2023/03/12 by Francesc Castella, Castella, Francesc, Kim Tuan Do +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2303.06751
openalex publication_date 2023/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct a new Euler system for the Galois representation Vf,χ attached to a newform f of weight 2r≥ 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 Vf,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(Vf,χ,s) vanishes at the center), our results show that the Bloch-Kato Selmer group of Vf,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 Vf,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.