Cycle Integrals of Meromorphic Hilbert Modular Forms

When considering the Doi-Naganuma lift, Zagier introduced certain Hilbert cusp forms $\omega_m^{\operatorname{cusp}}$ corresponding to positive integers $m$. A similar construction for negative $m$ yields meromorphic Hilbert modular forms $\omega_m^{\operatorname{mero}}$ with singularities along Hirzebruch-Zagier divisors. We extend the $\xi$-operator of Bruinier-Funke to Hilbert modular surfaces and define a $\xi$-preimage $\Omega_m^{\operatorname{cusp}}$ of $\omega_m^{\operatorname{cusp}}$. Using this we show that the cycle integrals of $\omega_m^{\operatorname{mero}}$ along real analytic cycles are related to cycle integrals of $\Omega_m^{\operatorname{cusp}}$ along Hirzebruch-Zagier divisors. The latter can be calculated using a new theta lift. This is joint work with C. Alfes-Neumann, B. Depouilly and M. Schwagenscheidt.