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The central question is :

Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us ?

Explanation:

"Suppose we have the solutions to a curve over local fields, I mean fix a curve, the degree of the curve is '$\rm{n}$' ( say ) , then let us name the curve as $C(X,Y,Z.....)$ .


NOTE: The degree of the curve and the Variables of the curve can be choosed by the person at the answering time, I have mentioned Degree as $n$ and curve in $m$ variables in order only to allow flexible manipulations, suppose the answer is known upto degree $2$, then answer it by taking degree as $2$, and also the number of variables is choosen by person, covering the maximum extent of work done up to the present day.

So let us call the set of points over local field as $C(\mathbb{F_P})$ for each prime $P$ , and let us call that set as local set. And now actual question is how can we find the points in the global set, I mean global set contains the solutions of curve over some number field $K$, for simplicity assume the degree of extension of $[K:\mathbb{Q}]$ is one, which means assume curve $C$ over $\mathbb{Q}$. Then is there any procedure to talk about global Set. As far as my knowledge is concerned, the only thing that talks about such entities is Hasse local-global principle . But the principle only talks about the existence of solutions to over $\mathbb{Q}$ if we have solutions over $\mathbb{F_P}$ and real numbers, but doesnt speak about the explicit generation of points.

Rather some traditional methods like descent, are not always successful in producing a generator.

But is there any morphism of the form $\rm{\phi}:C(\mathbb{F_P})\mapsto C(\mathbb{Q})$,or for the sake of brevity call the set $L(C)$ to be defined as $L(C)=\{C(\mathbb{F_P}), \forall P\}$, so is there any morphism $\rm{\phi}:L(C)\mapsto C(\mathbb{Q})$,such that given an input local part to that machine or morphism, it must produce global part ? , I mean effectively give the points on curve .

To be Blunt " Are there any methods other than Descent that give global points when they are fed with local points ? " .


Note(2): Users might have noticed that I have taken much care in both in formatting while writing and punctuations and studying the concepts they have advised me, so is any user thinks to down-vote this question, is requested to suggest the reason, which will contribute to a fruitful change in my case, and I have been striving to improve my language,so thanks a lot,I am in debt with all who served my hunger of learning mathematics, and patiently bearing my foolishness.

Thanking you,

Yours Cordially,

Iyengar.

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15  
There is so much confusion in this question that I wouldn't know where to begin sorting it out (and I am certainly not able to give "a perfect answer", which is what is being requested here). But you might want to read about the Chabauty method. McCallum and Poonen have written a very nice introduction. –  Alex B. Oct 23 '11 at 8:09
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Concering punctuation and formatting, you should put a space after punctuation marks, not before them... –  Hans Lundmark Oct 23 '11 at 9:31
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I rarely comment on formal linguistic errors in posts, but since you mention that you try to take care with these things, here's some feedback: In English, only proper names and the first word of a sentence are capitalized, not ordinary nouns like "User" or words like "Requested". Sentences are separated by periods, not commas; commas only separate clauses within a sentence. Your "Note(2)" would usually be something like three sentences. As Hans remarked, punctuation is followed by a space. Also, I'd suggest using boldface more sparingly. –  joriki Oct 23 '11 at 9:48
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@iyengar: Your response, "How could you imagine a person to know everything without learning?" is both pointless, and insulting. I do not imagine such a thing, nor do I imply such a thing. I do not ask you to be an expert, I ask you to look around and see what the standards are and try to abide by them. You seem unwilling and unable to do that, and instead prefer to blame me. Well, you know what? I won't waste my time fixing your prose and your questions, or trying to answer them any more. That way, you won't feel chastised, and I won't have to stand here and be falsely accused by you. –  Arturo Magidin Oct 25 '11 at 3:18
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@iyengar: I downvoted because the question was poorly written, poorly thought-out, poorly organized, poorly phrased, and poorly punctuated, and because your attitude when this was pointed out was "Don't blame me, it's your fault for expecting too much of poor little me." Nothing to do with what soap you use, everything to do with your attitude and continued unwillingness to abide by the standards of the community. Just because you say "Please" doesn't turn your demands into requests, and doesn't excuse your insults either. –  Arturo Magidin Oct 25 '11 at 13:05
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1 Answer 1

up vote 4 down vote accepted
+100

For any concrete interpretation of the word curve, a curve should be uniquely determined by the collection of its set of points modulo $p$ for all large enough primes. e.g. for affine irreducible plane curves you can determine from the sets of all points on all the mod $p$ reductions of the curve, a unique irreducible polynomial equation $P(x,y)=0$ for the curve, up to a constant multiple. And the same for projective curves, or curves in n-dimensional affine or projective space.

Assuming that principle is correct, the question is the same as asking for a procedure to find all rational points on a given curve. This is an ancient and unsolved problem. For curves with no rational points there is a conjectured algorithm to prove that in each case:

http://arxiv.org/abs/math/0507329

I think that one can prove, using a version of the Lefshetz principle from algebraic geometry or an ultraproduct of the mod $p$ prime fields, that the statement "$P=0 \leftrightarrow Q=0$" is true (over the algebraic closure) for any polynomial $Q$ cutting out the same sets of mod $p$ points as $P$, which for irreducible polynomials means that $Q = P$ up to a constant.

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But Hasse-principle just talks about the Existence of points,but doesn't speak about constructing global solutions.So is there any way of knowing the Global points when local solutions are with us,But please answer in a language Which I can understand,I think you know my background and my struggle to learn things by myself,so I am not complete expert as other people,nor I don't have anyone to teach me,so I need to crack my brain to understand complicated terms,But I beg to Step-down your intensity and tell me in naive language.@zyx –  Iyengar Oct 28 '11 at 14:58
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The local-global principle is not really a way of finding solutions and local solution sets may not help. It is similar to the Sieve of Erastosthenes, where primes are found as the remnant after eliminating multiples of 2, of 3, then 5, etc. The solutions are what is left over after imposing an infinite sequence of local conditions. This is not a way to discover the solutions (in finite time) but can sometimes be used to prove that no solutions exist: if there is a congruence condition preventing solutions it can be found in finitely many steps by sequentially testing all possibilities. –  zyx Oct 28 '11 at 21:02
    
For elliptic curves satisfying additional conditions (complex multiplication, positive analytic rank, finite Sha or BSD, etc), there are constructions by Darmon of some of the rational points using Heegner points, and a construction by Rubin around 1991 of one rational point (of infinite order) from a p-adic point. Both professors have webpages with papers on these questions and generalizations. Neither type of construction is assembling local information at different primes to determine the set of global points. –  zyx Oct 30 '11 at 19:26
    
Rubin's paper is called "p-adic L-functions and rational points on elliptic curves", published 20 years ago. I don't know which specific work of Darmon you were told of, but on his web page there are quite a few expository papers and papers on "rational points" and if you look in the sections on Heegner points or Heegner cycles it should be there. I don't know how relevant it is to the question as it was posted, because neither author uses a local-to-global strategy of patching together solutions at different primes. –  zyx Nov 1 '11 at 2:20
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