On the abundance of rigid meromorphic cocycles for quadratic forms in four variables

Let $(V,q)$ be an anisotropic quadratic space of dimension $4$ over the rationals and let $p$ be a prime such that the local quadratic space $V\otimes {\mathbb{Q}}_p$ is the orthogonal direct sum of two hyperbolic planes. We show that the group of $p$-adic rigid meromorphic cocycles attached to $V$ has infinite rank. A key ingredient in the proof is the computation of the group of invertible rigid analytic functions on products of certain rigid analytic subspaces of the projective line such as the $p$-adic upper half-plane.