Borcherds products for \(O(2,2)\) and the \(\theta\) operator on \(p\)-adic Hilbert modular forms

Abstract

A result of Bruinier and Ono shows that under certain conditions on the zeroes and poles of a meromorphic elliptic modular form \(f\) with respect to a prime \(p\), the quotient \(\theta f / f\) is a \(p\)-adic modular form of weight 2 in the sense of Serre, where \(\theta\) is the Ramanujan differential operator. We give another proof of the same result for \(p\)-adic modular forms in the sense of Katz, using the geometric interpretation of modular forms as sections of a line bundle over the modular curve. We also prove a new, analogous result for Hilbert modular forms. For a given prime \(p\), we characterize which Hirzebruch-Zagier divisors lie in the supersingular locus of the Hilbert modular surface modulo \(p\), yielding an application of the analogous result for Borcherds products.