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Say that a projective variety V over Q satisfies the local-global principle up to finite obstruction (#) if there are only finitely many isomorphism classes of projective varieties over Q that are not isomorphic to V over Q despite being isomorphic to V over every completion of Q..

In section 7 of Barry Mazur's 1993 article titled On the Passage From Local to Global in Number Theory, Mazur describes his attempt to prove that (#) for abelian varieties over Q implies (#) for all projective varieties over Q, and a partial result that he, Yevsey Nisnevich and Ofer Gabber achieved in this direction. Has there been further progress in this direction since 1993?

My understanding is that an effective version of (#) for genus 1 curves (an effective bound on certain Tate-Shafarevich groups) gives a finite algorithm (of a priori bounded running time) for determining whether a genus 1 curve has a rational point, and also that such an effective bound on Tate-Shafarevich groups is expected.

Is an effective version of (#) for general projective varieties over Q expected? If so, how does this relate to Hilbert's 10th problem over Q (which Bjorn Poonen has conjectured to be undecidable)?

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Automorphism schemes of projective varieties intervene in the method (prior to intervention of abelian varieties!), so unless one knows results on finite generation of their component groups (an open problem in most cases) there's no way one can make such an algorithm. Over global function fields the relatively new theory of pseudo-reductive groups has brought us to the same degree of progress as the number field case. As best I can tell, finiteness properties of component groups of Aut-schemes are intractable in general. – BCnrd Apr 25 '10 at 0:41
up vote 4 down vote accepted

"Has there been further progress in this area since 1993?"

So far as I know, there has been no direct progress. I feel semi-confident that I would know if there had been a big breakthrough: Mazur was my adviser, this is one of my favorite papers of his, and I still work in this field. Also, I just checked MathReviews and none of the citations to this paper makes a big advance on the problem, although two are somewhat relevant:

MR1905389 Thăńg, Nguyêñ Quoc On isomorphism classes of Zariski dense subgroups of semisimple algebraic groups with isomorphic $p$-adic closures. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 5, 60--62.

MR2376817 (2009f:14040) Borovoi, M.; Colliot-Thélène, J.-L.; Skorobogatov, A. N. The elementary obstruction and homogeneous spaces. Duke Math. J. 141 (2008), no. 2, 321--364.

I'm not sure what you mean by an effective bound on Shafarevich-Tate groups (henceforth "Sha"). It is certainly expected that the Sha of any abelian variety over a global field is finite. If this is true, then in any given case one can, "in principle", give an explicit upper bound on Sha by the method of n-descents for increasingly large n. (In practice, even for elliptic curves reasonable algorithms have been implemented only for small values of n.) I really can't imagine any algorithm having to do with Sha that has "a priori bounded running time". What do you have in mind here?

As to the final question, let me start by saying that it seems reasonable at least that the set of "companion varieties" (i.e., Q-isomorphism classes of varieties everywhere locally isomorphic to the given variety) of a projective variety V/Q is finite: as above, we believe this for abelian varieties, and Barry Mazur proved in this paper a lot of results in the direction that the conjecture for abelian varieties implies it for arbitrary varieties. (For instance, quoting from memory, I believe he proved the implication for all varieties of general type.)

Here is a key point: suppose you are given a variety V/Q and you are wondering whether it has rational points. If V is itself a torsor under an abelian variety (e.g. a genus one curve), then if you can compute Sha of the Albanese abelian variety of V, you can use this to determine whether or not V has a Q-rational point. In general, the connection between computation of sets of companion varieties of V and deciding whether V has a Q-rational point is less straightforward. If V is a curve, then there are theorem in the direction of the fact that finiteness of Sha(Jac(V)) implies that the Brauer-Manin obstruction is the only one to the existence of rational points on V. In particular, people who believe this (including Bjorn Poonen, I think), believe that there is an algorithm for deciding the existence of rational points on curves. But nowadays we know examples of varieties where the Brauer-Manin obstruction is not sufficient to explain failure of rational points.

So, in summary, it is a perfectly tenable position to believe that companion sets are always finite, even effectively computable, but still there is no algorithm to decide the existence of Q-points on an arbitrary variety.

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Thanks for your response. What I had in mind in referring to an effective bound on Sha and an a priori bound on running time is the strong from of the Birch and Swinnerton-Dyer conjecture - for example, I have the impression that in the analytic rank zero case it's possible to compute the size of Sha by computing the central critical value of the L-function attached to the elliptic curve to high precision (with running time bounded a priori) using the fact that elliptic curves are modular. But maybe I'm just confused. – Jonah Sinick Oct 29 '09 at 8:20
Ah, I see what you are saying. Yes, if the full BSD is given to you, you have a different way to compute the order of Sha. In the case where the analytic rank is zero, this seems to have a priori bounded running time. But if the analytic rank is greater than 1, how are you going to compute it (rigorously) without computing the Mordell-Weil rank instead? Anyway, I am far from an expert in the algorithmic aspects here. I hope someone else will weigh in. – Pete L. Clark Oct 29 '09 at 14:20
I don't have any idea of how one would treat the case with analytic rank > 1 - maybe it's not reasonable to expect an analogous approach in this case. Still, this different way of computing Sha is interesting even in the rank 0 and 1 cases in light of the fact that most elliptic curves over Q seem to have analytic rank 0 or 1. It would be interesting to identify a large and natural family of projective varieties over Q with the property that Hilbert's 10th problem is decidable for this class. (This would be analogous to Gromov's discovery of hyperbolic groups in relation to the word problem.) – Jonah Sinick Oct 29 '09 at 22:48

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