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Anonymous
Jacobi modular forms: 30 ans après 2016-10-02 22:33:51 Post No. 8387232
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Jacobi modular forms: 30 ans après 2016-10-02 22:33:51 Post No. 8387232
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Looking for other /sci/entists who might be interested to survey this free online course that just started, it's on Jacobi forms which are sort-of-modular, sort-of-elliptic functions: https://www.coursera.org/learn/modular-forms-jacobi
Pre-reqs are apparently knowing complex numbers and the group [math] SL_2(\mathbb{Z}) [/math], will post picture in next post
I'll be trying to work through all 12 weeks of it, just wondering if it's worthwhile to start some discussion threads on here about it or not
syllabus:
Jacobi modular forms: motivations
Jacobi modular forms: the first definition
Jacobi modular group and the second definition of Jacobi forms. Special values of Jacobi modular forms
Zeros of Jacobi forms. The Jacobi theta-series, the Dedekind eta-function and the first examples of Jacobi modular forms
The Jacobi theta-series as Jacobi modular form. The basic Jacobi modular forms
Theta-blocks, theta-quarks and the first Jacobi cusp form of weight 2
Jacobi forms in many variables and the Eichler-Zagier Jacobi forms
Jacobi forms in many variables and the splitting principle. Theta-quarks as a pull-back. Weak Jacobi forms in many variables
The Weil representation and vector valued modular forms. Jacobi forms of singular weight
Quasi-modular Eisenstein series. The automorphic correction of Jacobi forms and Taylor expansions
Modular differential operators. The graded ring of the weak Jacobi modular forms
Jacobi type forms and the generalisation of the Cohen-Kuznetsov-Zagier operator
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>To follow the course one has to know only elementary basic facts from the theory of modular forms (for example, the paragraphs 1-4 of the chapter VII of Serre’s “A Course in Arithmetic” are enough).
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final bump
as a bonus, lecture style is pretty comfy and upbeat
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sorry senpai, I'm too dumb for this shit, but I wish you the best of luck
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Man, I have finals in 3-4 weeks time otherwise I'd be keen. Perhaps after then.
Thanks OP.
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>>8388329
its ok senpai, i knew it would be a hard sell here. good luck with your studies
>>8388351
these courses usually stay up indefinitely, and a new 'session' starts every 4 weeks so hop in whenever if you're still interested. good luck with finals
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difficulty indicator: first homework
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Are these "Jacobi Modular Forms" just modular forms? Seems silly to use nonstandard terminology like that.
>>8388403
See Zagier's introductory lecture at SISSA, can be found on youtube.
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Hi calc 2 student here, this is basically all Dead Sea Scrolls to me but I'd love a laymen's explanation of exactly what you're doing and studying in terms of explaining the syllabus.
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>>8388403
I'm just starting to learn about them so I don't know all the details but:
Some examples include:
>generalizations of Jacobi theta functions
>the Weierstrauss p function
>the Fourier coefficients of a Siegel modular form rearranged in a clever way
These are functions in two variables, and if you specialize one of the variables you end up with a usual modular form which is nice. If you specialize the other variable you get an elliptic function on a lattice.
If you're interested in modular forms, they gives you a way to 'lift' elliptic modular forms to Siegel modular forms. The Jacobi forms of index 0 are basically integral-weight modular forms. The Jacobi forms of weight 1 are isomorphic to half-integral-weight modular forms. All of this is in the nicest way possible (most algebraic and analytic theory on one side commutes with the isomorphism)
Jacobi forms also show up in the theory of Heegner points, elliptic genera (Calabi-Yau varieties), string theory (quantum gravity).
>>8388411
No, they're automorphic forms on the Jacobi group (the modular group in a semidirect with [math] \mathbb{Z}^2 [/math]) instead, see pic related for the transformation laws
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>>8388420
woops:
The Jacobi forms of index* 1 are isomorphic to half-integral-weight modular forms.
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>>8388412
for laymen: Jacobi forms are functions of two variables (one in the complex plane, one in the upper-half of the complex plane) satisfying nice symmetry properties with respect to a certain group (specifically, the group of 2x2 matrices with integer coefficients and determinant 1, along with pairs of integers) acting on these planes
i don't know anything past the introductory material of the syllabus so i won't try to explain it
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>when you have no study buddies to explore the mathematical unknown with
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