Modular forms: Eisenstein series

This lecture is part of an online graduate course on modular forms.
We give two ways of looking at modular forms: as functions of lattices in C, or as invariant forms. We use this to give two different ways of constructing Eisenstein series.
For the other lectures in the course see • Modular forms

[@ac-dp3jk]

We are so lucky to have these lectures

[@brian8507]

In the last couple months.... he has risen to the best math content channel on youtube

[@mathematics5573]

who understands them? although, I agree it is a previlage to hear such an eminent mathematician explain and teaching

[@monny1815]

@@mathematics5573 Well, not to be rude, but you can't expect to learn this sort of math without doing any hard work, both in learning the prerequisites and in trying to understand the lectures, this is just the way it is. These are obviously not lectures made for the general public like other very good channels do, but they are gold for people with good mathematical maturity nonetheless.

[@mathematics5573]

@@monny1815 it was a rutorical question! See my maths.

[@mathematics5573]

@@monny1815"" I agree it is a previlage to hear such an eminent mathematician explain and teaching"" Do you not understand basic English?

[@MrGiuse72]

the first 5 minutes gave meaning to hours and hours of readings, tens of papers downloaded, tens of web pages researched.... you FINALLY clarified months of readings without ever getting to the point. I always felt a missing dot, elusive, enigmatically and I was never able to get it !!! OOOOOOOHHH Now the point is nailed there in front of me : modular forms map elliptic functions and this is the starting point !! THANK YOU THANK YOU THANK YOU

[@yilongwang6479]

That the two points of view of modular forms are equivalent and that they are different by zeta is amazing.

[@MrGiuse72]

ENLIGHTENING !!! finally a clarifying compendium of all the papers, bits and pieces I've collcted here and there that were never explicative of the topics underlying them. THANK YOU PROFESSOR ! GREAT DIDACTICS

[@SashaShmakov]

Richard, I would love it if you gave a lecture series on your proof of the moonshine conjectures in full detail (Monster VOAs, etc) ignoring the fact that this would lose a lot of the audience. I am a graduate student in arithmetic geometry and would love to hear this proof from the horse's mouth so to speak.

[@hausdorffm]

Let C be the set of complex numbers. Because an elliptic curve C/ depends only on w1,w2, we can write g(w1, w2) for a function defined on the set of all isomorphic classes of elliptic curves, .i.e., g(w1, w2) = g( C/ ). For any number c, two elliptic curves C/ and C/ are bi-holomorphic by correspondence z -> cz, and thus g(w1, w2) = g(cw1, cw2). Similarly, for any matrix (a,b,c,d) of SL(2,Z) , two elliptic curves C/ and C/ are bi-holomorphic by correspondence xw1+yw2 -> x(aw1 + bw2)+y(cw1 + dw2), and thus g(w1, w2) = g(w1 + bw2, cw1 + dw2). For an elliptic curves C/, we can find t so that an elliptic curve C/ is bi-holomorphic to an elliptic curve C/, where t is an element of Poincare upper half plane So, it is sufficient to consider function of the form g(w1, w2) = g(1, t) =: f(t). Then the condition g(w1, w2) = g(w1 + bw2, cw1 + dw2) is f( (at + b)/(ct + d) ) = f(t). If a holomorphic function on Poincare upper half plane satisfies f( (at + b)/(ct + d) ) = f(t) for any (a,b,c,d) of SL(2,Z), f is called a modular function. How to find a modular function? To find a modular function, we use modular forms. A modular form f(t)dt is a invariant differential form under the action of SL(2,Z) on Poincare upper half plane. The transition rule is given at 8:35. More generally, at 8:52 for weight k modular form f(t)(dt)^{k/2} is defined. So, a modular form is a section of certain line bundle over Poincare upper half plane. I am not sure, for any line bundle L, is there a line bundle L^{1/2}??? Using two distinct modular form of same weight, we can get a modular function as a ratio of modular forms. Different way of looking at modular form of weight k. By using g(w1, w2) = c^kg(c*w1,c* w2) instead of g(w1, w2) = g(cw1, cw2), we can deduce the transition rule of modular forms. To construct examples of modular form, we use Weierstrass function P(z, w1, w2 ), which satisfies c^2P(cz, cw1, cw2 ) = P(z, w1, w2 ). 14:44. Each coefficients of the Laurant expansion of Weierstrass function gives a modular form. 17:00 These coefficients are written by using Eisenstein series. I am not sure how to calculate the expansion 1/ (z -mw1 - nw2) ^2 = … of 19:01. What is the radius of convergence for the expansion? Typo? The last term of 19:01 should be 3z^2/( )^4 instead of 3z/ ( )^4? Eisenstein series is invariant under t -> t+1. There are two ways to get modular forms. These two methods are essentially same and differ by factor of Riemann zeta function. One method is coefficients of Laurent series of weierstrass function and the other is the ordinal method, that is the sum over the all action of SL(2,Z). However the sum does not converge, because there is an infinite subgroup of SL(2,Z) consisting of all linear map of the from t -> t +n for any n of Z such that dt is invariant under the action t -> t+ n. So, the sum over the subgroup is the sum of constant dt over all Z and it diverges. Thus, we consider quotient group of SL(2, Z) divided by the subgroup of form t -> t+ n. Then the sum converges and it slightly differ with coefficients of weierstrass function only in Riemann zeta function.

[@PIBIDQuixada2011]

Nice "modular" Sunday for you guys!!

[@user-jc2lz6jb2e]

16:33 "well actually they're not quite..." Wouldn't be a Borcherds video without a half truth lol.

[@NilodeRoock]

I started to notice them, yes. And the "Oops: PROBLEM"s.

[@thibaultleblanc2349]

Hi, at 4:50, professor Borcherds explains that the reason we want g to satisfy these two equations is to have it be a function of the set of isomorphism classes of elliptic curves. Does someone know the precise sense we should give to isomorphism of elliptic curve ? I was thinking maybe an isomorphism of Riemann surfaces, but then, is there anything telling us that such an isomorphism can only be either induces by mutltiplication with a given complew number or by the transformation by an element of SL_2(Z) ?

[@StNick119]

Is this the second lecture in the series? At the end of the first one he said he'd discuss level one modular forms in the next video.

[@nickpreetic4670]

Thank you sir

[@CHICKENCHICKENabdgnpt]

What is the d around 9 minutes in meant to mean? If I wanted to learn more about that to better understand this what would I have to search? Thank you!

[@yunjiangjiang6146]

Should I view (d\tau)^(k/2) as tensor product rather than wedge product, as is usually meant by differential forms?

[@markvp71]

It should be viewed as a simple product of k/2 times d\tau, just like in the denominator of the second derivative d^2x/dt^2.

[@reapingshadow2866]

Can someone explain me how can I represent an elliptic function with periods w1 and w2, but using the term tau=w2/w1? 5:38. I have calculated g(w1,w2) and g(1,w2/w1) and they were not equal. Did I make a miscalculation or I don't understand something essential?

[@tim7054]

By assumption g(w1,w2) = g(aw1, aw2) for any non-zero complex numbers a. Just put a=1/w1 should do it.

[@SG-kj2uy]

Does Professor Borcherds define an elliptic curve?

[@TheStempkik]

Yes at around 4:15 he says that an elliptic curve is C modulo a lattice.

[@SG-kj2uy]

@@TheStempkik Thanks!

[@tomaszgruszka3845]

3:26 that was more anger than I was ready for

[@yatima1158]

Can someone enlighten me on how elliptic curves are in correspondence with complex numbers (the upper half plane) through their period? I suppose this is a question of what is a period and how does it define elliptic curves uniquely?

[@reapingshadow2866]

Hello. There are doubly periodic functions, (aka elliptic functions). They have two periods, for example w1 and w2. The linear combinations of these w1, w2 periods form a lattice on the complex plain. Weierstrass p(z) function is an instance of these elliptic functions. p(z+w1)=p(z+w2)=p(z). It turns out if you take any z complex number and calculate p(z) and p'(z) (its derivative) they are going to make true the equation of an elliptic curve over the complex numbers.

[@silvascientist]

"is equal to some constant that I don't really CARE ABOUT" lol

[@ikarienator]

4:05 why is rescaling a lattice not going to change g? what is g exactly? Isn't g a function taking to complex numbers and returns a complex number?

[@faisalal-faisal1470]

The point here is that we want to look at functions defined on the set of isomorphism classes of elliptic curves. Since C/L is isomorphic to C/kL for any nonzero scalar k, g(C/L) should be equal to g(C/kL). So if we write for a basis of L, and write g(w1,w2) to mean g(C/L), then we want g(k w1, k w2)=g(w1, w2) for all k in C*. There is still an ambiguity here because we can change the basis for L, and this is what the GL_2(Z) and SL_2(Z) business is about (4:30).

[@ikarienator]

@@faisalal-faisal1470 Yeah the question is basically why is C/L isomorphic to C/kL? Why is it isomorphic to scaling at the same time but not shifting or any other kind of linear transformation?

[@faisalal-faisal1470]

​@@ikarienator The multiplication-by-k map C/L -> C/kL defined by z+L -> kz+kL is well defined and has an inverse given by multiplication-by-1/k. That's all that you need here. It's also true that if C/L and C/L' are isomorphic complex elliptic curves, then necessarily L'=kL for some k in C*, and that the isomorphism is given by scaling. You can find the proof in most books that discuss complex elliptic curves. The gist of the argument is that it's possible lift the isomorphism C/L->C/L' to a 'nice' map C->C, and this can be explained in a variety of ways (e.g. covering spaces).

[@mydearfriend007]

I read "Einstein" series.

[@iamtackler]

Pretty sure everyone does this on their first time reading Eisenstein, although most of the time people’s first contact with him is in abstract algebra with eisenstein’s criterion

[@martinepstein9826]

One of my favorite math/science jokes is this seemingly bog-standard chain post I saw on Facebook - An ideological professor bullies students into accepting his views on religion, but one brave student stands up and turns the professor's logic around, _destroying_ his argument and earning thunderous applause from his admiring classmates. Please like and share this inspiring story. By the way that student was EISENSTEIN.

[@annaclarafenyo8185]

@@martinepstein9826 Surprise twist, the student's first name was Sergei.

[@yunjiangjiang6146]

@@iamtackler that criterion was taught in high school but I still can’t prove it off hand after PhD

[@andreashartel5577]

I am a bit irritated by the use of the word elliptic curve at 2:00. Is it supposed to mean elliptic function? I consider elliptic functions to be those that are periodic in two periods, but I don't know how they are related to elliptic curves.

[@faisalal-faisal1470]

No. He literally means functions on the set of (isomorphism classes of) (complex) elliptic curves. Meaning: you give me an elliptic curve E, I give you a complex number f(E), and this complex number depends only on the isomorphism class of E. An example of such a function is the j-function --- which, once appropriately interpreted, is a modular form. He explains how this interpretation goes, i.e. how one can go from a function on elliptic curves to a function on the upper half plane that is SL_2(Z)-invariant.

[@faisalal-faisal1470]

P.S. Elliptic functions are related to elliptic curves (and in fact are the reason why elliptic curves are called "elliptic"). E.g. the differential equation (p')^2 = 4p^3 - g_2 p - g_3 satisfied by the Weierstrass p-function and its derivative p' defines an elliptic curve over C, and this elliptic curve is none other than C/L where L is the defining lattice for p.

[@andrewxiwu]

I think he is talking about elliptic curves over complex numbers: en.wikipedia.org/wiki/Elliptic_curve#Elliptic_curves_over_the_complex_numbers

[@hugolabella6417]

Yes, the nomenclature, in this case, is not great, but this how almost everyone calls things. Even though most people relate "elliptic" to ellipses or elliptic functions as you said, an elliptic curve is something very different, it is a way of calling certain curves that can be expressed with a cubic formula, or, equivalently, the zeros of some functions in two variables that has degree 3. I think it has some other restrictions but it refers to curves that look something like this Y^2 = X^3 + aX+b. It is a strange name to call something so specific and that seems so unrelated to other elliptic "stuff". In another of Bocherds' courses, he talks a bit about this and also complains about how easy it is to mess up with these names, if I remember correctly it was in the course on algebraic geometry and the chapter about elliptic curves, one of the first ones.

[@darryljohnson8516]

An elliptic curve is a smooth projective genus 1 algebraic curve. In complex geometry, this is a set of points carved out of C^2 by a certain degree 3 polynomial in two variables. Associated to any variety we have a coordinate ring of functions on that variety and associated to that coordinate ring is the quotient field. An elliptic function is an element of that quotient field; the way it links with the definition you've already seen is the fact that elliptic curves are topologically torii embedded in the complex projective plane. In short, an elliptic function is a function on an elliptic curve. A modular function is a function on the moduli space of elliptic curves.

[@migarsormrapophis2755]

yeeeeeeeeeeeeee

[@CSAN33]

Yeeeeeeeee