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Cycle Integrals of Meromorphic Hilbert Modular Forms

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Paul Kiefer
Universität Bielefeld
Wed, 20/03/2024 - 14:30 - 15:30
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

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.


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