Diagonal cycles and anticyclotomic Iwasawa theory of modular forms

Abstract

We construct a new Euler system for the Galois representation $V_{f,\chi }$ attached to a newform $f$ of weight $2r\ge 2$ twisted by an anticyclotomic Hecke character $\chi$. The Euler system is anticyclotomic in the sense of Jetchev–Nekovář–Skinner. We then show some arithmetic applications of the constructed Euler system, including new results on the Bloch–Kato conjecture in ranks zero and one, and a divisibility towards the Iwasawa–Greenberg main conjecture for $V_{f,\chi }$. In particular, in the case where the base-change of $f$ to our imaginary quadratic field has root number $+1$ and $\chi$ has higher weight (which implies that the complex $L$-function $L(V_{f,\chi },s)$ vanishes at the center), our results show that the Bloch–Kato Selmer group of $V_{f,\chi }$ is nonzero, as predicted by the Bloch–Kato conjecture; and if in addition a certain distinguished class ${\kappa }_{f,\chi }$ is nonzero, then the Selmer group is one-dimensional. Such applications to the Bloch–Kato conjecture for $V_{f,\chi }$ were left wide open by the earlier approaches using Heegner cycles and/or Beilinson–Flach elements. Our construction is based instead on a generalization of the Gross–Kudla–Schoen diagonal cycles.