Lecture 1 

Title: Split Kolyvagin Systems and the Bloch--Kato Conjecture for Conjugate-Dual Galois Representations

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 $Q$. 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. 

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Lecture 2

Title: Constructing Split Kolyvagin Systems from Special Cycles on Unitary Groups 

Abstract: In this lecture, we will apply the formalism of split Kolyvagin systems from the previous lecture to specific cases such as the twist of an elliptic curve by a Hecke character of an imaginary 

quadratic field, or the more general setting of conjugate-dual Galois representations arising in the context of the Gan--Gross--Prasad conjectures for unitary groups. In these cases, we will show how 

one can construct split Kolyvagin systems establishing sufficiently many norm relations (without Euler systems congruence relations) and what the consequences of these are on the Bloch--Kato 

conjecture in this setting. The latter uses very recent results of Reda Boumasmoud. Different aspects of this work are joint with various people including Reda Boumasmoud, Hunter Brooks, Jan Nekovar, Chris Skinner.