The depth-weight compatibility on the motivic fundamental Lie algebra and the Bloch-Kato conjecture for modular forms
Kenji Sakugawa
TL;DR
The paper links depth-weight compatibility of the depth-graded motivic Lie algebra to the Bloch-Kato conjecture for $p$-adic Galois representations attached to full-level modular cuspforms. By developing the framework of mixed elliptic smooth ${f Q}_p$-sheaves over ${ m M}_{1,1}$ and analyzing the depth-graded motivic Lie algebra, it identifies a canonical subalgebra that corresponds to the geometric part of the elliptic fundamental group. Under a natural depth-weight compatibility hypothesis, the authors prove non-vanishing of Eisenstein-cup products and deduce a non-trivial Euler-system argument (Kato-style) that yields the Bloch-Kato conjecture for the cuspform representations. The work further integrates Beilinson Eisenstein symbols with $p$-adic regulators to connect Eisenstein data to cuspform data, thereby providing a cohesive route from motivic depth to analytic–algebraic conjectures for modular forms. This approach advances understanding of BC via motivic and Euler-system methods and highlights deep connections between mixed Tate/mixed elliptic motives, period polynomials, and $p$-adic Galois cohomology.
Abstract
Let $p$ be a prime number and let $V$ be a continuous representation of $\mathrm{Gal}(\overline {\mathbf Q}/\mathbf Q)$ 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.