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.

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.

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.

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.

Recall that the description of of fields is given by the following theorem.

Matsumoto’s Theorem.
For any field

or equivalently in the context of presentations of groups, is the Abelian group with
Generators:
Relations:

1. (The Steinberg relation)
2. 
3. 

Definition. Let be a field and be an Abelian group. A Steinberg symbol on (with coefficients in ) is a -bilinear map such that

By Matsumoto’s theorem any Steinberg symbol gives a unique group homomorphism such that

Norm Residue Algebras.

Let be a field which contains a primitive -th root of unity and let be two given elements in . The dimensional vector space

with the following rules of multiplication:

is a central simple -algebra and it is called the norm residue algebra.

Theorem 1. Let be a central simple algebra of degree and let

be the minimal polynomial of over . If splits into distinct linear factors over , then

Corollary. Let . If either or has an -th root in , then

As a special case of the above statement we have

.

Theorem 2.

Proof.
We define the non-commutative binomial coefficients

where

It can be easily checked that . Now suppose that are elements of an arbitrary ring such that for some in the center of . Induction on shows that

.

In particular, for the generators and of , since and for all , we obtain that

Now by the same reason as the previous corollary we have

.

Theorem 3. Let be in . Then

Proof.
Let be the generators for and be the generators for . Define

Let be the algebra generated by and be the algebra generated by . Now , and

So and satisfy the relations for , thus . Similarly . Notice that and commute with and , hence we have a natural -algebra homomorphism

.

Since is simple, is injective. Since the dimensions of two sides are equal it must be an isomorphism.

Remark. We have already seen that . So by the above theorem we have

similarly

Here denotes the equivalence class of in the Brauer Group.

Define

The above remark says that is -bilinear. By Theorem 3 we observe that is a Steinberg symbol, hence we get a homomorphism

From Corollary it follows that

which shows that the image of is contained in

Since , the homomorphism annihilates , therefore it induces a homomorphism

which is called the norm residue homomorphism.

The following surprising theorem was proved by A. Merkurjev and
A. Suslin in 1982.

The Merkurjev-Suslin Theorem. Let be a field which contains an -th primitive root of unity. Then

is an isomorphism.

Norm Residue Homomorphism via Galois Cohomology.

The norm residue homomorphism can be described in terms of Galois cohomology. As a preliminary we need to recall the notion of the cup product in the cohomology of groups.

Let be a field and let be an integer coprime to char Set

The condition implies that has exactly elements. Assume that has an -th primitive root of unity, i.e. . Set and consider the following exact sequence of -modules:

The associated exact cohomology sequence is

As , the action of on is trivial, so . By Hilbert’s Satz we have , so the above sequence breaks up to the following exact sequences:

Hence the map induces an isomorphism , and the map induces an isomorphism between

and

By using that we obtain that Since acts trivially on it follows that is isomorphic to as -module, hence

The composition of the following maps

gives a -bilinear map which can be proved to be a Steinberg symbol, and the induced homomorphism is the norm residue homomorphism.

References:

1. Kersten, Ina Brauergruppen von Körpern. (German) [Brauer groups of fields] Aspects of Mathematics, D6.1.

2. Milnor, John Introduction to algebraic $K$-theory. Annals of Mathematics Studies, No. 72.

3. Rost, Markus Arbeitstagung 2007 – Norm residue homomorphism.

4. Tate, John Relations between $K\sb{2}$ and Galois cohomology. Invent. Math. 36 (1976), 257–274.

(Via Anton (my fellow-blogger) and his wife Masha- their seven year old daughter Ivanka’s notes from Hirzebruch’s opening lecture at the 50th Arbeitstagung here a couple of weeks ago)

He suggested the following (all revolving around the arithmetic Riemann-Roch theorem):

1. Papers by Gillet and Soule on arithmetic intersection theory

2. Papers by Bismut, Gillet and Soule on determinant of cohomology of an arithmetic variety (towards Riemann-Roch for this determinant.)

It seems that these papers are summarized in Soule’s book. Also there is a nice book by Faltings that modesty prevented him from referring it to me: Lectures on the arithmetic Riemann-Roch theorem.

Speaking of Arakelov theory, two final comments:

(1) The best place to get into the right frame of mind for these type of questions in an elementary setting is Neukirch’s book Algebraic number theory

(2) Our fellow student here Nikolai Durov has recently reworked the foundations of the entire theory from the point of view of generalized rings (including exotic objects like the field with one element ). His thesis is available here.

for elements () to define an element in the Bloch group. It appears that the condition naturally appears if one studies hyperbolic -manifolds.

Let be an abelian group. Let be a triangulated oriented -dimensional manifold. Let be an oriented -dimensional simplex of the triangulation.

Definition
An angle structure on is a collection of elements of , denoted , one for each edge satisfying relations:

We think of as the angle between the faces of meeting at .

We have

This, in particular, implies

Suppose each tetrahedron of the triangulation has an angle structure. The corresponding angles will be denoted , which means the angle between and , i.e. for a simplex we have elements listed below:

Let us define additionally

Therefore the notation is invariant with respect to even permutations of vertices of .

Definition
For each edge of let be the -simplices adjacent to with , and . Suppose the following condition is satisfied:

Then we call our manifold angled.

Suppose we have a finite set of tetrahedra with angle structure. Is it possible to glue them together and obtain an angled oriented manifold? We provide a necessary condition.

Theorem
In an angled oriented manifold the following condition is satisfied in the group :

the sum is over the -simplices which compose the fundamental class of the manifold.

Note that if is invertible in , then the terms and are .

Remark
Let and each tetrahedron is realized as an ideal tetrahedron in the hyperbolic -space with cross-ratio . Then its angles are , , . To make the product one can change angles to , , . Then we see that up to -torsion the sum of tensors is , which is well known. In other words, hyperbolic -manifolds provide elements in the Bloch group. However our approach seems to be more general.

The rest of this text provides a proof of the theorem.

Suppose is angled. Then we can construct elements with the property

for each oriented simplex . If is another such family then there is a family with

Fix a vertex . Let and be edges. Join and by a sequence of triangles for , , . Put

This does not depend on the choice of the sequence since for any oriented -simplex we have

Therefore there is a family with the property

If is another such family, there exists a family with

In particular for any -simplex we have

We see that we can replace with to make satisfying

If is another family with this condition then there is a family with the property

Put . Then for any oriented -simplex

Therefore there exists a family (this is different from the one used before) with property

This means that for any -simplex we have

Let us summarize the properties of :

Let be a -simplex. Then

Consider the following element in :

This element is invariant under cyclic permutations:

Moreover,

Since and , we can also write as

Let

For any oriented -simplex put

Then

We may rewrite

Therefore

Taking into account that

we obtain

Now we turn to .

Therefore

We see that depends only on the angles and its sum over the manifold is zero. If is invertible in then

This specializes to in the case of ideal hyperbolic tertrahedron.

To begin with:

Definition
A neutral Tannakian category over a base field is a rigid abelian tensor category with an -linear exact faithful tensor ( being the category of vector spaces over ) which one calls the fiber functor.

This definition is stolen from a paper of Breen in the Motives volume (Proc. Symp. Pure Math. Vol 55, part I); As Breen notes, the idea is to imagine as a fiber over ; is, therefore, a sort of a vector bundle over . Incidentally, this is the original definition given by Saavedra-Rivano in his thesis with Grothendieck. Deligne in his Grothendieck Festschrift paper gives an equivalent definition where the “primacy” is on dual objects than internal homs (required for the “rigid” part in the above definition.)

The main reason why one should care about Tannakian categories is this:

Theorem
Let be an affine group scheme over . Then the category of representations of is a Tannakian category. Moreover the automorphisms of the fiber functor (simply the forgetful functor)

The moral is simple: since the category of representations of an affine group scheme is Tannakian, just by studying the automorphisms of the fiber functor one recovers the affine group scheme itself (BTW, that by itself is a commutative but not necessarily co-commutative Hopf algebra.) One should compare the theorem above with the classical Pontryagin duality.

But for me (atleast), the real reason why Tannakian categories are interesting is because of the role they play in Grothendieck’s original conception of motives. Recall that the basic idea behind pure motives is this: let be the category of smooth projective schemes over some base (field) and be the category of graded vector spaces over a field of char 0 and the category of pure motives (ie, motives of smooth projective schemes) with some adequate relation on the cycles . In (the theory? fantasy? of) the original conception, we have three functors: the obvious contravariant one for good cohomology (Weil cohomology, for those in the know): and two other slightly mysterious functors: the functor of motivic cohomology that takes a sm. proj. scheme and associates to it a pure motive, and the realization functor which takes a motive and “realizes” it as a concrete cohomology theory such as deRham, etale with l-adic coeffs, …. Now the original idea was that if was a Tannakian category then one should consider the realization functor as the fiber functor- in which case one has some very interesting Galois groups as groups of automorphisms at hand (he motivic Galois groups.) In practice though, all of this is very hard to construct explicitly- I will return to some of these issues in a less name-dropping and leisurely way in later posts.