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 $2r\geq 2$ twisted by an anticyclotomic Hecke character $\chi$. The Euler system is anticyclotomic in the sense of Jetchev-Nekovar-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.