Timeline for Are theta functions cuspidal representations?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2017 at 0:24 | vote | accept | john mangual | ||
| Sep 19, 2017 at 20:03 | comment | added | paul garrett | @reuns, yes, indeed. A complication in the question as it stands is that the quadratic form is in an odd number of variables, so, as "half-integral weight" modular form, Hecke theory works differently than for integral-weight. | |
| Sep 19, 2017 at 4:04 | comment | added | reuns | For some homogeneous polynomial $P$ of degree $n$ there is another polynomial $Q$ such that $P(x_1,x_2,x_3) e^{-\pi (x_1^2+x_2^2+x_3^2) t} = C \ t^{-n} \ Q(\partial_{x_1},\partial_{x_2},\partial_{x_3}) e^{-\pi (x_1^2+x_2^2+x_3^2) t}$ whose Fourier transform is $C \ (-2i \pi/t)^n t^{-3/2} Q(\xi_1,\xi_2,\xi_3) e^{-\pi (\xi_1^2+\xi_2^2+\xi_3^2) /t}$. If $P$ is harmonic then $P = Q$ and $\sum_m P(m) e^{2i \pi |m|^2 z}$ is a modular form of weight $n+3/2$ ? @paulgarrett | |
| Sep 19, 2017 at 0:44 | comment | added | Kimball | I'm confused (on several things about the question, but first): are you working with theta functions of 3 variables or $n$ variables? | |
| Sep 18, 2017 at 23:47 | history | edited | paul garrett | CC BY-SA 3.0 |
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| Sep 18, 2017 at 23:45 | comment | added | paul garrett | As @reuns comments, perhaps partly rhetorically, one should only take homogeneous harmonic polynomials, and one must know Hecke's identity about Fourier transforms of such harmonic functions $\times$ Gaussians. This gives the usual modularity for $z\to z+1$ and $z\to -1/z$... | |
| Sep 18, 2017 at 19:38 | answer | added | GH from MO | timeline score: 9 | |
| Sep 18, 2017 at 18:46 | comment | added | john mangual | It's definitely not clear what's being asked. My I'd like to know about the relationship between theta functions and cuspidal representations. | |
| Sep 18, 2017 at 18:40 | history | asked | john mangual | CC BY-SA 3.0 |