Recurrence formulae for the coefficients of mock theta functions of order 5 and 7

We compute recurrence formulae for the coefficients of Ramanujan’s mock theta functions of order 7 and 5. Our computations rely on a method developed by J.H. Bruinier and M. Schwagenscheidt, who obtained an equation for such coefficients by evaluating regularized theta lifts of harmonic Maass forms in two different ways

Cycle integrals of meromorphic Hilbert modular forms

Alfes C, Depouilly B, Kiefer P, Schwagenscheidt M. Cycle integrals of meromorphic Hilbert modular forms. arXiv:2406.03465. 2024.We establish a rationality result for linear combinations of traces of cycle integrals of certain meromorphic Hilbert modular forms. These are meromorphic counterparts to the Hilbert cusp forms $\omega_m(z_1,z_2)$, which Zagier investigated in the context of the Doi-Naganuma lift. We give an explicit formula for these cycle integrals, expressed in terms of the Fourier coefficients of harmonic Maass forms. A key element in our proof is the explicit construction of locally harmonic Hilbert-Maass forms on $\mathbb{H}^2$, which are analogous to the elliptic locally harmonic Maass forms examined by Bringmann, Kane, and Kohnen. Additionally, we introduce a regularized theta lift that maps elliptic harmonic Maass forms to locally harmonic Hilbert-Maass forms and is closely related to the Doi-Naganuma lift