Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	Dimitar Jetchev，EPFL
Inviter:
Title:	Split Kolyvagin Systems and the Bloch--Kato Conjecture for Conjugate-Dual Galois Representations
Time & Venue:	2019.3.26 15:30-16:30 MCM 110
Abstract:	In a series of two lectures, we will explain a novel formalism of Kolyvagin systems that are useful in the study of conjugate-dual Galois representations of the absolute Galois group of an imaginary quadratic field. These new Kolyvagin systems are constructed for split primes and do not assume that the residual Galois representations extend to representations of the absolute Galois group of Q Q . The latter has been a crucial assumption in the original Kolyvagin-type argument as well as the subsequent applications to anticyclotomic Iwasawa theory. Using split primes, one not only addresses this concern, but also avoids the explicit use of the Euler system congruence relation, a statement that is difficult to verify in higher-dimensions. As a consequence, we obtain, under various sets of hypotheses including the non-vanishing of the base class, a rank-one statement for the Bloch--Kato conjecture, a bound on the Tate--Shafarevich group and one divisibility in a Perrin-Riou style anti-cyclotomic main conjecture. This is joint work with Chris Skinner and Jan Nekovar.