Binary hermitian forms and optimal embeddings
Gordan Savin Michael Zhao
math.NT


Department of Mathematics, Salt Lake City, UT 84112
[Gordan Savin]savin@math.utah.edu

Department of Mathematics, Salt Lake City, UT 84112
[Michael Zhao]mzhaox@gmail.com


Binary Hermitian Forms and Optimal Embeddings
    Michael Zhao
    December 30, 2023
=============================================



 Fix a quadratic order over the ring of integers. 
An embedding of the quadratic order into  a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this 
correspondence is 
a discriminant preserving bijection between the isomorphism classes of embeddings and integral binary hermitian forms. 
 


§ INTRODUCTION


Let K be a field of any characteristic and fix L, a separable quadratic extension of K. Let n_L : L → K, n_L(l)=ll̅,  be the norm map.
In this paper we study relationship between embeddings of L into a quaternion algebra and vector spaces over L with a hermitian form, 
as well as an integral version of this problem. 
Let V be a vector space over L.  A hermitian form on V is a 
quadratic form h: V → K such that h(lv) = n_L(l) h(v) for all l∈ L and v∈ V. This definition is different than what is customary in 
the literature, where a hermitian symmetric form is a function s : V × V → L, linear in the first variable, 
and s(y,x)= s(x,y) for all x,y∈ V. However, given h, we show that there exists a unique s 
 such that h(v)=s(v,v) for all v∈ V, even if the characteristic of K is 2. 


 Let i: L → H be an embedding of L into a quaternion algebra H. Let n: H → K be the (reduced) norm. 
 Then H is naturally a vector space over L of dimension 2,  with a hermitian form h: = n  representing 1, since 
n(1)=1. Conversely, let (V,v,h) be a triple consisting of a 2-dimensional vector space V over L, a non-degenerate hermitian form h on V, and a point v∈ V such that 
h(v)=1. Then one can define a quaternion algebra H_V such that H_V=V, as vector spaces over K,  v is the identity element of H_V,  
 and l ↦ l v is an embedding of L into H_V, where 
l v is the scalar multiplication inherited from V. The reduced norm is, of course, equal to the hermitian form h. This discussion can be placed in a categorical context; 
the following two categories are isomorphic: 
 one whose objects are pairs (i,L) where i is an embedding of L into a quaternion algebra, the other whose objects are triples (V,v,h) of 
pointed, non-degenerate, binary hermitian spaces. 
We now observe that different choices of the point v representing 1 give isomorphic objects. Indeed, if u∈ V is another element in V representing 1, then u is a unit element in the quaternion algebra 
H_V arising from the triple (V, v, h). Right multiplication by u gives an isomorphism  of the triples (V, v,h) and (V, u, h). 
As a consequence, there is a bijection between isomorphism 
classes of embeddings of L into quaternion algebras and non-degenerate binary hermitian spaces (V,h) representing 1. 

As the next result, we establish an integral version of this bijection. More precisely, assume that K is the field of fractions of a Dedekind domain A, and B  the integral 
closure of A in L.  An order O in H is an A-lattice, containing the identity element and closed under the multiplication in H. An embedding i: L → H is 
called optimal if i(L)∩ O = i(B). If this is the case, then O is naturally a projective B-of rank 2, such that the norm n takes values in A and represents 1. 
Conversely, let Λ⊂ V be a lattice, i.e. a projective B-module of rank 2, such that h is integral on Λ, i.e. 
h(Λ) ⊆ A. Such a pair (Λ, h) is called an integral binary hermitian form. 
Assume that there exists v∈Λ such that h(v)=1. We show that Λ is an order (denoted by O_Λ) in H_V, and thus we have 
constructed a quaternionic order and an embedding of B into it, given by b↦ bv. Just as in the field case, different choices of  v give isomorphic objects and, as a consequence, 
there is a bijection between isomorphism 
classes of embeddings of B into quaternionic orders and non-degenerate integral binary hermitian forms  (Λ,h) representing 1. 

We now turn to the question when a hermitian form represents 1, which is especially interesting in the integral case. To that end we establish a relationship 
between invariants  of the objects studied. The determinant d(Λ, h) of an integral binary hermitian form (Λ, h) is a fractional ideal in K generated by 
(s(v_i, v_j)) for all pairs (v_1,v_2) of elements of Λ.  
We note that d(Λ, h) is not necessarily an integral ideal. Integrality of h implies only that s takes values in the different ideal of B over A in L. 
 Since the discriminant D_B/A is the norm of the inverse different, the product D_B/A· d(Λ, h) is an integral ideal. This was 
 previously observed in <cit.> where the discriminant of an integral binary hermitian form is defined as this product. Thus we follow the same convention 
 and define the discriminant of  (Λ,h)  to be the integral ideal  Δ(Λ, h) =D_B/A· d(Λ, h). If h represents 1, so Λ is 
 the order O_Λ, then we have an identity 

    Δ( O_Λ)= Δ(Λ, h)
 
of integral ideals where Δ( O_Λ) is the discriminant of the order O_Λ. 
If K=ℚ and L an imaginary quadratic extension, then these invariants can be refined to be integers, and this is an equality of two integers. 
The following is the most interesting result in this paper: 
 




  Assume L is a tamely ramified, imaginary quadratic extension of ℚ. Let B be the maximal order in L. 
Let Δ<0 be a square free integer. 
There is a bijection between the isomorphism classes of embeddings of B into quaternionic orders of discriminant Δ and integral binary 
B-hermitian forms of discriminant Δ. 

 

The key here is to show that an integral binary 
hermitian form (Λ, h) of discriminant Δ represents 1. 
The condition Δ is square free assures that h represents 1 on 
Λ⊗ℤ_p for all p. The condition Δ <0 implies that h is indefinite on Λ⊗ℝ. 
The theorem follows from the integral version of the Hasse-Minkowski theorem for indefinite 
quadratic forms in 4 variables. 






§ HERMITIAN FORMS
 

Let K be a commutative field of any characteristic. Let L be a separable quadratic K-algebra. In particular, there exists an 
involution l ↦l̅ on L such that the set of fixed points is K. A vector space over L is a free L-module. 
We go over some basic results about hermitian forms on vector spaces. 


 Let V be a vector space over  L.  A hermitian symmetric form is a
function s: V × V → L which is L-linear in the first
argument and s(x,y) = s(y,x).



Let h: V → K be the function defined by h(x)=s(x,x) for all x∈ V. This is a quadratic form, if we consider V a vector space over K. in particular, 
the function b(x,y) = h(x + y) - h(x) - h(y) is K-bilinear. 
If we let n_L(l) = l l, for l∈ L, then  h(lx) = n_L(l) h(x). 
These two facts are a sufficient characterization of hermitian forms for our
purposes, due to the

  Let V be a vector space over L. 
Let h: V → K be a quadratic form, where V is considered a vector space over K, satisfying h(lx) = n_L(l) h(x)
for all l ∈ K. Then there is a unique hermitian symmetric form s(x,y) such that h(x)
= s(x, x).



Assume that s exists. Then, for all x, y ∈ V and l ∈ L,  

    s(x,y) + s(y,x)     = b(x,y) 
    
    l s(x,y) + l  s(y,x)     = b(lx, y).

We view this as a 2× 2 system in unknowns s(x,y) and s(y,x). If l≠l̅ then this system has a unique solution given by 

    s_l(x,y) = lb(x,y) - b(lx, y)l - l
 
where the subscript l indicates that the solution, so far, depends on the choice of l. 
(Note that l - l is invertible, even when L≅ K^2.) We claim that s_l(x,y) is independent of l. Indeed, let
l = c + d π, where L=K[π], and c,d∈ K. Then


    s_l(x, y)     = l b(x, y) - b(lx, y)l - l
        = c b(x, y) + d π b(x, y) - b(cx + d π x, y)c + d
    π - c - dπ
        = d π b(x, y) - d b(π x, y)dπ - dπ
        = s_π(x, y).

Hence s_l = s_π and we can write s for any of the s_l. It is then easy to check
that s(x, x) = s_l(x, x) = h(x).

Now we check that s(x,y) is linear in the first variable. It is clear that 
s(x_1 + x_2, y) = s_l(x_1 + x_2, y). Let l∈ L ∖ K. Then 

    s(l x, y)     = s_l(lx, y)  
        = l b(l x,y) - b(n_L(l) x, y)l - l
        = l b(l x,y) - n_L(l) b(x, y)l- l
        = l l b(x, y) - b(lx,y)l -l
        = l s_l(x,y) = l s(x,y),

which shows s is linear in the first argument. 
It remains to check that s(x, y) = s(y, x). 
Note that b(l
x, l y) = n_L(l) b(x, y). Replacing x with lx, we find that b(n_L(l)
x, l y) = n_L(l) b(lx, y), hence b(lx, y) = b(x, l y). Then

    s(y, x) = s_l(y, x)     = l b(y, x) - b(l y, x)l -l
        = l b(x, y) - b(lx, y)l - l
        = s_l(x, y) = s(x, y),

or s(x, y) = s(y, x), as desired. Uniqueness is clear. 


Thus, we can make the following definition.

 Let V be a vector space over L. 
A hermitian form is a quadratic form h: V → K, where V is a considered a vector space over K, such that h(lv) = n_L(l) h(v) for all l ∈ L, and for all v ∈ V, 
 



§ QUATERNION ALGEBRAS
 

Let H be a quaternion algebra over K, i.e. a central simple algebra over K of dimension 4. 
We know that by general theory (e.g. <cit.>), every element x∈ H 
 satisfies a reduced characteristic polynomial of degree two,
associated to its action on the algebra via left-multiplication. The reduced trace (x) is
the sum of the roots of the reduced characteristic polynomial of x. Let the
reduced norm n(x) be the product of the roots.
 The trace furnishes H with an involution
x := (x) - x such that n(x)=xx.  Let L be a separable quadratic field
extension of K and i: L → H an embedding. Since L is separable, the restriction of the involution of H to L is non-trivial. 
The norm n is multiplicative, i.e. n(xy)=n(x)n(y) for all x,y∈ H. Hence  the norm is a hermitian
form on H. This hints at a correspondence between embeddings of L into quaternion algebras 
and hermitian forms. The next section will formalize this idea. We finish this section with the following structural result. 



Let L be a separable quadratic field
extension of K. If i: L → H is an embedding, then there exists u ∈ H such that u̅ =-u and 
 θ∈ K^× satisfying H = L + Lu, u^2 = θ, and um = mu
  for all m ∈ L (cf. <cit.>). For every v=x+yu ∈  L + Lu, 
  
    n(v)= n_L(x) - n_L(y)θ,
 
  where n_L: L → K is the norm. 

 By Proposition <ref>, there exists a unique, L-hermitian bilinear form 
s(v,w) : V × V → L such that s(v,v)=n(v) for all v∈ H. 
 Take u≠ 0 with s(u, 1) = 0. Then H=L + Lu.  For any
l ∈ L ∖ K, 
    0 = s(u, 1) = s_l(u, 1) = lb(u, 1) - b(lu, 1)l - l.
 Since b(u, 1) = n(u + 1) - n(u) - n(1) = (u) and b(lu, 1) = (lu), hence
l(u) = (lu). If (u) ≠ 0, then l∈ K, a
contradiction. Hence (u) = 0 and (lu) = 0. The first of these implies
that u = -u, and so 0 = (lu) = lu + ul = lu - ul, i.e. that lu = u l, or u l= l u. Finally, since (u) =
0, u^2 = -n(u) ∈ K. Simplicity of H forces θ to be non-zero. The formula for n(v) follows from orthogonality of 1 and u. 


Note that the hermitian symmetric form s, attached to n, in the basis 1,u is given by a diagonal matrix with 1 and -θ on the diagonal. In particular, 
invertibility of θ implies that the hermitian form is non-degenerate. 




§ CORRESPONDENCE OVER A FIELD


Here L continues to be a separable quadratic extension of K.
The correspondence alluded to before 
Proposition <ref> can be formalized as an isomorphism of the
categories L of non-degenerate pointed hermitian spaces and
L of embeddings of L into quaternion algebras over K. 
We will define these terms more precisely, and give a proof of the isomorphism.



Let L be the category whose objects are pairs (i, H)
where H is a quaternion algebra over K, and i : L → H is an embedding of L into H. 
The morphisms f : (i, H) → (i', H') are morphisms f : H → H' of K-algebras
satisfying f ∘ i = i'. Note that any f must be an isomorphism of H with
H'.






Let L be the category whose objects are triples (V, v,
h), called pointed hermitian spaces, where V is a
two-dimensional L-vector space, h: V → K a non-degenerate
hermitian form, and v ∈ V such that h(v) = 1. The morphisms from (V,
v, h) to (V', v', h') are isomorphisms f: V → V' with f(v) = v'
and h(x) = h'(f(x)).



We can now define a functor F: L →
L, due to the following easy lemma: 

Let F: L → L be
given by F(i, H) = (H, 1, n), where n is the norm form of H, and 1
is the identity element of H. Given a morphism f: (i, H) → (i', H'),
define F(f) := f. Then F is a functor.



Now we construct a functor G: L →
L by defining a multiplication · on a given (V, v, h),
motivated by Proposition <ref>. Recall that there is a unique
hermitian symmetric form s (v,w) on V with h(x) = s(x, x) for all x∈ V.   
Embed L into V via l ↦ lv where lv denotes the scalar multiplication. 


  * v is the identity element, 

  * for x ∈ V,  and l∈ L, l · x := lx, the scalar multiplication, 

  * for y ∈ L^⊥ = { w  |  s(v, w) = 0}, y · (lv) := ly,
for l ∈ L.

  * for y ∈ L^⊥, y · y := -h(y)v.

  * conjugation on V is lv+y=lv -y,  for l∈ L and y ∈ L^⊥,

 This gives V = L ⊕ L^⊥ a structure of quaternion algebra, as in Proposition <ref>, 
 since we can choose u to be any element of L^⊥, and θ = -h(u).
  Non-degeneracy of s implies that θ≠ 0. The norm form is h. 
Let H_V denote this algebra. Then we have  


Define G(V, v, h) := (i_v, H_V), where i_v(l)=lv for l∈ L, and for a morphism f: (V,
v, h) → (V', v', h'), define G(f) := f. Then G is a functor from 
L to L.


It is clear that the functors F and G are inverses of each other. Thus, we have 



The categories L and 
L are isomorphic. 


Let (V,v, h) be a pointed binary hermitian space and H_V the quaternion algebra with the identity element v arising from it. 
Let u∈ V be another element in 
V such that h(u)=1. Then v,u, understood as elements in H_V,  satisfy v· u=u. Thus, the right multiplication by u gives an isomorphism of the
pointed symmetric spaces (V,v,h) and (V, u, h).  This shows that the isomorphism classes of objects in the category L  do not depend 
on the choice of the point representing 1.  Thus we have the following: 


  
The functors F and G give a bijection between the isomorphism classes of embeddings of L into quaternion algebras and the isomorphism classes 
of non-degenerate binary hermitian spaces (V,h) such that h represents 1. 
 



§ CORRESPONDENCE OVER A DEDEKIND DOMAIN


Assume that K is a field of fractions of a 
 Dedekind domain A. Let B be the integral closure of A in L. 
 It turns out an 
integral version of the above isomorphism can be established, replacing the field
with a Dedekind domain, quaternion algebras with orders, and binary hermitian
forms on a vector space with integral binary hermitian forms on a B-projective
module of rank 2. Let i: L → H, be an embedding of L into H. 
Recall that a lattice Λ in the quaternion algebra H is a finitely generated A-submodule containing 
a K-basis of H.  If the lattice Λ is a B-module, then it is also projective of rank 2. 
An order in H  is a lattice containing the identity element and closed under the multiplication. 




Let B be the category whose objects are pairs (i,
𝒪) where i : B → is an embedding into an order
 of quaternion algebra. The morphisms from  (i, 𝒪) to 
(i', 𝒪')  are isomorphisms f : 𝒪→𝒪' of A-algebras satisfying f ∘ i = i'.



 A binary B-hermitian form is a pair (Λ, h) where Λ is a 
projective B-module of rank two, and h: Λ⊗_A
K → K is an L-hermitian form. The form is integral if h(Λ) ⊂ A. 



In the above definition we have identified Λ with Λ⊗ 1 ⊂ V = Λ⊗_A K.  



Let B be the category whose objects are triples (Λ,v, h), 
called pointed integral hermitian spaces, where (Λ, h) is an integral, binary B-hermitian form, with h non-degenerate, 
 and v ∈Λ with h(v) = 1. The morphisms from (Λ, v, h) to (Λ', v', h') are B-module isomorphisms f: Λ→Λ' preserving the pointed hermitian structure, i.e. for x ∈Λ, h(x) = h'(f(x)) and f(v) = v'.




We will prove that the two categories are isomorphic. In one direction we have 
a functor given by the following: 


Let P: B → B be defined by P(i, 𝒪) := (𝒪, 1, n), where n is
the norm of the quaternion algebra ⊗_A K, and for a morphism
f: (i, 𝒪) → (i', 𝒪'), P(f) := f. Then P is a
functor between these two categories.



In order to define a functor Q: B →
B, we need to define a multiplication on a pointed
B-hermitian module (Λ, v, h). On V = Λ⊗_A K, as in the previous section, we have a 
quaternion algebra structure  H_V arising from h and the 
associated L-hermitian bilinear form s: V × V → L, such that v is the identity. 
We need the following: 


The B-module Λ, regarded as a
subset of H_V (recall V = H_V as sets), is closed under the multiplication on
H_V i.e. it is an order in H_V denoted by 𝒪_Λ 


We first reduce to the case of A and B principal ideal domains. 
Let  be any prime of A, S_ = A -, and A_ = S_^-1 A,
B_ = S_^-1 B the localizations of A and B respectively. 
 Then B_ is a semi-local Dedekind domain, hence it is a principal ideal domain. 
Let Λ_ = S_^-1Λ. If we show, for x,
y ∈Λ , that x · y ∈Λ_, for all , then using Λ =
⋂_Λ_, we  conclude that x · y ∈Λ. 

Hence we can assume that B is a principal ideal domain. Then Λ is a free B-module of rank 2. Note that
v is a primitive element of the lattice Λ since h(v) = 1. Hence v  is  member of a B-basis (v,w) of Λ 
 <cit.>.  Let  v^⊥ = w - s(w, v) v, then s(v, v^⊥)= 0.
Let γ = s(w, v) and θ = v^⊥· v^⊥ = -h(v^⊥).
Then to check x · y ∈Λ, we just need to check that

    w · w = (v^⊥ + γ v)· (v^⊥ + γ v)

is an element of Λ_. But we can compute

    w · w     = (v^⊥ + γ· v) · (v^⊥ + γ· v) 
        = v^⊥· v^⊥ + v^⊥·γ· v + γ· v ·
    v^⊥ + γ· v ·γ· v 
        = θ· v + γ· v^⊥ + γ· v^⊥ +
    γ^2 · v 
        = (θ + γ^2) v + (γ + γ) v^⊥
        = (θ + γ^2 - (γ)γ) v + (γ) w 
        = (θ - n(γ)) v + (γ) w

Now note that n(γ) = γγ= h(γ· v). Hence,

    -θ = s(v^⊥, v^⊥)     = s(w, w) + s(-γ v, w) + s(w, -γ v) + 
    s(-γ v, -γ v) 
        = h(w) + h(γ v) - γ s(v, w) - γ s(w, v) 
        = h(w) + n(γ) - 2 n(γ) = h(w) - n(γ),

since s(v,w) = γ. Hence θ - n(γ) = -h(w),
which is in A. Finally,

    (γ) = γ + γ    = s(w, v) + s(v, w) 
        = s(v + w, v + w) - s(v, v) - s(w, w) 
        = h(v + w) - h(v) - h(w),

which is also in A. Thus, the product from V preserves Λ. 


 Then we have
the lemma

Let Q: B → B be
defined by Q(Λ, v, h) = (i_v, Λ),  i_v(b)=b v for  b∈ B, and 
for a morphism f : (Λ, v, h) → (Λ', v',h'), define Q(f) := f. Then Q is a functor.


Again, it is easy to check that the functors P and Q are inverses of each other. In particular: 

The categories 
B and  B are isomorphic. 


Arguing in the same way as in the case of fields, Corollary <ref>, 
 two pointed binary integral B-hermitian modules (Λ, v, h) and (Λ, u, h) are isomorphic. Thus the isomorphism classes 
in B are the same as the isomorphism classes of binary integral hermitian spaces (Λ, h) such that h represents 1. 

  
The functors P and Q give a bijection between the isomorphism classes of embeddings of B into quaternionic orders of and the isomorphism classes 
of binary integral B-hermitian spaces (Λ,h) such that h represents 1. 
 

All that remains is to examine in which situations there is  existence of a
point with h(v) = 1. This can be done, in the field and domain cases, by using
local-global principles.



§ DISCRIMINANT RELATION


We are in the setting of the previous section. 
 Let Λ be a lattice in H. The fractional ideal d(Λ) in K generated by 
((u_i u_j)), for all quadruples  (u_1, … , u_4) of elements in Λ is a 
square. 
 
  
Assume firstly that A is a principal ideal domain. Then d(Λ) is a principal ideal generated by ((u_i u_j)) where 
 (u_1, … , u_4) is any A-basis  of Λ. Let Λ' be another lattice and (u'_1, … , u'_4) its basis. 
 Let T: H → H be the linear transformation such that T(u_i)= u_i' for all i. Then 
 
    d(Λ')= (T)^2 d(Λ).
 
 Hence, in order to show that d(Λ) is a square, it suffices to do so for one lattice. Write H=L + L u, as in Proposition <ref>.  Let 
 Λ= B + Bu. Assume that B= A[π] where π is a root of the polynomial x^2 + ax +b. Pick 1, π, u, π u as a basis of Λ. Then the matrix of traces is 
 
    (
        2     -a     0     0 
    
        -a     a^2 - 2b     0     0 
    
        0     0     -2θ    aθ
    
        0     0     aθ    -2 b θ
    ),

  and its determinant is (a^2 - 4b)(4b - a^2) θ^2 = -(a^2-4b)^2 θ^2. 
  Then d(Λ)= (D_B/Aθ)^2 (as ideals) where D_B/A is the discriminant of B over A. The general case 
 is now treated using localization. Let  be a maximal ideal in A. Then, one easily checks, 
 d(Λ)_𝔭= d(Λ_𝔭). Since A_𝔭 is a local Dedekind ring, hence a principal domain, d(Λ_𝔭) must be 
 an even power of 𝔭. 
 

 
The discriminant of an A-lattice Λ in H is the fractional ideal Δ(Λ) in K such that Δ(Λ)^2=d(Λ). 
 

If K=ℚ then we can refine the notion of Δ(Λ) to be a rational number, the generator of this ideal. 
We pick Δ(Λ) to be the positive generator if 
H⊗_ℚℝ is the matrix algebra, and negative otherwise. 


 Let Λ be a projective B-module of rank 2 and h a hermitian form on Λ, not necessarily integral. 
 Let s be the bilinear hermitian form on Λ such that 
s(x,x)=h(x), for all x∈Λ. Let d(Λ, h) be the fractional ideal in K generated by 
 (s(v_i,v_j)) for all pairs  (v_1, v_2)  of elements in Λ. 
 

If K=ℚ and L a complex quadratic extension we refine d(Λ,h) to be the positive generator of this ideal if h is a positive  definite 
hermitian form on Λ⊗_ℤℝ and negative otherwise. In other words this sign is the sign of (s(v_i,v_j)), for 
any basis (v_1, v_2)  of Λ⊗_ℤℝ. 



  
 Let Λ be a lattice in H that is also a B-module. Then we have the following identity of fractional ideals in K: 
 
    Δ(Λ)= D_B/A· d(Λ, n).
 
 If K=ℚ and L a complex quadratic extension, this is an identity of rational numbers. 
 
 
 
This is an identity of two fractional ideals, so it can be checked by localization at every prime ideal . 
 The localizations A_𝔭 and B_𝔭 are a local and a semi-local, respectively, Dedekind domains. 
Thus they are principal ideal domains <cit.>.  Hence, by localization, the proof can be reduced to the case of principal ideal domains. So we shall assume that 
A and B are both principal ideal domains. In that case d(Λ, n) is a principal ideal generated by (s(v_i,v_j)) where v_1, v_2 is a B-basis 
of Λ.  Let Λ' be another lattice in H that is also a B-module. Let v_1', v_2' be a B-basis of Λ'. 
Let T: H → H be the L-linear map defined by T(v_i)=v'_i. Then 
 
    d(Λ',n)= n_L(_L(T))d (Λ, n).
 
Now note that if we view T as a K-linear map, then _K(T)= n_L(_L(T)). Since Δ(Λ')= _K(T)Δ(Λ), it follows that both sides of 
the proposed identity change in the same way, if we change the lattice. Hence it suffices to check the identity for one lattice. Take Λ = B + B u, then both sides 
are ideals generated by D_B/Aθ,  where -θ= u^2, so the identity of ideals holds. 
The last statement is an easy check of signs, since the discriminant of complex quadratic fields is negative. 

 
We finish this section with the following proposition which defines the discriminant of an integral binary hermitian form, and establishes its integrality. 

  
Let Λ be a projective B-module of rank 2 and h an integral hermitian form on Λ. Then 
Δ (Λ, h)= D_B/A· d(Λ , h) is an integral ideal, called the discriminant of (Λ, h). 
 
 By localization, we can assume that A is local. Let v_1,v_2∈Λ and consider the entries of the matrix (s(v_i,v_i)). Since 
s(v,v)=h(v) the diagonal entries are integral. However, the off-diagonal entries need not be integral. Indeed, for every l∈ B, such that l≠ l, 
we have the formula 

    s(v_1, v_2) = lb(v_1,v_2) - b(lv_1, v_2)l - l.
 
If  splits or is inert in B, then there exists l such that l - l is a unit. Hence (s(v_i,v_j)) is in A. If  ramifies, then we can take 
l to be the uniformizing element in B. In this case  B=A[l ] and the discriminant D_B/A is precisely the norm of l -l. It follows that 
D_B/A·(s(v_i,v_j)) is integral. 
  










§ REPRESENTING ONE
 
Here we address the question when a binary hermitian form
represents 1, integral or otherwise. We first address the field case. The case of
integral representations is significantly more complicated; a local-global
principle is provided by <cit.>, which will reduce the
problem of an integral representation of 1 to the problem of integral
representations over p-adic rings. 




 §.§ Over a Field
 Assume that K is a number field. 
 Any binary L-hermitian form h is a quaternary
quadratic form over K. Recall that h is positive semi-definite at a real 
place of K (i.e. an embedding of K into ℝ) if h is always greater than or equal to 0 on V⊗_Kℝ. We say that h is 
positive semi-definite or indefinite if it is positive semi-definite or
indefinite at all real places of K. Then we have the following


The quadratic form h is positive semi-definite or indefinite if and only if
h represents 1. 

 
Consider the 5-dimensional space 
V ⊕ K with the quadratic form f(v,x)=h(v)-x^2. Since V⊕ K is 5-dimensional, by the Hasse-Minkowski theorem, it represents 0 if and only if it represents 0 at all 
real places. If h is positive semi-definite or indefinite, the form f is 
semi-define at all real places, so it represents 0 at all real places. Hence f 
represents 0, and if x ≠ 0 in the representation, then h represents 1 by dividing by x.
Otherwise, h represents 0, and so it represents any element of K. Conversely,
if it represents 1, then it does so at every real place. Hence it cannot be
negative semi-definite, and thus, it must be positive semi-definite or
indefinite.




 §.§ Over integers
 
We assume that K=ℚ and L a quadratic imaginary field. Thus, A=ℤ and B is the ring of integers in L.  Let D=D_B/A be the discriminant of 
L. We shall assume that L is unramified at 2, in order to avoid discussing the difficult 2-adic theory of integral hermitian and quadratic forms. 






 Let Δ<0 be a square-free integer. Let Λ be a projective B-module of rank 2 and 
h an integral hermitian form on Λ of the discriminant Δ. Then there exists v∈Λ such that h(v)=1 

 
  Let ℤ_p be the p-adic completion of ℤ. Let B_p=B⊗ℤ_p and Λ_p=Λ⊗ℤ_p. 
Note that the subscript p denotes the completion and not localization in this proof. 
Since h is assumed to be indefinite, 
by <cit.>, it suffices to 
  prove that for every p there exists v∈Λ_p such that h(v)=1.
  Assume that p does not divide D i.e. p does not ramify in L. 
 Recall  that  _p(Δ) =0 or 1, by the assumption on Δ. 
  We claim that there exists u∈Λ such that h(u)=x where _p(x) =0. Indeed, 
 if not,  then v↦ h(v)/p is an integral form.  But the discriminant of h/p is Δ/p^2, not an integer, contradicting 
   Proposition <ref>.  Furthermore, since p does not ramify in L,  
 the norm map n_L: B_p^×→ℤ_p^× is surjective. Hence we can rescale u by an element in 
 B_p^×  to get v such that h(v)=1. 
 
 
 Now assume p divides D, so p is tamely ramified in L. In particular, p is odd, and we can write 
 B_p=ℤ[π] where π is a uniformizer of B_p. We can assume that π =-π and, 
 for the price of changing the uniformizer in ℤ_p, that ππ=p. 
  Pick a B_p-basis 
 (v_1, v_2) of Λ_p and write any v∈Λ_p as a sum v=xv_1+ yv_2 where x,y∈ B_p. Then 
 
    h(v) = (
      
        x     y
      )(
      α   γ
    γ   β)(
      x
    y)
 
  where α,β∈ℤ_p. 
By the assumption, we have  _p(Δ) =0 or 1.  If _p(Δ)=1 then _π (γ)≥ 0, and it is easy to see that the matrix can be diagonalized, by 
a change of B_p-basis in Λ_p. Hence, in this case, 
the quadratic form h on Λ_p is equivalent to a diagonal form 
 
    a_1x_1^2+ a_2 x_2^2 + a_3 px_3^2 + a_4 px_4^2
 
 for some  a_i  in ℤ_p^×. Assume now that _p(Δ)=0. Then γ = (a+bπ)/π, for some a∈ℤ_p^× and 
 b∈ℤ_p.  Writing x = x_1 + x_2π and y = y_1 + y_2 π,  the 4× 4 matrix representing the quadratic form h in the variables 
 (x_1, x_2, y_1, y_2) is 
 
    (α    0     b/2     a/2 
    
    0    α p     a/2     p b/2 
    
    b/2     a/2    β    0 
    
    a/2     p b/2     0    β p
    ).

The determinant of this matrix modulo p is a^4/16 ∈ℤ_p^×. 
Recall that, since p is odd, every ℤ_p-integral form can be diagonalized. Since 
  the determinant is p-adic unit, the form is 
 integrally equivalent to a diagonal form 
  
    a_1x_1^2+ a_2 x_2^2 + a_3 x_3^2 + a_4 x_4^2
 
for some  a_i  in ℤ_p^×. But in each of the two cases the diagonal quartic, quadratic form 
 represents 1 by <cit.>. This completes the proof of lemma. 
 

The following is a combination of the previous lemma, Corollary <ref> and Theorem <ref>. 
We remind the reader that an integral binary B-hermitian form is a B-hermitian 
form h on a projective B-module Λ of rank 2, such that h(Λ) ⊆ℤ. 

 
Let L be a quadratic imaginary extension of ℚ of odd discriminant D <0. Let B be its maximal order. Let Δ<0 be  
 a square free integer. Then there is a natural bijection between isomorphism classes of embeddings of B into quaternionic orders 
of the discriminant Δ and isomorphism classes of integral binary B-hermitian forms of the discriminant Δ. 

 



§ ACKNOWLEDGEMENTS
 
We would like to thank Tonghai Yang for suggesting the relationship between optimal embeddings and integral binary hermitian forms, and to Dick Gross for a conversation on this subject. 
The first author has been supported by an NSF grant DMS-1359774. The second author was supported by REU and UROP funding at the University of Utah. 






amsplain

