Diagonal cycles and anticyclotomic Iwasawa theory of modular forms
Year: 2025
Type: article
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–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.
DOI: 10.4171/jems/1735
Authors: Francesc Castella, Kim Tuan Do
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4416784698