Seminar on Arithmetic Algebraic Geometry [The modular form method in Iwasawa theory」

2014-10-16 10:00 - 2014-10-17 16:00
Faculty of Science Bld. No.3-210
Kei Hagihara (Hokkaido University)
Abstract: In 1970s, Ribet invented an elegant method for the construction
of suitable unramified extensions of number fields using modular forms,
giving a proof of the converse of Herbrand's theorem.

His ingenious method (the modular form method) has become one of the two
main techniques in Iwasawa theory along with the Euler system method, as
is seen from the proof of the Iwasawa main conjecture due to Mazur-Wiles
and the recent achivement of Skinner-Urban in Iwasawa theory for GL_2.

In this talk, we give the central ideas in this method through the
explanation of

- Ribet's proof of the converse of Herbrand's theorem, and

- Wiles's approach to the Iwasawa main conjecture over the rational field
using the Hida family of elliptic modular forms.