User ilya nikokoshev - MathOverflow

What is a topos?

2009-10-04T22:20:36Z

According to Higher Topos Theory math/0608040 topos is

a category C which behaves like the category of sets, or (more generally) the category of sheaves of sets on a topological space.

Could one elaborate on that?

Most interesting mathematics mistake?

2009-10-17T14:28:43Z

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincare's 3d sphere charaterization or the search to prove that Euclid's parallel axiom is really ~~necessary~~ unnecessary.

But I also think there are less famous mistakes worth hearing about. So, here's a question:

What's the most interesting mathematics mistake that you know of?

This question is community wiki, meaning neither the question nor the answers receive points (which are reserved for "hard" questions). So please post as much as you like (indeed please post one answer per post so that others can upvote the ones easier), vote a lot and vote freely.

(should there be a tag 'not-math-related' or similar?)

Examples for Decomposition Theorem

2009-10-22T21:59:48Z

There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.

Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth scheme but more complicated for others, and by $IC_i$ one denotes the complex constructed from a pair ($Y_i$, $\mathcal L_i$) of subvariety together with the local system as $IC_i := j_{!*}\mathcal L_i$.

Now for a projective morphism $f: X\to Y$ turns out you can decompose in the derived category $$f_*IC = \oplus IC_i[n_i].$$ The special beauty of this decomposition theorem is in its examples. Here are some I think I know:

For a free action of a group G on some X, you get the decomposition by representation of G.
For a resolution of singularities, you get $f_*\mathbb Q = IC_Y \oplus F$ (and $F$ should have support on the exceptional divisor.)
For a smooth algebraic bundle $f_*\mathbb Q = \oplus\, \mathbb Q[-]$ (spectral sequence degenerates)

There are many known applications of the theorem, described, e.g. in the review

The Decomposition Theorem and the topology of algebraic maps* by de Cataldo and Migliorini,

but I wonder if there are more examples that would continue the list above, that is, "corner cases" which highlight particularly specific aspects of the decomposition theorem?

Question: What are other examples, especially the "corner" cases?

A single paper everyone should read?

2009-10-23T18:42:05Z

Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues.

Do you have such an example?

Let's try to go in the direction of papers that can actually be read online or accessible with little effort, e.g. in major libraries, so that people could actually follow your advice and read about it immediately.

And as usual let's do one per post and vote freely, vote a lot.

What does "supersingular" mean?

2009-10-19T19:39:37Z

Are supersingular primes and supersingular elliptic curves related?

(this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post)

Why no abelian varieties over Z?

2010-01-05T22:59:34Z

Motivation

I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form

the set $\{$ objects $\dots$ over field $K$ with good reduction everywhere except set $S$ $\}$ is finite/empty

One interesting thing he mentions is about abelian schemes in the most natural case $K = \mathbb Q$, $S$ empty. I think according to the definition we have a trivial example of relative dimension 0.

Question

Why is the set of non-trivial abelian schemes over $\mathop{\text{Spec}}\mathbb Z$ empty?

Reference

This is proven in Il n'y a pas de variété abélienne sur Z by Fontaine, but I'm asking because: (1) Springer requires subscription, (2) there could be new ideas after 25 years, (3) the text is French and could be hard to read (4) this knowledge is worth disseminating.

Proof of Bloch-Kato conjecture of K-theory?

2010-01-01T21:15:16Z

Wikipedia says:

this circle of ideas is distinct from the Bloch–Kato conjecture of K-theory, extending the Milnor conjecture, a proof of which was announced in 2009

What exactly is the K-theory conjecture of Bloch-Kato and has it been proven?

Answer by Ilya Nikokoshev for Intuition for Primitive Cohomology

2010-02-08T19:45:05Z

Is it true that they are dual to homology classes that can't be expressed as $X'\cdot H$, where $H$ stands for the hyperplane in $\mathbb C\mathbb P^n$?

Answer by Ilya Nikokoshev for Looking for an introductory textbook on algebraic geometry for an undergraduate lecture course

2010-02-06T12:14:54Z

You can also take a look at the question A learning roadmap for algebraic geometry.

Answer by Ilya Nikokoshev for A specific branched cover of S^2 as a subgroup of Pi_1

2010-02-04T22:41:24Z

In standard topological terms, the exact sequence that relates homotopy groups of the base $B$, fiber $F$ and total space $E$ of topological fibration gives

$$\pi_1(F) \to \pi_1(E) \to \pi_1(B) \to \pi_0(F),$$

that is,

$$ 0\to \pi_1(T-4) \to \pi_1(S^2-4) \to \mathbb Z_2.$$

The middle map works by taking a loop above and pushing it to the base; the right map works by taking a loop on $S^2-4$, lifting it as a path, and taking 0 or 1 depending on whether the resulting path is closed or not.

I believe the OP described the fundamental group of the torus as $\left< a, b, c, d, e, f| [a, b]cdef = 1\right>$ where $a, b$ are two circles of the torus and $c, d, e, f$ are four loops around the holes. For $S^2-4$ my suggestion would be to use $\left< C, D, E, F| CDEF = 1\right> = \left< C, D, E\right>$ where $c$ is over $C$ etc (rather then $g, h, x, w$).

Now, since the loop around $c$ on torus has to wind twice around it when projected to the sphere (think about complex $z\mapsto z^2$ map) it's easy to see that $c\mapsto C^2$, $d\mapsto D^2$, $e\mapsto E^2$, $f\mapsto F^2$.

What about $a$ and $b$? Careful observer should note that it's a bit tricky to define the loops. E.g, if you move $a$ parallel to itself, you'll get a new $a'$ which would differ by something like $cd$ depending on which points are where and depending on how you draw the basepoints on the loops.

For the exact calculations one should fix the torus to be $\mathbb R\times \mathbb R/\mathbb Z\times\mathbb Z$ so that fixed points of $z\mapsto z$ are the vertices $c = (0, 0)$, $d = (1/2, 0)$, $e = (0, 1/2)$, $f = (1/2, 1/2)$. Moreover, you should now select some basepoint and draw the cycles around $c, d, e, f$ so that $cdef = 1 $.

Unfortunately, from the picture it's not easy to say where some simple loops like horizontal or vertical go. While in homology they seem to be $C+D$ and $D+F$, one has to draw them really carefully with the basepoints. I couldn't do that, but here's something different instead.

I tried to exhibit some expressions $a, b$, which may be not exactly the cycles above, but which nevertheless satisfy $[a, b]cdef \mapsto 1$. In other words, these $a, b$ will be generators, but different ones.

I was able to make $a = CDEC^{-1}, b = CCDC^{-1}$ work:

$$CDEC^{-1}CCDC^{-1}(CDEC^{-1})^{-1}(CCDC^{-1})^{-1}CCDDEE(CDECDE)^{-1} = $$

$$ = CDEC^{-1}CCDC^{-1}CE^{-1}D^{-1}C^{-1}CD^{-1}C^{-1}C^{-1}CCDDEE(CDECDE)^{-1} = $$

$$ = CDEC^{-1}CCDE^{-1}D^{-1}D^{-1}DDEE(CDECDE)^{-1} = $$

$$ = CDECDE (CDECDE)^{-1} = 1 $$

I think this is more-or-less the explicit map you're asking for!

Finally, note that the map $ \left< C, D, E, F| CDEF = 1\right> \to \mathbb Z_2$ is given by counting all the letters modulo 2 (consistent because $F = 1 = 3 = (CDE)^{-1}$), so the image of the map discussed above should contain exactly expressions with even number of letters.

Answer by Ilya Nikokoshev for reference on examples of (g, K)-modules

2010-02-04T10:23:27Z

See also the question Unitary representations of $SL(2, \mathbb R)$.

Decomposition of k[G]

2009-10-30T21:55:07Z

There's a well-known decomposition of $L^2(G)$, a regular representation of compact complex group Lie $G$, called Peter-Weyl theorem.

Turns out for some reason I automatically think that there is a similar theorem that decomposes regular representation $k[G]$ of algebraic group $G$:

$$k[G] = \bigoplus_R \ R^* \otimes R$$

where sum goes over representations to $GL(n, k)$. For this to work I think we need $G$ to be a linear reductive group over, say, algebraically closed field $k$ of characteristic 0. Also, perhaps we need $\pi_1(G) = 1$.

But perhaps this is not true — the search hasn't given me a reference yet, but I wasn't able to provide a counterexample either.

Consider, for example, the multiplicative group $\mathbb G_m$. Then $k[\mathbb G_m] = k[x, x^{-1}]$ where each summand $k\cdot x^n$ is a separate representation of $\mathbb G_m$ into $\mathbb G_m = GL(1, k)$, specifically the one given by $a \mapsto a^n$. So the identity works.

So, is there such a theorem? What's a reference or a counterexample?

Answer by Ilya Nikokoshev for Is a free alternative to MathSciNet possible?

2010-02-03T22:19:34Z

I could be mistaken, but the way I understand it many people on MO are voting members of AMS, and combined together could ask it to change MathSciNet's current policy.

However, as others pointed out, MathSciNet works as a funding channel for AMS which would have to be either replaced by some other method of funding or leave AMS with less funds overall. A responsible proposal for making MathSciNet free would also somehow address the above alternative.

Answer by Ilya Nikokoshev for Degree 2 branched map from the torus to the sphere

2010-02-03T20:15:20Z

You can do a reverse construction: start with a sphere without 4 points; now add two points over each one in such a way that every time you go around one hole the two points get interchanged.

The same Riemann-Hurwitz calculation guarantees that you get a torus. If you have complex structure on you sphere without 4 points you get one on top as well; a beautiful fact is that you get all complex structures on a torus — in other words, all elliptic curves over $\mathbb C$ — that way.

Killing the torsion in homotopy

2010-01-05T02:38:34Z

Origin

This question was asked by John Baez in This Week's Finds in Mathematical Physics (Week 286). Therefore, please don't upvote this question (unless you really want to), but do upvote the answers.

Background/Motivation

For a CW complex (here for simplicity we'll have $\pi_1 = 0$), you can do the operation of "rationalizing", which will change its homotopy $\pi_n \to \pi_n \otimes \mathbb Q$. This works by attaching enough cylinders so that each original cell is killed, but its subdivisions are born instead.

Question

Does there exist a similar procedure of "killing the torsion" which would change the homotopy of 1-connected CW complex from $\pi_n$ to $\pi_n/\pi_n^{tors}$?

Thoughts

One encounters problems if one just tries to kill off the cell: the procedure might have changed higher homology (this doesn't happen in rationalizing since cylinders are simple). So I suspect the answer is "No", but how to construct a counterexample?

Answer by Ilya Nikokoshev for Switching Research Fields

2010-01-22T19:56:09Z

Quantum computations: there are some interesting books; when I was an undergrad I was quite happy about Classical and Quantum Computation by three wonderful mathematicians with big teaching talent.

You'll also be much more knowledgeable in that area if you learn basic quantum mechanics and statistical mechanics; all of this probably is a topic big enough for a separate question.

Generally, though, if you are interested in any field on a research level you should probably take a look at what people are doing: in many cities you can find a good seminar on at least one of those topics nearby. And (to me) looking at what kinds of questions people solve on arXiv or MathOverflow seems like a good way to "see where the wind is blowing".

Homotopy theory of schemes examples

2009-10-25T21:42:42Z

Is it possible give an example of (or explain) how the Voevodsky et al.'s homotopy theory of schemes computes higher Chow groups?

Beilinson conjectures

2009-10-26T22:50:11Z

Continuing an amazingly interesting chain of answers about motivic cohomology, I thought I should learn about the Beilinson conjectures, referred there.

I have found some references, and they seem to present the conjectures from different sides, e.g. there's the statement about vanishing but then there are also connections to motivic polylogarithms.

What I miss from these articles in a general picture that would allow us to start somewhere natural. So,

how would you describe an introduction into Beilinson conjectures in motivic homotopy?

Sorry for such a loaded question — I really don't know how to make it fit MathOverflow format better. One could theoreticlly post lost of specific questions on the topic, but to ask the right questions in this case you might need to know more than I do. Also, I know there are some technical developments, e.g. the language of derived stacks, and my hope would be that somebody could make a connection to these conjectures using some clear and suitable language.

Answer by Ilya Nikokoshev for Does the exponential function have a square root?

2010-01-17T08:28:17Z

Re: second question.

Yes, there are many of them on a half-line. Just take any smooth increasing function on $[0,1]$ with the property that $E(1)$ and $E'1(1)$ have desired values and extend it by the rule above. Since $E(x) > x$ the function will be increasing, thus invertible.

No, there are none on the whole line. For any number $x$, the sequence $\mathop{\text{ln}} \mathop{\text{ln}} \dots \mathop{\text{ln}} x$ cannot be continued indefinitely; at some point you encounter negative numbers.

Properties of monodromy of a fibration?

2009-10-22T18:38:57Z

Sorry for a loaded question.

I'm not an expert on those things, but I do know that a fibration gives rise to the representations of pointed fundamental group of the base on the cohomology of the fiber.

What are the properties of this map for different classes of fibrations? I think it's known what the image of this map can be. And the local properties are governed, at least in the complex case, by what type the manifold is.

And, most importantly, there is something about uppertriangularity. What exactly is that?

Homology class orthogonal to image of Chern characters?

2010-01-03T22:54:04Z

I had this simple question when formulating the Todd class question.

Does there exist an example of proper morphism $f:X\to Y$ together with nontrivial homology class $t\in H^*(X)$ such that for all coherent sheaves on $X$ the equality $f_*(\mathop{\text{ch}}(u)\cdot t) = 0$ holds?

"Every scheme as a sheaf" references?

2009-12-31T19:39:08Z

I have sometimes hard time reading papers that are written in the language of schemes being replaced by the functors they represent (I have especially homotopy scheme theory in mind).

I think the topic is connected to topoi and Grothendieck topologies, but for now I'm looking for something simple, just the working overview of the language of representable functors, what is a scheme, etc.

Why do Todd classes appear in Grothendieck-Riemann-Roch formula?

2010-01-03T22:46:49Z

Suppose for some reason one would be expecting a formula of the kind

$$\mathop{\text{ch}}(f_!\mathcal F)\ =\ f_*(\mathop{\text{ch}}(\mathcal F)\cdot t_f)$$

valid in $H^*(Y)$ where

$f:X\to Y$ is a proper morphism with $X$ and $Y$ smooth and quasiprojective,
$\mathcal F\in D^b(X)$ is a bounded complex of coherent sheaves on $X$,
$f_!: D^b(X)\to D^b(Y)$ is the derived pushforward,
$\text{ch}:D^b(-)\to H^*(-)$ denotes the Chern character,
and $t_f$ is some cohomology class that depends only on $f$ but not $\mathcal F$.

According to the Grothendieck–Hirzebruch–Riemann–Roch theorem (did I get it right?) this formula is true with $t_f$ being the relative Todd class of $f$, defined as the Todd class of relative tangent bundle $T_f$.

So, let's play at "guessing" the $t_f$ pretending we didn't know GHRR ($t_f$ are not uniquely defined, so add conditions on $t_f$ in necessary).

Question. Expecting the formula of the above kind, how to find out that $t_f = \text{td}\, T_f$ ?

You don't have to show this choice works (that is, prove GHRR), but you have to show no other choice works. Also, let's not use Hirzebruch–Riemann–Roch: I'm curious exactly how and where Todd classes will appear.

Answer by Ilya Nikokoshev for Polynomial representing all nonnegative integers

2010-01-10T19:40:16Z

The search turned up a 1981 paper by John S.Lew (in the Unsolved problems section)

Polynomials in Two Variables Taking Distinct Integer Values at Lattice-Points

which discusses related problems, and ends up stating this one. The author's problems are:

Problem A. Classify bijections $\mathbb N\times\mathbb N \to \mathbb N$.
Problem B. Classify bijections $\mathbb Z\times\mathbb Z \to \mathbb Z$.
Problem C. Classify surjections $\mathbb Z\times\mathbb Z \to \mathbb N$.

His main conjecture is that the only solutions to A are Cantor's $x+ \frac12(x+y-1)(x+y-2)$, which apparently goes to the time of Polya. Lew states C independently from empirical observations.

Answer by Ilya Nikokoshev for Why no abelian varieties over Z?

2010-01-06T00:52:29Z

Comments by Anweshi

The essential point is what Emerton mentioned, ie the analogy with Minkowski's theorem on number fields with ramification. The basic principle is that "arithmetic is geometry". Number rings are in a sense zero dimensional objects, elliptic curves one dimensional objects and abelian varieties correspond to higher dimensions. So we have Minkowski's theorem. And we ask, can we extend it to higher dimensions? Tate, after setting up the theory correctly as in his famous survey article on the arithmetic of elliptic curves, proved it rather trivially for elliptic curves(as Emerton mentions). Now the task is for abelian varieties.

Fontaine comes along, and proves that it is indeed the case. But the proof turns out to be much more complicated than expected. He built a whole lot of "Fontaine theory" around this. It goes into $p$-adic Hodge theory, $p$-adic Galois representations etc. He worked on it for some 15 years in isolation, it is said. The first major success of his theory was this theorem, and later it gained popularity. Now it is a major stream of research in arithmetic geometry.

References:

Neukirch, Algebraic number theory, for the general philosophy that "arithmetic is geometry".
Notes of Robert Coleman's course on Fontaine's theory of the mysterious functor
The Bourbaki expose of Bearnadette Perrin-Riou. Fonctions L p-adiques des représentations p-adiques, Astérisque 229, (1995).
Tate, The Arithmetic of Elliptic Curves, Survey Article, Inventiones.

It could be also worthwhile to have a look at the articles on finite flat group schemes in the volume Arithmetic Geometry of Cornell and Silverman, and in the volume Modular forms and Fermat's Last Theorem by Cornell, Silverman and Stevens. This is all intimately connected with them, as Emerton mentions. In fact, you can find a particular viewpoint by Fontaine on Finite Flat group Schemes.

There could be also be a simpler motivic explanation of this, without getting into the intricacies of Fontaine theory. The reason I think so is the following. I have heard the answer that there is no elliptic curve over $F_1$ because from the zeta functions the motives turn out to be mixed Tate. But, on the other hand, my own "proof" of this fact was that if there were an elliptic curve or abelian variety over $F_1$, it would be extensible to $Spec\ Z$ and there by Fontaine's theorem the only abelian scheme is the trivial one. Ever since I have wondered, whether it is possible to substitute Fontaine's theory arguments with motivic ones.

Emerton clarified to me in this connection: From a number theorist's point of view, p-adic Hodge theory is one of the key ingredients in the theory of motives, so these arguments are motivic, in a certain sense. (Perhaps one can say that p-adic Hodge theory encodes arithmetic properties of motives in a way analogously to the way that Hodge theory encodes geometric and analytic properties.)

Thus, by Emerton's answer, Fontaine theory seems to be thus a deeper part of motives. However, this "no abelian variety over Z" theorem of Fontaine was the first major application of Fontaine's theory. I imagined, if any results of Fontaine's theory were to be replaced by usual motivic arguments, then this ought to be the first candidate.

Before stopping, I must mention the intimate connection all this has with Iwasawa theory. Fontaine's theory is very much tangled with it, as could be seen in the expose of Perrin-Riou. However the more knowledgeable people should clarify on this.

This might be an apt place to mention the conference in honor of Fontaine. He is about to retire, after his great achievements.

Comment by Ilya

I think this should be indeed related to motives. (update: I think others provided some good references.)

Comments by Emerton

(1) There were earlier applications of Fontaine's results on finite flat group schemes; e.g. they played a role in Mazur's proof of boundedness of torsion of elliptic curves over $\mathbb Q$. I say this just to emphasize that Fontaine's theory didn't really develop in isolation. His theory is deep and technical, and it took people time to absorb it. But the theory of finite flat group schemes and $p$-divisible groups has a long history intertwined with arithmetic: there are results going back to Oda, Raynaud, and Tate; Fontaine generalized these; they were used by Mazur in his work, and by Faltings; Fontaine generalized further to $p$-adic Hodge theory (a theory whose existence was in part conjectured earlier by Grothendieck, motivated by, among other things, the work of Tate); ... . One shouldn't think of these ideas as being esoteric (despite the ``black magic'' label); they are and always have been at the forefront of the interaction between geometry and arithmetic, in one guise or another. (As another illustration, Fontaine's theory also closely ties in with earlier themes in the work of Dwork.)

(2) I'm not sure that there is any particular kind of usual motivic argument. The phrase motive conjures up a lot of different images in different peoples minds, but one way to think of what motivic means is that it is the study of geometry via structures on cohomology. From this point of view, $p$-adic Hodge theory is certainly a natural and important tool.

Here are some papers that give illustrations of $p$-adic Hodge theoretic reasonsing in what might be regarded as a motivic context:

Grothendieck, Un theoreme sur les homomorphismes de schemas abeliens, a wonderful paper. Although the results are essentially recovered and generalized by Delignes work in his Hodge II paper, it gives a fantastic illustration of how $p$-adic Hodge theoretic methods can be used to deduce geometric theorems.

Kisin and Wortmann, A note on Artin motives

Kisin and Lehrer, Eigenvalues of Frobenius and Hodge numbers

James Borger, Lambda-rings and the field with one element

These three are chosen to illustrate how $p$-adic Hodge theory arguments can be used to make geometric/motivic deductions. The paper of Borger is also an attempt in part to provide foundations for the theory of schemes over the field of one element, and illustrates how $p$-adic Hodge theory plays a serious role in their study.

Maulik, Poonen, Voisin, Neron-Severi groups under specialization, a terrific paper, which illustrates the possibility of using either $p$-adic Hodge-theoretic arguments or classical Hodge-theoretic arguments to make geometric deductions. (This is the same kind of complementarity as in Grothendieck's paper above compared to Deligne's Hodge II.)

Comments by Anweshi.

@Emerton, or anybody else: If there is something which does not make sense in my foray into "motivic" pictures, or something else which does not make sense, please feel free to erase and edit in whatever way you wish.

a further question by Thomas:

The great references given above let me ask about the current status of the many conjectures and open questions in Illusie's survey, e.g. finiteness theorems, crystalline coefficients, geometric semistability,... ?

ilya's comment: I think it would be very useful if somebody posted a question along the lines of what Thomas suggests, especially filling in some background from Illusie's paper (I would do it, but I don't have the paper itself).

** Anweshi's comment:** Fontaine's theory uses a great deal of crystalline cohomology. For instance please see Robert Coleman's notes referred above.

Nonnegative polynomial in two variables

2010-01-09T20:44:46Z

What can be said about the polynomials $f\in\mathbb Q[x, y]$ which are nonnegative on $\mathbb R\times \mathbb R$?

Motivation: this may lead to progress in the question about polynomial onto map $\mathbb Z\times \mathbb Z\to\mathbb N$, but I post it separately as it's interesting in itself.

Note: there are examples of polynomials nonnegative on $\mathbb Z\times \mathbb Z$, but not bounded from below on $\mathbb R\times \mathbb R$, e.g. $(x^2-x)y^2$, so this doesn't apply directly.

Homotopy groups of smooth manifolds?

2009-11-16T22:37:15Z

For a fixed $d$, is there a relationship between the homotopy groups of smooth $d$-manifolds?

The $d=1$ case is trivial, but I already don't know how to approach $d=3$ (I should have said that the case of $d=2$ is simple as well, since there is only a sphere to consider, but I don't know how to formulate the property of "having the same homotopy groups as $S^2$" in a simpler way).

Note about the discussion on the comments: it's unreasonable to expect an easy complete characterization of homotopy groups of $S^2$, even less for other manifolds. But I think one could try some partial relations. An interesting relationship would be: for some $d$, the groups $\pi_n$ can be determined from groups $\pi_m$ for $m<N<n$ (this is unlikely to be true though).

Answer by Ilya Nikokoshev for b^(n-1)=-1 mod n

2010-01-09T20:59:54Z

That would be equivalent to $2(n-1) = k\varphi(n)$ and $n-1\ne k'\varphi(n)$ by Fermat's little theorem for composite numbers.

The second condition is equivalent to being able to satisfy first with $k$ odd, so we could try $k = 3$. Thus we have $n = 3n' +1$ and $2n' = \varphi(3n' + 1)$. Now the trivial choice for $n' =1$ works!

Thus we find $n = 4$: $$(-1)^{(4-1)} = -1 (\mathop{\text{mod}} 4).$$

Answer by Ilya Nikokoshev for Why is an elliptic curve a group?

2010-01-08T22:27:25Z

Short answer: because it's a complex torus. Explanation below would take as through many topics.

Topological covers

The curve should be considered over complex numbers, where it can be seen as a Riemann surface, therefore a two-dimensional oriented closed variety. How to find out whether this particular one is a sphere, torus or something else? Just consider a two-fold covering onto $x$-axis and count the Euler characteristics as $-2 \cdot 2 + 4 = 0$ (don't forget the point at infinity.)

Complex tori

So this is a torus; now a torus with complex structure can be always defined as a quotient $\mathbb C/\Lambda$, where $\Lambda$ is the lattice of periods. It can be written as integrals $\int_\gamma \omega$ of any differential form $\omega$ over all elements $\gamma \in \pi_1$. The choice of differential form is unique up to $\lambda \in \mathbb C$.

Algebraic addition

A complex map of a torus into itself that leaves lattice $\Lambda$ fixed can be only given by a shift. Once you select a base point, these shifts are in one-to-one correspondence with points of $E$. We have unique distinguished point — infinity — so let's choose it as the base point. It follows that we now have an addition map $(u, v) \to u\oplus v$, though defined purely algebraically so far.

Geometric meaning

Now let's stop and ask ourselves: how to see this addition geometrically? For a start, consider map that sends $u$ to the third point of intersection with the line containing both $u$ and 0 (the infinity point). It's not hard to see that we fix 0 but change every class $\gamma$ in a fundamental group into $-\gamma$, so we must have the map $u\mapsto -u$ here.

Group theory laws

What would happen if you took a line through $u$ and $v$? By temporarily changing coordinates so that $u$ becomes the infinity point, one writes down that map as $(u, v) \mapsto -(u+v)$. Now if you took three points, there would be two different ways to add them; those would lead to $(u+v)+w$ and $u+(v+w)$ as complex numbers, which we know to be associative.

Logically proven

In the above, we worked over complex numbers, but we proved associativity which is a formal theorem about substitution of some rational expressions into others. Since it works over complex fields, it is required to work over all fields.

(In any case, the big discovery of mid-20th century was that you actually can take all of the intuition described above and apply it to the case of elliptic curves over arbitrary field)

Analytic computations (bonus)

Consider a line that passes through points $u$, $0$ and $-u$. This line is actually vertical, and $y$ is a well-defined function there which has two zeroes and one double pole at infinity. After a shift and multiplication of several such functions we'll be getting a meromorphic function on a complex torus with poles $p_i$ and zeroes $z_i$ having the property $\sum p_i = \sum z_i$. This method can give all such functions and only them; it's not hard to see that only meromorphic functions with this property are allowed on elliptic curve.

For example, $\wp'$-functions are the ones that have triple pole at 0 and single zeroes at points $\frac12w_1, \frac12w_2, \frac12(w_1+ w_2)$ where $w_1, w_2$ are generators of $\Lambda$.

Jacobian of a curve (bonus 2)

The formula above describes what types of functions are allowed on our curve. It is a good idea to organize this information into a curve: in this case, the information is that a single expression $p_1 + p_2 + \cdots + p_n - z_1 - \cdots - z_n$, considered a point of the curve, must vanish. For curves of higher genus, more relations are necessary; for $\mathbb C\mathbb P^1$, no relations beyond number of poles = number of zeroes are necessary. Those are relations in the group of classes of divisors (= Jacobian of a curve) mentioned in other answers.

In particular, elliptic curves coincide with their Jacobian and that's another explanation for the additive law.

Answer by Ilya Nikokoshev for What programming languages do mathematicians use?

2010-01-08T08:25:49Z

People in math seem to be pretty fond of Python (me included).

As an evidence, search on MathOverflow for posts where people mention the fact that they wrote a program, and it's nearly always either a special math framework (like Maple, Sage, Magma or other answers here) or Python.

And, by the way, Python is compiled to bytecode, which is run by VM. It's not much different from Java or precompiled JavaScript in that.

Comment by Ilya Nikokoshev

2010-03-02T10:04:37Z

+1. Quite an amazing paper!

Comment by Ilya Nikokoshev

2010-03-02T09:52:06Z

That looks like an interesting theorem! Could you provide a link for those new to it, please?

Comment by Ilya Nikokoshev

2010-02-10T18:12:25Z

Actually, I first write this as a statement, but then decided to change to question, since I didn't check it carefully. Will think a bit more and post!

Comment by Ilya Nikokoshev

2010-02-04T23:34:33Z

It's true, but the question was stated in geometric terms, so the answer should also include identification of letters with cycles. You can't have this for free: if your map is that simple, the expressions for cycles must be a bit more involved then mine :)

Comment by Ilya Nikokoshev

2010-02-04T19:34:10Z

@norondion, please don't take the closure personally. If the linked answer is not sufficient for some reason (e.g. it only deals with curves), you should just provide more background in the question body.

Comment by Ilya Nikokoshev

2010-02-04T19:30:25Z

I vote to close as duplicate. I also changed the title of the question linked to by David to make it reflect the question/answer more accurately.

Comment by Ilya Nikokoshev

2010-02-04T19:27:18Z

@Ben, I retitled your question to reflect more general second question; it includes the first one as well. Feel free to revert!

Comment by Ilya Nikokoshev

2010-02-04T13:05:45Z

They discuss the "plus/minus problem" as well: We prove polylogarithmic (C(log n)d) upper and lower bounds for the number of hypercubes needed to tile a hypercuboid in either of the above senses. In more than two dimensions this is a huge improvement on anything previously known. It was not even known whether an n × 1 × 1 cuboid could be tiled with fewer than n cubes in the plus/minus sense. (page 3)

Comment by Ilya Nikokoshev

2010-02-04T10:23:59Z

Thanks! This online reference is very useful.

Comment by Ilya Nikokoshev

2010-02-03T22:35:31Z

Your answer is very informative: I returned to change formatting and expand it a bit; feel free to revert!

Comment by Ilya Nikokoshev

2010-02-03T21:46:12Z

You're entitled to your opinion and generally have great control over the posts you make; sorry if my editing offended you. I'll move on. The site is a collaborative effort, however, so somebody else could still edit the post.

Comment by Ilya Nikokoshev

2010-01-22T20:01:35Z

No, only the poster or moderator can make CW.

Comment by Ilya Nikokoshev

2010-01-22T19:54:35Z

This should be a Community Wiki, see the FAQ.

Comment by Ilya Nikokoshev

2010-01-18T10:55:27Z

Your answers set a new high standard for MathOverflow, please keep going!

Comment by Ilya Nikokoshev

2010-01-17T09:08:29Z

This is a wonderful answer! I edited math a bit (there's more about typing math at mathoverflow.net/faq#latex), which could bring some mistakes, watch out :)