SMALL DISCRIMINANTS, CLASS NUMBERS, AND THE GEOMETRY OF NUMBERS

JACQUES MARTINET

Abstract. We discuss some problems related to discriminants and class groups of number fields, in connection with the Minkowski domains of dimension n = r_{1} + 2r_{2} attached to number fields of signature (r_{1}, r_{2}).

1. Some notation for Geometry of Numbers

First we consider pure geometry of numbers. Let E be a Euclidean space, of dimension n. The scalar product on E is denoted by x · y, and the norm of x ∈ E is N(x) = x · x, the square of the traditional norm ∥x∥.

Let Λ be a lattice in E, that is a discrete subgroup of E of maximal rank. Thus Λ has a basis B = (e_{1}, …, e_{n}) over ℤ, with which we associate its Gram matrix Gram(B) = (e_{i} · e_{j}), the determinant of which, clearly independent of B, is the determinant of Λ: det(Λ) = det Gram(B). The traditional notion (Minkowski) is the discriminant Δ(Λ) of Λ, the absolute value of the determinant of B with respect to an orthonormal basis for E. We have

det(Λ) = Δ(Λ)^{2} and N(x) = ∥x∥^{2} .

Introducing these squares simplifies many formulae. Moreover the notion of the determinant fits (up to sign) with that of the discriminant in number fields; see Section 2. Note that our lattice constants to be defined below will be the squares of the usual ones.

Let A ⊂ E. Recall that a lattice Λ is admissible for A if Λ ∩ A is reduced to {0} or is empty. We then define the lattice constant of A by

κ(A) = inf_{Λ admissible} det(Λ)

(κ(Λ) = +∞ if there are no admissible lattices for A). In the cases we shall consider, A will be an open , symmetric star body (with respect to the origin), that is, A is open in E, and for all x ∈ A and λ ∈ ℝ with |λ| ≤ 1, then λ x ∈ A; and moreover, our sets A will be of the form

Key words and phrases. Euclidean lattices, Minkowski domains October, 2010 January 2011.

A_{F} = {x ∈ E | F (x) < 1

where F is a distance-function, that is F is a continuous, non-negative real function, “homogeneous” of degree δ for some δ > 0, i.e., F satisfies the identity

∀ x ∈ E, ∀ λ ∈ ℝ , F (λx) = |λ|^{δ} F (x) .

[Conversely, modulo mild conditions on the boundary, every open star body is of the form A_{F} for some F.]

The definition of the lattice constant κ(A) of an open, symmetric star body A is often used in the following form: if det(Λ) < κ(A), then A contains a non-zero point of Λ, and even, for bounded A, if det(Λ) ≤ κ(A), then \overline{A} contains a non-zero point of Λ This last result extends to unbounded domains for which we may apply compactness results, or under some restrictions on Λ. This notably works for lattices related to modules inside number fields; see Section 2.

With an F as above we associate the following two functions on the space of lattices:

F (Λ) = inf_{x∈Λ∖\{0\}} F (x) and γ_{F} (Λ) = \frac{F (Λ)}{det(Λ)^{δ/2n}} .

Finally we set γ_{n,F} = sup_{Λ} γ_{F} (Λ) . The relation between κ and γ_{n,F} is easy to establish; it reads

κ(A_{F}) = γ_{n,F}^{−2n/d} .

An important example of distance-function is the norm, of degree 2. Then γ_{N}(Λ), denoted simply by γ(Λ), is the Hermite invariant of Λ, and γ_{n,N}, denoted simply by γ_{n}, is the Hermite constant for dimension n. We also denote by κ_{n} the lattice constant of the n-dimension unit ball; we have κ_{n} = γ_{n}^{−n}.

2. Connection with number fields

We denote by K a number field, of signature (r_{1}, r_{2}) and degree n = r_{1} + 2r_{2}, and order the embeddings σ_{i} : K → ℂ in such a way that σ_{j} be real for j ≤ r_{1} and (r_{j}, r_{j+r_{2}}) be complex conjugates for r_{1} + 1 < j ≤ r_{1} + r_{2}. We shall attach to K a Euclidean space K̂ and an open star body in K̂ (a Minkowski domain), and to each finitely generated, full submodule of K a lattice in K̂. Proofs for the results stated in this section can be read in the second chapter of [Mar].

Denote by K̂ the completion of K for any infinite norm on K identified with ℚ^{n}. This is an étale ℝ-algebra, canonically isomorphic to ℝ ⊗_{ℚ} K, thus isomorphic (non-canonically) toℝ^{r_{1}} × ℝ^{r_{2}}. Nevertheless the canonical involution of this last algebra (the identity on the real factors and the conjugacy on the complex ones) induces a canonical involution x ↦ \overline{x} on K̂, for which Tr_{K̂/ℝ} (x\overline{x}) is a scalar product. We provisionally define a Minkowski domain inside K̂ by |N_{K̂/ℝ} (x)| < 1.

To perform explicit calculations we shall identify K̂ with ℝ^{n}. this is done using coordinates x_{j} (j = 1, …, r_{1}) and y_{j}, z_{j} (j = 1, …, r_{2}), and embedding K in ℝ^{n} by

1_{K} ↦ 111 := (1, 1, …, 1) .

For z = a + bi ∈ ℂ, we have Tr_{ℂ/ℝ}(z\overline{z}) = 2(a^{2} + b^{2}). To take into account these factors 2 attached to all complex components, we define Minkowski domains in ℝ^{n} as follows.

Definition 2.1. Let

F_{r_{1},r_{2}} = 2^{−r_{2}} · |x_{1}| ⋯ |x_{r_{1}}| · (y_{1}^{2} + z_{1}^{2}) ⋯ (y_{r_{2}}^{2} + z_{r_{2}}^{2}) .

The Minkowski domain for signature (r_{1}, r_{2}) (r_{1} + 2r_{2} = n) in ℝ^{n} is

A_{r_{1},r_{2}} = {x ∈ ℝ^{n} | F_{r_{1},r_{2}}(x) < 1} .

Its lattice constant is denoted by κ_{r_{1},r_{2}}.

We observe that 111 belongs to the boundary of the domain. Moreover, the introduction of the term 2^{r_{2}} allows us to compare different domains in the same dimension.

Proposition 2.2. Let r_{1}, r_{2}, r′_{1}, r′_{2} such that r_{1} + 2r_{2} = r′_{1} + 2r′_{2}. If r′_{1} ≥ r_{1}, we have the inclusion

A_{r_{1},r_{2}} ⊂ A_{r′_{1},r′_{2}} .

Proof. This results by induction from the inclusion A_{r_{1}−2,r_{2}+1} ⊂ A_{r_{1},r_{2}} (r_{1} ≥ 2), which follows from the arithmetico-geometric inequality |y_{1}z_{1}| ≤ \frac{y_{1}^{2}+z_{1}^{2}}{2} . □

We now turn to the construction of lattices from submodules of K. By a module in K, we mean a finitely generated ℤ-submodule of K of rank n = [K : ℚ]. Such a module M has a basis (ω_{1}, …, ω_{n}) over ℤ, and a discriminant

d_{K}(M) = det(Tr_{K/ℚ}(ω_{i}ω_{j})) ∈ ℚ^{×}

(for short, d_{K} when M = ℤ_{K}, the ring of integers of K), which has the sign of (−1)^{r_{2}}. Since M is discrete in K, it is its own closure in K̂, and may consequently be identified with a lattice in K̂. Since the twist x ↦ \overline{x} multiplies determinants by (−1)^{r_{2}}, we have

det(M) = |d_{K}(M)| .

We now turn to the norm map N_{K/ℚ} on M∖{0}. Let q be a positive integer such that q ω_{i} ∈ ℤ_{K} for all i. Then we have q M ⊂ ℤ_{K}, which shows that the norms of elements of M are integral multiples of q^{−n}. In particular, |N_{K/ℚ}(x)| attains a minimum on M∖{0}. Rescaling the image of M in K̂, we obtain a lattice on which the norm function has minimum 1. Its image in ℝ^{n} is then an admissible lattice for A_{r_{1},r_{2}} having at least one pair ±x on its boundary.

Definition 2.3. We say that an admissible lattice for A_{r_{1},r_{2}} is algebraic if it is the image in ℝ^{n} of a module in some field of signature (r_{1}, r_{2}).

Remark 2.4. It results from the discussion above that for all domains A_{r_{1},r_{2}}, there exist admissible lattices which are algebraic, and from Proposition 2.2 that non-algebraic lattices also exist for domains with r_{2} > 0. It can be shown that non-algebraic lattices also exist in the case of A_{2,0}; see Section 5. The status of domains A_{n,0} is not known from n = 3 onwards.

Recall that an order in a number field K is a subring of K of maximal rank (i.e., containing a basis of K/ℚ) which is a finitely generated

ℤ-module. (This last condition is equivalent to the fact that all elements of an order are integral over ℤ. Hence the integral closure ℤ_{K} of ℤ in K is the

unique maximal order of K.) With every module we attach its associated order

𝔒(M) = {λ ∈ K | λ𝔒 ⊂ 𝔒} ,

the largest subring of K which stabilizes M. Thus modules in K are nothing but fractional ideals over some order, the largest of which is the associated order. (Fractional ideals of an order 𝔒 the associated order of which is 𝔒 itself are called proper.) The famous Dirichlet unit theorem asserts that the set 𝔒^{×} of invertible elements of 𝔒 is the direct product of the cyclic group μ_{𝔒} of roots of unity which belong to 𝔒 and a free Abelian group of rank r_{1} + r_{2} − 1.

Given an order 𝔒, a fractional ideal of 𝔒 is a non-zero sub-𝔒-module of K which is finitely generated over 𝔒 (or over ℤ, this amounts to the same). The rule

𝔞𝔟 = {∑ x_{i}y_{i} | x_{i} ∈ 𝔞, y_{i} ∈ 𝔟}

defines a multiplication on the set of fractional ideals of 𝔒, which is associative and commutative, with neutral element 𝔒. The invertible (fractional) ideals constitute a group, denoted by I_{𝔒}, which contains as a subgroup the set of principal fractional ideals, denoted by P_{𝔒}. The quotient group Cl_{𝔒} is the class group of 𝔒. This is a finite group. We also write I_{K}, P_{K} and Cl_{K} when 𝔒 = ℤ_{K}.

An important problem concerning (unbounded) star bodies is that of the rank of points belonging to the boundary (which can be zero). We shall prove (Theorem 4.12 below) that for every minimal-admissible lattice for a Minkowski domain, there exists a minimal-admissible having the same determinant and containing 111. For algebraic lattices, Dirichlet’s units theorem shows that the boundary of the domain contains a set of lattice points of rank at least r_{1} + r_{2}, thus the maximal value n in the totally real case. In some cases a better lower bound holds. For A_{2,0}, this rank is 1 except when the order 𝔒 is the ring of Eisenstein or of Gaussian integers, which contain roots of unity of order 4 or 6.

The main application of the lattice constants of Minkowski domains is the following theorem (which results from the very definition of lattice constants) and the corollary which follows, proved using the “trick of the inverse class” (thus, the result extends to classes of invertible ideal in orders).

Theorem 2.5. Let M be a module in a number field K of signature (r_{1}, r_{2}). Then there exists non-zero x ∈ M such that

|N_{K/ℚ}(x)| ≤ \sqrt{\frac{|d_{K}(M)|}{κ_{r_{1},r_{2}}}} . □

Theorem 2.6. Every ideal class of an order 𝔒 ⊂ K contains an integral ideal 𝔞 of norm

N_{K/ℚ}(𝔞) ≤ \sqrt{\frac{|d_{𝔒}|}{κ_{r_{1},r_{2}}}} . □

Minkowski, applying his lower bound based on the volume for the lattice constant of a convex set to

B_{r_{1},r_{2}} = {x ∈ K̂ | |x_{1}| + ⋯ + |x_{r_{1}}| + 2|z_{1}| + ⋯ + 2|z_{r_{2}}| < 1}

and using the inclusion

n2^{r_{2}/n}B_{r_{1},r_{2}} ⊂ A_{r_{1},r_{2}} ,

obtained the famous lower bound

\sqrt{κ_{r_{1},r_{2}}} ≥ (\frac{π}{4} )^{r_{2}} \frac{n^{n}}{n!} ,

with which he could prove Kronecker’s conjecture, namely that for any number field of degree n ≥ 2, one has |d_{K}| > 1. This result was first written in a letter to Hilbert of December 22nd, 1890 ([MBH]), then in a letter to Hermite of January 15th, 1891, partially published by Hermite in C.R. Acad. Sc. Paris. In this last letter, Minkowski remarks that using the ball of largest possible radius inside A_{1,1} yields a slightly better result than using B_{1,1}: our pretty good knowledge of the Hermite constants compensate the disadvantage of having a small volume. The optimal radius is easily calculated using the arithmeticogeometric inequality and gives the following lower bound.

Theorem 2.7. We have

κ_{r_{1},r_{2}} ≥ n^{n}κ_{n} = (\frac{n}{γ_{n}} )^{n} . □

The improvement on Minkowski’s original lower bound for κ_{r_{1},r_{2}} is particularly significant for totally imaginary domains^{1} in dimensions 2 (it is then optimal), 4, 6, and 8.

The exact values of κ_{r_{1},r_{2}} and the corresponding critical lattices (see Definition 4.4 below) are known for n = 2 and 3: κ_{0,1} = 3, κ_{2,0} = 5, κ_{1,1} = 23, and κ_{3,0} = 49. In all cases the critical lattices correspond to the ring of integers of the fields of minimal discriminant. Such a result conjecturally holds in small dimensions. Thus one conjectures that the lattice constants are 113, 275, and 725 for n = 4, and 1609, 4511, and 14641 = 11^{4} for n = 5.

The best known results for A_{4,0} and A_{5,0} are those of Noordzij ([Noo], κ_{4,0} ≥ 500) and Godwin ([God], κ_{5,0} > 3251.2 …). Asymptotic lower bounds improving on Minkowski’s have been obtained by C. A. Rogers in the totally real case (The product of n real homogeneous linear forms, Acta Math. 82 (1950), 185–208) and by H. P. Mullholand in general (On the product of n complex homogeneous linear forms, J. London Math. Soc. 35 (1960), 241–250.

3. Small discriminants of number fields

In this section we briefly discuss the following question: what are the first smallest discriminants (in absolute value) in a given signature (r_{1}, r_{2}) ? That one must take into account not only the degree n but also the signature (r_{1}, r_{2}) is clearly shown both by the structure of Minkowski’s lower bound and by experimental data. More precisely the aim is, given a real number B > 0, to find reasonably small bounds for the coefficients of polynomials that may be used to define all fields of signature (r_{1}, r_{2}) with |d_{K}| ≤ B.

1I first saw this trick in work of Descombes and Poitou devoted to approximation of complex numbers by elements of imaginary quadratic fields. This problem is related to admissible lattices for A_{0,2} containing the image of an imaginary quadratic field, like approximation of real numbers by rational numbers is related to admissible lattices for the domain A_{2,0}.

3.1. Bounds for defining polynomials. The basic idea to handle a number field K (for which we refer to [Mar1]) consists in using the orthogonal projection in K̂ to 111^{⊥}, and to apply results from geometry of numbers to the projection of ℤ_{K} (or more generally, of any order in K). One then bounds in terms of |d_{K}| the size of the projection of a convenient element of ℤ_{K}, hence of a pull back θ ∈ ℤ_{K} not in ℚ. This is the method used in [D-F] for cubic fields,^{2} formalized in general by Hunter in [Hun]. In this paper, he obtains a bound in terms of the discriminant d_{K} for the conjugates of some integer θ ∈ K not in ℚ. For a prime degree n, such a θ generates the field. Hunter found this way the smallest discriminants of quintic fields of any of the three signatures.

When n is not a prime, n′ = [ℚ(θ) : ℚ] may be a a strict divisor of n. One can then either consider successive minima, or prove a relative form of Hunter’s theorem, which is more efficient for computation. Such a generalization is proved using this time orthogonal projections in K̂ onto K̂′.

[Given a k-dimensional subspace in a Euclidean space E, equipped with an orthogonal basis (e_{1}, …, e_{k}), the projection p_{F^{⊥}} of E to F^{⊥} is given by the formula

p_{F^{⊥}}(x) = x − ∑_{i=1}^{k} \frac{x·e_{k}}{e_{k}·e_{k}} x;

just observe that p_{F^{⊥}} maps F to {0} and is the identity on F^{⊥}. This implies that the norm of the projection is

N(p(x)) = N(x) − ∑_{i=1}^{k} \frac{(x · e_{i})^{2}}{e_{i} · e_{i}} .

For instance, if E = ℝ^{n} and F = ℝ111, x = (x_{1}, …, x_{n}) is mapped onto

N(p(x)) = ∑_{i} x_{i}^{2} − \frac{1}{n} ( ∑_{i} x_{i})^{2} = \frac{1}{n} ∑_{1≤i<j≤n} (x_{j} − x_{i})^{2} .]

To state this extension of Hunter’s theorem, we need some notation. We consider a subfield K′ of K, of degree n′, and denote by I(K′) the set of embeddings σ : K′ → ℂ.

2The minimal discriminant of cubic fields in both signatures (namely, −23 and +49) were found by Philipp Furtwängler (Zur Theorie der in Linearfaktoren zerlegbaren ganzzahlingen ternären kubischen Formen, Dissertation, Göttingen, 1896; advisor: Felix Klein).

Definition 3.1. For θ ∈ K, let

S_{K/K′}(θ) = ∑_{σ∈I(K′)} ∑_{τ,τ′|σ}|τ(θ) − τ′(θ)|^{2} ,

and let S_{K}(θ) = S_{K/ℚ}(θ).

(The notation τ | σ means that τ ∈ I(K) extends σ ∈ I(K′).)

In analogy with S_{K}, given complex numbers z_{1}, …, z_{n}, we set

S(z_{1}, …, z_{n}) = ∑_{i<j}|z_{j} − z_{i}|,

and for a polynomial f ∈ ℂ[X] of degree n, we set S_{f} = S(z_{1}, …, z_{n}) where the z_{i} are the roots of f.

Remark 3.2. (1) S_{K/K′} is invariant under translations by elements of K′.

(2) If θ belongs to an intermediate field L, we may consider both S_{L/K′} and S_{K/K′}. Then S_{K/K′}(θ) = [K : L]^{2}S_{L/K′}(θ). In particular, S_{K}(θ) = [K : ℚ]^{2} if θ ∈ ℤ.

(3) The condition S_{K} > 0 (a necessary condition for f to have n distinct real roots) holds if and only if f^{(n−2)} has two distinct real roots.

Proposition 3.3. With the notation of Definition 3.1, let m = \frac{n}{n′} be the relative degree [K : K′], and for θ ∈ K and σ ∈ I(K′), let Tr_{σ,k/K′}(θ) = ∑_{τ|σ} τθ. We have

m ∑_{τ∈I(K)} |τ(θ)|^{2} − ∑_{σ∈I(K′)} |Tr_{σ,k/K′}(θ)|^{2} = S_{K/K′}(θ) .

Proof. This is an application of the following well known formula in quadratic affine spaces (E, q) involving n + 1 point O, M_{1}, …, M_{n}:

∑_{i<j} q(\overrightarrow{M_{i}M_{j}}) = m ∑_{i} q(\overrightarrow{OM_{i}}) − q(∑_{i} \overrightarrow{OM_{i}}) .

□

Theorem 3.4. Let K be a number field of degree n and let K′ be a strict subfield of K, of degree n′. Then there exists an integer θ ∈ K, θ not in K′, such that

S_{K/K′}(θ) ≤ m γ_{n−n′} |\frac{d_{K}}{m^{n′}d_{K′}} |^{1/(n−n′)} .

(Recall that γ_{n−n′} is the Hermite constant for dimension n − n′).

Proof. Just make explicit the projection in K̂ to K̂′; the details can be read in [Mar1]. □

Corollary 3.5. (Hunter) Every number field of degree n > 1 contains an element θ not in ℚ, the conjugates θ_{i} of which satisfy the condition

n ∑_{i=1}^{n} |θ_{i}|^{2} − Tr_{K/Q}(θ)^{2} = ∑_{i<j}|θ_{j} − θ_{i}|^{2} ≤ nγ_{n−1} (\frac{|d_{K}|}{n})^{1/(n−1)} .

Note that γ_{n−1}^{n−1} = κ_{n−1}^{−1}, a rational number. We may thus write the inequality above as

S_{K}(θ) ≤ (n^{n−2}γ_{n−1}^{n−1} |d_{K}|)^{1/(n−1)},

say, S_{K}(θ) ≤ (α_{n}d_{K})^{1/(n−1)}, with a rational coefficient α_{n}.

This inequality is optimal when n = 2 (γ_{1} = 1). When n = 3, this is optimal if and only if K is cyclic (the projection must be a hexagonal lattice). Optimality is not expected in larger degrees. At any rate, the Hermite constant is not significant for the problem of small discriminants. Other bodies could have been used, but the loss of information mainly comes from the bad quality of the bounds of the elementary symmetric functions of characteristic polynomial.

We now indicate briefly how one can use crude inequalities deduced from theorem 3.4 to prove the following theorem:

Theorem 3.6. (Hermite) Let B > 0. Then there are only finitely number fields (up to isomorphism, or inside a given algebraic closure \overline{ℚ} of ℚ) such that |d_{K}| ≤ B.

Proof. Minkowski’s Theorem 2.6 shows that the discriminants of number fields tend to ∞ with the degree. Hence it suffices to consider number fields having a given degree n.^{3}

For every non-trivial divisor m of n, one must consider the possibility that K may contain a strict subfield K′ ≠ ℚ with [K : K′] = m (including m = n). The transitivity formula of discriminants then shows that |d_{K′}| ≤ B^{1/m},^{4} so that we know by induction that there are only finitely many fields K′ ⊂ \overline{ℚ} to consider, and we may moreover assume that K/K′ is primitive.

Fix such a field K′. Using the invariance of S_{K/K′} under translations by integers of K′, we may choose Tr_{σ,K/K′}(θ) among m^{[K′:ℚ]} values (and indeed less, using the action of roots of unity over ℚ, it suffices to consider the ⌈\frac{n+1}{2}⌉ values 0, 1, …, ⌊\frac{n}{2}⌋).

3Hermite proved his theorem before geometry of numbers was invented, and used inequalities from the theory of positive definite quadratic forms. Proofs using geometry of numbers are simpler. The same comments apply to Dirichlet’s unit theorem.

4We have d_{K} = d_{K′}^{m} only if K/K′ is unramified; otherwise, a “relative Stickelberger congruence” will show that |d_{K}/d_{K′}| ≥ 3; one can also bound |d′_{K}| using inequality 2.12 of [Mar1].

Fix now the value Tr_{σ,K/K′}(θ) for a θ as in Theorem 3.4. Then Theorem 3.4 and Proposition 3.3 bound the absolute values of the conjugate of θ over K′, hence the conjugates of the coefficients of the minimal polynomial of θ over K′. There are thus only finitely many possible characteristic polynomials of K over ℚ. □

Notation 3.7. For θ ∈ K, we denote by s_{1}, …, s_{m} the elementary symmetric functions of the conjugates of θ over K′. Thus the characteristic polynomial of θ over K′ is

f(X) = X^{m} − s_{1}X^{m−1} + ⋯ + (−1)^{m} .

The proof above can hardly be effectively applied to list the small discriminants. In practice, one must use various tricks to bound more efficiently the coefficients of the polynomials. In the case of totally real fields, that we shall illustrate below with various examples, the absolute values which occur in the Theorems above disappear, which allows one to write more precise inequalities. Otherwise it is difficult to take into account the signature, except that one will find naturally the smallest discriminants when there is at most one real place. Actually, the minimal discriminants are known for all signatures up to n = 7, but in degree 8, only for fields which are either totally imaginary (Diaz y Diaz)^{5}or totally real (Pohst-Martinet-Diaz y Diaz)^{6}; and in this last case, the authors had to use local corrections; see Appendix an-geo.

Remark 3.8. Theorem 3.4 bounds S_{K/K′} in terms of the relative discriminant of some θ ∈ K. Conversely we can use the arithmetico-geometric inequality to bound this discriminant in terms of S_{K/K′}. We obtain

|N_{K′/ℚ}(d_{K/K′}(θ))| ≤ (\frac{S_{K/K′}(θ)}{n(m − 1)/2})^{n(m−1)/2} .

3.2. Some evaluations for totally real fields. We now restrict ourselves to the case when K′ = ℚ and K is totally real. and consider an integer θ ∈ K, with characteristic polynomial

f(X) = X^{n} − s_{1}X^{m−1} + s_{2}X^{m−2} + ⋯ + (−1)^{n} .

and conjugates θ_{1} ≤ θ_{2} ≤ ⋯ ≤ θ_{n}. When applying Hunter’s theorem, we shall assume that t_{1} ∈ \{0, 1, …, \frac{n}{2}\} and that t_{3} > 0 if t_{1} = 0.

5Petits discriminants des corps de nombres totalement imaginaires de degré 8, J. Number Theory 25 (1987), 34–52.

6The minimum discriminant of totally real octic fields, J. Number Theory 36 (1990), 145–159

We have

S_{K}(θ) = ∑_{1≤i<j≤n}(θ_{j} − θ_{i})^{2} = (n − 1) ∑_{i=1}^{n} θ_{i}^{2} − 2 ∑_{1≤i<j≤n} θ_{i}θ_{j} = (n − 1) t_{1}^{2} − 2n t_{2} .

Example 3.9. Totally real cubic fields.

For real cubic fields, Corollary 3.5 reads S_{K} ≤ \sqrt{4d_{K}} , and we have t_{1} = 0 or t_{1} = 1. We use the formula d_{g} = −4p^{3} − 27q^{2} for the discriminant of g = X^{3} + pX + q, which we apply with g = f if t_{1} = 0, and after the translation X ↦ X + \frac{1}{3} if t_{1} = 1.

Take B = 169. By Theorem 3.5, it suffices to consider fields K with S_{K} ≤ 26. Using the condition d_{f} > 0, getting rid of reducible polynomials, and discarding polynomials having a too large discriminant (warning: one must take care of square divisors of d_{f} which might produce fields for which d_{K} is a strict divisor of d_{f}), we find the following results:

• s_{1} = 0 (S ≡ 0 mod 6). Then s_{2} = 3, S = 18, s_{3} = 1 (d_{k} = 81); or s_{2} = 4, S = 24, s_{3} = 1 (d_{K} = 229) or s_{3} = 2 (d_{K} = 148); or S ≥ 30.

• s_{1} = 1 (S ≡ 2 mod 6). Then s_{2} = 2, S = 14, s_{3} = 1 (d_{k} = 49); or s_{2} = 3, S = 20, s_{3} = 1 (d_{k} = 148); or s_{2} = 4, S = 26, s_{3} = −1 (d_{k} = 169), or s_{3} ∈ {1, 2, 3} and d_{K} > 169; or S ≥ 32.

We note that two polynomials (f, with S = 20, and g, with S = 24) have discriminant 148. We check that if θ is a root of f, then θ^{2} − θ − 2 is a root of g. We have thus proved:

Proposition 3.10. The first three values of S_{K} for a totally real cubic field are S = 14, 18, 20, attained exactly on the fields having the smallest three discriminants, namely d_{K} = 49, 81, 148, defined by the polynomials X^{3} − X^{2} − 2X + 1, X^{3} − 3X − 1, and X^{3} − X^{2} − 3X + 1, respectively. The fourth field has d_{K} = 169 and S = 24, and is defined by the polynomial X^{3} − X^{2} − 4X − 1.

Using a method of Davenport, we shall prove in Section 6 that the lower bounds S = 14 and d = 49 are attained uniquely among admissible lattices for A_{3,0} on the ring of integers of field with discriminant 49.

Example 3.11. Totally real quartic fields.

In degree 4 the sign of the discriminant does not suffice to decide whether a polynomial has 0 or 4 real roots. We first recall a result which solves this problem. For a proof, see [D-F].^{7}

Proposition 3.12. Let f = X^{4} −aX^{3} +bX^{2} −cX +d ∈ ℝ[X]. Then f has 4 real roots if and only if the following three conditions are satisfied:

(1) d_{f} > 0.

7However in [D-F] condition (2[?]) is written with a wrong sign

(2) 3a^{2} − 8b > 0.

(3) b^{2} − a^{2}b + \frac{3}{16}a^{4} + ac − 4d > 0.

We now consider a polynomial f = X^{4} − s_{1}X^{3} + s_{2}X^{2} − s_{3}X + s_{4} ∈ ℚ[X]. Theorem 3.5 reads S_{K} ≤ (32 d_{K})^{1/3}. Applied with the two known discriminants obtained as extensions of K_{0} = ℚ(\sqrt{5}), namely 725 = 5^{2} · 29 and 1125 = 3^{2} · 5^{3} (corresponding to the maximal real subfield of ℚ(σ_{15})), we obtain for S_{K} the bounds 28 and 32, respectively.

We have S_{f} ≡ 3s_{1}^{2} mod 8, that is (S_{f} mod 8, s_{1}) = (0, 0), (3, 1), or (4, 2). For each a priori possible choice S for S_{f}, we bound s_{4}^{2} by the arithmeticogeometric inequality between the squares of the roots of f, which gives

|s_{4}| < (\frac{S+s_{1}^{2}}{16} )^{1/2}. For s_{3}, we use the four bounds of type

|θ_{1}θ_{2}θ_{3}| < (\frac{S + s_{1}^{2} − θ_{4}^{2}}{12} )^{3/2} < (\frac{S + s_{1}^{2}}{12} )^{3/2}

and observe that the θ_{i} cannot have the same sign, and thus bound |s_{3}| by

2 (\frac{S+s_{1}^{2}}{12} )^{3/2} if s_{4} > 0 and by 3 (\frac{S+s_{1}^{2}}{12} )^{3/2} if s_{4} < 0. These somewhat crude estimates suffice to classify the polynomials associated with fields having S_{K} ≤ 32 ; ω stands for the golden ratio \frac{1+\sqrt{5}}{2} :

S_{K} = 27: d_{K} = 725, f = X^{4} −X^{3} −3X^{2} +X +1 = N_{K/ℚ(\sqrt{5})}(X^{2} −ωX −1). S_{K} = 28: d_{K} = 725, f = X^{4}−2X^{3}−2X^{2}+3X+1 = N_{K/ℚ(\sqrt{5})}(X^{2}−X−ω^{2}). S_{K} = 32: we find three polynomials, with discriminants

1957 (f = X^{4} − 4X^{2} + X + 1), a primitive field;

2048 = 2^{11} (f = X^{4} − 4X^{2} + 2, K = ℚ(\sqrt{2 + \sqrt{2}}));

2304 = 2^{8} · 3^{2} (f = X^{4} − 4X^{2} + 1, K = ℚ(\sqrt{2}, \sqrt{3})). We observe that we have not found the Abelian field ℚ(ζ_{15})^{+}, of discriminant 1125, which shows the necessity of also classifying the imprimitive fields. This field needs S_{K} ≥ 35, and the minimal polynomial of ζ_{15}+ζ_{15}^{−1} is actually f = X^{4} − X^{3} − 4X^{2} + 4X + 1, for which S_{f} = 35.

Using the bound in Theorem 3.4, or using class field theory, we check that for d_{K} < 1400 (the bound which ensures S_{K} < 35), there are only the two imprimitive fields quoted above. We have thus proved:

Proposition 3.13. Let K be a totally real field of degree 4. Then:

(1) Either K is one of the two fields with discriminants 725 and 1125 quoted above, or we have d_{K} ≥ 1400.

(2) The three smallest values for S_{K} are S_{K} = 27, attained on the field with discriminant 725, S_{K} = 32, attained on one of three fields with discriminants 1957, 2048 or 2304, and S_{K} = 35, attained in particular on the field with discriminant 1125. □

Example 3.14. Totally real quintic fields. Here Corollary 3.5 reads

S_{K} ≤ (500|d_{K}|)^{1/4}. The maximal real subfield ℚ(ζ_{11})^{+} of ℚ(ζ_{11}) has discriminant 11^{4} = 14641, and may be defined by the polynomial X^{5} − X^{4} − 4X^{3} + 3X^{2} + 3X − 1, the minimal polynomial of −(ζ_{11} + ζ_{11}^{−1}), for which S = 44.

To complete after a PARI run

4. More on Minkowski’s domains

In this section, after having introduced in the first subsection some notions which apply to general distance-functions and open star bodies (always with respect to the origin, and assumed to be symmetric) in a Euclidean space E, we consider the distance-functions F = F_{r_{1},r_{2}} on ℝ^{n}, n = r_{1} + 2r_{2} (Definition 2.1) and the corresponding Minkowski domains A_{r_{1},r_{2}} defined by F (x) < 1, first for any signature, then in the totally real case.

4.1. Automorphisms and isolation phenomena.

Definition 4.1. We say that u ∈ GL(E) is an automorphism of a star body A (resp. of a distance-functions F) if u(A) = A (resp. if F (u(x)) = F (x) for all x ∈ E). Notation: Aut(A), Aut(F).

Clearly, Aut(A_{F}) = Aut(F). These automorphism groups are closed subgroups of GL(E), hence Lie groups, thus have a well-defined dimension. As in any topological group, their connected components are closed, normal subgroups.

Example 4.2. Quadratic domains.

Let q be a non-degenerate quadratic form. Then the automorphism group of the distance-function |q| is the orthogonal group O(q) of q. Its connected component has index 2 if q is definite, 4 if q is indefinite.

Proposition 4.3. Let A be an open star body (with respect to the origin). If there exist admissible lattices for A, then any automorphism of A has determinant ±1.

Proof. Assume that Λ is admissible for A, and that there exists u ∈ Aut(A) of determinant not ±1. Changing u into u^{−1} if need be, we may assume that det(u) < 1. Then Λ_{n} = u^{n}(Λ) is admissible for all n, and we have lim_{n→∞} det(Λ_{n}) = 0, which contradicts the existence of a lower bound for the determinant of an admissible lattice of any neighbourhood of 0. □

Definition 4.4. We say that a lattice Λ is minimal-admissible for an open star body A if Λ is admissible for A but the lattices λA are not if λ < 1, that Λ is extreme for A if Λ is admissible for A and its determinant is a local minimum on the set lattices which are admissible for A, and that Λ is critical (or absolutely extreme for A) if its determinant is the smallest possible among that of all lattices which are admissible for A (that is, if det(Λ) = κ(A), the lattice constant of A).

Proposition 4.5. Let A be an open, symmetric star body (with respect to the origin). Assume that admissible lattices for A exist. Then there also exist critical lattices for A.

Proof. Since the set of admissible lattices for A is not empty and A is a neighbourhood of the origin, we have 0 < κ(A) < +∞. For n > 0, there exists an admissible lattice Λ_{n} with κ(A) ≤ det(Λ_{n}) ≤ κ(A) + \frac{1}{n}. On this family, the determinants are bounded from above (by det(Λ_{1})) and the Euclidean distance to 0 is bounded from below (by the radius of a small enough ball centred at the origin). By Mahler’s compactness lemma, one can extract from (Λ_{n}) a sub-sequence which converges to a lattice Λ. We have det(Λ) = κ(A), and A ∩ (Λ∖{0}) is empty since ∁A is closed. □

It is easy to prove that if A is a bounded star body, then admissible, hence also critical lattices exist, and that extreme lattices have n independent points on the boundary of A; see [Cas2], V.6. (If A is convex, extreme lattices even have at least \frac{n(n+1)}{2} points on the boundary of A; theorem of Swinnerton-Dyer; see [Cas2], V.8).

Nothing similar holds for unbounded star bodies. First, it may happen that no admissible lattice exist (by Proposition 4.3, this is the case for the set A = {(x, y) ∈ ℝ^{2} | x^{2}|y| < 1}). A less trivial example is provided by the indefinite quadratic domains of dimension n ≥ 5 (conjecture of Oppenheim, now a theorem of Margulis).^{8}

The notion we introduce below will allow us to prove in some cases the existence of points on the boundary.

Definition 4.6. We say that a distance-function F or its corresponding star body A_{F} is of compact type (Mahler said automorphic) if the following property holds: for every B > 0, there exists a compact subset C_{B} of E such that for every x ∈ E with F (x) < B, there exists u ∈ Aut(F) which maps x into K_{B}.

8This results from the more precise statement: every non-degenerate, indefinite real quadratic form of dimension n ≥ 3 which is not proportional to an integral form takes on ℤ^{n} arbitrary small values of each sign, since by the Hasse-Minkowski theorem, indefinite integral quadratic forms of dimension n ≥ 5 represent zero.

Proposition 4.7. Let F be a distance-function of compact type. Then for every minimal-admissible lattice Λ for A_{F}, there exists a sequence u_{k} of automorphisms of F such that the sequence u_{k}(Λ) converges to a lattice having a a point on the boundary of A_{F}.

[This lattice is minimal-admissible and has the same determinant as Λ.]

Proof. Since Λ is minimal-admissible, it contains for all n > 0 a point x_{n} such that F (x_{n}) < det(Λ) + \frac{1}{n}. Let B = det(Λ) + 1. There exists a compact set K_{B} ⊂ E and for every n, an automorphism u_{k} of F such that u_{k}(x_{n}) ∈ K_{B}. By successive extractions of sequences, we construct a map φ : ℕ → ℕ such that u_{φ(k)}(Λ) tends to a lattice Λ′ and u_{φ(k)}(x_{n}) tends to a point x ∈ E. Then Λ′ is admissible for A_{F} and x belongs to ∂A ∩ Λ′. □

Corollary 4.8. If A_{F} possesses admissible lattices, then it possesses critical lattices having at least one pair ±x of points on its boundary.

□

Proposition 4.7 does not suffice to prove that every lattice Λ which is extreme for a domain A of compact type has points on ∂A. All we can say is that such a lattice exists in the closure of the set {u(Λ) | u ∈ Aut(F)}. The notion we introduce in the definition below will allow us to conclude in some situations.

Definition 4.9. We say that an admissible lattice Λ for an open star body A is isolated if there exists a neighbourhood of A on which every admissible lattice is of the form λu(Λ) for some λ ≥ 1 and some u ∈ Aut(A).

[Clearly, “ isolated ” ⟹ “ extreme ” ⟹ “ minimal-admissible ”.]

It may happen that we are able to show that the sequence u_{k}(Λ) in Proposition 4.7 converges to an isolated lattice. Then the sequence must be stationary, and consequently, the initial lattice Λ itself does have a point on the boundary of A.

4.2. The automorphism groups of Minkowski’s domains. We construct automorphisms of the Minkowski domain A_{r_{1},r_{2}}, that we write in the form

∏_{k≤r_{1}}|x_{k}| ∏_{k≤r_{2}}(y_{k}^{2} + z_{k}^{2}) (= ∏_{k≤r_{1}}|x_{k}| ∏_{k≤r_{2}}|y_{k} + iz_{k}|^{2} , i^{2} = −1) < 2^{r_{2}} .

Definition 4.10. Let λ_{k} (k ≤ r_{1}) and μ_{k} (k ≤ r_{2}) be r_{1} + r_{2} strictly positive real numbers, and let ω_{k} (k ≤ r_{2}) be r_{2} real numbers mod 2π, submitted to the condition ∏ λ_{k} ∏ μ_{k}^{2} = 1. Let G^{∘} ⊂ GL_{n}(ℝ) be the group of transformations x_{k} ↦ λ_{k}x_{k}, y_{k}+iz_{k} ↦ μ_{k}e^{iω_{k}}(y_{k}+iz_{k}), and let G ⊂ GL_{n}(ℝ) be the group generated by G^{∘}, the change of signs of the x_{k} and the z_{k}, the r_{1}! permutations of the x_{k}, and the r_{2}! permutations of the y_{k} + iz_{k}.

It is clear that G is a group of automorphisms of A_{r_{1},r_{2}} and that G^{∘} ≃ ℝ^{r_{1}+r_{2}−1} × (R/2πℤ)^{r_{2}}, of dimension n − 1, is its connected component. This information suffices for what we need in the sequel. For the sake of completeness, we nevertheless state the following result, leaving its proof to the reader.

Proposition 4.11. The group G of Definition 4.10 is the group of all automorphisms of A_{r_{1},r_{2}}. □

We observe that the Abelian group G^{∘}, modulo its maximal compact subgroup, has dimension r_{1} + r_{2} − 1, the Dirichlet number of fields K of signature (r_{1}, r_{2}), which is the rank of the unit group of any order 𝔒 ⊂ K. Taking into account the unit 1, we see that an algebraic lattice will have at least r_{1} + r_{2} independent points on the boundary of A_{r_{1},r_{2}}. In the totally real case, this is the maximal value. Otherwise, the maximal value will be r_{1} + r_{2}, except if the order 𝔒(M) = {λ | λM ⊂ M} contains roots of unity distinct from ±1.

In the easy case of A_{0,1} (the disc of radius \sqrt{2}), minimal-admissible lattices are isometric to a lattice having a basis (e_{1}, e_{2}) with N(e_{2}) ≥ N(e_{1}) = 2. The boundary of A_{0,1} generally contains only the pair ±e_{1}, except on the close subset N(e_{1}) = N(e_{2}). Then we may assume that ∠(e_{1}, e_{2}) ∈ [\frac{π}{3} , \frac{π}{2} ]. The endpoints of this interval correspond to algebraic lattices (Eisenstein and Gaussian integers). Lattices corresponding to angles in the open interval (\frac{π}{3} , \frac{π}{2} ) are not algebraic.

We now use the automorphisms constructed above to prove in a precise form that Minkowski’s domains are of compact type (see Subsection 4.1).

Theorem 4.12. (1) Let B > 0 and x ∈ ℝ^{n} with 0 < F_{r_{1},r_{2}}(x) ≤ B. Then there exists an automorphism of A_{r_{1},r_{2}} which maps x into the compact interval {λ111}, 0 ≤ λ ≤ 2^{−r_{2}/n}B^{1/n} of ℝ111.

(2) If Λ is a minimal-admissible lattice for A_{r_{1},r_{2}}, there exists a sequence u_{k} of automorphisms of A_{r_{1},r_{2}} such that u_{k}(Λ) tends to a lattice Λ′ containing 111.

Proof. (1) Let x ∈ ℝ^{n} such that F_{r_{1},r_{2}}(x) ≤ B, and let x_{i}, y_{j}, z_{j} be the components of x (notation Definition 4.10). We may assume by changing signs that the components are non-negative, hence strictly positive, and using plane rotations in the r_{2} planes ⟨y_{j}, z_{j}⟩, that y_{j} = z_{j} for j = 1, …, r_{2}. Set

λ_{i} = \frac{F (x)^{1/n}}{x_{i}} for i = 1, …, r_{1} and μ_{j} = \frac{F (x)^{1/n}}{\sqrt{y_{j}^{2} + z_{j}^{2}}} for j = 1, …, r_{2} .

Then x_{i} ↦ λ_{i}x_{i}, (y_{j}, z_{j}) ↦ μ_{j}(y_{j}, z_{j}) is an automorphism of A_{r_{1},r_{2}}, which transforms x into a vector with equal components x_{i} and equal components y_{j}, z_{j}. If r_{1} = 0 or r_{2} = 0, we are done. Otherwise, we perform the transformation x_{i} ↦ λx_{i}, (y_{j}, z_{j}) ↦ μ(y_{j}, z_{j}) with λ = (y_{1}x_{1}^{−1}) and μ = \frac{x_{1}}{y_{1}} λ.

(2) For k = 1.2, …, there exists x^{(k)} ∈ Λ with F (x^{(k)}) ≤ 2^{r_{2}}(1 + \frac{1}{k}). By (1), there exists u_{k} ∈ Aut(A_{r_{1},r_{2}} which maps x^{(k)} to an y_{k} ∈ 2^{r_{2}/n}[1, ε_{k}]111 with ε_{k} = O(\frac{1}{k}). We can extract from the sequence u_{k}(Λ) a converging sub-sequence, the limit of which contains an y which belongs to ∩_{k}2^{r_{2}}[1, ε_{k}]111 = {2^{r_{2}} 111}. □

4.3. The function S for totally real Minkowski domains. Let x = (x_{1}, …, x_{n}) ∈ ℝ^{n}. We assume that the components of x satisfy x_{1} ≤ x_{2} ≤ ⋯ ≤ x_{n}. In Definition 3.1 we introduced the function

S(x) = ∑_{1≤i<j≤n} (x_{j} − x_{i})^{2} .

We shall need later, in particular in section 6, to estimate the size of S for a given difference x_{n} − x_{1}. (Note that S is invariant under translations.) This is the aim of the proposition below.

Proposition 4.13. We have

\frac{n}{2} (x_{n} − x_{1})^{2} ≤ S ≤ k(n − k)(x_{n} − x_{1})^{2}

with k = \frac{n}{2} if n is even and k = \frac{n−1}{2} or k = \frac{n+1}{2} if n is odd.

The minimum is attained exactly when x_{i} = \frac{x_{1}+x_{n}}{2} for 1 < i < n, and the maximum when x_{i} takes k times the value x_{1} and n − k times the value x_{n}.

Proof. One can write S − (x_{n} − x_{1})^{2} as a sum of non-negative terms (x_{i} − x_{j})^{2}(1 < i < j < n and of n − 2 terms (x_{i} − x_{1})^{2} + (x_{n} − x_{i})^{2} which are minimum exactly when x_{i} = \frac{x_{1}+x_{n}}{2} , and the other terms are then zero.

The minimum is attained at the only point where all partial derivatives vanish, so that the maximum is attained on the boundary of the domain [x_{1}, x_{n}]^{n}. By induction, we see that all x_{i} are equal to x_{1} or x_{n}, so that the maximum is of the form k(n − k)(x_{n} − x_{1})^{2}, and k(n − k) is maximum exactly for k ∈ [\frac{k−1}{2}, \frac{k+1}{2}]. □

In practice, the x_{i} will never take many equal values, so that the bounds in the proposition above will be strict. We shall prove some better estimates when x belongs to a lattice Λ which is admissible for A_{n,0} and contains 1.

4.4. Totally real Minkowski domains. In this subsection, we consider 2-dimensional lattices in ℝ^{n} having a basis of the form (111, x) and no points in A_{n,0}; otherwise stated, we have F_{n,0}(y) ≥ 1 for all ℤlinear combination of 111 and x. We write (x_{1}, …, x_{n}), and assume that x_{1} ≤ x_{2} ≤ ⋯ ≤ x_{n}. (We can reduce ourselves to this case since permutations of the coordinates are automorphisms of A_{n,0} which preserve 111.) We observe that the x_{k} are not integral, so that each x_{k} lies in some interval J_{k} = (m_{k}, m_{k} + 1), and even that x_{k} cannot be too close to an endpoint of J_{k}. This we now give a precise form.

Fix an index i. We observe that we have x_{j} − m_{i} ≤ m_{j} − m_{1} + 1 if x_{j} ≥ m_{i}, and m_{i} − x_{j} ≤ m_{i} − m_{j} if x_{j} < m_{i}, and similar upper bounds for |m_{i} + 1 − x_{j}|. Since the products of the |x_{j} − m_{i}| and |m_{i} + 1 − x_{j}| are ≥ 1, we obtain this way upper bounds for x_{i} − mi and m_{i} + 1 − x_{i}. We state the result in the proposition below.

Proposition 4.14. Set

a_{i,j} = m_{j} − m_{i} + 1 if x_{j} ≥ m_{i} and a_{i,j} = m_{i} − m_{j} if x_{j} < m_{i}, and

b_{i,j} = m_{j} − m_{i} if x_{j} > m_{i} and b_{i,j} = m_{i} − m_{j} + 1 if x_{j} ≤ m_{i}

(thus a_{i,i} = b_{i,i} = 1). Then we have

0 < x_{i} − m_{i} < ∏_{j} a_{i,j}^{−1} and 0 < m_{i} + 1 − x_{i} < ∏_{j} b_{i,j}^{−1} . □

Proposition 4.14 gives us upper bounds for all differences x_{j} − m_{j} and m_{j} + 1 − x_{j}, that can then be used in the proof above. This “round 2” calculations produce improvements on the result above. We produce this way two families (a_{i}^{(k)}) and (b_{i}^{(k)}) , k ∈ ℕ, of increasing sequences contained in (0, 1), such that, for all i, one has m_{i} + a_{i}^{(k)} < x_{i} < m_{i}+1−(b_{i}^{(k)} . These sequences have limits a_{i}, b_{i}, and we have a_{i} ≤ x_{i} ≤ b_{i} for i = 1, …, n. One can even use infinitely many inequalities, those we obtain by calculating as above |pm_{j} − qx_{j}| and |p(m_{i} + 1) − qx_{j}| for convenient integers p, q (depending on i and j). The difficulty is to guess what “convenient” means. The Markoff chain for A_{_{2},0}, described in Section 5 below, is one of the few cases for which this difficult problem was solved. Examples of calculations of these limits are displayed in Subsection 5.1.

Remark 4.15. When classifying the possible values of S(x) (always assuming that ⟨111, x⟩ ∩ A_{n,0} = ∅), we may replace x by x + k for any k ∈ ℤ, and thus fix the integral part of one x_{k}, say, assume that 0 < x_{1} < 1. Moreover, using the symmetry x ↦ −x, we may assume one more restriction, for instance, x_{n} − x_{n−1} > x_{2} − x_{1}.

If we fix an upper bound for S, then x_{n} − x_{1} is bounded, and choosing x_{1} in (0, 1), we see that the x_{k} must be chosen in only finitely many intervals (m_{k}, m_{k+1}).

Proposition 4.16. Still under the hypothesis ⟨111, x⟩ ∩ A_{n,0} = ∅, we have:

(1) The components of x do not lie in the union of two consecutive intervals J_{k}.

(2) If n − 1 components of x lie in a same interval J_{k}, then S > \frac{n−1}{4} (2^{n} − 1)^{2}.

(3) If the components of x lie in the union of three consecutive intervals J_{k}, no (n−1) of them lying in the same interval, then one of the following properties holds:

(a) n = 4, m_{1} = m_{2}, and m_{3} = m_{4} = m_{2} + 2.

(b) n = 5, m_{1} = m_{2}, m_{3} = m_{2} + 1, and m_{4} = m_{5} = m_{1} + 2.

(c) n ≥ 6.

Proof. (1) By translation, we may assume that all x_{i} belong to (−1, +1), which contradicts the lower bound ∏_{i} x_{i} ≥ 1.

(2) By translation and symmetry x ↦ −x, we may assume that x_{i} ∈ (0, 1) for i < n, which implies x_{i}(1 − x_{i}) ≤ \frac{1}{4}, hence x_{n}(x_{n} − 1) > 4^{n−1}, thus x_{n} > \frac{1+\sqrt{1+4^{n}}}{2} > \frac{2^{n}+1}{2}, whence the result, since x_{n} − x_{i} > \frac{2^{n}−1}{2} for i < n.

(3) If n = 3 we may assume that −1 < x_{1} < 0 < x_{2} < 1 < x_{3} < 2. We then have x_{2}(1 − x_{2}) ≤ \frac{1}{4} and x_{i}|1 − x_{i}| < 2 for i = 1, 3, hence ∏_{i} x_{i} ∏_{i}|1 − x_{i}| < 1, a contradiction. We may now assume that n ≥ 4.

Suppose that n−2 components of x lie in the same interval, say, (0, 1), a hypothesis which holds for n = 4. Then the product ∏_{i} x_{i} ∏_{i}(1 − x_{i}) on these components is bounded from above by \frac{1}{4^{n−1}} . Using the symmetry x ↦ 1 − x, we may assume that the remaining two components are not negative, so that one of the three cases holds: (1) x_{1} ∈ (0, 1), x_{n} ∈ (1, 2); (2) x_{n−1} ∈ (1, 2), x_{n} ∈ (2, 3); (3) x_{n−1}, x_{n} ∈ (2, 3). Bounding the products of the x_{i}|1 − x_{i}| for the remaining two components, we obtain the inequalities 4^{n−2} ≤ 4, 4^{n−2} ≤ 12, and 4^{n−2} ≤ 26, respectively. The first two are impossible, and the third one holds only if n = 4 and if x_{,}x_{2} ∈ (0, 1) and x_{3}, x_{4} ∈ (2, 3).

[Such systems exist: take (x, y) such that ⟨(1, 1), (x, y)⟩ is admissible for A_{2,0}, e.g., a lattice coming from the ring of integers of a quadratic subfield of a quartic field; then we may take x_{1} = x_{2} = x and x_{3} = x_{4} = y.]

A similar proof applies in dimension 5: there must be clearly two pairs of components in the same interval, and we find as above that these pairs must be contained in the extreme intervals. □

Remark 4.17. An other way for bounding the x_{i} is to use the polynomial

f = ∏_{i} (X − x_{i}) = X^{n} − t_{1}X^{n−1}t_{2}X^{n−2} + ⋯ + (−1)^{n}t_{n} .

For all m ∈ ℤ, we have |f(m)| = |∏_{i} (X −m_{i})| ≥ 1, and f(m) has the sign of (−1)^{n−n_{m}} where n_{m} is the number of roots of f which are smaller than m. An example occurs in Subsection 6.1. Amazingly, the quality of the inequalities obtained by this method is not invariant under integral translations. One should center the set {x_{i}} close to small integers (0, ±1, …).

More generally, for any vector x = (x_{1}, …, x_{n}) ∈ Λ, x ∉ ℝ111, we must have ∏_{i=1}^{n} |(qx_{i} + p)| ≥ 1 for all p, q ∈ ℤ, q ≠ 0 (and indeed, q > 0 suffices), so that we have |f(\frac{p}{q})| ≥ \frac{1}{q^{n}} for all pairs of integers p and q > 1.

Definition 4.18. For q ≥ 1, let

E′_{q} = {x ∈ ℝ^{n} | ∀ p ∈ ℤ, ∏_{i=1}^{n} |(qx_{i} + p)| ≥ 1} and E_{q} = ∩_{k≤q} E′_{k} .

We moreover identify x ∈ E′_{q} with the relative lattice Λ = ⟨111, x⟩.

If Λ is admissible for A_{n,0}, every x ∈ Λ∖ℤ111 belongs to the intersection of the E_{q}. In particular the set of admissible lattices for A_{2,0} is the intersection of the E_{q}.

Note that if x ∈ E′_{q}, the components of x are not rational numbers, the denominator of which divides q. This proves:

Proposition 4.19. If Λ is admissible for A_{n,0}, then every x ∈ Λ∖ℤ111 has irrational components. □

5. A glance at dimension 2

To classify the possible values of the determinants d of minimaladmissible lattices Λ for A_{n,0} we may restrict ourselves to lattices Λ containing the point 111 = (1, …, 1); if moreover we can prove an isolation theorem for such lattices, we then obtain a classification of minimal-admissible lattices such that det(Λ) = d.

The remaining of this section is devoted to A_{2,0}. Then a lattice Λ containing 111, say, Λ = ⟨111, x⟩, is admissible for A_{2,0} if and only if x ∈ ∩_{q≥1} E_{q}.

In a first subsection, we prove that the first two minima of admissible lattices are attained on the lattices which are equivalent to the algebraic lattices associated with ℤ[\frac{1+\sqrt{5}}{2}] and ℤ[\sqrt{2}], with determinants 5 and 8, respectively. This is a small part of Markoff’s theorem, which describes all admissible lattices of determinant d < 9, that we state in a second subsection.

5.1. The first two minima for A_{2,0}.

Lemma 5.1. Let Λ ∈ E_{1}.

(1) Λ has a basis (111, x), where x = (x_{1}, x_{2}) satisfies the conditions

−1 < x_{1} < 0 and m < x_{2} < m + 1

for some integer m ≥ 1.

(2) Set u = x_{1} + x_{2} and v = x_{1}x_{2}, so that x_{1} and x_{2} are the roots of f(X) = X^{2} − uX + v. Then either f defines x uniquely, and then u = m, or there are two choices for x, one with u < m and one with u > m.

Proof. There exists x ∈ Λ such that (111, x) is a basis for Λ, and such an x is unique up to a transformation x ↦ ±x + k 111 for some k ∈ ℤ. Replacing x by x+k 111 for a convenient k ∈ ℤ, we may assume that −1 < x_{1} ≤ 0 and x_{2} ≥ x_{1}. Set m = ⌊x_{2}⌋. Then we have m ≤ x_{2} < m + 1, but the cases x_{1} = 0 and x_{2} = m must be excluded (if e.g., x_{2} = m, then x − m111 has a zero component), so that we have −1 < x_{1} < 0 and m < x_{2} < m + 1, and moreover m ≥ 1 by Proposition 4.16. Then x is unique up to its replacement by m111 − x.

Finally if x_{1} + x_{2} > m, then (m − x_{1}) + (m − x_{2}) < m. □

We now investigate the set E_{1}, choosing x ∈ E_{1} satisfying the hypotheses of Lemma 5.1.

Theorem 5.2. (1) Let m and x = (x_{1}, x_{2}) be as in Lemma 5.1. Then the set of determinants d of lattices Λ = ⟨111, x⟩ with x ∈ E_{1} fills the interval [m^{2} + 4, m^{2} + 4m].

(2) The endpoints of the intervals above are obtained exactly on the lattices associated with the order of discriminant d.

(3) The set of values of d for x ∈ E_{1} is

{5} ∪ [8, 12] ∪ [13, ∞) .

Proof. Recall that the determinant of Λ is

d = (x_{2} − x_{1})^{2} = (x_{1} + x_{2})^{2} − 4x_{1}x_{2} = u^{2} − 4v .

The conditions x_{1} ∈ (−1, 0) and x_{2} ∈ (m, m + 1) are equivalent to the four inequalities

f(−1) > 0, f(0) < 0, f(m) < 0, f(m + 1) > 0 , and the condition x ∈ E_{1} to

∀ k ∈ ℤ, |f(k)| ≥ 1 .

Hence the conditions

f(−1) ≥ 1, f(0) ≤ −1, f(m) ≤ −1, f(m + 1) ≥ 1

are necessary for Λ to belong to E_{1}, and they are indeed also sufficient, since we have f(x) > f(−1) on (−∞, −1), f(x) < max(f(1), f(m)) on (0, m) and f(x) > f(m + 1) on (m + 1, ∞). These conditions read

(a): u + v ≥ 0 ; (b): v ≤ −1 ; (c): u ≥ \frac{v+m^{2}+1}{m} ; (d): u ≤ \frac{v+m^{2}+2m}{m+1} .

Using (a) and choosing u ≤ m, we obtain the upper bound

d = u^{2} − 4v ≤ u^{2} + 4u ≤ m^{2} + 4m = (m + 2)^{2} − 4 ,

which is attained uniquely if u = m and v = −m, and then Λ is associated with the order of discriminant d = m^{2} + 4m.

Similarly, using (b) and choosing u ≥ m, we obtain the lower bound

d = u^{2} − 4v ≥ u^{2} + 4 ≥ m^{2} + 4 ,

attained uniquely if u = m and v = −1, and then Λ is associated with the order of discriminant d = m^{2} + 4.

Taking u = m and letting v run through [−m, −1], we obtain polynomials taking convenient values at −1, 0, m, m + 1, with determinants running through the whole interval I_{m} = [m^{2} + 4, m^{2} + 4m]. This proves (1) and (2).

To prove (3), just observe that I_{1} = [5, 5], I_{2} = [10, 12] and I_{3} = [13, 21], and that I_{m} and I_{m+1} overlap for m ≥ 3 (I_{4} = [20, 32], etc.).

□

Since the lattice L = ⟨111, (\frac{1−\sqrt{5}}{2}, \frac{1+\sqrt{5}}{2})⟩ is isolated among lattices of determinant d ≤ 8, we have:

Corollary 5.3. The lattice constant of A_{2,0} is κ_{2,0} = 5, and is attained only on lattices equivalent to the lattice of ℤ[\frac{1+\sqrt{5}}{2}]. Other minimaladmissible lattices have a determinant d ≥ 8. □

Remark 5.4. (1) The end points of the intervals I_{m} above are quadratic surds having a purely periodic continued fraction, namely [ \overline{m}] on the left and [ \overline{m 1} ] on the right. ([ \overline{m 1} ] = [ \overline{1} ] if m = 1 !)

(2) The orders defined by these endpoints need not be maximal. An example is provided by I_{4}, the endpoints of which both have conductor 2.

The next step in the study of admissible lattices for A_{2,0} consists in analyzing E_{q} for q = 2, 3, . . .. To this end we consider conditions x_{1} ∈ (−1 + \frac{i}{q}, −1 + \frac{i+1}{q}) and x_{2} ∈ (m + \frac{i}{q}, m + \frac{i+1}{q}) for i = 0, …, q − 1, that is, q^{2} pairs of intervals, among which q are fixed and \frac{q(q−1)}{2} are exchanged by x ↦ m111 − x, and by Theorem 5.2, we may assume that m ≥ 2.

This we now do for q = 2, dividing I = (−1, 0) and J = (m, m + 1) (now for m ≥ 2) into sub-intervals I′ = (−\frac{1}{2}, 0), I′′ = (−1, −\frac{1}{2}), and J′ = (m, m+ \frac{1}{2}), J′′ = (m+ \frac{1}{2}, m+1), and making use of the conditions |f(−\frac{1}{2})| ≥ \frac{1}{4} and |f(m+ \frac{1}{2})| ≥ \frac{1}{4} besides conditions (a) to (d) introduced in the proof of Theorem 5.2.

Theorem 5.5. Let Λ be a lattice having a basis of the form (111, x), x = (x_{1}, x_{2}), with −1 < x_{1} < 0 < x_{2} and x ∈ E_{2}. Then up to a symmetry, one of the following conditions holds:

(1) 1 < x_{2} < 2, and then (x_{1}, x_{2}) = (\frac{1−\sqrt{5}}{2}, \frac{1+\sqrt{5}}{2}) and det(Λ) = 5 ;

(2) −\frac{1}{2} < x_{1} < 0 and 2 < x_{2} < \frac{5}{2}, and then (x_{1}, x_{2}) = (1 − \sqrt{2}, 1 + \sqrt{2}) and det(Λ) = 8 ;

(3) −1 < x_{1} < −\frac{1}{2} and 2 < x_{2} < \frac{5}{2}, or −\frac{1}{2} < x_{1} < 0 and \frac{5}{2} < x_{2} < 3, and then the set of values taken by det(Λ) is the interval [\frac{221}{25}, \frac{480}{49}] (= [8.84, 9.7959 …]) ;

(4) −1 < x_{1} < −\frac{1}{2} and \frac{5}{2} < x_{2} < 3, and then the set of values taken by det(Λ) is the interval [10, 12].

(5) x_{2} > 3, and then det(Λ) ≥ 13.

In particular we have

\{det(E_{2})\} ∩ (0, 13) = \{5\} ∪ \{8\} ∪ [\frac{221}{25}, \frac{480}{49}] ∪ [10, 12] .

Proof. We keep the notation u = x_{1} + x_{2}, v = x_{1}x_{2}. The first and last assertions result from Theorem 5.2, so that we may restrict ourselves to the case when m = 2, and we moreover know that we then have 8 ≤ d ≤ 12.

(2) The lower bounds f(−\frac{1}{2}) ≥ \frac{1}{4} and f(\frac{5}{2}) ≥ \frac{1}{4} read (a′): u + 2v ≥ 0 ; (d′): 5u − 2v ≤ 12 ;

together with (b), they imply u = 2 and v = −1, which proves (2).

(3) Using the transformation x ↦ m111 − x we may assume that −\frac{1}{2} < x_{1} < 0 and \frac{5}{2} < x_{2} < 3, which implies the four conditions

(a′): −u − 2v ≤ 0 ; (b): v ≤ −1 ; (c′): −5u + 2v ≤ −13 ; (d): 3u − v ≤ 8.

We now consider the six pairs of values of (u, v) obtained by replacing inequalities by equalities in two systems (a′), (b), (c′), (d). We must excludes (a′) + (b), for which (u, v) = (2, −1) implies x_{2} = 1 + \sqrt{2} < \frac{5}{2}, and (c′) + (d), for which (u, v) = (3, 1) implies x_{2} = \frac{3+\sqrt{13}}{2} > 3. The remaining four cases are listed below together with the corresponding value of d = u^{2} − 4v, a function of (u, v) with non-vanishing partial derivatives in the domain we consider, so that the extremal values are attained on one of the four pairs (u, v) below:

(\frac{13}{6}, −\frac{13}{12}), d = \frac{325}{36} = 9.0277 …; (\frac{11}{5}, −1), d = \frac{221}{25} = 8.84 ;

(\frac{16}{7}, −\frac{8}{7}), d = \frac{480}{49} = 9.7959 …; (\frac{7}{3}, −1), d = \frac{85}{9} = 9.4444 ….

These four values of (u, v) are the vertices of a closed convex set in the plane ⟨u, v⟩, all points of which produce an admissible determinant for E′_{2}. As a consequence, the values of d fill the interval [\frac{221}{25}, \frac{480}{49}] . This proves (3), since this interval is included in E_{1}.

(4) There remains to consider the case when −1 < x_{1} < −\frac{1}{2} and \frac{5}{2} < x_{2} < 3. The conditions are now

(a): −u − v ≤ 0 ; (b′): u + 2v ≤ −1 ; (c′): −5u + 2v ≤ −13 ; (d): 3u − v ≤ 8.

Using (a) + (d) and (b′) + (c′), we prove that −2 ≤ v ≤ −\frac{3}{2}. Since the domain we consider is invariant under x ↦ 2111 − x, we may assume that u ≤ 2 (resp. u ≥ 2) to obtain an upper (resp. a lower) bound for d. Hence we have 10 ≤ d ≤ 12, and 2^{2} − 4v runs through the whole interval [10, 12] when v. runs through [−2, −\frac{3}{2}]. This proves (4) and completes the proof of the theorem. □

Various values of d which appear in Theorem 5.5 (5, 8, \frac{221}{25}, \frac{4}{80}49, 10, 12, 13) or in its proof (\frac{85}{9}) are indeed the determinants of admissible lattices for A_{2,0}. This we prove in the next subsection, in connection with the arithmetic of real quadratic orders.

Consideration of E_{2} beyond m = 2 does not throw much light on the Markoff spectrum. So we now sketch the study of E_{3}, first in the range d ∈ (0, 13), for which we must choose m = 2, then for d ∈ [13, 20], where we must choose m = 3.

If −\frac{1}{3} < x_{1} < 0, we have f(−\frac{1}{3}) ≥ \frac{1}{9} and f(0) ≤ −1, that is u+3b ≥ 0 and v ≤ −1, hence d = u^{2} − 4v ≥ 9v^{2} − 4v ≥ 13, which contradicts Theorem 5.2. Similarly, using the transformation x ↦ 2111 − x, we see that we must exclude the range 2 < x_{2} < \frac{7}{3}. We are thus left with four pairs of intervals, which reduce to three using the transformation above: (−\frac{2}{3}, −\frac{1}{3}) × (\frac{7}{3}, \frac{8}{3}) and (−1, −\frac{2}{3}) × (\frac{8}{3}, 3), which are invariant, and (−\frac{2}{3}, −\frac{1}{3}) × (\frac{8}{3}, 3), which is not.

5.2. Quadratic orders and the Markoff spectrum.

Definition 5.6. The set S of determinants of minimal-admissible lattices for A_{2,0} is called the Markoff spectrum.

[The usual definition is the square roots of these determinants d, which appear in diophantine approximations in the form |θ − \frac{p}{q}| ≤ \frac{1}{\sqrt{d} q^{2}} .]

The results of Markoff stated in the next subsection will show that S ∩ [0, 9] consists of a discrete subset of (5, 9) together with its unique limit point 9. Theorem 5.2 shows that (12, 13) is a gap in the Markoff spectrum, and indeed a maximal gap, since both 12 and 13 are determinants of algebraic lattices coming from quadratic orders. Besides the infinite series of gaps produced by the Markoff chain, other gaps exist, both in (9, 13) and in (13, +∞). We have proved above that [\frac{480}{49}, 10] is a gap, and it can be shown that 13 is isolated in the Markoff spectrum (the maximal gap at the right of 13 is (13, \frac{9\sqrt{3}+65}{22} = 13.418 …); see [C-F], Ch. 1, Lemma 9).

However a theorem of Marshall Hall asserts that every large enough real number belongs to S. It has been proved independently by Freiman and by Schecker^{9} that the Hall’s ray contains [21, +∞). Finally Freiman^{10} proved that the Hall ray is

[μ^{2}, +∞) for a = \frac{2221564096 + 283748\sqrt{462}}{491993569} (μ^{2} = 20.389...) .

Proposition 5.7. The Markoff spectrum is closed.

Proof. Let (d_{n}) be a convergent sequence of elements of the Markoff spectrum and let d be its limit. Every n is of the form d_{n} = det(Λ_{n}) for some minimal admissible lattice Λ_{n}. By Mahler’s compactness lemma, we can extract from (Λ_{n}) a sub-sequence converging to a lattice Λ. This lattice is minimal-admissible and has determinant d.

[Other viewpoint: the sets E_{q} are closed, and so is ∩_{q} E_{q}.] □

Let us now give some complements on orders in a quadratic field (real or not, this does not matter at this stage). Let K be a quadratic field, and let om ∈ K such that (1, ω) is a ℤ-basis for the ring of integers ℤ_{K} of K. In practice, if K has discriminant d_{0}, we take ω = \frac{1+\sqrt{d_{0}}}{2} if d_{0} is odd, and ω = \sqrt{d_{0}} if d_{0} is even. For every integer f ≥ 1, (1, fω) is a ℤ-basis for an order 𝔒_{f} ⊂ ℤ_{K}, and the assignment f ↦ 𝔒_{f} is one-to-one. The integer f is called the conductor of 𝔒_{f}, the discriminant of which is d = d_{0}f^{2}.

A basic property of quadratic orders is that every proper fractional ideal 𝔞 of a quadratic order 𝔒 is invertible, that is, there exists a fractional ideal 𝔟 such that 𝔞𝔟 = 𝔒. Thus every module in K can be viewed as an invertible fractional ideal of an order, a property which fails in higher degrees.

9 see H. Schecker, Über die menge der Zahlen, die als Minima quadratischer Formen auftreten, J. Number T. 9 (1977), 121–141

10 In the unpublished Russian paper Approximations and the geometry of numbers, Moscow (1975), cited by Conway and Guy; in [C-F], p. 55, one must replace μ by μ + 4.

[In the general theory of orders in a separable algebra K/K_{0} relatively to a Dedekind domain R with quotient field K_{0}, “invertible” amounts to “projective”, and in a few important cases, “projective” implies “locally free” (these orders are called clean). This last property holds when K is commutative (by standard techniques of commutative algebra) and when the order is maximal (theorem of Auslander-Goldman); this also holds when for group rings R[G] (theorem of Swan). Given an order 𝔒 of R in K and a maximal order 𝔐 containing 𝔒, the annihilator of 𝔐/𝔒 is an ideal in 𝔒 and in 𝔐, ‘the conductor 𝔣 of 𝔒 in 𝔐. Every integral ideal prime to 𝔣 is invertible, and every locally free ideal is equivalent to an ideal prime to 𝔣.]

For any order 𝔒 in any number field K of signature (r_{1}, r_{2}), the class group (group of invertible fractional ideals modulo Principal fractional ideals) is finite. This can be proved using Theorem 2.6: Every class of 𝔒 contains an integral ideal of norm bounded from above by \sqrt{\frac{d_{𝔒}}{κ_{r_{1},r_{2}}}} .

More precisely, for any class c ∈ Cl_{𝔒}, denote by N_{c} the minimum of the norms of integral ideals in c (which depends only on the pair {c, c^{−1}}). The theorem above bounds N = max_{c} N_{c}, but the collections of all N_{c} is useful to construct modules.

Given an invertible fractional ideal 𝔞 in 𝔒, we have

min_{x∈𝔞∖{0}} |N_{K/ℚ}(x)| = N(𝔞)

and d_{𝔞} = d_{𝔒} N(𝔞)^{2}, and replacing 𝔞 by an equivalent ideal 𝔟 = α 𝔞 multiplies the norm form on 𝔞 by N_{K/ℚ}(α), so that once rescaled to minimum 1, the algebraic lattices deduced from 𝔞 and 𝔟 are equivalent. In this scaling, taking 𝔞 ∈ c of norm N_{c}, we obtain a lattice of determinant

d_{c} = \frac{d_{𝔒}}{N_{c}^{2}} ,

and since all modules in a quadratic field arise from invertible ideals in orders, we obtain:

Proposition 5.8. The discriminants of algebraic, minimal-admissible lattices for A_{2,0} are the rational numbers d_{c} = \frac{d_{𝔒}}{N_{c}^{2}} where 𝔒 runs through the set of orders in a real quadratic field, c runs through the set of classes of 𝔒, and N_{c} is the smallest norm of an integral ideal in c. □

The discriminants of quadratic orders are easy to characterize: these are the integers d which are not a square and are congruent to 0 or 1 modulo 4. Among these discriminants, those of maximal orders are those which are square-free except for a divisor 4 when d is congruent to 8 or 12 modulo 4.

Since κ_{2,0} = 5, we can use the bound max N_{c} ≤ \sqrt{\frac{d}{5}} to evaluate the N_{c}. However, Theorem 5.5 shows that if we exclude the first two discriminants 5, 8 of real quadratic orders, we can use the better bound

max N_{c} ≤ \sqrt{\frac{d}{221/25}} = \sqrt{\frac{d}{8.84}} .

We now give some elements of the spectrum obtained using class groups. Given a prime p , we adopt the notation (p) = 𝔭_{p}^{2} if p ramifies in 𝔒 and (p) = 𝔭_{p}𝔭′_{p} if p splits in 𝔒.

[Warning: if p divides the conductor, then there is in 𝔒 one maximal ideal 𝔭_{p} above p, and this ideal is not invertible. For instance, in 𝔒 = ℤ[\sqrt{5}], we have 𝔭_{2} = (2, 1 + \sqrt{5}), hence 𝔭_{2}^{2} = (2)𝔭_{2} though 𝔭_{2} ≠ (2). The norm of an integral ideal 𝔞 can be defined as the cardinality of the residue ring 𝔒/𝔞, but the norm need not be multiplicative when primes dividing the conductor are involved.]

• There are 30 orders of discriminant D < 100, among which 25 have class number h = 1 and 5 have h = 2, namely those of discriminant 40, 60, 65, 85 (maximal) and 96 (with f = 2). The values of N_{c} for the unique non-trivial class are 2 for the first three and 3 for the last two, which produces lattices of determinant 10, 15, \frac{65}{4} = 16.25, \frac{85}{9} = 9.44 … and \frac{32}{3} = 10.66 …; we met the determinants 10 and \frac{85}{9} in Theorem 5.5.

• The first determinant for which max N_{c} ≥ 4 is D = 145 = 5 · 29, the discriminant of a field K in which 2 and 3 split. The class group is cyclic of order 4, generated by c = cl(𝔭_{2}). Since N(\frac{13+\sqrt{145}}{2}) = 6, we have cl(𝔭_{3}) = c^{±1}, so that norm 4 is needed to represent c^{2}. We construct this way a lattice of determinant d = \frac{145}{16} = 9.0625, fairly close to the Markoff limit 9.

• The first determinant for which max N_{c} ≥ 5 is D = 221 = 13 · 17. We have h = 2, and since 2 and 3 are inert, we need consider an ideal above 5 to generate the class group. We construct this way a lattice of determinant d = \frac{221}{25} = 8.84. We showed in Theorem 5.5 that this is the third element of the Markoff spectrum (after 5 and 8).

• Let K = ℚ(\sqrt{30}), of discriminant d_{K} = 120, and let 𝔒 be the order of discriminant D = 480 = d_{K} · 2^{2}. We have h_{K} = 2 (we need consider ideals of norm N ≤ 3, and since N(6 + \sqrt{30}) = 6, the ramified ideals 𝔭_{2} and 𝔭_{3} are equivalent). For further use we note that the fundamental unit of K, namely ε = 11 + 2\sqrt{30}, belongs to 𝔒, and that 𝔭_{5} is principal (we have N(5 + \sqrt{30}) = −5).

In 𝔒, 𝔭_{2} disappears, so that we must discard norms 2, 4 and 6, and since ε ∈ 𝔒, 𝔭_{5} is no longer principal. We have Cl_{𝔒} = {1, cl(𝔭_{3}), cl(𝔭_{5}), cl(𝔭_{3}𝔭_{5})}, so that norm 7 is needed to represent cl(𝔭_{3} 𝔭_{5}). We construct this way the lattice of determinant d = \frac{480}{49} = 9.7959 … which appears in Theorem 5.5. [Check: N(13 + 2\sqrt{30}) = 7^{2}, N(15 + 2\sqrt{30}) = 3 · 5 · 7.]

5.3. The Markoff chain. The description below of the admissible lattices for A_{2,0} of low determinant relies on the second chapter of Cassels’s book [Cas], where a chain of indefinite binary quadratic forms is constructed, a method which is easily translated into the language of lattices. This makes his book more suitable for our purpose than for instance that by Cusick and Flahive ([C-F]), in which only approximation theory is considered. Actually Cassels’s motivation is mainly the theory of diophantine approximations, but he establishes a dictionary which relies indefinite, binary quadratic forms and approximations of real numbers by rational numbers.

The quadratic forms we shall consider do not represent zero, and will be written in the following two ways:

f(x, y) = ax^{2} + bxy + cy^{2} = a(x − θy) (x − θ′y)

with real, irrational θ, θ′, and discriminant D = b^{2} − 4ac > 0; we have |a(θ′ − θ)| = \sqrt{D}. The minimum of f (not necessarily attained) is m = inf_{(x,y)≠(0,0)} |f(x, y)| ≥ 0.

In a crude form, Markoff’s theorem reads as follows:

Theorem 5.9. (Markoff) There exists a sequence (Λ_{n}) of algebraic lattices and an increasing sequence (m_{n}) of positive integers (the Markoff numbers) such that:

(1) lim_{n→∞} m_{n} = +∞.

(2) det(Λ_{n}) = 9 − \frac{4}{m_{n}^{2}} .

(3) Every admissible lattice Λ for A_{n,0} of determinant d < 9 is of

the form Λ = λ u(Λ_{n}) for some λ ≥ 1 and some u ∈ Aut(A_{n,0}). One has d_{1} = 5, d_{2} = 8, and the lattices Λ_{1}, Λ_{2} are the images of ℤ[\frac{1+\sqrt{5}}{2}] and ℤ[\sqrt{2}], respectively.

[Whether the sequence (m_{n}) is strictly increasing is not known.]

The translation in terms of approximations runs as follows. We say that two irrational numbers θ, θ′ are equivalent if θ′ = \frac{aθ+b}{cθ+c} for some a, b, c, d ∈ ℤ with ad−bc = ±1. Then for any irrational number θ, the inequality |θ− \frac{p}{q}| < \frac{1}{\sqrt{5} q^{2}} has infinitely many solutions, and the constant \frac{1}{\sqrt{5}} is optimal if θ is

equivalent to \frac{1+\sqrt{5}}{2}; otherwise, the inequality |θ − \frac{p}{q}| < \frac{1}{\sqrt{8} q^{2}} has infinitely many solutions, and the constant \frac{1}{\sqrt{8}} is optimal if θ is equivalent to \sqrt{2}; otherwise, the inequality |θ − \frac{p}{q}| < \frac{1}{\sqrt{d_{3}} q^{2}} has infinitely many solutions, etc.

If m is not attained uniquely at (1, 0) and is non-zero, so that we may rescale f to minimum 1, we obtain inequalities |θ − \frac{x}{y}| |θ′ − \frac{x}{y}| ≤ \frac{1+ε}{|a|} \frac{1}{y^{2}} . If \frac{x}{y} can be made arbitrarily close to θ, we obtain inequalities of the form |θ− \frac{x}{y}| ≤ \frac{1+ε}{\sqrt{d} y^{2}} . To obtain the sharper inequalities stated above, we shall have to restrict to forms for which θ, θ′ are conjugate quadratic irrational numbers and having moreover a minimum f attained on both positive and negative sides.

The construction of the sequence m_{n} needs the consideration of the diophantine equation (the Markoff equation)

m_{1}^{2} + m_{2}^{2} + m_{3}^{2} = 3m_{1}m_{2}m_{3}

in positive integers m_{1}, m_{2}, m_{3}. Any m_{i} is called a Markoff number. In practice we shall consider solutions (m, m_{2}, m_{1}) with m ≥ m_{2} ≥ m_{1} (and indeed m > m_{2} > m_{1} except for the two singular solutions (1, 1, 1) and (2, 1, 1)) and attach to each solution (m, m_{2}, m_{1}) an integral, indefinite, binary quadratic form f_{m} of minimum m attained on both sides (that is, f represents both m and −m), Then \frac{1}{m} f has minimum 1, and we can associate with f an admissible lattice for A_{2,0} containing 111.

[The notation should be f_{m,m_{1},m_{2}}, but see Conjecture 5.10 below.]

Given a non-singular solution (m, m_{2}, m_{1}), we can view m as a root of the quadratic X^{2} − 3m_{1}m_{2}X + m_{1}^{2} + m_{2}^{2}. Then the other root

m′ = 3m_{1}m_{2} − m = \frac{m_{1}^{2}+m_{2}^{2}}{m}

is smaller than min(m_{1}, m_{2}), so that all solutions of the Markoff equation stem from a singular solution by applying successively the inverse procedure, which yields to solutions (m′_{2}, m, m_{1}) and (m′_{1}, m, m_{2}) such that m′_{i} ≥ m ≥ m_{j}. Note that given a Markoff number m, there may a priori exist several pairs (m_{1}, m_{2}) with m ≥ m_{2} ≥ m_{1}. Extensive calculations did not produce any such example, which supports the conjecture below, which apparently appeared for the first time in a 1913 paper of Frobenius: Conjecture 5.10. For every Markoff number m, there exists only one solution (m, m_{2}, m_{1}) with m ≥ m_{2} ≥ m_{1} to the Markoff equation.

We now define a form f_{m} attached to (m, m_{1}, m_{2}). Let k ≡ \frac{m_{1}}{m_{2}} mod m chosen in (0, m). Exchanging m_{1} and −m_{2} changes k into m − k. We chose k ≡ \frac{m_{1}}{m_{2}} or − \frac{m_{2}}{m_{1}} such that 1 ≤ k ≤ \frac{m}{2}. Clearly m divides k^{2} + 1, so that setting k^{2} + 1 = m ℓ, we define an integer ℓ. Finally let

f_{m}(x, y) = mx^{2} + (3m − 2k)xy + (ℓ − 3k)y^{2} .

The key result in the theory of the Markoff chain is the following theorem, for a proof of which we refer to Chapter 2 of [Cas].

Theorem 5.11. (Markoff) The minimum of the form f_{m} is equal to m and is attained both by f and −f. The discriminant of f_{m} is

D = 9m^{2} − 4 .

The sequence (m_{n}) in Theorem 5.9 is the sequence of Markoff numbers organized in increasing order. □

Remark 5.12. The gcd of the first two coefficients of f_{m} is (2, m), and if m is even, then k and ℓ are odd. Hence for m odd (resp. even), the primitive form associated with f_{m} is g_{m} = f_{m} (resp. g_{m} = \frac{1}{2}f_{m}). For even m, setting m′ = \frac{m}{2}, g_{m} reads

g_{m} = m′x^{2} + (3m′ − k)xy + \frac{ℓ−3k}{2}y^{2} .

Its discriminant is D′ = 9m′^{2} − 1 = \frac{D′}{4} . In both cases, the discriminant of the lattice Λ attached to f_{m} is d = \frac{D}{m^{2}} = 9 − \frac{4}{m^{2}} (d = 5, 8, \frac{221}{25} = 8.84, …).

We now attach to a Markoff number m (more precisely, to a Markoff triple (m, m_{1}, m_{2}) with m ≥ m_{2} ≥ m_{1}) the monic polynomial h(X) canonically associated with the primitive form g_{m}, namely m g_{m}(\frac{X}{m}, 1) or m′g_{m}(\frac{X}{m′} , 1), that is

h(X)=X^{2}+(3m−2k)X +m(ℓ−3k) or h(X)=X^{2}+(3m′−k)X + \frac{m′(ℓ − 3k)}{2}

according as m is odd or even, and we denote by 𝔒_{m}, or simply 𝔒 the order generated by a root of h in the splitting field K of h. To prove various properties of the discriminant d_{𝔒} of 𝔒, we need some more results on the Markoff numbers, including a congruence recently proved by Zhang^{11}.

Lemma 5.13. Let (m, m_{1}, m_{2}) be a Markoff triple with m ≥ m_{2} ≥ m_{1}. (1) m, m_{1}, m_{2} are pairwise coprime.

(2) m and m_{1}±m_{2} are coprime except for a common divisor 2 when m is even.

(3) (Frobenius) The odd prime divisors of m, 3m − 2 and 3m + 2 are congruent to 1 modulo 4.

(4) (Zhang) m is congruent to 1 modulo 4 or to 2 modulo 32.

Proof. (1) If (m, m_{1}, m_{2}) is not singular, it has a neighbour (m_{2}, m_{1}, m′) with m′ < m_{1}. By induction we reduce ourselves to the obvious case of a singular triple.

(2) The Markoff equation may be written in the form m^{2} + (m_{2} ± m_{1})^{2} = m_{1}m_{2}(3m ± 2), which shows that the gcd of m and (m_{2} ± m_{1}), since it is coprime with m_{1}m_{2}, must divide 3m ± 2, hence 2.

11Ying Zhang, Congruence and uniqueness of certain Markoff numbers, Acta Arith. 128 (2007), 295–301; arXiv:math/0612620v2 [math.NT] 29 Dec 2006

(3) The equation above shows that 3m ± 2 or \frac{3m±2}{4} divides the sum of two coprime integers, hence both 3m + 2 and 3m − 2 have no divisor p ≡ 3 mod 4. The same proof works for m, writing the Markoff equation as m(3m_{1}m_{2} − m) = m_{1}^{2} + m_{2}^{2}.

(4) Only the second congruence needs a proof. Since m is even, m_{1} and m_{2} are odd, so that m_{1}^{2} + m_{2}^{2} ≡ 2 mod 8, hence m ≡ 2 mod 4 because m divides this sum, and since m_{1} ≡ m_{2} ≡ 1 mod 4, the Markoff equation modulo 8 reads 3m ≡ m^{2} + 2 ≡ 6, i.e. m ≡ 2 mod 8. Then \frac{3m−2}{4}, which is an odd divisor of m^{2} + (m_{2} − m_{1})^{2} by (2), is congruent to 1 modulo 4. This shows first that m ≡ 2 mod 16 and then that 3m + 2 ≡ 8 mod 16, i.e. that \frac{3m+2}{8} is odd. Writing finally the Markoff equation in the form (\frac{m_{1}+m_{2}}{2} )^{2} + (\frac{m}{2})^{2} = 2m_{1}m_{2} (\frac{3m+2}{8}), we see that we have \frac{3m+2}{8} ≡ 1 mod 4, i.e. m ≡ 2 mod 32. □

Proposition 5.14. The discriminant d_{𝔒} of an order 𝔒 attached to a Markoff triple is a product of primes p ≡ 1 mod 4 (or p = 8), and is congruent to 1 modulo 4 or to 8 modulo 16. Its conductor is odd.

Proof. The assertions on odd numbers are consequences of Lemma 5.13, (3). If d_{O} is even, it is of the form D′ = m′^{2} − 1 with m′ = \frac{m}{2}. By Lemma 5.13, (4). one has m′ ≡ 1 mod 16, hence D′ ≡ 8 mod 16. Write d_{𝔒} = d_{0}f^{2} where d_{0} is a fundamental discriminant. Then d_{0} is even, hence must be congruent to 8 or 12 modulo 16. This shows that the first congruence holds, which implies f ≡ 1 mod 2. □

Proposition 5.15. The fundamental unit of the Markoff orders, except those of the first two, have norm +1.

Proof. The fundamental units of the first two orders, namely \frac{1+\sqrt{5}}{2} and 1 + \sqrt{2} are of norm −1. Let now 𝔒 be a Markoff order of discriminant Δ > 8.

Recall the following well known and easy to prove result: consider an order in a real quadratic field K = ℚ(\sqrt{μ}), μ > 1. with fundamental unit ε = u + v\sqrt{μ}, u, v > 0, u, v ∈ ℤ or \frac{1}{2} + ℤ. For k > 1, let ε^{k} = u_{k} + v_{k}\sqrt{μ}. Then we have v_{k} > v except if μ = 5 and k = 2, where (\frac{1+\sqrt{5}}{2})^{2} = (\frac{3+\sqrt{5}}{2}).

If d_{𝔒} is odd, then η = \frac{3m+\sqrt{9m^{2}−4}}{2} is a unit of 𝔒, and since d_{𝔒} = 9m^{2} − 4, the result above shows that η is the fundamental unit of the order except if 9m^{2} − 4 = 5.

If d_{𝔒} is even, we consider the unit η = 3m′ + 2\sqrt{(m′^{2} − 1})/4. This time this is of the form u + v\sqrt{μ} with v = 2. One easily checks that η is a k-th power for some k ≥ 2 if and only if μ = 2 and k = 2, where (1+\sqrt{2})^{2} = 3+2\sqrt{2}, or k = 3 and μ = 5, where (\frac{1+\sqrt{5}}{2})^{3} = 2+\sqrt{5}. □

Remark 5.16. The conductor of 𝔒 may be non-trivial. The first occurrence is produced by m = 34, thus m′ = 12 and d_{𝔒} = (8·13)·5^{2}. In this case, the fundamental unit of the order, namely η = 51 + 10\sqrt{26}, is the square of the fundamental unit ε = 5 + \sqrt{26} of the maximal order, of norm −1.

6. Dimension 3

In a first subsection, we determine following Davenport the lattice constant of A_{3,0}, then describe the extension due to Swinnerton-Dyer of Davenport’s results in a second subsection. Finally we state some conjectures on totally real Minkowski’s domain of dimension n ≥ 3.

6.1. The lattice constant of A_{3,0} (after Davenport). In this subsection, following Davenport’s [Dav1], we consider a 2-dimensional lattice L generated by 111 = (1, 1, 1) and a second vector x = (x_{1}, x_{2}, x_{3}), and set S = S(x) (Definition 3.1).

Lemma 6.1. Assume that L ∩ A_{3,0} = {0}. Then either L or its image under an orthogonal automorphism of A_{3,0} has a basis (111, x) with −1 < x_{1} < 0 < x_{2} < 1 < 2 < x_{3} < 3, or S > 15.84 .

Proof. Using permutations of the coordinates and base changes x ↦ ±x+k, k ∈ ℤ, we may assume that x_{1} < x_{2} < x_{3} and 0 < x_{2} < 1. If x_{1} or x_{3} belongs to (0, 1), we have S ≥ \frac{49}{2} > 24 by Proposition 4.16, (2). Hence we may assume that x_{1} < 0 and x_{3} > 1, and by Proposition 4.16, (3), we have x_{3} > 2 or x_{1} < −1. Changing x into 1−x, we may assume that x_{3} > 2.

If x_{1} < −2, or x_{1} < −1 and x_{3} > 3, or x_{3} > 4, we have S > \frac{3}{2} 4^{2} = 24 by Proposition 4.13. We are thus left with three possible cases:

(a) −1 < x_{1} < 0 and 2 < x_{3} < 3;

(b) −1 < x_{1} < 0 and 3 < x_{3} < 4;

(c) −2 < x_{1} < −1 and 2 < x_{3} < 3, and we bound S from below in the last two cases, using Proposition 4.14.

In case (b), we have x_{1} < −\frac{1}{4}, x_{3} > 3 + \frac{1}{12}, hence S > \frac{3}{2} (\frac{10}{3})^{2} = \frac{50}{3} = 16.666 . . ..

In case (c), we have x_{1} < −1 − \frac{1}{8}, x_{3} > 2 + \frac{1}{8}, hence S > \frac{3}{2} (\frac{13}{4})^{2} = \frac{507}{32} = 15.84375.

[Using four times Proposition 4.14, we obtain S > 16.13.] □

Remark 6.2. The smallest two values of S for integers of cubic fields (14 and 18) occur using an x of the form (a) above. Polynomials having roots in the intervals described above are obtained by translation from those given in Example 3.9: there is in each case a unique choice, namely X^{3} −2X^{2} −X +1 and X^{3} − 3X^{2} + 1. The next value (S = 20) needs ⌈x_{3}⌉ − ⌊x_{1}⌋ > 4.

Proposition 6.3. Under the hypotheses of Lemma 6.1, and assuming that x_{1} < x_{2} < x_{3}, we have S ≥ 14, and equality holds if and only if (x_{1}, x_{2}, x_{3}) is the image under a transformation x ↦ ±111 + kx (k ∈ ℤ) of the roots of the polynomial f = X^{3} − X^{2} − 2X + 1.

[The roots of f are θ_{1} = −2 cos \frac{2π}{7} < θ_{2} = −2 cos \frac{4π}{7} < θ_{3} = −2 cos \frac{6π}{7} .]

Proof. After having performed the translation x ↦ x − 111, Lemma 6.1 shows that we may assume that we have

−2 < x_{1} < −1 < x_{2} < 0 < 1 < x_{3} < 2 .

Set f(X) = (X − x_{1})(X − x_{2})(X − x_{3}) = X^{3} − t_{1}X^{2} + t_{2}X − t_{3}. Calculating f at −1, 0, 1, we obtain the inequalities

(1) 1 + t_{1} + t_{2} + t_{3} ≤ −1 ; (2) − t_{3} ≤ −1 ; (3) 1 − t_{1} + t_{2} − t_{3} ≤ −1 . The combinations (1)+(3) and (1)+(2) then read t_{2} ≤ −2 and −t_{1} ≥ t_{2} + 3, that we may write t_{2} = −2 − h and t_{1} = −1 + h − h′ with h, h′ ≥ 0. This implies

S = 14 + 2(h + 2h′ + (h − h′)^{2}) ≥ 14 ,

which shows that S ≥ 14, and that equality holds if and only if h = h′ = 0, i.e. t_{2} = −2, t_{1} = 1. Then (2) and (3) show that t_{3} = 1 (the inequalities above now read max(1, 1 − 2h + h′) ≤ t_{3} ≤ 1 + h′), and f(−X) is the polynomial of the proposition. □

Theorem 6.4. The lattice constant of A_{3,0} is κ_{3,0} = 49, attained uniquely (among lattices on which x_{1}x_{2}x_{3} attains the value 1) on lattices equivalent to the image of ℤ[ζ_{7} + ζ_{7}^{−1}].

[The restriction is not necessary, but a proof of this needs an isolation result.]

Proof. Let Λ be an admissible lattice for A_{3,0} containing 111, and let p be the orthogonal projection to F = ℝ111^{⊥}. Since N(111) = 3, we have det(p(Λ)) = \frac{1}{3} det(Λ). By the results of Subsection 3.1, the projection of x = (x_{1}, x_{2}, x_{3}) ∈ E has norm \frac{S(x)}{3}. By Proposition 6.3, p(Λ) is admissible for the sphere of square radius R^{2} = \frac{14}{3}. Since the lattice constant κ_{2} of the unit disc is equal to \frac{3}{4}, we obtain

det(Λ) ≥ 3 · \frac{3}{4} · (\frac{S}{3} )^{2} = (\frac{S}{2} )^{2} ≥ 49 .

There remains to characterize the admissible lattices Λ of determinant 49 which contain 111. Since equality must hold everywhere in the inequalities above, p(Λ) must be critical for the disc of square radius \frac{14}{3}. Hence there are in Λ three vectors v_{1}, v_{2}, v_{3} the components x_{1}, x_{2}, x_{3} of which are permutations of the roots θ_{1}, θ_{2}, θ_{3} of f, and moreover there exists relation ∑ ±p(v_{i}) = 0 between their projections which lifts to a relation ∑ ±v_{i} = λ111 for some λ ∈ ℤ. There cannot be a transposition among the permutations, for if, say, v_{1} = (θ_{1}, θ_{2}, θ_{3}) and v_{2} = (θ_{1}, θ_{3}, θ_{2}), no combination ∑ ±v_{i} belongs to ℝ111. Hence the three permutations are the three possible circular permutations, the v_{i} add to −1, and v_{1}, v_{2}, v_{3} constitute a basis for the lattice which represents ℤ_{K}. □

6.2. The next minima of A_{3,0} (after Swinnerton-Dyer). This subsection relies on Swinnerton-Dyer’s paper [SwD]; see also [SwD2].

6.3. Conjectures on totally real Minkowski’s domains. The behaviour of the first eighteen minima for A_{3,0} found by Swinnerton-Dyer suggests the following conjecture:

Conjecture 6.5. Let n ≥ 3 odd. There exists a sequence M_{1}, …, M_{n} of modules in totally real number fields of degree n such that the sequence Λ_{n} of lattices in ℝ^{n} associated with m_{n} has the following properties:

(1) Every admissible lattice for A_{n,0} is of the form λ u(Λ_{n}) for some n ≥ 1, λ ≥ 1, and u ∈ Aut(A_{n,0}).

(2) The determinants of the Λ_{n} constitute an increasing sequence which tends to infinity.

The conjecture is false for n = 2, and similarly, its analogue for relative quadratic extensions might well be false: we cannot exclude the existence of some “Markoff-like chain”. This is the reason for the hypothesis “n odd”.

Note that the comments made by Swinnerton-Dyer in [SwD] indicate that he believed the conjecture above is true in dimension n = 3, though he does not state it explicitly. Note also that part (1) implies part (2) thanks to Cassels and Swinnerton-Dyer’s paper [Cas-SwD].

7. Dimension 4

Difficulties related to the possible existence of several components of an x ∉ ℝ111 belonging to a same interval (m, m + 1) occur from dimension 4 onwards. As was noted by Godwin in [God], we can obtain lower bounds for S by assuming that such components are equal. A precise statement is given in the next proposition, the easy proof of which is left to the reader.

We thus consider x = (x_{1}, …, x_{n}) and assume that p components x_{i} lie in some interval (m, m + 1), and more precisely in some interval [α, β] ⊂ (m, m + 1) (which can be constructed using Proposition 4.14 under the hypothesis that x lies outside A_{n,0}), and the remaining q = n − p components are off this interval. Let a = \frac{1}{q} ∑_{x_{i}∉(m,m+1)} x_{i} be the average of the x_{i} off (m, m + 1).

Proposition 7.1. Under the hypotheses above, S = ∑_{i<j}(x_{j} − x_{i})^{2}, viewed as a function of the x_{i} belonging to (m, m + 1) attains its minimum when these x_{i} are equal, to α, to a or to β according to whether a < α, α ≤ a ≤ β or a > β. □

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