Kim Tuan Do: Construction of anticyclotomic Euler Systems

In this talk, we present the construction of three distinct Euler systems: one over an imaginary quadratic field $K$, and two over an imaginary biquadratic field K_0. All of them are anticyclotomic in the sense of Jetchev--Nekovar--Skinner.

In the first setting, we focus on the two-dimensional Galois representation V_{f,\chi} attached to a newform f twisted by an anticyclotomic Hecke character \chi over K, originating from the speaker's thesis and generalized in joint work with Castella. We compare this framework with classical Euler systems of Heegner points.

In the biquadratic case, we discuss two Euler system constructions over K_0. The first is the construction for V_{f,\chi} and the second is a recent Euler system attached to the one-dimensional Galois character O_\chi for a self-dual Hecke character \chi over K_0. This latter construction yields new evidence for the Bloch--Kato conjecture in analytic rank 2 for certain CM abelian surfaces (generalizing work of Castella for CM elliptic curves) and establishes the full Iwasawa--Greenberg main conjecture over K_0 (generalizing Rubin's classical work on elliptic units).