The depth-weight compatibility on the motivic fundamental Lie algebra and the Bloch-Kato conjecture for modular forms

2024/02/20 by Kenji Sakugawa, Sakugawa, Kenji
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2402.13406

openalex publication_date 2024/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a prime number and let V be a continuous representation of Gal( \mathbf Q/\mathbf Q) on a finite dimensional \mathbf Q_p-vector space, which is geometric. One of the Bloch-Kato conjectures for V predicts that the rank of the Hasse-Weil L-function of V at s=0 coincides with the rank of Blcoh-Kato Selmer group of V^\vee(1). In this paper, we prove that the depth-weight compatibility on the fundamental Lie algebra of the mixed Tate motives over \mathbf Z implies the Bloch-Kato conjecture for the p-adic Galois representations associated with full-level Hecke eigen cuspforms.

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