Timeline for Hecke operators that lower level
Current License: CC BY-SA 4.0
13 events
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| Dec 10, 2018 at 15:49 | comment | added | François Brunault | @ramanujan_dirac I don't know of linear operators other than those given by double cosets. The only other thing I can think of would be, since you are working in weight 0, to take products (or quotients) of the functions $f(d\tau)$. There is also an obvious multiplicative version of the trace map (more accurately called a norm map). But these are not linear operators. | |
| Dec 10, 2018 at 13:42 | comment | added | user35360 | @FrançoisBrunault: Thanks again for the helpful comment. If I could trouble you with a related question, are there any operators that raise level apart from the scaling map I mentioned? I am studying operators that map modular forms of various levels but none of the ones I have found in literature satisfy my needs. | |
| Dec 9, 2018 at 21:36 | history | edited | user35360 | CC BY-SA 4.0 |
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| Dec 9, 2018 at 21:34 | comment | added | François Brunault | @ramanujan_dirac I see, then the trace maps are well-defined on meromorphic forms, and they preserve weakly holomorphic modular forms (as general double cosets do). | |
| Dec 9, 2018 at 21:28 | history | edited | user35360 | CC BY-SA 4.0 |
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| Dec 9, 2018 at 21:27 | comment | added | user35360 | @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. | |
| Dec 9, 2018 at 21:18 | comment | added | François Brunault | 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. | |
| Dec 9, 2018 at 21:16 | comment | added | Sylvain JULIEN | Maybe composing an automorphism of your space as I suggested with the inverse map you consider would generate all desired maps. | |
| Dec 9, 2018 at 21:09 | comment | added | user35360 | @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. | |
| Dec 9, 2018 at 21:00 | comment | added | Sylvain JULIEN | 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)) $? | |
| Dec 9, 2018 at 20:58 | history | edited | user35360 | CC BY-SA 4.0 |
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| Dec 9, 2018 at 20:38 | history | edited | user35360 | CC BY-SA 4.0 |
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| Dec 9, 2018 at 20:01 | history | asked | user35360 | CC BY-SA 4.0 |