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As everyone know that Hasse principle (I am referring to Hasse Local-Global Principle) doesn't work for cubics, but today my question is concerned about:

Is there any method or any known theorem, that proves or fixes the Hasse principle in some sense, I mean is there any proven way of applying the Hasse principle to cubics, as we have the "Tate-Shafarevich group" with us, which measures the failure of the principle, can we look forward to fix it by utilizing the TS-group (Tate-Sha), to be precise:

Is there any way to fix the Hasse-principle for cubics, as we have the error term with us, (I mean the TS-group), and can we adjust it to apply it for cubics, and is there any work done in such way, and how can we comment the existence global solution for an cubic equation if the equation has local solutions with us?

Thanks a lot,

(Please, any downvote should be followed by a comment explaining it, so that I can correct the problem).

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You might be interested in the article by W. Aitken and F. Lemmermeyer, Counterexamples to the Hasse principle, in the current issue of the American Mathematical Monthly, August-September 2011. –  Gerry Myerson Sep 14 '11 at 22:47
@Iyengar: The rank of your favorite group appears only in the stronger form of BSD. You have so far failed to note that if the Mordell-Weil theorem were not true, then even the weaker form of BSD wouldn't make any sense. But for learning the proof of the Mordell-Weil theorem, it is essential to know some algebraic number theory. This was one of the many reasons I asked you earlier to learn that subject. Once again, please learn it earnestly; otherwise you will only keep running around in circles with cataract-like haziness. –  George Sep 16 '11 at 5:59

1 Answer 1

up vote 4 down vote accepted

The Hasse principle basically addresses the question of whether an algebraic variety defined over a global field $K$ has any $K$-rational points at all. For cubic equations(curves and varities), this was a topic of much research. The most commonly studied obstruction to Hasse principle is the Manin obstruction, or the Brauer-Manin group. See for example the last chapter of Y. I. Manin's book "Cubic forms : algebra, geometry and arithmetic", preview available for free on google books. Others too have taken up this line of thought and come up with various other results -- for instance, in some cases the Manin obstruction is insufficient to explain the nonexistence of rational points. This is an active topic of research. Refer to the works of David Harari, Bjorn Poonen, Alexei Skorobogatov, among others, for examples of results obtained in the last few decades.

It appears to me that you have a misunderstanding about definitions. The Tate-Shafarevich group is an object attached to an elliptic cuve $E$ over a number field $K$. It can be equivalently described as equivalence classes of all torsors of $E$ that have $K_v$-rational points for all places $v$ for $K$, but no $K$-rational point, modulo the equivalence relation of equivalence as $E$-torsors, the isomorphisms being defined over $K$. Here it so happens that the given elliptic curve is the Jacobian variety of the torsor, the latter being a genus $1$ curve over $K$ possibly without a $K$-rational point. The definition of an elliptic curve $E/K$ is that it is a genus $1$ curve $E$ defined over $K$ with a specified rational point(the origin), and therefore you do not need to consider the Hasse principle for the elliptic curve itself; it is a moot question; and your statement of "fixing" the Hasse principle for elliptic curves is meaningless.

Lastly, the phrase "error term" is used in analytic expression involving some asymptotic results or approximations, such as in the prime number theorem. In the algebraic context that phrase is strange. You should rather say "obstruction", "obstruction group", etc..

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