arXiv subjects
Publications and source records attributed to Sören Sprehe.
We confirm a conjecture of Darmon--Vonk on the antisymmetry of real quadratic singular moduli. The proof relies on a careful analysis of rigid meromorphic cocycles à la Darmon--Gehrmann--Lipnowski for the orthogonal group of a quaternion algebra. Moreover, we prove the modularity of a generating series of Kudla--Millson divisors in the spirit of Gross--Kohnen--Zagier.
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.