Congruence restricted modular forms (Q2490444)

For \(f(z)\) given on the upper half-plane by an exponential series \[ f(z)= \sum^\infty_{n=n_0}a_ne^{2\pi i(n+x)z},0\leq k<1, \] let \[ f(z; r,t)=\sum_{\substack{ n=n_0\\ n\equiv r\pmod t}} a_ne^{2\pi i(n+x)z}\;(r,t\in \mathbb Z,t>0), \] a corresponding ``congruence restricted'' exponential series. The author points out that in recent years ``several people who work with modular forms have made and used the following observation: often, when \(f(z)\) is a modular form on a congruence subgroup \(\Gamma\) of level \(N\), \(f(z;r,t)\) turns out to also be a modular form on a congruence subgroup of level \(N'\), where \(N|N'\). Furthermore, the modular form \(f(z;r,t)\) inherits the weight and the multiplier system of \(f(z)\).'' The operative word here is ``often'', since the survival of modularity under congruence restrictions is a principle, but not a theorem. To apply the principle one must discover conditions that guarantee its applicability and, in fact, the author's main result, Theorem 4.1, does exactly that. This theorem deals with modular forms, on the Hecke congruence group \(\Gamma_0(N)\), of arbitrary integral weight \(k\) and arbitrary multiplier system (MS) \(v\) in weight \(k\). The (quite natural) condition that the author proves sufficient involves two classes of subgroups of \(\Gamma_0(N)\), which, to my knowledge, have never been discussed before and likely will reward further close study. They are: (i) the congruence groups \[ \Gamma_{0,n}(N)=\left\{{ab\choose cd}\in \Gamma_0(N),\;a\equiv d\pmod n\right\}, \] with \(n,N\in \mathbb{Z}^+\); (ii) the groups \(S_{v,t}\subset \Gamma(1)\), of more complicated structure. (Of the second class I note only that \(t\in \mathbb Z^+\) and \(v\) is an arbitrary MS on \(\Gamma_0(N)\), in weight \(k\). It is not at all clear when \(S_{v,t}\) has finite index in \(\Gamma(1))\). Theorem 4.1 can be stated as follows. Suppose \(N,t\in \mathbb Z^+\), \(k,r\in \mathbb Z\) and assume \(f(z)\) is a modular form with respect to \((\Gamma_0(N),k,v)\), with \(v\) an MS on \(\Gamma_0(N)\) in weight \(k\). If \(S_{v,t}\) is of finite index in \(\Gamma(1)\), then \(f(z;r,t)\) is a modular form with respect to \((S_{v,t},k,v)\). Furthermore, if \(f(z)\) is a cusp form (respectively, entire form), then \(f(z;r,t)\) is a cusp form (entire form). In Proposition 3, the author shows that \(S_{v,t}\) is a subgroup of \(\Gamma_0(N)\). The Main Theorem makes evident the interest in determining when (i.e., for which \(r,t,v,N)\) \(S_{v,t}\) is a congruence group, or at least of finite index in \(\Gamma(1)\).