User shanmukha_srinivasan - MathOverflow

Blueprint of L-functions and need for introducing them ( Hasse-Weil L-functions )

2012-11-06T18:17:38Z

Dear All,

This question may appear elementary to all the experts in number theory , but forgive me. I really wanted to know how did the $L$-functions came into existence, especially the Hasse-Weil L-functions . Do they have some specific meaning in their formulation or they are just framed heuristically to build some thing else , as scaffolding .

I do know that Zeta functions and L-functions of the curve act as spies in collecting the secret information about the local part of curves and embed that information inside them, but its really a great trouble in understanding the formulation. I referred to many books and they have started saying " Let $L(s,E)$ be the ...." in an assuming manner .

I just wanted to know , why should one consider $$\zeta_{C/\mathbb{F_q}}(u)=\exp \bigg(\ \sum_{n=1}^{\infty}\frac{ | C(\mathbb{F_{q^n}})|}{n} u^{n} \bigg)$$ where $C$ is a projective curve with non-negative genus over finite field $\ \mathbb{F_q}$. Here are my pointers :

I didn't understand about the reason behind introducing exponential function on the right side .
I understood that there is some measure of points taking a ratio of the cardinality ( on R.H.S ) of the solutions, but why is the ratio needed ? I got this doubt when I looked at some other heuristic consideration $\prod\frac{N_p}{p}$ ( Where $N_p$ is the cardinality of solution set at some prime $p$ ) , why is the need to take the ratio ? Isn't it not sufficient to look at just $N_p$ ? We get the cardinality directly, why should we find the ratio again by dividing it with $p$ ?

Similarly , why is the formulation of local part of $L$-series ( Hasse Weil L-function ) appear as $L_p(T)=1-a_pT+pT^2$ when the curve has good reduction at $p$ ( here $a_p=p+1-N_p$ and has some other formulation like $L_p(T) = 1-T$ and $1+T $ when the curve has split multiplicative and non-split multiplicative reductions at $p$ respectively , and $L_p(T)=1$ when the curve has additive reduction at $p$.

How was the quadratic equation on R.H.S ( i.e $1-a_pT+pT^2$ ) formulated ? Was it a scaffolding to get some heuristic output later , or it has a specific meaning derived from something, or what ? Same with $1-T$ and $1+T$ .

Please do explain me , I am sorry my learned friends, if I have wasted your time, but every book I referred starts with Let, and I thought that its just a setting . If you want me to suggest some book that does the same task of explaining what I asked, you are welcome to suggest me .

Cordially,

Shanmukha Srinivasan.

Why are Galois Representations so important in Number theory ?

2012-08-03T09:49:31Z

Dear everyone,

Motivation :

From the past few days, I have been reading about the Galois Representations . I was really amazed to see that every seminal idea in the theory of elliptic curves have Galois Representation as an ingredient !!

I would be very happy listening to :

What made Galois representations so famous ? ( especially in number theory ), I was wondering, may be Galois representations are having some special symmetries that can facilitate the problem solving more easily.
What are the special properties of Galois representations ?

Case Study :

To describe the application of Galois representation in a beautiful manner, I came accross a paper of Skinner and Urban , where they relate the ranks of Selmer Groups to the non-vanishing of $L$-functions. They use this Galois representations as a major ingredient, but due to extensive use of Algebraic Geometry , I was not able to understand the quintessence of the paper. It was so difficult to read. But on the other hand, I know how can one relate the volumes of lattices ( groups ) to the $L$-functions, using Siegel's formula. But I didn't come across any such track in that paper ( The word Siegel is not found in that paper ) . May be they have used some other different approach. I would be very happy in listening to that , as an application of Galois Representation.

Any other good applications of Galois Representations are welcomed with high appreciation.

My Background :

I know number theory ( Mass formula and other things ) and rudimentary theory of Elliptic curves.

Epilogue :

I thank everyone for sparing your time in answering / reading my questions and other questions at MO, in-spite of your hectic schedule.

-Shanmukha.

Equivalence between statements of Hodge conjecture

2012-07-28T07:07:55Z

Dear everyone,

I was unable to obtain the equivalence between the two statements of the Hodge conjecture. I searched for some previous questions that others asked here, to check whether someone has asked the same thing previously. But I didn't find any such instance, that is why I am asking. We know that Hodge conjecture gives some relation between the topological cycles and algebraic cycles. But I have read two different variations of the same conjecuture. I number my pointers.

A fantastic description given by Prof.Dan Freed (here), which an undergraduate student can also understand.
A bit tough description given by Prof.Pierre Deligne (here), with lot of technical terms and constructions.

So I was befuddled in asking myself that how can one obtain equivalence between the both statements.

Dan Freed's Version :

He considers a Topological cycle ( boundary less chains that are free to deform ) on a projective manifold. Then he says that the topological cycle is homologous to a rational combination of algebraic cycles, if and only if the topological cycle has a rotation number Zero.

P.Deligne's Version :

On a projective non-singular algebraic variety over $\mathbb{C}$ , and Hodge Class is a rational combination of classes $\rm{Cl(Z)}$ of algebraic cycles.

So now I have the following queries for my learned friends.

How can one explain that both the statements are equivalent to each other ? One speaks about the rotation number and another doesn't even speak about it. How can one say that both the statements are valid ? I infact know that both the statements are valid ( as both the speakers are seminal mathematicians ) But how ?
So can anyone explain me what the Rotation number has to do with the Hodge Conjecture ? I obtained some information about the rotation number from Wiki. But I am afraid , to decide whether Freed is speaking about the same rotation number ( given in wiki ) in his talk ? or something different ?

I would be really honored to hear answers for both of them . Thank you one and all for sparing your time reading my question.

Urge/reason for inventing interior product ( Grassmann algebra )

2012-07-23T08:42:41Z

Hello everyone,

I wanted to lecture on Grassmann and his works , and I have been reading the collected works of Grassmann " Die Lineale Ausdehnungslehre ". There Grassmann introduced something called " Interior product " ( Left and Right interior products ) . So I was completely stuck up there, the Bourbaki papers don't speak on the reason or urge in creating such manipulations, but its understood by someone who really mastered them.

Can anyone suggest me a good definition of the left and right interior products and explain the purpose of introducing them along with the intuition ?

I would be really honored to hear that.

Thank you.

Interplay between Riemann and Swinnerton-Dyer

2012-07-13T09:04:37Z

Hello everyone,

After reading RH ( Riemann's Hypothesis ) and Swinnerton-Dyer conjecture, I asked myself why can't RH hold for $L$-Functions ( Hasse-Weil L-function ). In particular the GRH imposes the location of finding zeroes of a $L$-function with $\chi$ character.

So in particular Swinnerton-Dyer's conjecture states that

The underlying Mordell-Weil group of an elliptic curve has an infinite cardinality if there is a zero at $s=1$. i.e. $$E(\mathbb{Q})=\infty \iff L(E,1)=0.$$ $$E(\mathbb{Q})<\infty \iff L(E,1)\neq 1.$$ So can we predict that Hasse-Weil L-function of an elliptic-cruve satisfies the GRH. After normalizing can we put it this way :

There are infinitely many zeroes in the critical strip of Hasse-Weil L-function $L(E,s). $ i.e. Let $\mathfrak{K} $ be the number of zeroes of the Hasse-Weil L-function. Then $\mathfrak{K}=\infty \iff s=1+it$ ? . ( Assume that $E$ has infinitely many points, other wise $L(E,s)\neq0$.

I have some more set of questions concerning the significance of zeroes . They can be stated as

Are there any zeroes existing in the critical strip of $L(E,1+it)$ ? .
We know that $\rm{Rank(E(\mathbb{Q}))}= \rm{ord}_{s=1} L(E,s).$ So what about the significance of order of vanishing for other zeroes which are located at $s=1+it$ . Do they have some interesting relation with the properties of elliptic curves ? .

Are there any interesting results that are published in this direction so far ?

Thank you.

Heuristics for the Hodge Conjecture

2012-07-02T08:58:56Z

W. V. D. Hodge is famous for his Hodge conjecture, one of the Millennium prize problems. Hodge might have had some rough heuristics or ideas that led him to the formulation of the conjecture.

I am looking for the history and background behind the formulation of Hodge Conjecture. How did Hodge arrive at his conjecture?

Hodge Conjecture ( What I understood after reading Dan Freed's article ) :

On a complex projective manifold $\mathbb{X}$, a topological cycle $\mathfrak{C}$ ( on $\mathbb{X}$ ) is homologous to a rational combination of algebraic cycles iff $\mathfrak{C}$ has rotation number $0$. ( Rotation number $e^{i\theta}$ on differential form, volumes are invariant under rotation).

Hodge Conjecture (Deligne's description):

On a projective non-singular algebraic variety over $\mathbb{C}$, any Hodge class is a rational combination of classes $\rm{cl(\mathbb{Z})}$ of algebraic cycles.

The things that interest me:

How are Freed's version and Deligne's version versions equivalent?
How did Hodge arrive at that conclusion? Were there heuristic reasons or intuitive arguments that gives him some hope for a conjecture in that direction? .
How can one state an analogue of the Hodge conjecture in number theory? Are there any attempts to formulate an analogue in that case?

I am curious to hear answers, even if highly technical in nature.

Effective way of finding generators on the curve and the rank conjecture

2012-06-27T17:49:11Z

Hello everyone,

I have never heard of a polynomial time running algorithm that finds the generators of elliptic curves efficiently. I do know that Nagell-Lutz theorem is useful in computing the torsion part in $$E(\mathbb{Q})=\mathbb{Z}^{\phi} \oplus E_{\rm{Torsion }}(Q).$$ So what about the count of $\phi$ and the effective generation of points.

I do know 2-descent , that doesn't not work perfectly always. Regarding the 3-descent , only some curves having CM are said to pass through it. But are there any latest advancements in the area of computing the generators of the curve, if so please give some references.

I also read the rank conjecture, for expressing the rank of the underlying abelian group $E(\mathbb{Q})$ in terms of the order of vanishing of the taylor expansion of associated $L$-function. So I again got stuck.

Why do one go for a rank, if one has a point that has infinite order ? .

Rank 1-curves usually mean that they have a point of infinite order on them that can be used to generate all other points by successive chord and tangent methods.
Rank 2- Curves have 2 such points of infinite order that can be used to generate all other points.

So my question is why should one bother about $n$ points ( of infinite order ) if we have one point of infinite order. To express an analogous statement, suppose think that you are given a secret map that will lead you to some treasure. After the hard journey, you at-last said " Eureka" and you have found a " Machine X" . Its written on the Machine X, that it will produce dollar notes, as many as you want. So its limit is infinity. You can extract infinite number of dollar notes from the machine. You took that machine and packed it and again saw the secret-map. There are in fact some other markings on the map that will lead you to a place that contains the same Machine X.

Do you again go to those places searching for another Machine-X if you have a Machine-X that will produce infinite amount of money. I think the analogy is clear. So if we already have a point that produces infinite points why to again bother about other points that give rise to infinite points.

So if I am right, does the task of finding $\phi$ number of generators reduce to the task of finding one generator ? . That will make the above equation look like this $$E(\mathbb{Q})=\mathbb{Z} \oplus E_{\rm{Torsion }}(Q).$$

( Which is nothing but the Rank-1 situation ).

Thank you.

Why should I believe in the Siegel's and Hasse's rationale ?

2012-06-26T09:36:30Z

Hello everyone,

I was deeply attracted by the Hasse and Siegel's theorems while studying $p$-adic analysis. While reading a paper B.J. Birch and H.P.F. Swinnerton-Dyer - Notes on elliptic curves. I, Journ. reine u. angewandte Math. 212 (1963), 7-25 the authors emphasize that their central conjecture related to some cubic curves ( more popularly called as elliptic curves ) is based upon the work of Siegel and Tamagawa.

In that paper Peter points out that C.L. Siegel has done some work showing that "densities of rational points on a quadric surface can be expressed in terms of the densities of $p$ -adic points ". I was totally surprized that how can one get an intuition in what the author is pointing to.

More-over the Hasse-principle also tells the same thing. Existence of global solutions can be decided by looking at the local solutions. I can't understand the whole point. Suppose let us take a polynomial $f(x,y)=0$.

So my question is how can you estimate and find the solution set $(x,y)$ to $f$ seeing the solution set $(x^{\prime},y^{\ \prime})$ of $f(x,y)=0 \mod p$ . So I am sure that $(x,y) \neq (x^{\prime},y^{\ \prime})$ in all cases. So are we merging all the obtained local $(x^{\prime},y^{\ \prime})$ solutions and thereby constructing $(x,y)$ ?.

Thank you.

Comment by Shanmukha_Srinivasan

2012-12-21T11:30:32Z

@AndyPutman, @AndresCaicedo, @MichaelGreinecker, @QiaochuYuan, @ChandanSinghDalawat : Well, why didn't you leave a single comment ? as an explanation for the reason behind closing this question ? Well, I have posted the same, above, and I don't really know why can't you understand a formal protocol, even after you read the above comment. Please do post the comment, for calling it as 'off-topic'. Why are some people present here , behave in such an arrogant manner ?

Comment by Shanmukha_Srinivasan

2012-12-20T17:20:55Z

Now , can someone say the reason behind closing this question ? Well, Mark Sapir, Steven Landsburg, Franz Lemmermeyer, Alexandre Eremenko, Dmitri Pavlov, well when you are closing some question, you must leave some comment or notice. I think that is a common protocol. Hope you know it !

Comment by Shanmukha_Srinivasan

2012-12-20T17:15:59Z

@Quid : Thank you, I made it.

Comment by Shanmukha_Srinivasan

2012-12-20T17:02:22Z

@quid : I opened it just now. I didn't find any community wiki check box, well does it have something to do with reputation or some special privilege ? BTW, why is this one getting down votes ? Any particular reason ?

Comment by Shanmukha_Srinivasan

2012-12-20T16:37:24Z

Well, can I know why this question got closed as not real without any intimation ? Anyway I have opened a [meta thread](meta.mathoverflow.net/discussion/1497/…) here. BTW, Thank you @Qfqfq. @HW : how can one make this one as Community-Wiki ? I am new , please do explain .

Comment by Shanmukha_Srinivasan

2012-12-09T18:39:32Z

I surely suspect that the '4' down-votes noticed on the question is purely due to vendetta . What a pity that people are separated by narrow things , I have nothing more to say !

Comment by Shanmukha_Srinivasan

2012-12-09T17:05:25Z

@Angelo : Well, you can read that I counted you among the persons who don't derive pleasure in down-voting. Please re-read my comments, especially this line " I hope you are not one among them....." , well any how I am shifting to MO[meta] for further discussion ! Mazel Tov !

Comment by Shanmukha_Srinivasan

2012-12-09T05:29:05Z

@ everyone : Is there any chance to re-open this question ?

Comment by Shanmukha_Srinivasan

2012-12-09T05:28:42Z

@Angelo : Well, I have flagged your comment as offensive , I didn't like your tone of expressing your views on OP. I am sure that , OP knows about the books you have suggested and he knows the order of reading . Of course, I hope that you wont be too offended, if I point out sincerely that, isn't this question better than the other questions ? which are ask for some "date" , and inquiring, when someone wrote something etc .

Comment by Shanmukha_Srinivasan

2012-12-09T05:10:31Z

(Cont'd) Of course, @Charles Matthews, is a senior mathematician , and has been assisting Wiki projects , and I am sure that he gauges the meaningfulness of a question before posting and then post it ! If this is not a real question, I am sure that he himself wouldn't have posted it, and nevertheless he wont get '6' up-votes . I don't agree completely in closing this question, that too with a tag 'not-real'. Of-course it makes much sense, but one has to re-read it one more time, and I am sure that he understands the nub of the question !

Comment by Shanmukha_Srinivasan

2012-12-09T05:04:41Z

@Angelo : Well, there are people whom , I noticed , who get more pleasure in closing down or voting down some question, rather than trying to re-construct the question ( or re-interpret ) in a meaningful manner or helping the OP, to the extent they can ! I hope you are not one among them. I saw the question of @Charles Matthews, makes more sense, and he wanted to know whether there is some room for improvement in the EGA/SGA material, and the question is well posed , but we have to step into his shoes and think , then we understand the dire need to formulate such questions ! (Cont'd)

Comment by Shanmukha_Srinivasan

2012-12-03T16:41:31Z

@Mozibur Ullah : There is nothing impolite in asking about the down-votes, and your silence leads to more of such down-votes, as its said, a stitch in time , saves nine ! so its important that you ask , accept suggestions and re-frame your question accordingly as per the environment, I saw many worthy questions down-voted , for small reasons, and hence I asked it . Anyway I totally agree with @Tom Leinster, and I am just repeating what he said

Comment by Shanmukha_Srinivasan

2012-12-03T11:33:51Z

Well, why is this question down-voted ? Is it due to improper question framing ?

Comment by Shanmukha_Srinivasan

2012-11-29T19:32:27Z

Dear agustin, Welcome to MO. This is a research level question and answer website, and its not intended to be used for solving home work problems. Asking home work problems here is not a crime, don't worry with down votes, but it don't suits the atmosphere , and more over , one must be roman when he is Rome . There are many other websites that are meant for home work questions , for example : [Math.SE](math.stackexchange.com) and also read the [How to Ask](mathoverflow.net/howtoask) section in MO. Mazel Tov !

Comment by Shanmukha_Srinivasan

2012-11-29T19:11:01Z

But doesn't MO prevent the moniker collisions by stating " User name already in use " when an user is trying to register with same name which is already existing ? @Dmitri