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

Let p be a prime number and let V be a continuous representation of Gal( Q̄/ Q) on a finite dimensional Qₚ-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⁽1). In this paper, we prove that the depth-weight compatibility on the fundamental Lie algebra of the mixed Tate motives over Z implies the Bloch-Kato conjecture for the p-adic Galois representations associated with full-level Hecke eigen cuspforms.