Diagonal cycles and anticyclotomic Iwasawa theory of modular forms

TL;DR

This work constructs a novel anticyclotomic Euler system for the conjugate-self-dual Galois representation $V_{f,\chi}$ attached to a modular form twisted by an anticyclotomic Hecke character. The construction generalizes Gross–Kudla–Schoen diagonal cycles and uses diagonal-triple products to produce cohomology classes with explicit norm relations across ring class fields, enabling applications to Bloch–Kato in ranks $0$ and $1$ and to divisibilities in the anticyclotomic Iwasawa main conjecture. Depending on the root number and the infinity-type of $\chi$, the paper derives nonvanishing results or one-dimensional Selmer groups, connecting to derivative or central values of Rankin–Selberg $L$-functions via explicit reciprocity laws and Gross–Zagier type formulas. The framework blends CM-patching, triple-product $p$-adic $L$-functions, and Iwasawa-theoretic machinery to give a broad set of rank-one and divisibility results, with definite/indefinite dichotomies reflecting the sign in the functional equation and yielding new evidence for the Bloch–Kato conjecture in analytic rank $1$ and main-conjecture divisibilities without level-raising hypotheses.

Abstract

We construct a new Euler system for the Galois representation $V_{f,χ}$ attached to a newform $f$ of weight $2r\geq 2$ twisted by an anticyclotomic Hecke character $χ$. 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 $V_{f,χ}$. In particular, in the case where the base-change of $f$ to our imaginary quadratic field has root number $+1$ and $χ$ has higher weight (which implies that the complex $L$-function $L(V_{f,χ},s)$ vanishes at the center), our results show that the Bloch-Kato Selmer group of $V_{f,χ}$ is nonzero, as predicted by the Bloch-Kato conjecture; and if in addition a certain distinguished class $κ_{\,f,χ}$ is nonzero, then the Selmer group is one-dimensional. Such applications to the Bloch-Kato conjecture for $V_{f,χ}$ 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.

Theorems & Definitions (97)

• Theorem A: Theorem \ref{['maintheorem2']}
• Theorem B: Theorem \ref{['thm:BK-def-1']}
• Theorem C: Theorem \ref{['thm:BK-def']}
• Remark A
• Proposition A: LLZ-K
• Definition B
• Theorem C
• proof
• Proposition A
• proof