Cycle integrals of meromorphic modular forms and coefficients of harmonic Maass forms

Higher Siegel theta lifts on Lorentzian lattices, harmonic Maass forms, and Eichler-Selberg type relations

“…Finally, we note how one could obtain the Fourier expansion of the lift. Note that for j = 0 we recover the lift studied by Bruinier [4] and Bruinier-Schwagenscheidt [7], and for n = 2 the lift used centrally in [2].…”

“…In a recent breakthrough paper, Bruinier, Ehlen, and Yang [5] made a major advance towards the Gross-Zagier conjecture by proving that a certain two-variable Green function evaluated at CM points in one variable and an average over CM points in the other variable takes algebraic values. A pivotal result that the authors there used was the connection between the Green function and a "higher" Siegel theta lift on a lattice of signature (2,1). A further example lies in [2], where Alfes-Neumann, Bringmann, Schwagenscheidt, and the author used the higher Siegel theta lift 1 on lattices of signature (1,2) to investigate traces of cycle integrals of a certain cusp form.…”

“…A pivotal result that the authors there used was the connection between the Green function and a "higher" Siegel theta lift on a lattice of signature (2,1). A further example lies in [2], where Alfes-Neumann, Bringmann, Schwagenscheidt, and the author used the higher Siegel theta lift 1 on lattices of signature (1,2) to investigate traces of cycle integrals of a certain cusp form. By relating the lift to the Fourier coefficients of certain modular objects called harmonic Maass forms, the results also gave an elegant proof of the rationality of such traces of cycle integrals, which had previously been shown in [3].…”

“…This situation lies at the interface of [2,5] which included iterated raising operators but restricted to the case of n = 2, and [7] where no iterated raising operators were included, but n was unrestricted. We borrow heavily techniques from each of these papers, and obtain a description of the lift Λ reg as a series involving Gauss hypergeometric functions 2 F 1 , as well as a constant term of a Fourier expansion involving coefficients of f , a theta function, and a harmonic Maass form.…”

Locally harmonic Maass forms and periods of meromorphic modular forms

“…Moreover, it is meromorphic on H and has poles of order k precisely at the Γ-translates of the CM point τ P defined by P (τ P , 1) = 0. It was shown in [3,4,23] that certain linear combinations of geodesic cycle integrals of f k,P (z) are rational. Motivated by these results, in the present work we investigate the rationality of the periods…”