My family problems, the poverty in India ( I always used to think of helping people in atleast getting food, there are people who even eat their human wastes for hunger, see how worse it is ), and also the illiteracy, I wanted to change all of them, but couldn't change, so all those things get stacked up in my mind and that is why I sound arrogant. Its a result of my grief when someone tries to discourage and condemn me.

I can say only one thing which is sure " I (NOT ONLY ME ANYONE) CANT FIND A SINGLE PERSON WHO WILL STILL SAY THE SAME WORDS IF HE IS PRESENT IN MY SITUATION, I MEAN HE CANT APPLY ADVISES TO HIMSELF IF HE IS PRESENT IN MY SITUATIONS ".

I can surely say that I am much better compared to others who don't even know what a computer is, Its surely a great thing ( For me ) to atleast know all these things and to atleast understand and speak of such things, coming from my background. I know every mathematician present in this century and their works, Its surely impossible in major cases for a person like me with my background. I can say that anyone who is present in my situation can't at least imagine all these things,

I am far better as I atleast in knowing these things, so I some how get confidence, but its not my mind that is arrogant, but its my english language which appears to be arrogant, some times when language is not properly used it sounds harsh.

So Last but not least " People who have shoes never know the pain thorn gives when stepped on "," only people present in darkness know the value of light", but people present in light always never know how bad is darkness.

Its easy to condemn others but its difficult if you have same situation and apply the same words to yourself.

So I kneel down and apologize anyone who thinks that my comments are harsh, think them only as my grief that make my english turn harsh. I need to copy 100 pages from Neukirch book once I have done it I surely start reading it, I surely take the advice of reading from basics which I was doing and will be doing.

Thanks a lot, yours truly, Iyengar

]]>And regarding your words

"Ramanujan credits his family goddess Namagiri for many of his insights, and it seems that mathematics for trustgod also has a highly religious significance."

That happens surely sir, as many discoveries of number theory are just flashes in mind, like I want to say something , like Ramanujan the Eminent mathematician "Prof.Ramachandra kanakanahalli " was a great seminal mathematician and propounded outstanding results in "Analytical Number Theory", Ramanujanâ€™s taxi cab number 1729 = 9^3 +10^3 = 1^3 +12^3 has become a part of mathematics folklore. During his college days, Ramachandra had a similar counter with the number 3435. His college principal had a car with the number 3430 on the number plate. Ramachandra worked on the mathematical possibilities of this number and in the process he found that upon adding 5, the number 3435 is the only number with the unique property that when each digit was raised to a power equal to itself and the resulting number was added up,the sum equals the original number i.e. 3^3+4^4+3^3+5^5 =3435. Many inventions and discoveries in Number theory are just a mere flashes that occur in mind, even the work of persons I always refer to in my questions was discovered naively, "Birch and Swinnerton Dyer conjectures" were a result of naive computations and they were very lucky as it turned out to be a great thing gauging the cardinality of solutions. Many things in number theory conjectures etc. are a mere flashes from mind, one can't explain any reason why they occurred, they may occur to any one in any time. I can even add many beautiful patterns and formulations are merely a result of random computations that occurred when one is trying to compute something .

It shows the intervention of Supreme being always.

Regarding eduation system in India, I don't know about entire India, but I am sure that Indian education is worse compared to other education systems,there are many institutions that claim to serve students but cant do it practically, there are thousands of unrevealed geniuses in villages who are not recognized as they are no literate, but in my place its still worse, I used to Mug-up and rehearse mathematics upto tenth standard like if we have a problem like x^2+2=10; x^2=8; x=4. We used to just read it by heart multiple times like one does for literature, ( one can imagine students reading and rehearsing an english poem simultaneously like chorus "Humpty Dumpty sat on a wall...Humpty Dumpty sat on a wall.... humpty had a great fall" ) ,similarly we used to read ( x^2+2=10.....) and just remember things and vomit the same in exams, without knowing how they work, we used to just pass and go further.It was in my tenth standard when one day I thought why is 2*2=4, after thinking for sometime I understood the reason, then I realized that entire mathematics has the same sort of reasoning and logic, so I went to the top of building and started looking behind equations and analyzing the mathematics from (1-10th standards , I was lucky enough compared to people who are 80-yrs old and retired from a good job without knowing these things.

Later it was a sudden amusement for me, a sudden leap from basic to advanced level, Google and wikipedia are my Gurus all time, I slowly started learning each thing, at the beginning I used to learn Theory of relativity, Space time and gravity etc.., later on turned towards biology and later towards engines and mechanics, but atlast realized that the mathematics is the ultimate origin and attached myself to it . At the beginning I used to strike my head and it took me some weeks to compute calculations by hand, later I learnt that we can program a machine, so I learnt C,C++,Oops concepts and java, And tried to write some bits of codes to compute values of series, prime numbers etc..Later once I got an introduction to SAGE and other softwares there is not need to write my routines myself, ( even though my knowledge is useful to do private projects and earn a bit of money yearly ).

]]>If these impressions are correct, my advice to trustgod would be to stop daydreaming and get busy. The stuff about the Shafarevitch sha is, I think, precisely a daydream. The hard reality, as we all know, is that becoming a mathematician is a lot of slow, detailed, disciplined work, most of the time on more mundane things. Happily, the insights one accrues from such detailed concentration, even on lesser matters than B-SD, can be immensely pleasurable. "God is in the details." I think it's great that trustgod is acquiring these books (Neukirch, Hatcher, etc.) -- now I hope he will slowly and patiently read them, solve a lot of exercises, and patiently build his mathematics up brick by brick -- and not announce he will "master them in a year" (more daydreaming). There really is no other way.

]]>Anyway, since you seem to already have very strong views about mathematics, education, the world in general, which probably won't be shaken (see Angelo's reference to arrogance), I shall refrain from any further comments to you other than of strictly mathematical content.

]]>being able to prove theorems (new ones, but not necessarily big and hard conjectures) in modern arithmetic geometry would definitely show you were ready. And to get to that point, you need to be able to speak the language of that field. Just as learning a new language (Icelandic, for argument's sake) fluently takes years, so does mathematics. And unfortunately, Mathematics doesn't just come with new words and rules and grammar, there is very little to connect it with the language you already know, unlike the case of starting from Croatian (again, for argument's sake) and trying to learn Icelandic - there one already knows the *things* you want to talk about, and just need the vocabulary and grammar. Mathematics is very different. Understanding the declensions and conjugations of a family of Latin words in all their uses is easy compared to understanding the definition of, e.g. Sha. I will probably never in my life understand Sha, and the same for most mathematicians. This is not a thing to be ashamed of. Most mathematicians will never understand the proof of the Weil conjectures, even though they are arguably a high point of 20th century mathematics, and that is nothing to be ashamed of. Linus Torvalds once said

"A lot of big projects actually start out as something of an accident. Show me somebody who claims that he designed a big thing from scratch, and I'll show you a liar. Nothing starts out as a big project, and if it's complicated enough, nobody really is smart enough to know where it will end up."

The same is true for proving big conjectures. Even the proofs of the biggest conjectures started out doing something small. The proof of Fermat's last theorem started out as noticing 'coincidences' between modular forms and elliptic curves (ok, that is not entirely accurate, but approximately so). The key ingredient proof of Poincare's conjecture started out as trying to understand what metrics positive curvature 3-manifolds will admit, and the idea from that came from somewhere else.

If you want to do mathematics start small, prove to yourself that you can understand existing proofs, starting small (and I mean really understand); that you can do calculations; that you can anticipate proofs; that you can write your own proofs of known facts; that you can write new proofs of known facts. Then you will be at a point where you know how to prove new theorems. Unfortunately, the area you have chosen is very hard (don't let that dissuade you), and there is a lot to learn, because people have been studying solutions to equations since DIophantus, and Fermat and Gauss, and lots of people last century have worked very hard to get to where we are today.

And proofs of hard conjectures might well come from other areas of mathematics, or even consist of a new field of mathematics altogether! We don't know how this will happen. Just rehashing old idea will probably not result in a breakthrough, because it is likely that others have thought of that already.

This is all I have to say, and I don't promise to respond to replies.

]]>Actually there are many institutes in world that just announce and claim that they are helping the people by spreading education and satisfying the needs of students. But we can barely find such institutions really helping people, there are many people in the world who have strong aim and zeal to learn but no further opportunities. Moreover I must accept that quality of education and standards and technologies are very poor compared to other parts of the world. But thats my problem sir, no body can help me .

Once I get a job and this month and live near internet cafe and I study continuously from basics. I follow your advice sincerely sir. Thanks a lot for supporting me, but how can I prove you that I know basics ? , I just asked the quotient group to know the intuition sir. As asking without knowing is different than seeking a different perspective of notion. Its not that I don't know quotient group, but I wanted to know another perspective, as the books doesn't give good intuitions that people here provide, they just cover basic topics, and in the book I referred there was only one line about quotient group and directly stated its definition.

And my notion will be understood by the people who have never attended any classes and started reading things by own. You have gone to university and your lecturer might have told how does quotient group works, and it gets fixed in your mind ( I am not referring to you sir, but to any person who have attended university ) . Listening from a professor ( that too from a good professor ) is different as student never get any confusion then. But imagine you are learning things from yourself from calculus, textbooks cant serve the entire purpose. There are quite few books which gives the actual meaning and intuition and reason behind introducing such terms. But imagine you are learning things by yourself, you have read something but you think that you understood something but if you proceed to advanced things someday you get a small doubt about basic things, it doesn't mean that you don't know basic things but it means that the foundations were not proper. ( Foundation in the sense I can solve problems, solving problems is not an issue , but now-a-days people judge other people by the marks and grades which are not at all a gauge for talent) , solving a problem never requires an intuition. suppose if some one knows the basic algebra if I give some student a problem like this if I tell them find the values of 2x=4 , then students do as follows x=4/2 --> x=2. But that doesn't need any intuition, simply one needs to know the formula and multiplication table, Student can simply proceed and pass the test even if he don't know why is 2*2=4, students can simply ( remember )mug up the multiplication tables . After some days of thinking he understands that adding 2 twice gives 4 and multiplication is nothing but repetitive addition. I can challenge that I saw quite many people working with multiplications without knowing how it operates deep inside ( I dont know about other countries but it happens here, from childhood teachers teach us only important problems and just tell us to mug them up like we mug up social and other languages, I also used to do the same but later changed, our lecturers never used to tell such intuitions ). Thats what I refer to as " knowing things happening behind equations ", I challenge there are many people who just simply learn things without bothering what happens behind them. I was searching for such intuitions.

As I don't have a lecturer I must depend upon internet and hit my head to understand all these things.

You can ask me some basic questions to test me in case if you want to do. So that you can know whether I am eligible for listening and asking about B.S.D or not.

Thanks a Lot for responding sir.

]]>I have great sympathy for people who are enthusiastic about something, and despite great personal difficulties work at it. However, what everyone is telling you is absolutely correct: you have to get the basics before you can read those papers, and that takes many years. Unfortunately, among all human activities mathematics is one of those requiring the longest time to learn. The idea that you can "abstractly think directly and get some raw ideas and later map it to mathematical language" is just plain wrong.

Anyway, I don't think you can be convinced, and this whole debate is probably completely fruitless. I hope I am mistaken.

If I may add some criticism, the mixture of obsequiousness and arrogance that you display does not help in getting people to take you seriously. ]]>

I think I said this somewhere before, but I will repeat it: if you are serious about learning from other people, then a good step would be to stop assuming that people on MSE don't answer your questions because there is nobody who is knowledgeable enough to answer them. If after several days or weeks and after much prodding from your part on various forums (including this one, where it is off-topic) there is still no answer, then you should start looking for reasons in your question, and not in the ignorance of everyone else who reads these forums.

I don't agree with Andy Putman that the only sensible option is to give up. But I agree that there is no short cut (this seems to be something that you are very reluctant to accept), and that you need to learn lots of basics before trying to solve some of the deepest problems in modern mathematics (let alone purporting to have solved them, as you have done in the past). When somebody simultaneously asks about the definition of a quotient group and about finiteness of sha, then this is just absurd, and from an answerers point of view completely puzzling: I have no idea what kind of answer I am supposed to give to either of these questions.

The sorts of questions you should be asking (or searching for, since I am sure that they have been asked before) are about learning road maps that start from 0 and get to where you want to get to, about book recommendations, etc. And before you do that, you should brace yourself for spending a couple of years acquiring all this basic knowledge, practice in solving exercises and so on. That's what we have all done (with the help of university curricula). And this does NOT mean coming back two weeks later and saying "ok, I have got the basics now, now tell me about Tamagawa numbers". If you don't believe in this incremental method, then I suggest that you stop seeking help from people who do, since you will keep hearing the same things over and over again.

All this has been said to you many times before, so I am not sure whether I am wasting my time yet again. We will see.

]]>If some body have something to tell other than this please do comment.

As said by Morrison, I never post again in MO. As every mathematician didn't use MO in the history, there is a scope to do something without using MO

]]>I'm sure that all of us sympathize with your personal situation, but I don't think there is a real solution. The kinds of things that you are interested in are the kinds of things that are basically impossible to learn about outside of a university setting. Before you could begin to understand any of these topics, you would need the knowledge that would be gained by an undergraduate math education followed by the first couple of years of graduate school. There is no short-cut, and I think you are wasting your time (and ours) trying to find one. I hate to tell someone to give up, but that seems to be the only possible advice.

best wishes,

Andy Putman ]]>

So I don't find any other choice rather than posting it here .

But there are many other people ( I don't want to mention those people's name here who don't wanted to help me ) , I have stopped asking research level questions, see my last question, I have struggled my-self to get an answer to it by myself. And you can't do anything as its my fate, if I had an proper internet and an expose to all recent journals, then probably I wouldn't have to depend on these sites and asking everyone and bothering everyone.

There are people who know answers to my questions, like generating global solutions based upon local solutions which is here , but didn't gave answers due to my bad formatting, but I was in learning stage then ( also now , but improved ) , so I request for an advice from your side.

I can barely say some names of people who concretely work on these areas at Math.SE, like Prof.Emerton, Prof.Alex Bartel etc.

So if you could advice me anything I would be happy . What should I do in present situation, not only you sir any one in this forum give an advice to me.

@ Alex Bartel : Sir you have answered to my question sir, but now I changed all formatting and cropped the unwanted one, so could you post a beautiful answer to the above questions sir, even though the answer posted was satisfactory but it didn't met all requirements, so I wanted to see an answer to it from you sir .

]]>I refer you to my post on August 26, http://tea.mathoverflow.net/discussion/1080/2/questions-posted-by-trust-god/, in which I asked you not to use MathOverflow any further. It is indeed that case that the software suspension recently expired --- this was an oversight, and we did not intend this is be interpreted as revoking our explicit requests that you do not use the site. Your recent questions were promptly deleted by a moderator.

I'm very sorry that this is the current situation, but it is. Regardless of whether you find yourself technically allowed by the software to post, please make no further posts on MathOverflow.

yours sincerely,

Scott Morrison (for the MathOverflow moderators)

]]>The answer to the question I saw is: no, we can't say that sha(E/K) = class group of K, because it isn't.

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