Complex analysis: Classification of elliptic functions

This lecture is part of an online undergraduate course on complex analysis.
We give 3 description of elliptic functions: as rational functions of P and its derivative, or in terms of their zeros and poles, or in terms of their singularities.
We end by giving a brief description of the analogy between elliptic and trig functions.
For the other lectures in the course see • Complex analysis

[@ShenghuiYang]

What a great journey 🚅 on complex analysis! Great prequel of the modular form series. Thanks professor!

[@PaPa-kr5yt]

Dear Prof. Borcherds, I just watched the whole this masterful series and really get intrigued on this. Thanks for the lecture. However this is the one I have lost. Could you recommend good references or literatures on this topic?

[@caspermadlener4191]

I like that for most complex functional equations, the family of functions satisfying the equations can more easily be build from the functions satisfying the weakest variant of the functional equation that still retains a variant of the properties that can be directly derived by doing some algebra with the equation.

[@MichaelFJ1969]

Is the sigma function related to the Jacobi elliptic function sn(x) ?

[@vladkovalchuk8299]

Do there exist generalizations of this story to general higher dimensional lattices?
These maps won't have to be angle-preserving, i.e. satisfying Cauchy-Riemann equation analogues, because there are not many of those thanks to Liouville's theorem in D>2. But perhaps they could satisfy another meaningful class of PDEs that allows for "large-enough" class of solutions to exist, e.g. some analogues of Cauchy-Riemann equations.
P.S. Thanks for a great lecture! The analogy with trigonometric functions is very illuminating!

[@annaclarafenyo8185]

Maybe the proper generalization is to n complex dimensions.

[@vladkovalchuk8299]

@@annaclarafenyo8185 Right, that is likely the most fruitful generalization. I guess that it would be related to a more general theory of Abelian varieties.
www.martinorr.name/blog/2015/10/06/periods-of-abelian-varieties/
I guess I am more plagued by the question of maps f:C^n ->C^n rather than f:C^n->C.

[@faisalal-faisal1470]

@@vladkovalchuk8299 The story here is really about meromorphic functions on the torus C/L, so the proper analogy would be about meromorphic functions on higher dimensional complex tori C^n/L (where L is a lattice in C^n). And there is indeed a very rich story here. It includes the theory of abelian varieties ('nice' complex tori). The Weierstrass sigma function given in the video, which is essentially all you need to create the p function and the general theory on C/L, is an example of something called a 'theta function'. Theta functions in turn have a very rich and well-developed theory. All of this also ultimately leads one into the world of modular and automorphic forms.

[@richarde.borcherds7998]

Yes, there are generalizations to some but not all higher dimensional lattices: see "Analytic theory of Abelian varieties" by H. P. F. Swinnerton-Dyer or "Abelian varieties" by David Mumford.

[@chrisprime2357]

Hi Prof. Borcherds! In 19:16, the implication should be true only if we assumed f is elliptic?
I think we are trying to show that given the singularities of an arbitrary f, if the sum of residues in each parallelogramic region is zero, then f is elliptic? So we shall not assume the mentioned implication is correct?

[@mathvisuallyexplained6839]

I didn't get why we quotient out by the lattice instead of removing it from the domain in the elliptic curve construction... The function is undefined them, how the quotient space is going to remove the problem?

[@kilogods]

Bro I laughed so hard when he said you need to say some magic words to make the sum and derivative operators commute, “locally uniformly convergent” he’s absolutely right about how difficult it is to remember the definition on the spot lmao XD

[@yalcncanklc3907]

At which point exactly?

[@caspermadlener4191]

That was two videos ago...

[@reapingshadow2866]

why did we define sine with period of 2 pi? It seems more fundamental with period 1, and only circle yields the period of pi.

[@omrizemer6323]

You said "by changing the p_i, it is possible to make the sum n_ip_i equal to zero". But if the n_i have a common factor, then changing the p_i by elements of the lattice can't make the sum zero always... Am I wrong?

[@SG-kj2uy]

I think you’re right. But if ni have a common divisor, say m, we first consider ni/m and then construct an elliptic function, say f(x). For the original ni, you consider f(x)^m.

[@omrizemer6323]

@@SG-kj2uy This won't work, because the sum (ni/m) p_i need not be an element of the lattice.

[@SG-kj2uy]

@@omrizemer6323 You’re right. My argument has something wrong. How about moving zeros slightly so that the two relations still hold and all of zeros are distinct? And we take the limit wrt the places of zeros.

[@Yuki-dw2dn]

The idea is, if \Sigma n_{i}p_{i}=\omega, then we replace it by \Sigma n_{i}p_{i}+1*0-1*\omega, where the sum is 0 now, and the product cancels zero at 0 and pole at \omega. Alternatively, we can just take the product of those \sigma(z-p_i)^{n_{i}} with a factor \sigma(z)\sigma(z-\omega)^{-1}.

[@chetanvuppulury7537]

Yeeeeeee

[@migarsormrapophis2755]

yeeeeeeeeeeeeeeeeeee

[@sleepdeprived-huh]

@@migarsormrapophis2755 yeeeeeeeeeeeeeeeeee