Pencils of Quadratic Forms and Hyperelliptic Function Fields

Let P(k) denote the set of equivalence classes of nonsingular pencils of quadratic forms of even order defined over a field k, char k ≠ 2. Let F(k) denote the set of k-isomorphism classes of hyperelliptic function fields defined over k. We define a map Φ: P(k) → F(k) and determine precisely when Φ is surjective and when Φ is injective. This extends a method used by A. Weil to study pairs of quadratic forms over finite fields.