As stated previously, the benefit of viewing the symmetric space X as a cone of positive definite Hermitian forms is that it makes computations more straightforward. To clarify this statement, we will now give a brief exposition of the work of Koecher on positivity domains ([Koe60]), which will give us a computable model for this cone. We follow the treatment given by Paul Gunnells (see for example, [Gun11] or [GY13]).
Let V be a finite-dimensional real vector space, equipped with an inner product h , i, and give V the standard topology induced by this inner product. For a subset C ⊂ V , let C denote its closure (with respect to the aforementioned topology), Int(C) its relative interior, and ∂C = C \ Int(C) its boundary.
We call a subset C ⊂ V a positivity domain if the following are satisfied: i. C is open and non-empty;
ii. hX, Y i > 0 for all X, Y ∈ C; and
iii. For each X ∈ V \ C there exists a non-zero Y ∈ C such that hX, Y i ≤ 0. Proposition 4.2.1. The cone C of positive definite Hermitian forms over F de- fined in the previous section (viewed as a subset of the full space V of Hermitian forms) is a positivity domain.
Proof. This follows immediately from the fact that C is self-adjoint. In fact, it is easy to see that any positivity domain is a cone (in the sense that it is convex and closed under positive real homotheties) and cannot contain any lines.
Now, let D be a discrete non-empty subset of C \ {0}. For each Φ ∈ C, let mD(Φ) = inf
X∈D{hΦ, Xi},
the minimum of Φ (with respect to D). In [Koe60] it is shown that mD(Φ) ≥ 0,
and furthermore that the infimum is achieved only on a finite set of points. We call this set the set of minimal vectors of Φ, and denote it by MD(Φ):
MD(Φ) := {X ∈ D; hΦ, Xi = mD(Φ)}.
We call a point Φ ∈ C perfect (with respect to D) if the linear span of its minimal vectors MD(Φ) is the full space V .
In the specific example of the cone C of positive definite Hermitian forms over F , we refer to perfect points as perfect forms. We have the following characterization of such forms:
Proposition 4.2.2. Let C denote the cone of positive definite Hermitian forms over F , and let D be a discrete non-empty subset of C. Then a point Φ ∈ C is per- fect if, and only if, it can be recovered uniquely from the data {mD(Φ), MD(Φ)}
(that is, if Φ0 ∈ C satisfies m
D(Φ0) = mD(Φ) and MD(Φ0) = MD(Φ), then
Φ0= Φ).
Proof. Let v1, . . . , vrand vr+1, . . . , vr+sdenote the set of real and complex
places of F respectively, and define an R-basis for V by giving each Vvi the basis
Bi=
{xi,1, xi,2, xi,3}; i ∈ {1, . . . , r}
{xi1, xi2, xi3, xi4}; i ∈ {r + 1, . . . , r + s}, where xi1= (1 00 0) , xi2= (0 00 1) , xi3= (0 11 0) , xi4= −α 00 α , and α2= −1. Now, let A =X i,j aijxij, and B = X i,j bijxij
be any two points in V. Then hA, Bi = r+s X i=1 (ai1bi1+ ai2bi2+ 2ai3bi3) + 2 r+s X i=r+1 ai4bi4= aˆbT, where aij = aij, and ˆ bij = bij; if j = 1, 2, 2bij; if j = 3, 4.
Given Φ ∈ C, let MD(Φ) = {P1, . . . , Pt} denote the set of minimal vectors
of Φ, and let Pk=
X
i,j
p(k)ij xij for each k, and Φ =
X
i,j
φijxij.
We obtain a linear system of equations
p(k)φˆT = mD(φ), k = 1, . . . , t,
as above. From this it is clear that we have a unique solution for Φ if, and only if, the p(k)form the rows of a matrix of rank 3r + 4s, which occurs if, and only
Throughout our work, we will fix a choice of discrete set D, taking the set Ξ consisting of points of the form q(x), x ∈ O2
F\ {0}, where
q(x) = (xvx∗v),
with each xv the image of x under the embedding F2 ,→ Fv2. Note that each
matrix xvx∗v has rank one, and thus q(x) ∈ C.
It is clear that a point Φ in a positivity domain C is perfect if, and only if, λΦ is also perfect for any λ ∈ R+, in which case mD(λΦ) = λmD(Φ). We
may therefore consider only those perfect forms Φ for which mD(Φ) = 1. Given
a discrete set D, we denote by Perf(D) the set of perfect points for D whose minimum is 1.
One of the benefits of studying positivity domains is that they exhibit a re- duction theory: given a positivity domain C, we will find that we can decompose C into a family of cones Σ which has only finitely many orbits under the action of certain discrete subgroups Γ of the automorphism group GC⊂ GL(V ) of C.
This is reminiscent of the classical situation, and the fundamental domains for the action of congruence subgroups Γ ⊂ SLn(Z) on the complex upper half-plane
h2.
In the case of binary Hermitian forms over a number field, this theory will utilize the discrete set Ξ we have described previously. More generally, given a positivity domain C, call a non-empty discrete set D ⊂ C \ {0} admissible if for any sequence (Φi) in C converging to a point in ∂C, we have mD(Φi) → 0.
Proposition 4.2.3. The set Ξ defined above is an admissible subset of the cone C of positive definite Hermitian forms over F .
Proof. See [Koe60], Lemma 11.
We are almost in a position to discuss the aforementioned reduction theory. Before we proceed, we need a few basic notions from the field of convex geometry.
A polyhedral cone in a real vector space V is a subset σ of the form σ = σ(v1, . . . , vt) =
nX
λivi; λi ≥ 0
o ,
where v1, . . . , vt ∈ V is a fixed set of vectors. We say that the set {v1, . . . , vt}
is a spanning set for σ. If σ admits a linearly independent spanning set, then we call σ simplicial. The dimension of a polyhedral cone σ is the dimension of its linear span; if d = dim(σ), we call σ a d-cone.
Fix an inner product space V , with positivity domain C, and let D ⊂ C \ {0} be an admissible subset. Given a perfect point Φ ∈ Perf(D), one can naturally define a polyhedral cone σ(Φ) = σ(P1, . . . , Pt), where {P1, . . . , Pt} = M (Φ) is
the set of minimal vectors of Φ. We call such a cone the perfect pyramid asso- ciated to Φ. By definition, it is a cone of dimension dimR(V ), although it need not be simplicial. Let Σ = ΣDdenote the set of perfect pyramids, together with
all their proper faces, as we range over all perfect points Φ ∈ Perf(D). Then Koecher proves in [Koe60] the following result:
Theorem 4.2.4. The perfect pyramids have the following properties: (i) Any compact subset of C meets only finitely many perfect pyramids. (ii) Two different perfect pyramids have no interior point in common. (iii) Given a perfect pyramid σ, there are only finitely many perfect pyramids
σ0 such that σ ∩ σ0 contains a point of C. By part (ii), this must lie on the boundaries of σ and σ0.
(iv) The intersection of any two perfect pyramids is a common face of each. (v) Let σ be a perfect pyramid and τ a codimension one face of σ. If τ does
not lie completely in the boundary ∂C, then there exists precisely one other perfect pyramid σ0 such that σ ∩ σ0= τ .
(vi) S
σ∈Σσ ∩ C = C.
By a facet of a perfect pyramid σ, we shall mean a codimension one face. If two perfect pyramids σ and σ0 meet in a facet τ as in condition (v) above, we say that σ and σ0 are neighbours.
We call Σ the Koecher fan, and the cones in Σ the Koecher cones.
Let GC⊂ GL(V ) denote the group of automorphisms of V which fix the cone
C, and let Γ ⊂ GC be a discrete subgroup which preserves the admissible set D.
In [Koe60], Section 5.4 it is shown that Γ admits a properly discontinuous action on C. Then:
Theorem 4.2.5. We have an explicit reduction theory for Γ in the following sense:
(i) There are finitely many Γ-orbits in Σ.
(ii) Every X ∈ C is contained in a unique cone in Σ.
(iii) If σ ∈ Σ does not lie completely in the boundary ∂C, the stabilizer Sσ:= {γ ∈ Γ; γ(σ) = σ}
If we choose representatives σ1, . . . , σk of the orbits of Γ ∈ Σ, and let Ω = Ω(Γ) = k [ i=1 (σi∩ C),
then the intersection of each cone σ ∈ Σ with C has a Γ-translate which is contained in Ω. This is not quite a fundamental domain, as we have non-trivial stabilizers to worry about, but since these are finite groups, it doesn’t in practice cause a problem.