Meromorphic Modular Forms with Rational Cycle Integrals

Abstract

We study rationality properties of geodesic cycle integrals of meromorphic modular forms associated to positive definite binary quadratic forms. In particular, we obtain finite rational formulas for the cycle integrals of suitable linear combinations of these meromorphic modular forms.

Locally harmonic Maass forms and periods of meromorphic modular forms

“…Eventually, we remark that the methods of this work can be used to study the rationality of periods of certain linear combinations of the meromorphic modular forms f k,P (z), similar to [23]. For example, in analogy to Theorem 2.4 from [23], one can show that certain linear combinations of Hecke-translates of f k,P (z) have rational periods.…”

“…In other words, the function H 1−k,n (τ ) is essentially the n-th period of the meromorphic modular form z → H k,k−1 (z, τ ). In [23] we showed that the locally harmonic Maass form F 1−k,D (τ ) from [5] can be viewed as the D-th trace of cycle integrals of the function z → H k,k−1 (z, τ ), which inspired the above definition of H 1−k,n (τ ).…”

“…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…”

“…where the integral is defined using the Cauchy principal value as in [23], Section 3.5, if a pole of f k,P lies on the positive imaginary axis. To get a first idea how these periods look, we consider the quadratic form We notice that the even periods of f k,−3 (z) for 0 < n < 2k − 2 seem to vanish and the odd periods for k = 6 appear to be rational numbers, but the outer periods r 0 (f k,−3 ) and r 2k−2 (f k,−3 ) and the odd periods of f 6,−3 (z) do not seem to be particularly nice.…”

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

“…For the first one, we use the fact that tr f k,A (D) can be written as a special value of the iterated raising operator applied to a locally harmonic Maass form F 1−k,D , which was first introduced by Kane, Kohnen, and one of the authors [4] and whose precise definition in the vector-valued setup is recalled in Section 3. Namely, [15,Corollary 4.3] implies that In the second step, we write the function R k−1 2−2k (F 1−k,D ) as a regularised theta lift, following Borcherds [3]. Namely, in Theorem 3.2 we show that…”