How to decrease the levels of modular integrals (Q1128128)

A modular integral is a generalization of the concept of modular forms (see \textit{M. I. Knopp} [Duke Math. J. 45, 47--62 (1978; Zbl 0374.10014)]). It is well known that the level of a modular form is a very important parameter for the investigation of modular forms. In [Prog. Math. 85, 479--491 (1990; Zbl 0732.11020)], \textit{H. M. Stark} dealt with the problem of finding the lowest level of a given modular form, which is known to have some level, and gave a nice result. In this paper the authors discuss the same problem for modular integrals. Their main result is the following. Theorem. Suppose that \(f(z)\) is in the set of all modular integrals \(\text{MI}(N',k,\chi)\), for some \(N'\) with character \(\chi\pmod{N'}\) and that \(f\) is not identically zero. Let \(N\) be the greatest common divisor of all positive integers \(\tilde N\) such that \[ g\mid_k \begin{pmatrix} 1&b\\ 0&1\end{pmatrix}=g+q_{\begin{pmatrix} 1&0\\ -\tilde Nb&1\end{pmatrix}}\mid_k \begin{pmatrix} 0&-1\\ \tilde N&0\end{pmatrix}\quad \text{for all}\;b\in \mathbb Z, \] where \(g = f\mid_k \begin{pmatrix} 0&-1\\ \tilde N&0\end{pmatrix}\) and \(q_A\) is the rational period functions of \(f\). Then \(N\mid N'\), \(\chi\) is definable (mod \(N\)) and \(f(z)\) is in \(\text{MI}(N,k,\chi)\).