What are nice applications of Tate-Poitou duality?
6 Answers
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One of the main themes of current number theory research, instigated by the work of Wiles and Taylor--Wiles on the Shimura--Taniyama modularity conjecture for elliptic curves, is the study of modularity (or more generally automorphy) of representations of Galois groups. The basic tool for proving modularity is the so-called Taylor--Wiles method, which involves introducing certain well-chosen auxiliary primes to aid in analyzing a generalized Selmer group (a certain Galois cohomology group) which controls the infinitesimal structure of the space of Galois representations.
In applying this method, one relies on a certain formula (due to Greenberg and Wiles) which expresses the ratio of the orders of a Selmer group and the corresponding dual Selmer group in terms of a product of local terms, and the proof of this formula is an application of the Poitou--Tate exact sequence. See Washington's article in "Modular forms and Fermat's Last Theorem" for an exposition of the Greenberg--Wiles formula.
There are lots of similar applications of Poitou--Tate in Iwasawa theory. Whenever one studies Selmer groups, one confronts the problem of trying to analyze a global object which is defined in terms of local conditions. The Poitou--Tate exact sequence provides a relationship between local and global Galois cohomology groups, and so is fairly ubiquitous in the study of Selmer groups.
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Take a look at the recent paper of Mazur and Rubin, "Ranks of twists of elliptic curves and Hilbert's 10th problem"
Lemma 3.2, one of the indispensable lemmas of the paper, is a direct application of the Poitou-Tate exact sequence.
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$\begingroup$ +1, that's actually an excellent example, both in terms of the importance of the result and the importance of the role Poitou-Tate plays there. $\endgroup$– Alex B.Commented Oct 17, 2010 at 11:57
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There are loads of applications and what you consider "nice" hugely depends. One application is e.g. Tate's proof that the Birch and Swinnerton-Dyer conjecture is invariant under isogenies. That and more applications are contained in Milne's book on "Arithmetic Duality Theorems".
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Three examples of the use of the Poitou-Tate duality:
All parity results (Dokchitsers, Mazur-Rubin, Nekovar, ...) for elliptic curve use this duality somewhere.
The duality is also crucial for Euler systems. An Euler system produces a collection of "derived" global cohomology classes. Using the pairing induces by Poitou-Tate, one can show that these classes are not in the Selmer group but orthogonal to it. This then bounds the size of the Selmer group from above.
The Poitou-Tate duality is an analogue of the Poincaré duality. For instance, Tate reduced the conjecture of Birch and Swinnerton-Dyer over a global field of characteristic $p$ to the finiteness of Sha. This argument uses Poincaré duality on the corresponding ellitic surface over the finite field. Analogous in Iwasawa theory, one uses the (Cassels-)Poitou-Tate duality in the computation of the leading term of the characteristic series of the Selmer group of the elliptic curve.
I would suspect there are more examples in "Cohomology of Number Fields".
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It has applications to the Brauer-Manin obstruction for torsors under connected linear algebraic groups, see Sansuc, J.-J., Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80 for number fields and Diego Izquierdo, Dualité et principe local-global sur des corps locaux de dimension 2, http://www.math.ens.fr/~izquierdo/Localdim204052016.pdf
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I think that Galois cohomology of elliptic curves by Coates and Sujatha is a good place to see some of the applications to Iwasawa theory as mentioned by prof Emerton.