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

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.