Iwasawa theory of Galois representations (IwTheGaR)
Iwasawa theory of Galois representations
Start date: Oct 1, 2008,
End date: Sep 30, 2012
"Galois representations have become one of the central objects of study in number theory and arithmetic geometry. Since Wiles' proof of the Taniyama-Shimura conjecture (hence the completion of the proof of Fermat's last theorem) using Iwasawa theoretical ideas (via Hida's and Mazur's theory of deformations of Galois representations), tremendous amount of progress have been made in the study of Galois representations. Despite the great achievements in the last decade, many important questions in this area of research remain open, such as the ""main conjectures"" of Iwasawa theory in various contexts, posed in great generality by Greenberg, for various Galois representations and for their various deformations. Once the main conjectures are proven, many interesting arithmetic data may be extracted about the Galois representation one starts off with. The main technical tool to attack the main conjectures is the machinery of Euler systems and Kolyvagin systems. This project aims for a study of Euler systems and Kolyvagin systems attached to Galois representations and attached to their Iwasawa theoretical deformations; as well as related themes within the Euler system theory. Some progress have already been made by the applicant in his thesis where he was able to prove that Kolyvagin systems attached to many Galois representations could be deformed along the ""cyclotomic direction"", namely in the most classical Iwasawa theoretical setting. Currently, he is involved with a relevant project about the construction of Kolyvagin systems for the nearly ordinary deformations of Galois representations attached to Hilbert modular forms, out of Euler systems constructed by Olivier Fouquet in his thesis. Should this project succeed, it already would be an important step further beyond the treatment of Mazur-Rubin and the applicant's thesis, where they have successfully dealt with the Kolyvagin system theory for the ""cyclotomic"" deformations, namely the simplest type."
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