Construction of anticyclotomic Euler systems using diagonal cycles

Abstract

In this thesis, we construct a new anticyclotomic Euler system for the four-dimensional Galois representation attached to two modular forms and a Hecke character of an imaginary quadratic field. To state the results more precisely, let $g$ and $h$ be newforms of weights $l\geq m$ of the same parity and let $\psi$ be a Hecke character of an imaginary quadratic field $K$ of infinity-type $(1-k,0)$ for some even integer $k\geq 2$. Assume that the product of the characters of $g$, $h$ and the CM-form attached to $\psi$ is trivial. Let $p$ be a prime which splits in $K$. We then study the $p$-adic $G_K$-representation $V:=V_g\otimes V_h(\psi^{-1})(1-c)$, where $c=(k+l+m-2)/2$. Combining a geometric construction using modified diagonal cycles in the product of three modular curves with a result of Lei--Loeffler--Zerbes, we obtain cohomology classes over ring class field extensions of $K$, and we prove that they form a split anticyclotomic Euler system in the sense of Jetchev--Nekov\'a\v{r}--Skinner. The bottom $\Lambda$-adic class of our Euler system is related to a one-variable specialization of the triple product $p$-adic $L$-function constructed by Darmon--Rotger via a reciprocity law proved by Bertolini--Seveso--Venerucci and Darmon--Rotger. This reciprocity law, together with the Euler-system machinery developed by Jetchev--Nekov\'a\v{r}--Skinner, allows us to deduce, under some additional hypotheses, different cases of the Bloch--Kato conjecture for the representation $V$ in analytic rank zero and one. As a different application, we also give two equivalent formulations of an Iwasawa--Greenberg Main Conjecture in this setting and prove one divisibility. When $h=g^\ast$, i.e., the modular form obtained by conjugating the Fourier coefficients of $g$, we obtain an Euler system for the three-dimensional $G_K$-representation $V':=\mathrm{ad}^0(V_g)(\psi^{-1})(1-k/2)\subset V$ and use it to derive similar applications towards the Bloch--Kato conjecture in analytic rank zero and one and towards a divisibility of an Iwasawa--Greenberg Main Conjecture.