Algebraic Number Theory on Hyperbolic 3-Manifolds Definitions and Some Basic Facts

• A quadratic space is a pair (V,B) where V is a finite-dimensional vector space over a field K and B : V × V → K is a symmetric bilinear form. • For quadratic spaces (V,B) and (V ′, B′), a map τ : V → V ′ is an isometry iff τ is a linear isomorphism and ∀~v, ~ w ∈ V : B(~v, ~ w) = B′(τ(~v), τ(~ w)). In this case V and V ′ are isometric. • The quadratic form determined by B and basis β := {~v1, . . . , ~vn} for V is qB,β : V → K, (x1, . . . , xn) 7→ n Σ i,j=1 B(~vi, ~vj)xixj .