Timeline for When does a modular form satisfy a differential equation with rational coefficients?
Current License: CC BY-SA 3.0
28 events
| when toggle format | what | by | license | comment | |
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| Jul 14, 2016 at 17:34 | vote | accept | Dror Speiser | ||
| Jul 13, 2016 at 23:48 | answer | added | Mark | timeline score: 5 | |
| Jun 12, 2015 at 21:58 | comment | added | ACL | @DrorSpeiser: You're right, sorry. I misunderstood your question. | |
| Jun 12, 2015 at 19:46 | comment | added | Dror Speiser | @ACL: I don't understand... are we talking about the same thing? the algebraic coefficients in my question are functions, not complex numbers. | |
| S Mar 31, 2014 at 16:25 | history | bounty ended | CommunityBot | ||
| S Mar 31, 2014 at 16:25 | history | notice removed | CommunityBot | ||
| Mar 30, 2014 at 15:39 | comment | added | ACL | @Dror: In general, of course no (think of a solution to $y'=y\sqrt{-1}$). But since you assume that the form has a rational $q$-expansion, yes. Form a differential equation $E$ of high degree with indeterminate coefficients. Check that $f$ is a solution of $E$ by looking at the expansion. That $f$ is an actual solution is a linear system with rational coefficients in the indeterminate coefficients of the equation. It has a solution in $\mathbf C$ by assumption. Hence it has a solution in $\mathbf Q$. (You can replace $\mathbf Q$ by the field generated by the coefficients of the $q$-expansion). | |
| Mar 30, 2014 at 9:06 | comment | added | Dror Speiser | @ACL: I don't know :) is it? And if so, can it be of the same degree? | |
| Mar 26, 2014 at 12:46 | comment | added | ACL | @DrorSpeiser: If a form $F$ is solution of a differential equation with algebraic coefficients, then isn't it also solution of a differential equation with rational coefficients ? | |
| Mar 26, 2014 at 12:29 | comment | added | Dror Speiser | @Noam: I bet all modular forms satisfy such an equation. I know nothing of differential Galois theory, but maybe there is an analogue of compositum of fields, so that if some eigenforms satisfy some equations, then a combination of them will also. And then we would want to show that eigenforms satisfy the condition with the rational functions having coefficients in the same number field of the eigenforms q-expansion coefficients. | |
| Mar 26, 2014 at 1:21 | comment | added | Noam D. Elkies | Could it be that every modular form satisfies such an equation? Is there a counterexample, or better yet a non-vacuous necessary condition? | |
| Mar 25, 2014 at 23:15 | history | edited | Dror Speiser | CC BY-SA 3.0 |
Motivation and addition of ag tag. Because an answser probably has to use ag.
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| S Mar 23, 2014 at 15:17 | history | bounty started | Dror Speiser | ||
| S Mar 23, 2014 at 15:17 | history | notice added | Dror Speiser | Draw attention | |
| S Oct 11, 2013 at 1:56 | history | bounty ended | CommunityBot | ||
| S Oct 11, 2013 at 1:56 | history | notice removed | CommunityBot | ||
| S Oct 3, 2013 at 0:10 | history | bounty started | Dror Speiser | ||
| S Oct 3, 2013 at 0:10 | history | notice added | Dror Speiser | Draw attention | |
| Apr 6, 2013 at 19:22 | history | bounty ended | Dror Speiser | ||
| Apr 5, 2013 at 12:55 | history | edited | Dror Speiser | CC BY-SA 3.0 |
expanded on idea
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| Mar 31, 2013 at 0:06 | comment | added | Dror Speiser | @robot: hey, thanks for the link. I've seen this paper before, and unfortunately (for me) it suffers from the same problem every paper I've seen suffers from: it begins with any given modular function $t$, instead of constructing an interesting one. | |
| Mar 30, 2013 at 22:47 | comment | added | Vít Tuček | Perhaps this could be helpful. mmrc.iss.ac.cn/pub/mm25.pdf/7.pdf | |
| Mar 30, 2013 at 18:39 | history | bounty started | Dror Speiser | ||
| Mar 27, 2013 at 9:47 | comment | added | François Brunault | @Dror : I see, you're right, I was reading your question too quickly. | |
| Mar 27, 2013 at 8:43 | comment | added | Dror Speiser | @Francois: thanks for the reference. My french is a bit a rusty (non-existent), but I think the remark says what I wrote above: that the coefficients are in general algebraic. | |
| Mar 27, 2013 at 8:12 | comment | added | François Brunault | Here is a reference : F. Martin, E. Royer, Formes modulaires et périodes smf4.emath.fr/Publications/SeminairesCongres/2005/12/pdf/… (see Remarque 140). It seems the result you mention in the genus 0 case also works in general, the only thing is that $F$ is a multivalued function. | |
| Mar 27, 2013 at 2:23 | comment | added | Dror Speiser | I also asked this in math.stackexchange: math.stackexchange.com/questions/338453/… | |
| Mar 27, 2013 at 2:22 | history | asked | Dror Speiser | CC BY-SA 3.0 |