By Wai Kiu Chan

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22. Otherwise × ∼ ∼ 1 , 2 , 3 = A ⊥ α for some α ∈ Z2 . We may then assume that α, 4 , 5 = A ⊥ β for × ∼ some β ∈ Z2 . ✷ 7 Quadratic Forms over Z Unless stated otherwise, (V, Q) is always a quadratic space over Q and all lattices considered in this section are Z-lattices. Since ±1 are the only units in Z, the discriminant of a nondegenerate lattice in V is well-defined as a rational number. 1 Preliminaries A lattice L in V is called integral if B(L, L) ⊆ Z. For an integral lattice L, L# contains L and d(L) = [L# : L]2 d(L# ).

We shall define a norm if no confusion arises, on O(V ). Let {x1 , . . , xn } p , or simply be a basis for V . The norm which we are about to define is with respect to the basis {x1 , . . , xn } unless stated otherwise. Let M be the Zp -lattice spanned by this basis. We first define the norm on V . For any x ∈ V , express it as x = α1 x1 + · · · + αn xn , αi ∈ Qp , and define the norm of x by x = max |αi | i where | | is the p-adic valuation on Qp . It is not hard to see that which satisfies the following three properties: (i) x ≥ 0 for all x ∈ V and x = 0 ⇔ x = 0; 56 is a real-valued function (ii) αx = |α| x for all α ∈ Qp and x ∈ V ; (iii) x + y ≤ max{ x , y }, with equality sign holds when x = y .

For any p ∈ T , σ(Lp ) = σp (Lp ) = φ(Kp ). As a result, σ(Lp ) = φ(Kp ) for all p and hence K = φ−1 σ(L). 3 Let L be a lattice on an indefinite quadratic space of dimension ≥ 4 over Q. If a is represented by gen(L), then a is represented by L. Proof. 3 does not hold for ternary lattices. For example, let L be the Z-lattice corresponding to the quadratic form −9x2 + 2xy + 7y 2 + 2z 2 . In terms of symmetric matrix −9 1 0 L∼ = 1 7 0 . 0 0 2 Since d(L) = −27 , Lp is unimodular for all primes p ≥ 3.

### Arithmetic of Quadratic Forms by Wai Kiu Chan

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