By Wai Kiu Chan
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Extra resources for Arithmetic of Quadratic Forms
If the Strong Approximation Theorem held for O+ (V ), then there would be an isometry σ ∈ O+ (V ) such that σ(Mp ) = σp (Mp ) for all p ∈ T and that σ(Mp ) = Mp for all p ∈ T . Then σ(M ) = L and this means that gen(L) has only one class. But from the class number formula obtained in the next section we know that there exists indefinite Z-lattice whose class number is not 1. Therefore, the Strong Approximation Theorem does not hold for O+ (V ). However, there is a weak approximation for O+ (V ).
Therefore, L is a metric space and the distance between σ and τ is σ − τ . The addition and scalar multiplication are continuous operations on L. However, we also have multiplication in L to consider. We find that σ(x) ≤ σ x for all σ ∈ L and x ∈ V . This shows that στ ≤ σ τ for any σ and τ in L. Therefore, multiplication is continuous on L. For any σ ∈ O(V ), we have det(σ) = ±1. Therefore σ ≥ 1. Then σ = 1 if and only if σ ≤ 1, and this is equivalent to σ(M ) ⊆ M . Hence σ(M ) = M since the discriminant of σ(M ) is the same as that of M .
Vn } for V (m ≤ n). Let n Zp pt vi M =N⊕ i=m+1 for some integer t. If t is large enough, then n(M ) ⊆ (a). Let L be a lattice on V which contains M and n(L) ⊆ (a). Then d(M ) = [L : M ]2 d(L) and d(L) ∈ s(L)n ⊆ (2−1 a)n . Since [L : M ] is an integer, it must be bounded and hence there exists a lattice K which is maximal with respect to inclusion such that K ⊇ M and n(K) ⊆ (a). 16 Let L be an (a)-maximal lattice on a nondegenerate quadratic space V over Qp . For any b ∈ Q× p , b is represented by L if b is represented by V and b ∈ (a).
Arithmetic of Quadratic Forms by Wai Kiu Chan