≤ d < m ≤ n − 1.

(4) (\frac{n|D_{k}|^{d(n−d)/2}}{Res_{s=1}ζ_{k}(s)} \frac{∏_{j=n−d+1}^{n} Z_{k}(j)}{∏_{j=2}^{d} Z_{k}(j)})^{2/(n[k:ℚ])} ≤ γ_{n,d}(k) ≤ (\frac{2^{r_{1}+r_{2}}|D_{k}|^{1/2}}{V (n)^{r_{1}/n}V (2n)^{r_{2}/n}} )^{2d/[k:ℚ]}.

Here Z_{k}(s) = (π^{−s/2}Γ(s/2))^{r_{1}}((2π)^{1−s}Γ(s))^{r_{2}}ζ_{k}(s) denotes the zeta function of k, D_{k} the discriminant of k and r_{1} (resp. r_{2} ) the number of real (resp. imaginary ) places of k.

We particularly write γ_{n}(k) for γ_{n,1}(k). Newman ([N, XI]) and Icaza ([I]) also considered γ_{n}(k) based on Humbert’s reduction theory. Tables below show the known explicit values of γ_{n}(k) (cf. [BCIO], [G-L], [N]).

\begin{matrix}n & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ γ_{n} & 2/\sqrt{3} & \sqrt[3]{2} & \sqrt{2} & \sqrt[5]{8} & \sqrt[6]{64/3} & \sqrt[7]{64} & 2 \end{matrix}

\begin{matrix}d & −1 & −2 & −3 & −7 & −11 & 2 & 3 & 5 \\ γ_{2}(ℚ(\sqrt{d})) & \sqrt{2} & 2 & \sqrt{6}/2 & \sqrt{21}/3 & \sqrt{22}/2 & 2/\sqrt{2\sqrt{6} − 3} & 2 & 2/\sqrt[4]{5} \end{matrix}

By using the Voronoi theory, Coulangeon [Co] proved that γ_{n}(k) is an algebraic number for all n if the class number of k is equal to one.

2. Fundamental Hermite constants. Thunder’s definition shows that γ_{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\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 π of G in [W1]. This constant, say γ_{π}^{G}, was named a generalized Hermite constant attached to π, because γ_{n,d}(k) is equal to γ_{π_{d}}^{GL_{n}} of the d-th exterior representation π_{d} of GL_{n}. A strongly k-rational representation is used for embedding k-rationally Q\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\G provided that Q is maxmal. This new definition depends only on G, Q and does not need a strongly k-rational representaton π. We write γ(G, Q, k), or simply γ_{Q}, for this new constant. Two constants γ_{π}^{G} and γ_{Q} have a relation of the form γ_{π}^{G} = (γ_{Q})^{c_{π}}, where c_{π} is a positive rational number depending on π. In other words, γ_{Q} is considered as an essential part of γ_{π}^{G} in the sense that it is independent of any embedding of Q\G into a projective space. In this section, we first recall the definition of γ_{Q}, and then we state some properties of γ_{Q}.

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(𝔸) the adele group of R and X_{k}^{∗}(R) the module of k-rational characters of R. For a ∈ R(𝔸), define the homomorphism ϑ_{R}(a) from X_{k}^{∗}(R) into the group ℝ_{+} of positive real numbers by ϑ_{R}(a)(χ) = |χ(a)|_{𝔸} for χ ∈ X_{k}^{∗}(R), where | · |_{𝔸} stands for the idele norm of the idele group of k. Then ϑ_{R} gives rise to a homomorphism from R(𝔸) into Hom(X_{k}^{∗}(R), ℝ_{+}). The kernel of ϑ_{R} is denoted by R(𝔸)^{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} ⊂ 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(𝔸).

We define the height function H_{Q} on G(𝔸). Since Q is maximal, X_{k}^{∗}(M_{Q}/Z_{G}) is of rank one and has a generator α̂_{Q} such that α̂_{Q}|_{S} is contained in the closed cone generated by the simple roots with respect to (P, S) over ℝ. Define the map z_{Q}: G(𝔸) ⟶ Z_{G}(𝔸)M_{Q}(𝔸)^{1}\M_{Q}(𝔸) by z_{Q}(g) = Z_{G}(𝔸)M_{Q}(𝔸)^{1}m if g = umh, u ∈ U_{Q}(𝔸), m ∈ M_{Q}(𝔸) and h ∈ K. This is well defined and a left Z_{G}(𝔸)Q(𝔸)^{1}-invariant. Then the function H_{Q}: G(𝔸) ⟶ ℝ_{+} is defined by H_{Q}(g) = |α̂_{Q}(z_{Q}(g))|_{𝔸}^{−1} for g ∈ G(𝔸).

We set Y_{Q} = Q(𝔸)^{1}\G(𝔸)^{1} and X_{Q} = Q(k)\G(k). Then X_{Q} is regarded as a subset of Y_{Q}. Since Z_{G}(𝔸)^{1} = Z_{G}(𝔸) ∩ G(𝔸)^{1} ⊂ M_{Q}(𝔸)^{1}, z_{Q} maps Y_{Q} = Q(𝔸)^{1}\G(𝔸)^{1} to M_{Q}(𝔸)^{1}\(M_{Q}(𝔸) ∩ G(𝔸)^{1}). Namely, we have the following commutative diagram:

Y_{Q}  ⟶^{z_{Q}} M_{Q}(𝔸)^{1}\(M_{Q}(𝔸) ∩ G(𝔸)^{1})
↓  ↓
Z_{G}(𝔸)Q(𝔸)^{1}\G(𝔸) ⟶^{z_{Q}} Z_{G}(𝔸)M_{Q}(𝔸)^{1}\M_{Q}(𝔸)

Since both vertical arrows are injective, H_{Q} is restricted to Y_{Q}. Let B_{T} = {y ∈ Y_{Q}: H_{Q}(y) ≤ T } for T > 0. We can prove the following.

Proposition. For T > 0 and any g ∈ G(𝔸)^{1}, B_{T} ∩ X_{Q}g is a finite subset of Y_{Q}. Hence, one can define the function

Γ_{Q}(g) = min{T > 0: B_{T} ∩ X_{Q}g ≠ ∅} = min_{y∈X_{Q}g} H_{Q}(y)

on G(𝔸)^{1}. Then the maximum

γ(G, Q, k) = max_{g∈G(𝔸)^{1}} Γ_{Q}(g)

exists.

The existence of the maximum is a result of the reduction theory due to Borel–Harich-Chandra and Harder. The constant γ_{Q} = γ(G, Q, k) is called the fundamental Hermite constant of (G, Q) over k. An interesting thing is a similarity between the definitions of γ_{n} and γ_{Q}. Namely, γ_{n} is represented as

γ_{n} = max_{\begin{matrix}g∈GL_{n}(ℝ) \\ | det g|=1 \end{matrix}} min\{T > 0: B_{T}^{n} ∩ gℤ^{n} ≠ \{0\}\} ,

where B_{T}^{n} denotes the ball of radius T with center 0 in ℝ^{n}. On the other hand, by definition,

γ_{Q} = max_{g∈G(𝔸)^{1}} min{T > 0: B_{T} ∩ X_{Q}g ≠ ∅} .

Thus X_{Q} plays a role of the lattice ℤ^{n} and B_{T} is an analogue of the ball B_{T}^{n}. In some cases, it is more convenient to consider the constant

γ̃(G, Q, k) = max_{g∈G(𝔸)} min_{y∈X_{Q}g} H_{Q}(g) .

If k is an algebraic number field, then γ̃(G, Q, k) is always equal to γ(G, Q, k) as the natural map Y_{Q} ⟶ Z_{G}(𝔸)Q(𝔸)^{1}\G(𝔸) is bijective. The next example shows a relation between γ(GL_{n}, Q, k) and γ_{n,d}(k).

Example 1. Let V_{n,d}(k) = ⋀^{d} k^{n} be the d-th exerior product of k^{n} and V_{n,d}(𝔸) = V_{n,d}(k)⊗_{k}𝔸 the adele space of V_{n,d}(k). We fix a k-basis e_{1}, ⋯, e_{n} of k^{n}, and identify the group of linear automorphisms of k^{n} with GL_{n}(k). For 1 ≤ d ≤ n − 1, Q_{d}(k) denotes the stabilizer of the subspace spanned by e_{1}, ⋯, e_{d} in GL_{n}(k). A k-basis of V_{n,d}(k) is formed by the elements e_{I} = e_{i_{1}} ∧ ⋯ ∧ e_{i_{d}} with I = {1 ≤ i_{1} < i_{2} < ⋯ < i_{d} ≤ n}. Let k_{v} be the completion field of k at a place v and | · |_{v} the usual normalized absolute value of k_{v}. The local height H_{v}: V_{n,d}(k) ⊗_{k} k_{v} ⟶ ℝ_{+} is defined by

H_{v}(∑_{I} a_{I}e_{I}) = \{ \begin{matrix}(∑_{I} |a_{I}|_{v}^{2/[k_{v}:ℝ]})^{[k_{v}:ℝ]/2} & (v is infinite) \\ sup_{I}(|a_{I}|_{v}) & (v is finite) \end{matrix}

Note that |a|_{v} = a\overline{a} if k_{v} = ℂ. Then the global height H_{n,d}: V_{n,d}(k) ⟶ ℝ_{+} is defined to be the product of all H_{v}, i.e.,

H_{n,d}(x) = ∏_{v} H_{v}(x)

for x ∈ V_{n,d}(k). This is immediately extended to the subset GL(V_{n,d}(𝔸))V_{n,d}(k) of the adele space V_{n,d}(𝔸) by

H_{n,d}(Ax) = ∏_{v} H_{v}(A_{v}x)

for A = (A_{v})_{v} ∈ GL(V_{n,d}(𝔸)) and x ∈ V_{n,d}(k). Especially, for g ∈ GL_{n}(𝔸) and x_{1}, ⋯, x_{d} ∈ k^{n}, the height H_{n,d}(gx_{1} ∧ ⋯ ∧ gx_{d}) of gx_{1} ∧ ⋯ ∧ gx_{d} is defined. Then there exists the following maximum:

γ̂_{n,d}(k) = max_{g∈GL_{n}(𝔸)} min_{\begin{matrix}x_{1},··· ,x_{d}∈k^{n} \\ x_{1}∧···∧x_{d}≠0 \end{matrix}} \frac{H_{n,d}(gx_{1} ∧ ⋯ ∧ gx_{d})}{| det g|_{𝔸}^{d/n}}.

In the case that k is an algebraic number field, Thunder’s γ_{n,d}(k) is defined by

γ_{n,d}(k) = γ̂_{n,d}(k)^{2/[k:ℚ]}

It is immediate to see that

\frac{H_{n,d}(g^{−1}e_{1} ∧ ⋯ ∧ g^{−1}e_{d})}{| det g^{−1}|_{𝔸}^{d/n}} = H_{Q_{d}}(g)^{gcd(d,n−d)/n}

for g ∈ GL_{n}(𝔸), and hence

γ̂_{n,d}(k) = γ̃(GL_{n}, Q_{d}, k)^{gcd(d,n−d)/n} .

In general, Z_{GL_{n}}(𝔸)GL_{n}(𝔸)^{1} is an index finite normal subgroup of GL_{n}(𝔸), but it is not necessarily equal to GL_{n}(𝔸) if k is a function field. Let Ξ be a complete set of representatives for the cosets of Z_{GL_{n}}(𝔸)GL_{n}(𝔸)^{1}\GL_{n}(𝔸). If we put

γ̂_{n,j}(k)_{ξ} = max_{g∈Z_{GLn} (𝔸)GL_{n}(𝔸)^{1}ξ} min_{\begin{matrix}x_{1},··· ,x_{d}∈k^{n} \\ x_{1}∧···∧x_{d}≠0 \end{matrix}} \frac{H_{n,d}(gx_{1} ∧ ⋯ ∧ gx_{d})}{| det g|_{𝔸}^{d/n}}

= \frac{1}{| det ξ|_{𝔸}^{d/n}} max_{g∈GL_{n}(𝔸)^{1}ξ} min_{\begin{matrix}x_{1},··· ,x_{d}∈k^{n} \\ x_{1}∧···∧x_{d}≠0 \end{matrix}} H_{n,d}(gx_{1} ∧ ⋯ ∧ gx_{d})

for ξ ∈ Ξ, then

γ̂_{n,d}(k) = max_{ξ∈Ξ} γ_{n,d}(k)_{ξ},

and in particular, for the unit element ξ = 1,

γ̂_{n,d}(k)_{1} = γ(GL_{n}, Q_{d}, k)^{gcd(d,n−d)/n}.

If k is a number field, Z_{GL_{n}}(𝔸)GL_{n}(𝔸)^{1} = GL_{n}(𝔸) holds, and hence one has

γ̃(GL_{n}, Q_{d}, k) = γ(GL_{n}, Q_{d}, k) = γ_{n,d}(k)^{n[k:ℚ]/(2 gcd(d,n−d))}.

We summarize the properties of γ(G, Q, k).

Theorem 1. Assume the exact sequence

1 ⟶ Z ⟶ G ⟶^{β} G′ ⟶ 1

of connected reductive groups defined over k satisfies the following two conditions:

• Z is central in G.

• Z is isomorphic to a product of tori of the form R_{k′/k}(GL_{1}), where each k′/k is a finite separable extension and R_{k′/k} denotes the functor of restriction of scalars from k′ to k.

Then γ(G, Q, k) is equal to γ(G′, β(Q), k).

Theorem 2. If k/ℓ is a finite separable extension, then γ(R_{k/ℓ}(G), R_{k/ℓ}(Q), ℓ) is equal to γ(G, Q, k).

Theorem 3. Let R and Q be two different maximal standard k-parabolic subgroups of G, Q^{R} = M_{R} ∩ Q a maximal standard parabolic subgroup of M_{R} and M_{Q}^{R} = M_{R} ∩ M_{Q} a Levi subgroup of Q^{R}. We write α_{Q}^{R} for the ℤ-basis α_{Q^{R}} of X_{k}^{∗}(M_{Q}^{R}/Z_{R}). Then ℚ-vector space X_{k}^{∗}(M_{Q}^{R}/Z_{G}) ⊗_{ℤ} ℚ is spanned by α_{Q}^{R} and α̂_{R}|_{M_{Q}^{R}}. If we take ω_{1}, ω_{2} ∈ ℚ such that

α̂_{Q}|_{M_{Q}^{R}} = ω_{1}α_{Q}^{R} + ω_{2}α̂_{R}|_{M_{Q}^{R}} ,

then one has an inequality of the form

γ(G, Q, k) ≤ γ̃(M_{R}, Q^{R}, k)^{ω_{1}}γ(G, R, k)^{ω_{2}} .

Example 2. We illustrate that Theorem 1 and Theorem 3 are generalizations of the duality relation (2) and Rankin’s inequality (3) in §1, respectively. We use the same notations as in Example 1. First, we consider the automorphism β: GL_{n} ⟶ GL_{n} defined by β(g) = w_{0}(^{t}g^{−1})w_{0}^{−1}, where

w_{0} = ( \begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 0 & 0 & ⋯ & 1 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 1 & ⋯ & 0 & 0 \\ 1 & 0 & ⋯ & 0 & 0 \end{matrix} ) ∈ GL_{n}(k).

Since β(Q_{d}) = Q_{n−d}, Theorem 1 deduces

γ(GL_{n}, Q_{d}, k) = γ(GL_{n}, Q_{n−d}, k).

If k is a number field, this implies the duality relation (2). Next, for i, j ∈ ℤ with 1 ≤ i < j ≤ 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} × GL_{n−j}, M_{Q} = GL_{i} × GL_{n−i} and M_{Q}^{R} = GL_{i} × GL_{j−i} × GL_{n−j}. It is easy to see

ω_{1} = \frac{n}{j} \frac{gcd(i, j − i)}{gcd(i, n − i)} , ω_{2} = \frac{i}{j} \frac{gcd(j, n − j)}{gcd(i, n − i)} .

Theorem 3 deduces

γ(GL_{n}, Q_{i}, k) ≤ γ̃(M_{Q_{j}}, Q_{i}^{Q_{j}}, k)^{\frac{n}{j} \frac{gcd(i,j−i)}{gcd(i,n−i)}} γ(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).

Let τ(G) (resp. τ(Q)) be the Tamagawa number of G (resp. Q) and ω_{𝔸}^{G} (resp. ω_{𝔸}^{U_{Q}} and ω_{𝔸}^{M_{Q}}) the Tamagawa measure of G(𝔸) (resp. U_{Q}(𝔸) and M_{Q}(𝔸)). The modular character δ_{Q}^{−1} of Q(𝔸) is defined by the relation dω_{𝔸}^{U_{Q}}(m^{−1}um) = δ_{Q}(m)^{−1}dω_{𝔸}^{U_{Q}}(u) for u ∈ U_{Q}(𝔸) and m ∈ M_{Q}(𝔸). We define constants ê_{Q} and C_{G,Q} as follows:

• δ_{Q}(m) = |α̂_{Q}(m)|ê_{Q} ; 𝔸 for all m ∈ M(𝔸).

• dω_{𝔸}^{G}(g) = C_{G,Q}^{−1}δ_{Q}(m)^{−1}dω_{𝔸}^{U_{Q}}(u)dω_{𝔸}^{M_{Q}}(m)dν_{K}(h) for all g = umh, u ∈ U_{Q}(𝔸), m ∈ M_{Q}(𝔸) and h ∈ K. Here ν_{K} denotes the Haar measure of K normalized so that ν_{K}(K) = 1. By an argument of the mean value theorem, we can show the following theorem.

Theorem 4. One has an estimate of the form

(C_{G,Q} · D_{G,Q} · E_{Q} · \frac{τ(G)}{τ(Q)})^{1/ê_{Q}} ≤ γ(G, Q, k) ,

where D_{G,Q} and E_{Q} are given as follows:

D_{G,Q} = \{ \begin{matrix}\frac{[X_{k}^{∗}(Z_{G}) : X_{k}^{∗}(G)]}{[X_{k}^{∗}(Z_{Q}) : X_{k}^{∗}(M_{Q})]} & (ch(k) = 0), \\ \frac{(log q)^{rank X_{k}^{∗}(G)}[Hom(X_{k}^{∗}(G), q^{ℤ}) : Im ϑ_{G}]}{(log q)^{rank X_{k}^{∗}(M_{Q})}[Hom(X_{k}^{∗}(M_{Q}), q^{ℤ}) : Im ϑ_{M_{Q}}]} & (ch(k) > 0), \end{matrix}

E_{Q} = \{ \begin{matrix}ê_{Q}[X_{k}^{∗}(Z_{Q}/Z_{G}) : X_{k}^{∗}(M_{Q}/Z_{G})] & (ch(k) = 0), \\ (1 − q_{0}^{−ê_{Q}}) & (ch(k) > 0). \end{matrix}

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 |α̂_{Q}(M_{Q}(𝔸) ∩ G(𝔸)^{1})|_{𝔸} of the cyclic group q^{ℤ} generated by q. Moreover, this inequality is strict if ch(k) > 0.

We note that γ(G, Q, k) ∈ q_{0}^{ℤ} if ch(k) > 0.

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 §1. If ch(k) > 0, we obtain q_{0} = q^{n/ gcd(d,n−d)} and

(\frac{q^{(g(k)−1)(d(n−d)+1)}(q − 1)(1 − q^{−n})}{h_{k}} \frac{∏_{i=n−d+1}^{n} ζ_{k}(i)}{∏_{i=2}^{d} ζ_{k}(i)})^{1/ gcd(d,n−d)} < γ(GL_{n}, Q_{d}, k) ,

where g(k) denotes the genus of k, h_{k} the divisor class number of k and ζ_{k}(s) the congruence zeta function of k. On the other hand, from the definition of γ̂_{n,d}(k) and Thunder’s theorem on an analogue of Minkowski’s second convex bodies theorem ([T1]), it follows that

1 ≤ γ(GL_{n}, Q_{d}, k) ≤ γ̃(GL_{n}, Q_{d}, k) ≤ q^{ndg(k)/ gcd(d,n−d)} = q_{0}^{dg(k)} .

If g(k) = 0, i.e., k is a rational function field over 𝔽_{q}, this implies γ(GL_{n}, Q_{d}, k) = γ̃(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} · \frac{(q − 1)(q^{2n} + a_{1}q^{n} + q)}{(q + a_{1} + 1)(q^{2n} − q^{n+1})} < γ(GL_{n}, Q_{1}, k) ≤ γ̃(GL_{n}, Q_{1}, k) ≤ q^{n} ,

where h_{k} = a_{1} + q + 1. Combining this with the Hasse–Weil bound |a_{1}| ≤ 2\sqrt{q}, we have γ(GL_{n}, Q_{1}, k) = γ̃(GL_{n}, Q_{1}, k) = q^{n} provided that h_{k} ≤ q − 1.

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 γ(G, Q, k).

Theorems 1 – 4 and Example 3 were proved in [W5]. Furthermore, we can add a small result on γ̃(GL_{n}, Q_{1}, k).

Theorem 5. We define the constant Δ_{k} as follows:

Δ_{k} = \{ \begin{matrix}|D_{k}| & (k is an algebraic number field of absolute discriminant D_{k}). \\ q^{2g(k)−2} & (k is a function field of genus g(k) and constant field 𝔽_{q}). \end{matrix}

If ℓ is a separable extension of k with degree r, then

\frac{γ̃(GL_{n}, Q_{1}, ℓ)}{Δ_{ℓ}^{n/2}} ≤ r^{−nrs_{k}/2} · \frac{γ̃(GL_{nr}, Q_{1}, k)}{Δ_{k}^{nr/2}} ,

where s_{k} denotes the number of infinite places of k.

This theorem was first proved in [O-W] in the case of k = ℚ. See [W6] for a genral case.

Remark. By definition, γ(G, Q, k) measures the existence of rational points in B_{T}. Namely, if T ≥ γ(G, Q, k), then B_{T} ∩X_{Q}g ≠ ∅ for any g ∈ G(𝔸)^{1}, especially g = e. If k is an algebraic number field, the cardinality of B_{T} ∩ X_{Q}g is increasing in proportion to the volume of B_{T} as T → ∞. More precisely, one has

lim_{T →∞} \frac{♯(B_{T} ∩ X_{Q}g)}{ω_{Y} (B_{T})} = \frac{τ(Q)}{τ(G)}

for all g ∈ G(𝔸)^{1}, where ω_{Y} = ω_{𝔸}^{Q}\ω_{𝔸}^{G} is the Tamagawa measure on Y. The volume ω_{Y} (B_{T}) is equal to (C_{G,Q}D_{G,Q}E_{Q})^{−1}T ê_{Q}. See [W4] for details.

3. Behavior of fundamental Hermite constants under isogenies. 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.

Theorem 6. Let

1 ⟶ F ⟶ Ĝ ⟶^{β} G ⟶ 1

be a separable central k-isogeny of a connected reductive k-group G and Q a maximal kparabolic subgroup of G. Then

γ(Ĝ, Q̂, k)^{d_{β}} ≤ γ(G, Q, k) ,

where Q̂ = β^{−1}(Q) and d_{β} = [X_{k}^{∗}(MQ̂/ZĜ) : X_{k}^{∗}(M_{Q}/Z_{G})].

Proof. We note that XQ̂ = Q̂(k)\Ĝ(k) is isomorphic with X_{Q} = Q(k)\G(k) as F is central in G. Since X_{k}^{∗}(F) is a torsion group, β gives rise to an isomorphism between Hom(X_{k}^{∗}(Ĝ), ℝ_{+}) and Hom(X_{k}^{∗}(G), ℝ_{+}). This implies that β(Ĝ(𝔸)^{1}) is contained in G(𝔸)^{1}. By the similar reason, β(Q̂(𝔸)^{1}) is contained in Q(𝔸)^{1}. Therefore, one has the following commutative diagram:

XQ̂ ⟶_{≅}^{β} X_{Q}
↓  ↓
YQ̂ ⟶^{β} Y_{Q}

The central isogeny

1 ⟶ F/F ∩ ZĜ ⟶ MQ̂/ZĜ ⟶^{β} M_{Q}/Z_{G} ⟶ 1

yields the exact sequence

1 ⟶ X_{k}^{∗}(M_{Q}/Z_{G}) ⟶^{β^{∗}} X_{k}^{∗}(MQ̂/ZĜ) ⟶ X_{k}^{∗}(F/F ∩ ZĜ).

Since X_{k}^{∗}(M_{Q}/Z_{G}) = ℤα̂_{Q} and X_{k}^{∗}(MQ̂/ZĜ) = ℤα̂Q̂ by definition, β^{∗}(α̂_{Q}) is equal to d_{β}α̂Q̂. Moreover, by definition of the map z_{Q}, it is easy to see that the following diagram is commutative:

Ĝ(𝔸) ⟶^{z_{Q}} ZĜ(𝔸)MQ̂(𝔸)^{1}\MQ̂(𝔸)
β↓  ↓^{β}
G(𝔸) ⟶^{z_{Q}} Z_{G}(𝔸)M_{Q}(𝔸)^{1}\M_{Q}(𝔸)

Then, for any g ∈ Ĝ(𝔸), one has

H_{Q}(β(g)) = |α̂_{Q}(z_{Q}(β(g)))|_{𝔸}^{−1} = |α̂_{Q}(β(zQ̂(g)))|_{𝔸}^{−1} = |α̂Q̂(zQ̂(g))|_{𝔸}^{−d_{β}} = HQ̂(g)^{d_{β}}.

From β(Ĝ(𝔸)^{1}) ⊂ G(𝔸)^{1} and XQ̂ ≅ X_{Q}, it follows that

max_{g∈G(𝔸)^{1}} min_{x∈X_{Q}} H_{Q}(xg) ≥ max_{g∈Ĝ(𝔸)^{1}} min_{x∈X_{Q}} H_{Q}(β(xg)) = max_{g∈Ĝ(𝔸)^{1}} min_{x∈X_{Q}} HQ̂(xg)^{d_{β}}.

This leads us to γ(G, Q, k) ≥ γ(Ĝ, Q̂, k)^{d_{β}}. □

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

1 ⟶ F ⟶ G̃ ⟶^{β} G ⟶ 1

is the simply connected covering of G defined over an algebraic number field k. We put

G(𝔸_{∞}) = ∏_{\begin{matrix}w \\ infinite \end{matrix}} G(k_{v}) × ∏_{\begin{matrix}v \\ finite \end{matrix}} K_{v} ,

where K_{v} denotes the v-component of the fixed good maximal compact subgroup K of G(𝔸). It is known that G(k)G(𝔸_{∞}) is a normal subgroup of G(𝔸) and G(k)G(𝔸_{∞})\G(𝔸) = G(k)\G(𝔸)/G(𝔸_{∞}) is a finite set ([P-R, Proposition 8.8]). Let Ξ_{G} be a complete set of representatives of G(k)G(𝔸_{∞})\G(𝔸). For each ξ ∈ Ξ_{G}, we set

γ(G, Q, k)_{ξ} = max_{g∈G(k)G(𝔸_{∞})ξ} min_{x∈X_{Q}g} H_{Q}(x) ,

and especially

γ(G, Q, k)_{pr} = max_{g∈G(k)G(𝔸_{∞})} min_{x∈X_{Q}g} H_{Q}(x) .

From G(𝔸) = G(𝔸)^{1}, it follows that

γ(G, Q, k) = max_{ξ∈Ξ_{G}} γ(G, Q, k)_{ξ} .

Theorem 7. Being the notations and assumptions as above, we have

γ(G̃, Q̃, k)^{d_{β}} = γ(G, Q, k)_{pr} ,

where Q̃ = β^{−1}(Q) and d_{β} = [X_{k}^{∗}(MQ̃) : X_{k}^{∗}(M_{Q})].

Proof. Since Γ_{Q}(g) = min_{x∈X_{Q}} H_{Q}(xg) is left G(k)-invariant and right K-invariant, it is sufficient to prove that G(k)β(G̃(𝔸))K = G(k)G(𝔸_{∞}). By the proof of [P-R, Proposition 8.8], one has [G(𝔸) : G(𝔸)] ⊂ β(G̃(𝔸)) ⊂ G(k)G(𝔸_{∞}). This implies G(k)β(G̃(𝔸))K ⊂ G(k)G(𝔸_{∞}). To prove the converse, we must show G(k_{w}) ⊂ β(G̃(k_{w}))K_{w} for any infinite place w. This is obvious if k_{w} = ℂ. Thus we assume K_{w} = ℝ. Since G̃(ℝ) is connected as a real Lie group ([B-T, (4.7)]), β(G̃(ℝ)) coincides with the topologically connected component G(ℝ)^{0} of G(ℝ). Then G(ℝ) = G(ℝ)^{0}K_{w} follows from the maximality of K_{w} (cf. [P-R, Proposition 3.10]). □

In general, if β_{1} : G_{1} ⟶ G is a central finite covering of G, we have G(k)G(𝔸_{∞}) ⊂ G(k)β(G_{1}(𝔸))K ⊂ G(𝔸). Therefore, there exists a subset Ξ_{G,G_{1}} of Ξ_{G} such that

γ(G_{1}, Q_{1}, k)^{d_{β_{1}}} = max_{ξ∈Ξ_{G,G_{1}}} γ(G, Q, k)_{ξ} ,

where Q_{1} = β_{1}^{−1}(Q) and d_{β_{1}} = [X_{k}^{∗}(M_{Q_{1}}) : X_{k}^{∗}(M_{Q})].

As a corollary of Theorems 1 and 7, we obtain the following.

Corollary. If k is an algebraic number field and Q_{d} the maximal parabolic subgroup of GL_{n} given as in Example 1, then

γ(SL_{n}, Q_{d} ∩ SL_{n}, k)^{n/ gcd(d,n−d)} = γ(P GL_{n}, Z_{GL_{n}}\Q_{d}, k)_{pr}

In particular, if the ideal class group I_{k} = k^{×}𝔸_{∞}^{×}\𝔸^{×} of k satisfies I_{k} = I_{k}^{n}, then

γ(SL_{n}, Q_{d} ∩ SL_{n}, k)^{n/ gcd(d,n−d)} = γ(P GL_{n}, Z_{GL_{n}}\Q_{d}, k) = γ(GL_{n}, Q_{d}, k).

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Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043 Japan

  E-mail address: watanabe@math.wani.osaka-u.ac.jp