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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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 $\geq 2$. |
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