ON THE INTEGER POINTS OF AN AFFINE

TORIC VARIETY (GENERAL CASE)

B.Z. Moroz

1. Given an algebraic variety X defined over ℤ, the study of the distribution of integer points X(ℤ) in the real locus X(ℝ) is a classical subject in Diophantine Geometry. Our purpose here is to address this problem for a class of (integral models of) affine toric varieties. To be specific, let T be an algebraic torus defined over ℚ, and let X be an affine toric variety associated with T. In other words, we are given an integral representation

ρ : 𝔊 → GL(d, ℤ), d := dim T, 𝔊 := Gal(K | ℚ),

of the Galois group Gal(K | ℚ) of the splitting field K of T, and a 𝔊 -invariant strongly convex rational polyhedral cone σ_{ℚ} in the ℚ -vector space V = M ⊗ ℚ, where M is the ℤ[𝔊] -module defined by ρ. In this situation, one defines a group I(σ) of “fractional ideals”, and a class of varieties {X_{𝔅} | 𝔅 ∈ I(σ)} defined over ℤ, for each of which X_{𝔅} ×_{ℤ} ℚ ≅ X (see n^{∘}2). Let Y be the (Zariski-) open subset of X isomorphic to T, the so-called “big” orbit (or “embedded torus”). Although our variety X may be singular, Y is certainly smooth. In n^{∘} 5, we construct a homomorphism

ξ : Y (ℝ) → (ℝ_{+}^{∗})^{ν} × ℂ_{1}^{d_{1}} × (ℤ/2ℤ)^{ν}  (1)

with d_{1} + ν = d, ν = μ + r, where r is equal to the free rank of the group of units T (ℤ), μ ≥ 1, and ℂ_{1} := {z | z ∈ ℂ, | z |= 1}. The map ξ allows one to define a “height” ∥ · ∥: Y (ℝ) → ℝ_{+} in such a way that the set

U(w) = {a | a ∈ Y (ℝ), ∥ a ∥< w}, w ∈ ℝ_{+}

contains at most finitely many integer points. Indeed, the following theorem holds true. Let Y_{𝔅}(ℤ) := X_{𝔅}(ℤ) ∩ Y (ℚ). Given a “conductor” 𝔣 = (𝔣_{0}, 𝔣_{1}) with 𝔣_{0} ∈ ℤ, 𝔣_{0} ≥ 1, 𝔣_{1} ⊆ (ℤ/2ℤ)^{ν}, one can define the “ray”

Z_{𝔣} = {a | a ∈ Y_{𝔅}(ℤ), a = 1(mod 𝔣)}

in such a way that Z_{𝔣} = Y_{𝔅}(ℤ) for 𝔣 = (1, (ℤ/2ℤ)^{ν}) (see n^{∘} 5).

Theorem 1. If σ_{ℚ} ∩ M ≠ {0}, then

card (Z_{𝔣} ∩ U(w)) = c w(log w)^{β}(1 + O(1/ log w)),  (2)

Typeset by AMS-TEX

where c ≥ 0, 0 ≤ β < B + r, β ∈ ℤ; moreover, if Z_{f} ≠ ϕ then c > 0. Here B is a non-negative integer, to be defined in Lemma 3.

This paper is a sequel to [8], where theorem 1 has been proved for the subclass of “maximal” toric varieties (cf. also [7, 9]). In n^{∘} 4, the general problem considered here is essentially reduced to the leading special case studied in [8]; theorem 1 is proved in n^{∘} 5. In the next section, we set up our notation and make a few comments on the nature of the objects defined there; our concluding remarks are collected in n^{∘} 6.

2. Throughout this paper ℚ, ℝ, ℂ, and ℤ denote, respectively, the fields of rational, real, complex numbers, and the ring of rational integers; ℝ_{+}^{∗} is the multiplicative group of positive real numbers; in general, R^{∗} stands for the group of units of a (commutative) ring R; ℂ_{1} := {a | a ∈ ℂ, | a |= 1} is the unit circle in ℂ. A number field is, by definition, a finite extension of ℚ; given a number field L, let I(L), I_{0}(L), and S(L) be the group of fractional ideals of L, the monoid of integral ideals, and the set of prime ideals of L respectively. Given a (finite) group 𝔊 and a ℤ[𝔊]-module M, let M(𝔊) := {a | a ∈ M, ga = a for g ∈ 𝔊} denote the submodule of the 𝔊-invariants of M.

Let T be an algebraic torus defined over a number field k; it splits over a finite normal extension K | k, so that T ×_{k} K ≅ G_{m,K}^{d}, where d := dim T is the dimension of T. Let n := [K : k], and denote by 𝔒 and v the rings of integers of K and k, respectively; let 𝔊 be the Galois group of the normal extension K | k. The torus T is uniquely determined by the integral representation ρ : 𝔊 → GL(d, ℤ) acting on a ℤ[𝔊] -module M (of rank d over ℤ); let M̌ be the dual ℤ[𝔊]-module, corresponding to the contragredient representation ρ̌. Let V = M ⊗ ℚ and V̌ = M̌ ⊗ ℚ be the corresponding ℚ[𝔊] -modules; write, for brevity, gv := ρ̌(g)v, gv′ := ρ(g)v′ for v ∈ V̌ , v′ ∈ V, g ∈ 𝔊, so that (gv | gv′) = (v | v′). Let σ_{ℚ} be a 𝔊-invariant strongly convex rational polyhedral cone in the ℚ-vector space V, and σ̌_{ℚ} := {v | v ∈ V̌ , (v | u) ≥ 0 for u ∈ σ_{ℚ}} be the dual cone in V̌; let σ := σ_{ℚ} ∩ M, and σ̌ := σ̌_{ℚ} ∩ M̌. Then σ̌ = {∑_{i=1}^{l} m_{i}u_{i} | m_{i} ≥ 0, m_{i} ∈ ℤ}, where {u_{i} | 1 ≤ i ≤ l} is the minimal set of generators of σ̌ (cf. [4, § 1.3]). Let us fix a ℤ-basis {e_{i} | 1 ≤ i ≤ d} of M. Since σ̌ + (−σ̌) = M̌, there are two matrices

c = (c_{ij})_{\begin{matrix}1≤i≤d \\ 1≤j≤l \end{matrix}} and

c̃ = (c̃_{ij})_{\begin{matrix}1≤i≤l \\ 1≤j≤d \end{matrix}}

with c_{ij} ∈ ℤ, c̃_{ij} ∈ ℤ for all i, j, and such that u_{i} = ∑_{1≤j≤d}e_{j}c_{ji}, and e_{j} = ∑_{1≤i≤l}u_{i}c̃_{ij}.

Let C := c̃ · c; since c · c̃ = 1, it follows that C^{2} = C, c · C = c and C · c̃ = c̃. Note also that u_{i} = ∑_{j=1}^{l} C_{ij}u_{j} with C = (C_{ij})_{1≤i,j≤l}. On writing ge_{i} = ∑_{j=1}^{d} e_{j}r_{ji}(g), it follows that gu_{i} = ∑_{j=1}^{l} u_{j}r̃_{ji}(g) with r̃(g) := c̃r(g)c.

Definition 1. Given an Abelian group R, we let a_{i}^{g} = ∏_{j=1}^{d} a_{j}^{r_{ji}(g)} for a ∈ R^{d}, and

b_{i}^{g} = ∏_{j=1}^{l} b_{j}^{r̃_{ji}(g)} for b ∈ R^{l}; g ∈ 𝔊.

Definition 2. For an Abelian group R, let f : R^{l} → R^{d} be given by the equations

f(b) = a, a_{i} = ∏_{j=1}^{l} bc̃_{ji} ; j, 1 ≤ i ≤ d, with a ∈ R^{d}, b ∈ R^{l}.

Lemma 1. The map f : R^{l} → R^{d} is a surjective 𝔊 -homomorphism.

Proof. It follows from Definition 1 that f(b^{g}) = f(b)^{g} for b ∈ R^{l}, g ∈ 𝔊. Equation c · c̃ = 1 shows that f is surjective.

To define the affine toric variety X over k corresponding to σ_{ℚ}, let us choose a set x = {x_{ij} | 1 ≤ i ≤ l, 1 ≤ j ≤ n} of l · n independent variables, and a basis {ω_{i} | 1 ≤ i ≤ n} of K | k. Write, for brevity, gy_{i} := ∑_{j=1}^{n} x_{ij}gω_{j}, g ∈ 𝔊, 1 ≤ i ≤ l. The toric variety X is defined by the system of equations

gy = y^{g}, g ∈ 𝔊.  (3)

Each of the equations gy_{i} = l ; ∏_{j=1}y_{ν}^{r̃_{νi}(g)} is equivalent to a system of n polynomial

equations

P_{j}^{(i,g)}(x) = 0, 1 ≤ j ≤ n,

with P_{j}^{(i,g)}(x) ∈ k[x]. Let ℭ(σ) be the ideal of k[x] generated by the polynomials {P_{j}^{(i,g)}(x) | 1 ≤ j ≤ n, 1 ≤ i ≤ l, g ∈ 𝔊}, then X = Spec(k[x]/ℭ(σ)). The torus T corresponds to the cone σ_{ℚ}^{(0)} = {0}, and it may be defined by the system of equations

gt = t^{g}, g ∈ 𝔊, ∏_{i=0}^{d} t_{i} = 1, t := (t_{1}, …, t_{d}),

where again we introduce the independent variables z = {z_{ij} | 0 ≤ i ≤ d, 0 ≤ j ≤ n}, and let gt_{i} := ∑_{j=1}^{n} z_{ij}gω_{j}, t_{i}^{g} = ∏_{j=1}^{d} t_{j}^{r_{ji}(g)}, 1 ≤ i ≤ d, g ∈ 𝔊, t_{0} := ∑_{j=1}^{n} z_{0j}ω_{j}; thus T = Spec(k[z]/ℭ_{T}), the ideal ℭ_{T} being generated by the polyno-

mials {P_{j}^{(i,g)}, P_{j}^{(0)} | 1 ≤ j ≤ n, 1 ≤ i ≤ d, g ∈ 𝔊} arising from the defining equations (cf. [7-9]). Let Y be the open subset of X defined by the condition ∏_{i=1}^{l} y_{i} ≠ 0, where y_{i} := ∑_{j=1}^{n} x_{ij}ω_{j}, 1 ≤ i ≤ n. The variety Y is easily seen to be isomorphic to T (it is the “big” orbit alluded to in n^{∘} 1), the isomorphism φ : Y → T being defined by the equations φ(y)_{i} = ∏_{j=1}^{l} yc̃_{ji} ; j, 1 ≤ i ≤ d, ∏_{i=0}^{d} φ(y)_{i} = 1.

Let now I(σ) = {𝔅 | 𝔅 ∈ I(K)^{l}, g𝔅 = 𝔅^{g} for g ∈ 𝔊}, and let I_{0}(σ) := I(σ) ∩ I_{0}(K)^{l}. Choose 𝔅 in I(K), and suppose, for simplicity, that

𝔅_{i}  is an v-free module,  1 ≤ i ≤ l.  (4)

Let {ω_{j}^{(i)} | 1 ≤ j ≤ n} be an v-basis of 𝔅_{i}, and let gy_{i} := ∑_{j=1}^{n} x_{ij}gω_{j}^{(i)}, 1 ≤ i ≤ l. Now the system of equations (3) defines a variety X_{𝔅} = Spec(v[x]/J(𝔅)) over the ring of integers v, the ideal J(𝔅) being generated by the polynomials

{P_{j}^{(i,g)}(x) | P_{j}^{(i,g)}(x) ∈ v[x], 1 ≤ j ≤ n, 1 ≤ i ≤ l, g ∈ 𝔊},

where each of the equations gy_{i} = ∏_{ν=1}^{l} y_{ν}^{r̃_{νi}(g)} is equivalent to the system of n equations P_{j}^{(i,g)}(x) = 0, 1 ≤ j ≤ n. By construction, X_{𝔅} ×_{v} k ≅ X. Let A(v) be the category of commutative v-algebras. Clearly,

X_{𝔅}(A) = {α | α ∈ B, gα = α^{g}  for  g ∈ 𝔊}, A ∈ A(v),  (5)

where B_{i} := A ⊗_{v} 𝔅_{i}, B := (B_{1}, …, B_{l}), 1 ≤ i ≤ l. Let, by definition, Y_{𝔅}(v) := X_{𝔅}(v) ∩ Y (k).

If k = ℚ, then v = ℤ and therefore condition (4) holds automatically. On the other hand, making use of the Weil restriction of scalars Res_{k/ℚ}, [11], [5, p. 245-246], we note that X may be regarded as a toric variety for the algebraic torus Res_{k/ℚ}T defined over ℚ. Therefore, when investigating the distribution of integer points Y_{𝔅}(v) in the real locus Y (ℝ) in n^{∘} 5, we may (and shall) assume that k = ℚ. This assumption would not, however, simplify the matters in n^{∘} 3 and n^{∘} 4, where k is subject to no restrictions.

Remark 1. Let Sgr be the category of commutative semigroups. Consider the functor 𝔉_{𝔅} : A(v) → Sgr defined by

𝔉_{𝔅} : A ↦ {α | α ∈ B, gα = α^{g}  for  g ∈ 𝔊}, A ∈ A(v),

where B := (…, A ⊗_{v} 𝔅_{i}, …) as above. Under supposition (4), the scheme X_{𝔅} represents the functor 𝔉_{𝔅} in view of (5). If (4) does not hold, the v-scheme representing 𝔉_{𝔅} still gives an integral model of the toric variety X; this model may be of interest for its own sake.

Example 1. Clearly, T (A) = {α | α ∈ (B^{∗})^{d}, gα = α^{g} for g ∈ 𝔊} with B := A ⊗_{v} O, A ∈ A(v). In particular, T (v) = {α | α ∈ (O^{∗})^{d}, gα = α^{g} for g ∈ 𝔊}.

Example 2. Let T = Res_{K/ℚ}G_{m,K}, k = ℚ, 𝔅_{i} = O for 1 ≤ i ≤ d, d = l = n, σ_{ℚ} = {a | a ∈ V, g · a ≥ 0 for g ∈ 𝔊} (here a ≥ 0 means a_{i} ≥ 0 for 1 ≤ i ≤ d). Then T (ℤ) ≅ O^{∗}, Y_{𝔅}(ℤ) = O \ {0}; in particular, taking K = ℚ, d = 1 one obtains T (ℤ) = {1, −1}, and Y_{𝔅}(ℤ) = ℤ \ {0}.

Remark 2. Let I^{pr}(σ) = {(α) | α ∈ K^{∗l}, gα = α^{g} for g ∈ 𝔊}. If 𝔅′ = (α)𝔅 with (α) ∈ I^{pr}(σ), then the varieties X_{𝔅} and X_{𝔅′} are v-isomorphic. Since I(σ)/I^{pr}(σ) is a finite group (cf. n^{∘} 4), it follows that there are only finitely many pairwise non-isomorphic schemes X_{𝔅} for a given X.

3. In this section, we study the structure of the group of fractional ideals I(σ), and establish the first properties of the L-functions associated to homomorphisms of the monoid I_{0}(σ) of integral ideals. For 𝔭 ∈ S(K), let 𝔊_{𝔭} = {g | g ∈ 𝔊, g𝔭 = 𝔭} be the decomposition group at 𝔭, and let 𝔊^{(𝔭)} = {g𝔊_{𝔭} | g ∈ 𝔊} denote the set of the right cosets of 𝔊 modulo 𝔊_{𝔭}. Let p ∈ S(k), and 𝔭 | p, 𝔭 ∈ S(K); then p = ∏_{h∈𝔊^{(𝔭)}}(h𝔭)^{e(p)}. One defines the group

I^{(p)}(σ) = {𝔄_{a} | 𝔄_{a} = (…, ∏_{h∈𝔊^{(𝔭)}}(h𝔭)^{(u_{i}|ha)}, …), a ∈ M(𝔊_{𝔭})}

of the p-primary ideals, and the semigroup

I_{p}(σ) = {𝔄_{a} | 𝔄_{a} ∈ I^{(p)}(σ), a ∈ σ}

of the p-primary integral ideals (here M(𝔊_{𝔭}) denotes the submodule of G_{𝔭} -invariants in M). A straightforward calculation shows that

I(σ) = ∏_{p∈S(k)} I^{(p)}(σ),  and  I_{0}(σ) = ∏_{p∈S(k)} I_{p}(σ).  (6)

For 𝔅 ∈ I(σ), let N_{K/k}𝔅 := ∏_{j=1}^{l} N_{K/k}𝔅_{j}, and N𝔅 := ∏_{j=1}^{l} N𝔅_{j} (where, as usual, N := N_{K/ℚ}). It follows that

N_{K/k}𝔄_{a} = p^{(z|a)/e(p)} with z := (∑_{g∈𝔊}g) · ( ∑_{1≤i≤l}u_{i})  (7)

for 𝔄_{a} ∈ I^{(p)}(σ). Let σ(m) = {a | a ∈ σ, (z | a) = m}, and let

κ = min{m | m ∈ ℤ, m ≥ 1, σ(m) ≠ ϕ}.

Given a homomorphism χ : I_{0}(σ) → ℂ_{1} ∪ {0} such that χ^{−1}({0}) = ∏_{p|𝔣_{0}(χ)}I_{p}(σ) for

some 𝔣_{0}(χ) in I_{0}(k), one defines a Dirichlet series

L(σ; χ, s) = ∑_{𝔄∈I_{0}(σ)} χ(𝔄)N𝔄^{−s/κ}.  (8)

Let

L_{p}(σ; χ, s) = ∑_{𝔄∈I_{p}(σ)} χ(𝔄)N𝔄^{−s/κ} = ∑_{a∈σ_{𝔭}} χ(𝔄_{a})Np^{−s(z|a)/κe(p)},  (9)

where σ_{𝔭} := M(𝔊_{𝔭}) ∩ σ. Since the set σ(m) contains no more than (m + 1)^{l_{1}} elements, where l_{1} is the minimal number of generators of the semigroup σ, the power series (9) converges absolutely for Re s > 0. We have

L(σ; χ, s) = ∏_{p∈S(k)} L_{p}(σ; χ, s),  (10)

and both the Dirichlet series (8) and the Euler product (10) are easily seen to converge absolutely for Re s > 1 (cf. [7-9]).

Let 𝔓 ∈ I_{0}(σ). We say that 𝔓 is prime if 𝔄 | 𝔓 ⇔ (𝔄 = 1 or 𝔄 = 𝔓), and 𝔓 is said to be a strict prime if 𝔄 | 𝔓^{m} ⇒ (𝔄 = 𝔓^{n} for some n) for 𝔄 ∈ I_{0}(σ) and m, n ranging over the non-negative integers. Each prime 𝔓 lies in I_{p}(σ) for some p in S(k), and every ideal 𝔄 in I_{0}(σ) can be factorised into primes: 𝔄 = 𝔓_{1}^{n_{1}} … 𝔓_{s}^{n_{s}}, 𝔓_{i} ≠ 𝔓_{j} for i ≠ j, 𝔓_{i} is prime, n_{i} > 0, 1 ≤ i ≤ s (assuming 𝔄 ≠ 1), although such a decomposition does not need to be unique. It follows from a combinatorial theorem in the theory of linear Diophantine equations [10, theorem 2.5] that

L(σ; χ, s) = ∏_{𝔓∈P(σ)}(1 − χ(𝔓)N𝔓^{−s/κ})^{−1} ∏_{p∈S(k)} Q_{p}(Np^{−s/e(p)κ}),

where Q_{p}(t) ∈ ℂ[t], Q_{p}(t) = 1(mod t^{κ}), and P(σ) denotes the set of strict primes (cf. [7-9]). If σ generates M then the primes generate I(σ); on the other hand, if σ = {0} then I_{0}(σ) = {1} (and consequently L(σ; χ, s) = 1 for all χ and s). Henceforth, it will therefore be assumed that

σ ≠ {0}.  (11)

Let μ denote the ℤ-rank of M(𝔊), the submodule of 𝔊-invariants. The torus T is said to be isotropic if μ ≥ 1 (and anisotropic if μ = 0). Assumption (11) implies, in particular, that T is isotropic. For let b ∈ σ\{0}, and let a = ∑_{g∈𝔊}g·b; it follows from

the definition that a ∈ M(𝔊). Moreover, since σ is 𝔊-invariant, a ∈ σ; therefore a ≠ 0 (because σ_{ℚ} is a strongly convex cone).

4. Let us investigate the properties of the map f : R^{l} → R^{d} described in Definition 2, for R = I(K) and for R = K^{∗}. Let σ(k) := {α | α ∈ K^{∗l}, gα = α^{g} for g ∈ 𝔊}, and let I(T) = {𝔄 | 𝔄 ∈ I(K)^{d}, g𝔄 = 𝔄^{g} for g ∈ 𝔊}. It follows from (6) that I(T) = ∏_{p∈S(k)} I^{(p)}(T), where

I^{(p)}(T) := {ℭ_{a} | a ∈ M(𝔊_{𝔭}), ℭ_{α} = ∏_{h∈𝔊^{(𝔭)}}(h𝔭)^{ha}}

with 𝔮^{a} := (𝔮^{a_{1}}, …, 𝔮^{a_{d}}), a_{i} := (e_{i} | a), 1 ≤ i ≤ d, for a ∈ M, 𝔮 ∈ S(K) (cf. [7-9]). By Lemma 1, f is a surjective 𝔊-homomorphism.

Lemma 2. (i) When restricted to I(σ), the map f is an isomorphism, f : I(σ) ⟶^{≅} I(T); moreover, f(𝔄_{a}) = ℭ_{a} for 𝔄_{a} ∈ I^{(p)}(σ), ℭ_{a} ∈ I^{(p)}(T), a ∈ M(𝔊_{𝔭}), p ∈ S(k). In particular, f(I^{(p)}(σ)) = I^{(p)}(T). (ii) When restricted to σ(k), the map f is an isomorphism, f : σ(k) ⟶^{≅} T (k).

Proof. It follows from the definitions by a straightforward calculation (let us note that T (k) = {α | α ∈ K^{∗d}, gα = α^{g} for g ∈ 𝔊}).

As in the cited papers [7-9], we write T (v) ≅ ℤ^{r} × A with finite A, and let T_{∞}(k) = ∏_{p|∞}T (k_{p}), where p | ∞ means that p ranges over all the infinite places of

k. There is an isomorphism

η : T_{∞}(k) → (ℝ_{+}^{∗})^{μ} × T_{∞}^{1}(k),

with

T_{∞}^{1}(k) := (ℝ_{+}^{∗})^{r} × ℂ_{1}^{d_{1}} × (ℤ/2ℤ)^{ν}, ν ≤ μ + r;

moreover, if k is a totally real field then ν = μ + r, ν + d_{1} = d (see, for instance, [8, lemma 2]). Let 𝔣 = (𝔣_{0}, 𝔣_{1}) with 𝔣_{0} ∈ I_{0}(k), 𝔣_{1} ⊆ (ℤ/2ℤ)^{ν}. By composing η with the diagonal embedding T (k) ↪ T_{∞}(k) and then projecting on (ℤ/2ℤ)^{ν}, one obtains a homomorphism

ι : T (k) → (ℤ/2ℤ)^{ν}.

Write 𝔣_{0} = ∏_{p∈S(k)}p^{m_{p}}, m_{p} ≥ 0, and let α ∈ T (k). We say that α = 1(mod 𝔣) if

ι(α) ∈ 𝔣_{1} and α_{i} = 1(mod 𝔭^{m_{p}}) for 𝔭 | p, 𝔭 ∈ S(K), 1 ≤ i ≤ d; further, one writes α = 1(mod 𝔣) for α ∈ σ(k) if ι(f(α)) ∈ 𝔣_{1} and α_{i} = 1(mod 𝔭^{m_{p}}) for 𝔭 | p, 𝔭 ∈ S(K), 1 ≤ i ≤ l. By construction,

(α = 1(mod 𝔣) for α ∈ σ(k)) ⇔ f(α) = 1(mod 𝔣).

Write, for brevity, (α) := (…, (α_{i}), …) for α ∈ K^{∗m}, 1 ≤ i ≤ m. Let

I_{𝔣}^{pr}(σ) = {(α) | α ∈ σ(k), α = 1(mod 𝔣)},

and I_{𝔣}^{pr}(T) = {(α) | α ∈ T (k), α = 1(mod 𝔣)}. One notes that (cf. (14))

f(I_{𝔣}^{pr}(σ)) = I_{𝔣}^{pr}(T), and (f(α)) = f((α)) for α ∈ K^{∗l}.  (15)

Let I_{Φ}(σ) = ∏_{p∈Φ}I^{(p)}(σ), and I_{Φ}(T) = ∏_{p∈Φ}I^{(p)}(T) for Φ ⊆ S(k). By lemma 2, we

have

f(I_{Φ}(σ)) = I_{Φ}(T).  (16)

Let, in particular, Φ = {p | p ∈ S(k), p ∤ 𝔣_{0}}, and let

H_{𝔣}(σ) = I_{Φ}(σ)/I_{𝔣}^{pr}(σ), H_{𝔣}(T) = I_{Φ}(T)/I_{𝔣}^{pr}(T).

It follows from (15) and (16) that the map f : I(σ) → I(T) induces an isomorphism

f_{1} : H_{𝔣}(σ) ⟶^{≅} H_{𝔣}(T).

Furthermore, we have [8, relation (11)]:

T_{∞}^{1}(k)/T_{𝔣}(v) ≅ T_{𝔣} with T_{𝔣} := ℂ_{1}^{d_{1}+r} × (ℤ/2ℤ)^{ν_{0}}, ν_{0} ≤ ν,

where T_{𝔣}(v) := {α | α ∈ T (v), α = 1(mod 𝔣)}. There is a homomorphism g(𝔣) : I_{Φ}(T) → H(𝔣), where H(𝔣) := T_{𝔣} × H_{𝔣}(T) × B_{𝔣}, the group B_{𝔣} being defined by the relation (14) in [8]. By composing the maps f and g(𝔣) one obtains a homomorphism

f(𝔣) : I_{0}(σ) ∩ I_{Φ}(σ) → H(𝔣).

As before [8, n^{0} 4], one writes Gr(T) := ∪_{𝔣}Gr_{𝔣}(T) for the group of normalised

grossencharacters and notes that the group Gr_{𝔣}(T) of grossencharacters modulo 𝔣 is isomorphic to H^{∨}(𝔣), the group of characters of H(𝔣). We set Gr(σ) = {χ ∘ f | χ ∈ Gr T }, and write 𝔣(χ) = 𝔣(χ ∘ f^{−1}) for χ ∈ Gr(σ). A character χ in Gr(σ) may be regarded [loc. cit] as a homomorphism χ : I_{0}(σ) → ℂ_{1} ∪ {0} with χ^{−1}({0}) =

∏_{p|𝔣_{0}(χ)}I_{p}(σ).

Lemma 3. Let χ ∈ Gr(σ), and let B be the number of different 𝔊 -orbits in σ(κ). There are number fields k_{j}, 1 ≤ j ≤ B, such that k ⊆ k_{j} ⊆ K for every j and

L(σ; χ, s) = ∏_{j=1}^{B} L(χ_{j}, s)L_{1}(σ; χ, s),  (17)

where L(χ_{j}, s) is a Hecke L-function with a (normalised) grossencharacter χ_{j} of k_{j}, the function s ↦ L_{1}(σ; χ, s) being holomorphic for Re s > κ/κ + 1.

Proof. It follows from a theorem of Draxl’s [3, Satz 1] (see [7,8] for an elementary recast of Draxl’s argument).

In the course of the proof of lemma 3 [loc. cit.] one observes that the set

E(σ) = {χ | χ ∈ Gr(σ), χ_{j} = 1 for 1 ≤ j ≤ B}

of those characters χ in Gr(σ) which give rise to trivial χ_{j} for all j in (17) is a subgroup of Gr(σ).

Lemma 4. If the set σ(κ) generates M (as an Abelian group) then the group E(σ) is finite.

Proof. See [3, Satz 1]. We have essentially reduced our problem to the theory of “maximal” toric varieties of [7-9], corresponding to the cone

σ_{0} = {a | a ∈ M, g · a ≥ 0 for g ∈ 𝔊}.

All the results and comments in [8] concerning the distribution of integral and prime ideals are valid mutatis mutandis in the present more general situation, in particular, Theorem 1, Corollaries 1-3, and Remarks 5, 6. There is a natural homomorphism

f_{2}(𝔣) : I_{0}(σ) ∩ I_{Φ}(σ) → H_{𝔣}(σ).

For C ∈ H_{𝔣}(σ), let

N (C, w) = card {𝔄 | 𝔄 ∈ f_{2}(𝔣)^{−1}(C), N𝔄 < w}.

Proposition 1. If f_{2}(𝔣)^{−1}(C) ≠ ϕ then

N (C, w) = w P_{C}(log w) + O(w^{1−γ}), γ > 0,

with P_{C}(t) ∈ ℝ[t], P_{C}(t) ≠ 0, and the degree of the polynomial P_{C}(t) is lower than B.

Proof. It corresponds to (a weaker form of) Corollary 2 in [8] and can be proved in exactly the same way as this Corollary.

5. Let now k = ℚ. In this case, T_{∞}(ℚ) = T (ℝ); therefore (12) and (13) give an isomorphism

η : T (ℝ) → (ℝ_{+}^{∗})^{ν} × ℂ_{1}^{d_{1}} × (ℤ/2ℤ)^{ν}  (18)

with ν = μ + r, ν + d_{1} = d. Since T is isomorphic to Y, the manifold Y (ℝ) is homeomorphic to T (ℝ), and the isomorphism η defined by (18) gives rise to the homeomorphism

ξ : Y (R) → (ℝ_{+}^{∗})^{ν} × ℂ_{1}^{d_{1}} × (ℤ/2ℤ)^{ν}.  (1)

We write ξ(a) := (ξ_{1}(a), …, ξ_{ν+d}(a)) for a ∈ Y (ℝ), where ξ_{i}(a) ∈ ℝ_{+}^{∗} for i ≤ ν, ξ_{ν+i}(a) ∈ ℂ_{1} for i ≤ d_{1}, and ξ_{i}(a) ∈ ℤ/2ℤ for i > d. For a ∈ Y (ℝ), let

y_{i}(a) := ∑_{1≤j≤n} a_{ij}ω_{j}^{(i)}, gy_{i}(a) := ∑_{1≤j≤n} a_{ij}gω_{j}^{(i)}, g ∈ 𝔊, 1 ≤ i ≤ l,

and let Ny(a) := ∏_{\begin{matrix}g∈𝔊 \\ 1≤i≤l \end{matrix}} gy_{i}(a), ∥ a ∥:= max\{| Ny(a) |^{1/κ}, ξ_{j}(a), ξ_{j}(a)^{−1} | μ < j ≤

ν}. It follows from the definitions that y(a) ∈ σ(ℚ) for a ∈ Y_{𝔅}(ℤ); by a slight abuse of notation, we write a = 1(mod 𝔣) if y(a) = 1(mod 𝔣) for a ∈ Y_{𝔅}(ℤ). By definition (see n^{∘} 1),

U(w) = {a | a ∈ Y (ℝ), ∥ a ∥< w}, w > 0,

and

Z_{𝔣} = {a | a ∈ Y_{𝔅}(ℤ), a = 1(mod 𝔣)}

for 𝔣 = (𝔣_{0}, 𝔣_{1}), 𝔣_{0} ∈ ℤ, 𝔣_{0} ≥ 1, 𝔣_{1} ⊆ (ℤ/2ℤ)^{ν}.

Proof of Theorem 1. By definition,

Y_{𝔅}(ℤ) = {a | a ∈ ℤ^{nl}, g · y(a) = y(a)^{g} for g ∈ 𝔊, ∏_{1≤i≤l} y_{i}(a) ≠ 0}.

The map φ : a ↦ y(a) may be regarded as an embedding

φ : Y_{𝔅}(ℤ) ↪ 𝔅 ∩ σ(ℚ).

Clearly,

φ(Z_{𝔣}) = {α | α ∈ σ(ℚ), α = 1(mod 𝔣), (α) = 𝔅𝔄, 𝔄 ∈ I_{0}(σ)}.  (19)

Let, for brevity, C := f_{2}(𝔣)^{−1}(𝔅^{−1}), and let

e(𝔄, w) = card {α | α ∈ φ(Z_{𝔣}), w^{−1} < η_{j}(f(α)) < w for μ < j ≤ ν}.

Since the map φ is injective, it follows from (19) that

card (Z_{𝔣} ∩ U(w)) = ∑_{N𝔄<w_{1}} ψ(𝔄)e(𝔄, w), w_{1} := w^{κ}N𝔅^{−1},  (20)

where ψ(𝔄) = 1 if f_{2}(𝔣)(𝔄) = C and ψ(𝔄) = 0 otherwise. Suppose that f_{2}(𝔣)(𝔄) = C, then 𝔄𝔅 = (α_{0}) for some α_{0} in σ(ℚ) with α_{0} = 1(mod 𝔣). Therefore by lemma 1, (ii)

e(𝔄, w) = card{ϵ | ϵ ∈ T_{𝔣}(ℤ), 1/w < η_{j}(ϵf(α_{0})) < w for μ < j ≤ ν},

and, since T_{𝔣}(ℤ) ≅ ℤ^{r} × A_{𝔣} with finite A_{𝔣} we have:

e(𝔄, w) = c_{1}(log w)^{r}(1 + O(1/ log w)), c_{1} > 0,  (21)

with c_{1} independent of 𝔄. The assertion of the theorem follows now from (21), (20), and Proposition 1.

Remark 3. Suppose now that B = 1 and that E(σ) is a torsion subgroup of Gr (σ). Then it follows from Lemma 3 (cf. [7-9]) that

∑_{\begin{matrix}𝔄∈I_{0}(σ) \\ N𝔄<w^{κ} \end{matrix}} χ(𝔄) = g(χ)c_{2}w + O(w^{1−γ}), γ > 0, c_{2} > 0,

with g(χ) = 0 if χ ∉ E(σ), g(χ) = L_{1}(σ; χ, 1) for χ ∈ E(σ). In particular,

∑_{\begin{matrix}N𝔄<w^{κ} \\ 𝔄∈I_{0}(σ) \end{matrix}} χ(𝔄) = O(w^{1−γ}), γ > 0, if χ is of infinite order.

In this situation, one can prove an equidistribution theorem on the number of integer points Y_{𝔅}(ℤ) ∩ U in a smooth subset U of the real locus Y (ℝ); it turns out that card (Y_{𝔅}(ℤ)∩U) is asymptotically proportional to the (appropriately defined) Borel measure of U. Since all the tools, necessary to carry out this proof, have been described elsewhere (see [6] and references therein), we leave the details of this argument to the reader.

6. Recently, in a series of interesting papers (see, for instance, [1] and references therein), V.V. Batyrev and Yu. Tschinkel have obtained an asymptotic formula for the number of rational points of bounded height on a projective toric variety. Several authors study the distribution of integer points in the real locus of an affine homogeneous space of a semisimple group (cf. [2] and references therein). It is instructive to compare the results and methods of the cited papers with the approach adopted here; let us note, in particular, the important role played by Draxl L-functions in [1], and refer to the Introduction of [2] for a discussion of the general problem of counting the number of integer solutions of Diophantine equations (cf. also [6], n^{∘} 1).

Acknowledgement. This paper was written during my stay at the Max-Planck-Institut für Mathematik in Bonn. It is a pleasure to thank the Institute for its unfailing hospitality and financial support.

References

[1] V.V. Batyrev, Yu. Tschinkel, Manin’s conjecture for toric varieties, Journal of Algebraic Geometry, \underline{7} (1998), 15-53.

[2] M. Borovoi, Z. Rudnik, Hardy-Littlewood varieties and semisimple groups, Inventiones mathematicae, \underline{119} (1995), 37-66.

[3] P.K.J. Draxl, L-Funktionen algebraischer Tori, Journal of Number Theory, \underline{3} (1971), 444-467.

[4] W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, \underline{131} (1993), Princeton University Press.

[5] S. Lang, Number Theory III, Springer-Verlag, 1991.

[6] B.Z. Moroz, On the distribution of integral points on an algebraic torus defined by a system of norm-form equations, Quarterly Journal of Mathematics Oxford (2), \underline{45} (1994), 243-253.

[7] B.Z. Moroz, Exercises in analytic arithmetic on an algebraic torus, The F. Hirzebruch Festband, Israel Mathematical Conferences Proceedings, \underline{9} (1995), 347-359.

[8] B.Z. Moroz, On the integer points of some toric varieties, Quarterly Journal of Mathematics Oxford (2), \underline{48} (1997), 67-82.

[9] B.Z. Moroz, On the distribution of integer points in the real locus of an affine toric variety, Proceedings of the Symposium on sieve methods, exponential sums, and their applications in number theory, LMS Lecture Notes, \underline{237} (1996), 283-291.

[10] R.P. Stanley, Linear homogeneous Diophantine equations and magic labelings of graphs, Duke Mathematical Journal, \underline{40} (1973), 607-632.

[11] A. Weil, Adeles and algebraic groups, Birkhäuser (1982).

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