Antisymmetry of real quadratic singular moduli

Abstract

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.

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