Timeline for Weight, Index, and Congruence Subgroup of Classical Jacobi Theta Functions
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 21, 2018 at 20:10 | vote | accept | Benighted | ||
| Aug 20, 2018 at 8:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jul 21, 2018 at 7:07 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jun 21, 2018 at 6:51 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| May 22, 2018 at 6:42 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 22, 2018 at 5:20 | answer | added | user123589 | timeline score: 5 | |
| Apr 18, 2018 at 14:08 | comment | added | Jeff Harvey | Theta functions with half integral weight are best thought of as vector-valued forms which transform under the Weil representation of the metaplectic group which is a double cover of $SL(2, \mathbb{Z})$. | |
| Apr 18, 2018 at 3:49 | comment | added | Benighted | I see, thanks a lot. If we expand the definition of Jacobi forms to half-integral weight, do you know what the index of the theta functions I write is, as well as which congruence subgroups they transform under? Or perhaps these questions don't quite make sense. | |
| Apr 18, 2018 at 2:13 | comment | added | Jeff Harvey | Eichler and Zagier were certainly inspired by Jacobi theta functions, but the definition they give of Jacobi forms requires that the weight and index are integers. The Jacobi theta functions you give are weight one-half and so are not Jacobi forms by the definition given in Eichler-Zagier. | |
| Apr 18, 2018 at 1:50 | comment | added | Somos | That is a good answer, but even for N=1, specializing a Jacobi form doesn't mean you can get exactly a Jacobi theta function. Some adjustments have to be made. Eichler and Zagier should have given explicit details of how, for example, $\vartheta_3$ is a specialization of a Jacobi form. Did they? | |
| Apr 18, 2018 at 1:05 | comment | added | Benighted | Well because they look schematically like that general form in the first equation of my OP in the case of N=1. And in fact, just after Eichler and Zagier write that formula they mention that it's a generalization of what Jacobi studied for N=1. I figured they were referring to these theta functions. | |
| Apr 18, 2018 at 0:46 | comment | added | Somos | Why do you think that the Jacobi theta functions are Jacobi forms? | |
| Apr 17, 2018 at 21:15 | history | asked | Benighted | CC BY-SA 3.0 |