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 $V$ ha