Question

What are nice applications of Tate-Poitou duality?

Answer 1

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.

Answer 2

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.

Answer 3

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".

Answer 4

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".

Answer 5

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 of mentioned by prof Emerton.