can anyone tell me the reason behind the failure of hasse-local global principle for selmer curves, i think that adding some more will fix it,but is there any established reason behind the failure, thanking you , may god bless you
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17$\begingroup$ Let us keep gods out of this. $\endgroup$– Chandan Singh DalawatApr 17, 2011 at 3:20
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2$\begingroup$ I guess I don't really understand the question. KConrad's comment is the best I could have done by way of an answer, but that's the answer to why any genus one curve violates the Hasse principle (and also to why it doesn't!); it seems sort of tautological. And what does "some more will fix it" mean? $\endgroup$– Pete L. ClarkApr 17, 2011 at 8:08
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7$\begingroup$ Why the hostility Chandan? $\endgroup$– Steven GubkinApr 20, 2011 at 18:41
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5$\begingroup$ This site is for mathematical questions. We should be wary of religious or political controversies; they do not belong here. $\endgroup$– Chandan Singh DalawatApr 23, 2011 at 4:01
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10$\begingroup$ I agree - there would be no controversy if you didn't create one. $\endgroup$– Steven GubkinApr 24, 2011 at 21:51
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1 Answer
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I'm not sure exactly what you mean by Selmer curves, perhaps you mean those curves modeled by the diophantine equation originally studied by Selmer $$ax^3 + by^3 + cz^3 = 0.$$
Today, Selmer's examples are thought of in a more general context, relating to the study of the group of rational points on elliptic curves or more generally abelian varieties.
This survey article by Barry Mazur entitled "On the Passage from Local to Global in Number Theory" is perhaps a good place to start, it's one of the many places where the ideas your question seems to be reaching for are stated precisely.
In particular your hope that "adding some more will fix it" amounts to the statement of the Shafarevich-Tate conjecture.
It should be mentioned that curves like Selmer's are not rare. I'm not going to be as precise as I should, but Manjul Bhargava gave a talk this past Tuesday discussing how he and A. Shankar can show that if you pick a cubic plane curve "at random" it "should" have about a 64% chance to be everywhere locally soluble but not globally soluble.
Here "at random" has basically the interpretation discussed in this MO question, and "should" means under the hypothesis of the rank distribution conjecture, i.e. that 50% of all elliptic curves over (ordered by height) have rank 0, 50% have rank 1 and 0% have rank $\geq 2$.
answered Apr 16, 2011 at 17:44