A Hasse principle for quadratic forms over function fields

Abstract

We describe the classical Hasse principle for the existence of nontrivial zeros for quadratic forms over number fields, namely, local zeros over all completions at places of the number field imply nontrivial zeros over the number field itself. We then go on to explain more general questions related to the Hasse principle for nontrivial zeros of quadratic forms over function fields, with reference to a set of discrete valuations of the field. This question has interesting consequences over function fields of p p -adic curves. We also record some open questions related to the isotropy of quadratic forms over function fields of curves over number fields.