applications of Tate-Poitou duality - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:18:59Z http://mathoverflow.net/feeds/question/42488 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42488/applications-of-tate-poitou-duality applications of Tate-Poitou duality norondion 2010-10-17T11:11:01Z 2010-10-25T15:33:55Z <p>What are nice applications of Tate-Poitou duality?</p> http://mathoverflow.net/questions/42488/applications-of-tate-poitou-duality/42491#42491 Answer by stankewicz for applications of Tate-Poitou duality stankewicz 2010-10-17T11:39:44Z 2010-10-17T11:39:44Z <p>Take a look at the recent paper of Mazur and Rubin, "Ranks of twists of elliptic curves and Hilbert's 10th problem"</p> <p>Lemma 3.2, one of the indispensable lemmas of the paper, is a direct application of the Poitou-Tate exact sequence.</p> http://mathoverflow.net/questions/42488/applications-of-tate-poitou-duality/42492#42492 Answer by Alex Bartel for applications of Tate-Poitou duality Alex Bartel 2010-10-17T11:40:04Z 2010-10-17T11:40:04Z <p>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 <a href="http://jmilne.org/math/Books/index.html" rel="nofollow">Milne's book</a> on "Arithmetic Duality Theorems".</p> http://mathoverflow.net/questions/42488/applications-of-tate-poitou-duality/42502#42502 Answer by Emerton for applications of Tate-Poitou duality Emerton 2010-10-17T14:01:09Z 2010-10-17T15:02:13Z <p>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. </p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/42488/applications-of-tate-poitou-duality/43367#43367 Answer by Guillermo Mantilla for applications of Tate-Poitou duality Guillermo Mantilla 2010-10-24T10:05:04Z 2010-10-24T10:05:04Z <p>I think that <a href="http://www.amazon.co.uk/Cohomology-Elliptic-Institute-Fundamental-Publication/dp/8173192936%20/%22cohomology%20of%20elliptic%20curves%22" rel="nofollow">Galois cohomology of elliptic curves</a> by Coates and Sujatha is a good place to see some of the applications to Iwasawa theory of mentioned by prof Emerton. </p> http://mathoverflow.net/questions/42488/applications-of-tate-poitou-duality/43531#43531 Answer by Chris Wuthrich for applications of Tate-Poitou duality Chris Wuthrich 2010-10-25T15:33:55Z 2010-10-25T15:33:55Z <p>Three examples of the use of the Poitou-Tate duality:</p> <p>All parity results (Dokchitsers, Mazur-Rubin, Nekovar, ...) for elliptic curve use this duality somewhere. </p> <p>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.</p> <p>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. </p> <p>I would suspect there are more examples in "Cohomology of Number Fields".</p>