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DATE November 25 (Wed), 2020
TIME 14:00-15:30
PLACE Online
SPEAKER 켄타로 나카무라
HOST Kim, Chan-Ho
INSTITUTE Saga University
TITLE [Number Theory] Congruences of zeta elements and its application to Iwasawa main conjecture 2
ABSTRACT The abstract is

In his work on Iwasawa main conjecture for Hecke eigen cusp newforms,
Kazuya Kato constructed zeta elements, which are elements in Iwasawa
cohomology of p-adic Galois representations associated to Hecke eigen cusp newforms.
He also predicted that such elements exist for arbitrary families of p-adic global Galois
representations (generalized Iwasawa main conjecture).

In this series of lectures, I explain the results of my recent work
( on this conjecture, precisely,
the construction of zeta elements for universal deformations of
absolutely irreducible mod p odd Galois representations of rank two, and its application
to Iwasawa main conjecture (without p-adic L-function).

I (plan to) first recall Kato’s results on zeta elements for eigen cusp newforms and his generalized
Iwasawa main conjecture. Then, I explain my main result on zeta elements for universal
deformations. As a corollary of this result, we can obtain congruences for zeta elements
between congruent eigen cusp newforms. As an application of these congruences, we can
show that the validity of Iwasawa main conjecture (without p-adic L-function) for one eigen cusp new forms
are equivalent to that for arbitrary congruent eigen cusp new forms. I explain the outline of
the proof of this application. After that, I’ll explain the idea of the construction of
zeta elements for universal deformations. In our construction, many deep results in
p-adic (global and local) Langlands correspondence for GL_{2/Q} are crucial.
I’ll recall these results, and explain how to use these results to construct our zeta elements.

We review the p-adic local Langlands correspondence for GL2(Q_p) and the completed cohomology, the main ingredients of the proof of the main theorem.
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