The Arithmetic of Quaternion Algebra

64 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD
Proof. According to the definition of lattice (I.4), for a given lattice Y , there
exists a, b ∈ K × such that aY ⊂ X ⊂ bY . For almost every v /∈ ∞, av, bv are
units. Thus Xp = Yp, p.p. We shall prove V ↦→ (Vp) p/∈S is surjective. If (Zp) p/∈S
is a set of local lattices which are almost everywhere equal to Xp, we then set
Y = �
p/∈S (H ∩ Zp). We want to prove Y is a lattice, and Yp = Zp. There exists
a ∈ R such that aXp ⊃ Zp ⊃ a −1
p for every p /∈ S. It follows aX ⊃ Y ⊃ a −1 X,
hence Y is a lattice. Since S �= ∅, according to 1.4, H is dense in �
p/∈S Hp.
From this we have H ∩ (πZp) = Y is dense in πZp. in particular, Y is dense
in Zp, thus Yp = Zp if p /∈ S. We now prove Y ↦→ (Yp) p/∈S is injective. let
Z �
p/∈S (Yp ∩ H). We claim that Y = Z. It is true that Y ⊂ Z, and there exists
a ∈ R such that aZ ⊃ Y ⊃ Z. Let z ∈ Z. There exists y ∈ Y very near to z by
p-adic for every place p /∈ S, such that a is not a unit in Rp. In fact, we have
Yp = Zp if p /∈ S, and we utilize the approximation theorem 1.4. There exists
then y ∈ Y such that y − z ∈ aZ. We conclude that z ∈ Y . The proposition is
proved.
Definition 3.16. A property ⋆ of lattice is called a local property when a lattice
Y has the property ⋆ if and only if Yp has the property ⋆ for every p /∈ S.
Examples of local property: The properties for a lattice to be
1. an order,
2. a maximal order,
3. an Eichler order, i.e. the intersection of two maximal orders,
4. an ideal,
5. an integral ideal,
6. a two-sided ideal,
are the local properties. This can be deduced easily by the proposition 5.1. We
utilize that if I is an ideal, then its left order Ol(I) (cf. I.4, above Prop.4.2)
satisfies Ol(I)p = Ol(Ip) for all p /∈ S.
Definition 3.17. The level of Eichler order O is an integral ideal of R, denoted
by N such that Np is the level of Op ∀p /∈ S.
Corollary 3.5.2. Let I be an ideal of H,and O be an order of H. n(I) denotes
the reduced norm of I, and d(O) the reduced discriminant of O. Then we have
n(Ip) = n(I)p and d(Op) = d(O)p.
Proof. If (f) is a finite system of generators of I/R, by the definition (I. above
the lemma 4.7) n(I) is the R-ideal generated by n(f). Moreover (f) is also a
finite system of generators of Ip/Rp. It follows that n(Ip) = n(I)p. By the
definition in I,§4, above the lemma 4.9,
I ∗ = {x ∈ H|t(xf) ∈ R, ∀f}.
By the proposition 5.1 we obtain (Ip) ∗ = (I ∗ )p. Replacing I by O, and taking
the reduced norm, we see that
d(O)p = n(O ∗−1 )p = [n(O ∗ ) −1 ]p = n(O ∗ ) −1
p = n(O ∗−1
p = d(Op).