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I am working with weakly holomorphic modular functions (weight $=0$) $f \in M_0(\Gamma(N), \chi)$ of level $N$ with some character $\chi$ (We can ignore the character for now). Let $f \in M_0(\Gamma(N))$ and let $M|N$. Clearly $M_0(\Gamma(M)) \subset M_0(\Gamma(N))$.

Question: Can I construct Hecke operators that lower level, that is a map $T:M_0(\Gamma(N)) \to M_0(\Gamma(M)) $?

The Atkin-Lehner-Li theory gives an embedding $M_0(\Gamma(M)) \to M_0(\Gamma(N))$ via a "scaling map", $f(\tau) \to f(d \tau)$ where $d|(N/M)$, but I am looking for operators that go the other way. An obvious map is the inverse map $f(\tau) \to f(\tau/d)$ but are there other non-trivial and interesting operators that could be constructed or have been constructed in literature?

In more concrete terms, maps between modular functions with $q$-expansions, $$q^{1/N}(\sum_{m=-n}^{\infty} a_m q^m) \to q^{1/M}(\sum_{m=-n'}^{\infty} a'_m q^m)$$

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  • $\begingroup$ From a total non expert and provided I understood correctly : wouldn't a suitable definition of the automorphism group of your space of modular functions provide an affirmative answer to your question ? Unless you meant a map from $M_{0}(\Gamma(N)) $ to $M_{0}(\Gamma(M)) $? $\endgroup$
    – Sylvain JULIEN
    Commented Dec 9, 2018 at 21:00
  • $\begingroup$ @SylvainJULIEN: I did indeed mean a map from $M_0(\Gamma(N))$ to $M_0(\Gamma(M))$. I had made a typo which I have now corrected. $\endgroup$
    – user35360
    Commented Dec 9, 2018 at 21:09
  • $\begingroup$ Maybe composing an automorphism of your space as I suggested with the inverse map you consider would generate all desired maps. $\endgroup$
    – Sylvain JULIEN
    Commented Dec 9, 2018 at 21:16
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    $\begingroup$ Which modular functions do you mean exactly? The notation $M_0$ is not standard (modular forms of weight 0 are constant). Also $f(\tau/d)$ is not modular of level $M$. That being said, your are looking for trace maps, see e.g. Diamond-Shurman, A first course in modular forms, p. 166. It is an exercise to show these maps are the adjoints of the scaling maps wrt the Petersson scalar product. $\endgroup$
    – François Brunault
    Commented Dec 9, 2018 at 21:18
  • $\begingroup$ @FrançoisBrunault: My apologies. I mean weakly holomorphic (finite order poles at the cusps) or even meromorphic modular functions. I am not imposing holomorphicity. Thanks for your comment. I will check it out. $\endgroup$
    – user35360
    Commented Dec 9, 2018 at 21:27

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