A probable Hasse principle for pencils of quadrics

Abstract:

Let k be a global field, ${\text {char}}(k) \ne 2$. Although pencils of quadrics over k may fail to satisfy a local-to-global equivalence principle, the failures are exceptional in the precise sense of having limiting probability zero. The proof uses the classification of pairs of quadratic forms. It also requires knowing that a square class in a finite extension usually comes from k when it does so locally; the Galois-theoretic criterion for this is determined.