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 local 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.

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 local 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.