Title/Abstract/Reference
Kentaro Nakamura
Title: Local and global epsilon conjectures for the rank two case
Abstract: The Iwasawa main conjecture and the Bloch-Kato’s Tamagawa number conjecture predict a mysterious relationship between the special values of L-functions and the Galois cohomology (e.g. Selmer groups) associated to motif over number fields. As a unified generalization of these conjectures, Kato, (7), formulated a conjecture called the generalized Iwasawa main conjecture, which predicts that such a relationship exists for arbitrary p-adic families of p-adic global Galois representations. In particular, as a generalization of Euler system, he predicts the existence of zeta elements which are incarnations of L-functions in the world of the Galois cohomology. The global epsilon conjecture in the title of my talk is a conjecture the functional equation of zeta elements, and the local epsilon conjecture is a conjecture on the local factor of the functional equation satisfied by the zeta elements. The aim of my talk is to explain my results on these conjectures for the rank two case.
In my talk, I first briefly recall these conjectures and roughly state my main results. For the proof, we first need to recall the fundamentals of the theory of (phi,Gamma)-modules (in particular, its Galois cohomology and Iwasawa cohomology) and the Colmez`s theory of p-adic local Langlands correspondence for GL_2(Q_p). After recalling these topics, I will explain the proof of my main results.
Reference (with explanation):
(1) K. Nakamura: Local epsilon isomorhisms for rank two p-adic representations of Gal(¥bar{Q}_p/Q_p) and a functional equation of Kato’s Euler system, http://arxiv.org/abs/1502.04924.
This article (1) heavily depends on (in particular (2))
(2) P.Colmez: Repr’esentations de GL_2(Q_p) et (phi,Gamma)-modules, Ast’erisque 330(2010),61-153.
(3) K.Nakamura: A generalization of local epsilon conjecture for (phi,Gamma)-modules over the Robba ring, http://arxiv.org/abs/1305.0880.
For the explicit reciprocity law via (phi,Gamma)-modules,
(4) P.Colmez: Fontaine’s rings and p-adic L-functions, in his homepage http://webusers.imj-prg.fr/~pierre.colmez/Enseignement.html.
(5) K.Nakamura: Iwasawa theory of de Rham (phi,Gamma)-modules over the Robba ring, Journal of the Institue of Mathematics of Jussieu/volume 13/Issue 01/january 2014, pp 65-118.
The references of Kato’s conjectures (generalized Iwasawa main conjecture, global and local epsilon conjectures) are
(6) T.Fukaya, K.Kato: A formulation of conjectures on p-adic zeta functions in non commutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society. vol XII (Providence, RI), Amer. Math. Soc. Trails. See. 2, vol 219, Amer. Math. Soc., 2006, pp.1-85.
(7) K.Kato: Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via B_{dR}, Arithmetic algebraic geometry, Lecture Notes in Mathematics 1553, Springer-Verlag, Berlin, 1993.
(8) K.Kato: Lectures on the approach to Iwasawa theory for Hasse-Wil L-functions via B_{dR}, part II, unpublished preprint.
(the global and local epsilon conjecture is formulated in (8), but it is unpublished. The article (6) formulates all these conjectures, but for, more generally, non commutative case. In my articles (1) and (3), I recall local and global epsilon conjectures (precisely) and generalized Iwasawa main conjecture (roughly).)
Shanwen Wang
Title: Kato's Euler systems
Abstract: In this series of talks, we will explain the construction of Kato's Euler system and its variants.
Reference:
1. The Kato