Planet Math Blogseminars

Blackbox computation on the arXiv

2008-07-25T17:16:46+00:00

My paper on blackbox computation of A-infinity algebra structures, submitted to the Kadeishvili Festschrift issue of the Georgian Mathematics Journal, is now on the arXiv.

Computation of A-infinity algebra structures in group cohomology

2008-07-17T13:47:05+00:00

Since the thesis is now defended, and the degree provisionally granted, I present to you my doctoral thesis.

Lyubashenko-Manzyuk on the arXiv - some additional thoughts

2008-02-21T11:51:57+00:00

In the arXiv mailing from today, February 2nd 2008, there was a large chunk of A ∞-related preprints from Lyubashenko and Manzyuk. Most of these preprints are getting published in journals, or in Max-Planck-Institute report series.

For reference, the arXiv posts I’ll be talking about here are:
http://arxiv.org/abs/0802.2885
http://arxiv.org/abs/math/0211037
http://arxiv.org/abs/math/0306018
http://arxiv.org/abs/math/0312339
http://arxiv.org/abs/math/0701165

In general sweeps - acquired from scanning the abstracts - Lyubashenko and Manzyuk are talking about A ∞-categories as defined by Fukaya, and doing various basic constructions with them - demonstrating that various definitions of unitality coincide, showing a relation to Serre k-linear functors, constructing quotient categories with the same kind of structure et.c.

Now, one question that occurs to me quickly is the following:
Could an A ∞-algebra be considered to be a one-object A ∞-category? And if this is so, can the quotient constructions Lyubashenko and Manzyuk are using be extended (possibly significantly) to form a transferral of A ∞-(co)algebra structure across surjections in general?

And the way off target question: Does this somehow give us a way to transfer A ∞-(co)algebra structures along things like the restriction map in group (co)homology?

Blackbox computation of A-infinity structures

2008-02-06T15:48:05+00:00

In my doctoral thesis (soon to be released to the general public), I spend some time discussing a blackbox technique for computing A-infinity structures on Ext algebras using Kadeishvili’s proof of the minimality theorem as an inspiration for an algorithmic approach.

Basically, we build up a quasi-isomorphism from H *A to A by only ever defining the bits and pieces we really need to define, and figuring out what they should be by staring at the Stasheff morphism axioms and recursing down until everything we need to know is computed.

During my visit to Sydney last autumn, I implemented this as a module for the computer algebra system MAGMA. Now, as a part of my last minute thesis revisions, I have tracked down and fixed (hopefully all) bugs in the computation. Hence, beginning with release 2.14-10 of MAGMA, there will be an A-infinity computation module distributed with the system which at least computes the already known A-infinity structures on H *(C p n,Z/p) correctly. It can also be used for explorations in p-group cohomology.

Some examples (the examples are run in MAGMA 2.14-9, but with a development implementation of the A-infinity module):

> G := CyclicGroup(3);
> Aoo := AInfinityRecord(G,10);
> S<x,y> := Aoo`S;
> HighProduct(Aoo,[ x : i in [1..10]]);
0
> { #k : k in Keys(Aoo`m) | not IsZero(HighProduct(Aoo,k)) };
{ 2, 3 }
> { #k : k in Keys(Aoo`m) | not IsZero(HighMap(Aoo,k)) };
{ 2 }

This example shows us that the only higher products and quasi-isomorphism components that do not vanish in H *(C 3 ,Z/3 ) have arity 2 and 3. With some extra verification, we can use some of the latest arguments in my thesis to convince ourselves that the entire A-infinity structure on this cohomology ring follows from this particular computation.

> G := DihedralGroup(4);
> Aoo := AInfinityRecord(G,10);
> S<x,y,z> := Aoo`S;
> HighProduct(Aoo,[x,y,x,y]);
z
> HighProduct(Aoo,[x,y,x,y^2]);
y*z
> HighProduct(Aoo,[x,y,x,y^3]);
y^2*z
> HighProduct(Aoo,[x,y,x,y^4]);
y^3*z

These are essentially a small, but representative subset of the computations on the cohomology of the dihedral group with 8 elements that I presented in T’bilisi in 2006.

So - if you have a MAGMA installation, 2.14-x, and want to play with this before 2.14-10 is released, prod me and we’ll work something out.

Infinity preprints on the arXiv - November 28 - December 10 2007

2007-12-10T16:43:36+00:00

arXiv:0711.4499
Steffen Sagave - DG-algebras and derived A-infinity algebras

In this paper, the theory of A ∞-algebras as sketched in Kellers survey papers is developed to give bigraded (homological and “original”) resolutions of dg-algebras, and construct minimal models for dg-algebras using an extended notion of A ∞-algebras called derived A ∞-algebras. These take a bigraded structure into account.

Sagave proves a derived analogue of the minimality theorem applicable for any dg-algebra over any commutative ring.

arXiv:0712.0996

Valery A. Lunts: Formality of DG algebras (after Kaledin)

This paper, which states in the arXiv summary that it’s an early draft, and welcomes additional input, sets out to develop the background in order to prove a result stated in a paper by Dima Kaledin. Kaledin picks out a special cohomology class in a Hochschild cohomology ring, which acts as an obstruction to formality of the algebra studied. The core results deals with how this obstruction is realized in the Kaledin class and how formality of A ∞-algebras gets inherited in various constructions.

Anton Mellit

2007-11-10T12:40:18+00:00

I always use PARI when I need to do computations and I am a big fan of this little program. I believe that it is possible to do in PARI everything you can do with such big programs as Maple and Mathematica. Well… almost everything. Here I’d like to present some tricks to do things in PARI that seem impossible from first sight, or just convenient hints. Readers are very welcome to publish their own tricks in comments. This way we may create something like a library of tricks.

Weird functions

Ok, which kind of functions you can use in PARI? Polynomials in several variables, rational functions in several variables are o.k., power series (in several variables also!). There is one thing you can do in one variable and cannot do with several. It is

Factoring polynomials.

Suppose you want to factor something like this:

Here is a solution, which I heard from Don Zagier. You substitute in place of y some big number. I think he prefers to use some big prime numbers, but to me any big number will do the job. Then factor the resulting polynomial as a polynomial in x, then if it does not factor you know it is irreducible. If it factors you try to guess the factors as polynomials in x and y:

(11:22) gp > x^2+2*x*y+y^2
%1 = x^2 + 2*y*x + y^2
(11:23) gp > factor(%)
  *** factor: sorry, factor for general polynomials is not yet implemented.
(11:23) gp > subst(%,y,100000000)
%2 = x^2 + 200000000*x + 10000000000000000
(11:23) gp > factor(%)
%3 =
[x + 100000000 2]

(11:23) gp > %1/(x+y)
%4 = x + y

Algebraic functions.

Sometimes you need to use functions like . In PARI you have Mods. Mod is an object in a finite extension of something. Their primary use, I guess, is for number fields, so that Mod(x, x^2+x+2) means . But nobody stops us from using it like Mod(y, y^2-x^2-1). So let us try:

(11:24) gp > y0=Mod(y,y^2-x^2-1)
%5 = Mod(y, -x^2 + (y^2 - 1))
(12:12) gp > y0^2
%6 = Mod(y^2, -x^2 + (y^2 - 1))

Oops. We expected to get . This is the problem which everyone working with PARI should know about.

Side note: priority of variables

It is variable order. All variables are arranged in the order according to the time of first usage. Since x was used before y, it has ‘bigger priority’ than y. Therefore every expression in x and y is considered in first place as an expression in x, so the equation is treated like where is a kind of parameter. So this corresponds to . Therefore we should do it in a slightly different way:

(12:12) gp > y0=Mod(y,y^2-x0^2-1)
%7 = Mod(y, y^2 + (-x0^2 - 1))
(12:19) gp > y0^2
%8 = Mod(x0^2 + 1, y^2 + (-x0^2 - 1))

Don’t forget to use ‘lift‘ when you want to get your final answer in a readable form.

Transcendental functions

In general: what can we do with transcendental functions? Since they are transcendental you cannot get any algebraic statements about them, so there is nothing to ask. There are, of course, exceptions to this. For example sometimes one transcendental function algebraically depends on another transcendental function. Like sine and cosine. Therefore the previous approach works perfectly well. You should encode sine and cosine in the following way:

(12:19) gp > Cos
%9 = Cos
(12:26) gp > Sin
%10 = Sin
(12:26) gp > Cos0=Mod(Cos, Cos^2+Sin^2-1)
%11 = Mod(Cos, Cos^2 + (Sin^2 - 1))

Another exception is when you are expanding a transcendental function into a power series. This goes without problems:

(12:26) gp > sin(x)
%12 = x - 1/6*x^3 + 1/120*x^5 - 1/5040*x^7 + 1/362880*x^9 - 1/39916800*x^11 + 1/6227020800*x^13 - 1/307674368000*x^15 + O(x^17)

Another thing you may want to do with a transcendental function is to differentiate it. Well, we come back to our example with Sin and Cos.

(12:28) gp > D(f)=subst(deriv(lift(f),Sin)*Cos-deriv(lift(f),Cos)*Sin, Cos, Cos0)
(12:30) gp > D(Sin)
%13 = Mod(Cos, Cos^2 + (Sin^2 - 1))
(12:31) gp > D(Cos0)
%14 = -Sin

Now you can differentiate any combination of sine and cosine.

Differential equations

If you have a differential equation, like , you can encode it using variables y, dy and defining a differentiation operation like above which sends y to dy and dy to y:

(12:31) gp > D(f)=deriv(f,y)*dy+deriv(f,dy)*y

Numbers

There are several ways of dealing with algebraic numbers.

1. Using ‘Mod’. Just type Mod(x, x^2+x+2) when you need .
2. Using approximation. Simple approach: use (-1+sqrt(-7))/2 and in the end use very powerful algdep or lindep functions if you need the minimal equation:

(12:35) gp > (-1+sqrt(-7))/2
%15 = -1/2 + 1.322875655532295295250807877*I
(12:41) gp > algdep(%,2)
%16 = x^2 + x + 2

3. Using approximation, but with several embeddings of your number field in C.

Intersection theory

I have some experience computing some intersections of algebraic varieties. The approach is to use Mods to define algebraic varieties. Say, elliptic curve is Mod(y, x0^3+a*x0+b-y^2). For higher dimensional varieties one can use Mods with Mods inside. Then if you need to intersect something you simply get more equations. In the end you probably want to get points. Then the coordinates of these points will be some mods which give them as algebraic numbers. To find multiplicities solve all the equations in power series and look at the exponent of the main term. In the process of writing my thesis I was finding some points which are defined over some field of high degree (I think it was 12). If you try to use this approach don’t forget about the function ‘polcompositum‘, which helps, if you have numbers in different fields, to pass to the composite field.

Final remarks

The only thing that I not-so-like in PARI is its programming abilities. I think it is better to use some carefully designed standard scripting language for programming. That’s why I am working on integration of PARI with Python. This is still work in progress (look at some screenshots here).

If you know some other not-so-obvious tricks for PARI, you are very welcome to post them below.

Associahedral diagonals

2007-11-09T13:38:59+00:00

There is a paper by Loday on the arXiv since about a month ago, entitled The diagonal of the Stasheff polytope. The basic idea of the paper is to introduce a new operad AA ∞, built on the simplicial chains of Lodays triangulation of the associahedron, and using the relative simplicity of forming diagonals on simplicial complexes to generate a reasonably natural diagonal on this new operad.

With quasiisomorphisms from our familiar A ∞-operad to AA ∞ and back again, he then constructs a diagonal on A ∞, formed by going to AA ∞ and computing a simplicial diagonal, which finally gets deformed into a diagonal on A ∞.

Thus, this construction is based in a slightly different approach to diagonal computation than both the Saneblidze-Umble construction and the Markl-Schnider paper; but Loday conjectures equality between the two constructions based on a comparison of the results for the diagonal 5-ary operation.

If anyone out there has read more of the paper than I have (or if Loday himself is reading this sporadic blog), I would appreciate some nudges on how to internalize the deformation enough to fix it in computer code.

Partial Report on AIM Workshop: Towards Relative Symplectic Field Theory

2007-10-01T01:35:19+00:00

towards-relative-symplectic-field-theory-report.pdf

Computational projects

2007-09-27T03:09:41+00:00

A sneak preview of my current project — which will end up as about a third of my PhD thesis:

dynkin:~/magma> magma
Magma V2.14-D250907   Wed Sep 26 2007 13:19:51 on dynkin   [Seed = 1]
Type ? for help.  Type -D to quit.

Loading startup file “/home/mik/.magmarc”

> Attach(”homotopy.m”);
> Attach(”assoc.m”);
> Aoo := ConstructAooRecord(DihedralGroup(4),10);
> S := CohomologyRingQuotient(Aoo`R);
> CalculateHighProduct(Aoo,[x,y,x,y]);
z
> exit;
Total time: 203.039 seconds, Total memory usage: 146.18MB

A infinity at the Secret Blogging Seminar

2007-09-21T00:32:19+00:00

AJ Tolland links to some seminar notes he took from a seminar by Kevin Costello on A ∞-algebras in topological string theory.

Thanks to AJ for telling me what I got wrong in the first take.

Overview of the computation of A-infinity structures

2007-08-29T09:20:05+00:00

Ben Webster asked, in a comment to the post Question for the audience for pointers to literature on the computation of A ∞-structures. Since I’m still out traveling, and far from a university, I won’t give any real pointers, but rather stick to namedropping. It’s not very difficult, using the names, to dig out the relevant article references from MathSciNet.

I also suffer from being slightly new to the field. There are people out there with a much better overview of what has been done, and what does apply. (Jim Stasheff, I’m looking at you! :))

At the core of A ∞-structures on Ext algebras lies the so called minimality theorem, proven by A Whole Range Of People. It states, roughly, that if A is a dg-algebra, then the A ∞-structure on A given by m 1 =d and m 2 =⋅ induces an A ∞-structure on H *A, together with a quasiisomorphism H *A→A in such a way that any two such induced structures are quasiisomorphic to each other.

And with A taken as, for instance, End A(pS), for pS suitable resolution (i.e. dg-module quasiisomorphic to the A-module S), we can view Ext A(S,S)=H *(End A(pS)), and thus find an A ∞-structure on Ext induced by the obvious structure on the endomorphism dg-algebra of a chain complex.

Now, the trick is how to get such a structure. One of the earliest mentions I know of is the paper by Kadeishvili, where a purely algorithmic approach is taken. He gives a recursion, using the Stasheff axioms, where, in order to calculate a certain higher product for a certain input, you calculate a sum of products and compositions of lower products, ending up with something that is, inductively, known. This way, you end up with calculation of specific products reducing to a matter of taking preimages under certain differentials and a lot of bookkeeping.

This method is the one I’m basically talking about in my previous post.

Another method floating around uses Homotopy Perturbation Theory (or was it Homology Perturbation Theory - I keep forgetting). Some of the relevant names here are Huebschmann, Stasheff, Gugenheim, Johansson (Lennart, not Mikael!!!) and Lambe. The idea here is to find a strong deformation retract A→H *A, and work with the portions of that to find everything you need.

This kind of approach I have seen used by Berciano in her work with the KENZO module ARAIA-CRAIC for calculation of A ∞-coalgebra structures in topological contexts, and is also being used by Berciano-Umble in at least one recent preprint.

Finally, Merkulov has taken the basic idea of HPT and refined it for cases where a lot is already known. He requires a vector space splitting of the dg-algebra A=H⊕B⊕N, where H=H *A, H⊕B=kerd and N is the complement of H⊕B. Using this, and a few functions - a projection π down on H and a homotopy π∼Id, he gets enough data to be able to write down the A ∞-structure on H with each function and each component of the quasiisomorphism being given by sums of planar evaluation trees, with each internal edge being an application of the homotopy, and each vertex being a normal multiplication in the dg-algebra.

It has the good property of being generic and at the same time very explicit. However, unless this splitting is given, and the homotopy found, you can’t do very much with it.

I hope I haven’t now missed anyone important in the overview. Please correct me if I have.

Question for the audience

2007-08-23T12:29:17+00:00

The blog is in quite a summer lull, and I’m not doing near my share of keeping it up. I’m blaming my vacation and my marriage ceremony tomorrow.

Anyway, here is a question I was thinking about on my way home. About appropriate terminology for my own research. I’m putting quite a bit of thought into calculation of A ∞-structures on Ext-algebras in cases where the usual calculation methods - Homotopy perturbation theory, Merkulov’s method, et.c. - do not really work well since the required data about the endomorphism ring of the appropriate chain complex ends up being much too large. So far I have been calling this local computation, but it struck me that it might end up confusing those more used to local being used for .. say .. localization in various contexts.

On the way I thought about blind computation, to indicate the lack of information compared to the more global methods, but this doesn’t seem to be quite it either.

Thus a question to the readers (and writers?) of this blog: what would be a good word to describe my particular brand of computation of A ∞-structures on H *(A), for A a DG-algebra which gets viewed as a black box, capable of performing calculations, but not of displaying its internals in any good way?

Something truly outrageous!

2007-08-08T10:45:02+00:00

Faithful readers of Vivastgasse 7 may have noticed that we haven’t posted anything in a while now- the reasons are numerous, but it mostly boils down to the fact that two of us (Anton and I) are in the middle of resolving several quasi-bureaucratic/ quasi-academic things: Anton is right now in Ukraine, finishing up his thesis and busy planning his move to Paris in a couple of months for his first postdoc. I’m busy planning a move as well- off to Durham next month for five months. So both of us are sort of ‘out-of-commission’ right now. However, I do intend to post something soon (maybe this weekend?) either on schemes over the mysterious “field with one element” or better yet, an unpacking of our guest-blogger Sniggy Mahanta’s post on conformal field theories. (Thanks goes to AJ Tolland for pointing out some gross inaccuracies in that post!)

But here is the main reason for this (non-mathematical) post- to vent!

It turns out that some very well-meaning people (lead by Ali Nesin) started a mathematical summer camp in Sirince, Turkey- the idea of the camp was to provide motivated undergrads with exposure to mathematics beyond what is usually taught at the universities. I personally like these sorts of camps very much. I doubt it that I would have become a research mathematician had I not been exposed to such camps and REUs while I was an undergrad. (The summer of 2002 is particularly memorable- I attended the IAS/PCMI summer program on automorphic forms; I’ve been smitten by number theory since.) Our American colleagues will also look appreciate summer camps for high school students such as the Ross program at Ohio State and PROMYS at Boston University.

Anyway, coming back to the summer school in Turkey something really bizarre happened (almost Kafkaesque in nature) towards the middle of the program (which was to last till the end of this month)- it was shut down by the authorities for providing “education without permission”!!! (A complete account of the whole story is to be found in Alexandre Borovik’s blog)

This is simply unacceptable! While I have some guesses as to why such a thing may have happened, the academic nature of this blog prevents me from making conjectures of a political nature here. I will only say this: such behavior will only hurt the Turkish scientific aspirations in the long run, not to mention the fact that it puts Turkish political and educational authorities in a very bad light in the West. On behalf of all like-minded mathematicians and educators, I call on the Turkish authorities to immediately allow the reopening of the camp as well as issuing a public statement as to why they closed the camp in the first place. (Sorry, the oh-so-glib “educating without permission” simply doesn’t cut it! I also ask the readers of this blog to visit Save Mathematical Summer School blog (explicitly devoted to this issue) to sign a petition to the Turkish premier asking him to intervene.

Last week on the arXiv

2007-07-30T07:39:19+00:00

Some of the recent preprints seen in the arXiv mailings include:

Mikael Johansson: A partial A ∞-structure on the cohomology of C n×C m

Suppose k is a field of characteristic 2, and n,m≥2 powers of 2. Then the A ∞-structure of the group cohomology algebras H *(C n,k) and H *(C m,k) are well known. We give results characterizing an A ∞-structure on H *(C n×C m,k) including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.

Alastair Hamilton & Andrey Lazarev: Cohomology theories for homotopy algebras and noncommutative geometry

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A ∞,C ∞ and L ∞-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of C ∞-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.

Alastair Hamilton & Andrey Lazarev: Symplectic A ∞-algebras and string topology operations

In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincar\’e duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading `string topology’. Our method is based on obstruction theory for C ∞-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.

Yes, this post includes a bit of shameless self-promotion.

Some remarks on CFTs.

2007-07-26T17:44:29+00:00

Readers should be warned that the author is not an expert of CFT and, in fact, not even a novice in physics. What follows should be taken with a hefty pinch of salt.

Given any Riemann surface (as a target manifold) one is able to associate to it an or a super conformal field theory. The word super can just be construed as a -grading of the theory. This is a simplistic version as normally one should also take into account several other parameters like a B-field and so on. Within an SCFT there is a topological sector called a TQFT (topological quantum field theory) which is insensitive to the metric on the target space. There are axiomatic descriptions of this theory due to Atiyah and Segal and in its latest version possibly due to Costello. Roughly an -dimensional TQFT is a functor satisfying a lot of axioms from -manifolds with labelled boundaries (incoming and outgoing) to symmetric monoidal DG (differential graded) categories with some twisting.

Now people with a background on derived categories would like to see the triangulated structure of . Indeed, the homotopy category of a DG category resembles a triangulated category. It seems at this point it is possible to associate two models, namely, and to the theory after Witten. The model leads to a complicated Fukaya category (possibly as an -category) and the model leads to the derived (or DG) category of coherent sheaves on . There is a chiral ring that can be associated to each of these two categories. On the side it is the quantum cohomology ring and on the side it is its Hochschild homology . There is a Frobenius algebra structure on them; on at least when is Calabi-Yau.

The homological mirror symmetry conjecture by Kontsevich would predict the existence of an equivalence between the two categories arising out of the and models. This should naturally give rise to an isomorphism of Frobenius algebras after passing on to their chiral rings.

The construction of CFTs is a hard problem in general. Sometimes it is possible to get one’s hands on to the (possibly local) symmetries of a CFT, which are called chiral algebras and then one might be tempted to construct the CFT out of it. The chiral algebras as mathematical objects have the structure of a VOA (vertex operator algebra) . The construction of CFT from its chiral algebra seems to be possible if the representation category of the chiral algebra has finitely many irreducible objects (communicated by Liang Kong). This can also be phrased in the language of the partition function of the CFT as Gukov and Vafa do. Such a CFT is called rational (abbreviated RCFT). They enjoy some other very desirable properties, which make them particularly interesting. When the target space is a complex torus Gukov and Vafa argue that rationality of the CFT is related to the complex torus having complex multiplication. So in this case the RCFTs are plentiful but in higher dimensions they are supposed to be rather sparse (not dense in the moduli space).

Meng Chen studied the higher dimensional case of the same in her thesis and came up with her own geometric definition of the rationality of a CFT and related it to abelian varieties with large endomorphism rings. It is not clear if her geometric definition of rationality of a CFT is related to the, rather algebraic, notion of rationality presented above.

This is an intriguing connection between physics and number theory. In which direction the information would flow remains to be seen. There are some more interesting connections of this sort relating generating functions of some counting problems of (arithmetic) algebro-geometric nature to the partition functions of CFTs. If the author can weather this storm more postings on them will follow.

Sincere apologies for the inaccuracies and for straying from the main theme of the blog which is arithmetic algebraic geometry. The author just stumbled upon the thesis of Meng Chen recently, got fascinated and wanted to share this new-found knowledge.

Anton Mellit

2007-07-23T14:02:59+00:00

It seems when people talk about modular forms they tend to forget that they are very related to families of elliptic curves. Here I want to explain some simple way to understand the connection. We will consider modular forms for the full modular group .

So consider the simplest family of elliptic curves, the Weierstrass family:

Here and are meant to be some formal parameters. This indeed defines and elliptic curve over the ring , where is the base field, which is supposed to be of characteristic , and is the discriminant:

When we write we in fact mean the corresponding projective variety over with equation

Let us denote the affine chart with coordinate functions by and the point at infinity by since it is the zero point for the addition on the curve.

Now we are going to compute some Laurent series expansions at . First we choose local parameter . Indeed, has pole of order and has pole of order at , therefore has simple zero there. To find expansion of $x$ we solve the following equation in Laurent series:

Rewriting it as

we obtain a polynomial equation in which can be solved by Newton’s method starting with . We obtain

Let us compute the expansion of the invariant differential :

We see that it is possible to integrate this series formally and make it the new local parameter:

Then the expansions of $x$ and $y$ with respect to the new local parameter are:

We also consider the formal integral of :

Consider the power series

If we substitute this power series in place of and find

then we can easily verify that

i.e. we have found a solution for , .

This explains that we should in general put

and define in such a way that

In this way we obtain as a polynomial of , , but in fact it is true that this polynomial is the same polynomial that expresses the Eisenstein series of weight in terms of the Eisenstein series of weights and . So for us modular forms will be homogeneous polynomials of and where weight of is and weight of is .

To define the weight more geometrically let us consider the action of the multiplicative group on :

Then a modular form of weight is a function of which transforms like

If we consider not only functions of , but functions of then we obtain Jacobi forms of index .

Derivatives of modular forms

We want to apply this language to understand some natural operations on modular forms. The first operation is the Euler derivative . This simply takes a modular form of weight and sends it to . It is easy to see that this is exactly the action of the Lie algebra of the multiplicative group. Next we want to reconstruct the Serre derivative.

Suppose we have a derivation on . Let us try to lift it to obtain a derivation of the ring of functions on (which is generated by ). We would have , satisfying a relation

But note that we could simply apply to the Laurent series expansions of term by term (denote it by ) and get a solution to the relation above. Therefore the difference must satisfy

But we also have a solution to the equation above! Namely it is the operator which will be denoted simply by . Therefore we must have a Laurent series which satisfies

Using the fact that commute it is easy to obtain

We expect to be regular functions on . Clearly one can assume to contain only even powers of and to contain only odd powers of - this corresponds to being invariant under the involution . We see that the right hand side is a regular function on which contains only odd powers of . Therefore it is a product of and a polynomial in . So we write

Noting that gives

Next observation is that for any polynomial in we can express it as a derivative of an expression of the form

In fact is the formal integral of and is the formal integral of and these forms generate the first cohomology of . So,

It implies that

But we know that

Looking at the power series expansions we conclude that

So it is natural to consider a derivation for which and a derivation for which . In the former case we obtain

It is easy to see that we have got the Euler operator. In the latter case we obtain

Using our convention one can see that this is the Serre derivative :

It is important that we did not only obtain as a certain canonical derivation which lifts to a derivation on , but we also computed which can be interpreted as a formula which gives the Serre derivatives of all the Eisenstein series.

In the end I would like to mention that using this approach and studying the Gauss-Manin connection one can explain some other things which appear in the theory of modular and quasi-modular forms and seem mysterious, like Bol’s identity and Rankin-Cohen brackets.

The main idea is: “the ring of modular forms, or the ring of quasi-modular forms come naturally equipped with an elliptic curve over it.“

Also here is a useful formula for values of modular forms. If is a modular form of weight and a curve has periods , then

On the left we have the values of as a polynomial of and on the right we have its value as a function on the upper half plane. There is a corresponding formula relating values of quasi-modular forms and periods of differentials of second kind.

Opening the Infinite Seminar

2007-07-17T16:16:45+00:00

There are more and more mathematical research group blogs entering the scene lately. I would like to try and start one more, this one with a focus on various closely related subjects - the * ∞ algebras.

We know of the A ∞ algebras. We know of the L ∞ algebras. And we know of E ∞ ring spectras. I most probably forget about several highly interesting similar constructions, which I however expect will end up being more than welcome here.

The tagline I expect the prevalent theme here to follow, though, will be “…up to homotopy”. This seems, to my mind, to be a unifying factor of most * ∞-theories I have seen so far. Thus, as the technical host of discussions, I wish you all welcome.

If you feel you can contribute more than just as a regular reader and commenter, and might even wish to write posts, then please create a user and drop me a note on mik@math.uni-jena.de. As soon as I see the user created, I will be able to grant further privileges. Hopefully, this way, we can get enough of the conversation going here to make this as vibrant and interesting a group blog as all the other examples out there.