Note on conjectures of Beilinson-Bloch-Kato for cycle classes (Q1964706)

Let \(X\) be a smooth projective variety over a number field \(K\). For integers \(i,n\), such that \(i-2n \leq -2\), conjectures of Beilinson and Bloch-Kato predict that the order of vanishing of the \(L\)-function \(L(h^i(X),s)\) associated to the \(i\)-th motive \(h^i(X)\) of \(X\) at the point \(s = i+1-n\) is equal to the dimension of the \(\mathbb Q\)-vectorspace \(H^{i+1}_{\mathcal M}(X,\mathbb Q(n))_{\mathbb Z}\) as well as to the dimension of the \(\mathbb Q_p\)-vectorspace \(H^1_f (K,W)\). Here \(H^*_{\mathcal M}(X,\mathbb Q(n))_{\mathbb Z}\) denotes the integral part of the motivic cohomology groups and \(H^1_f(K,W)\) is the Selmer group contained in \(H^1(K,W)\), \(W= H^i(\overline{X},\mathbb Q_p(n)),\) defined by \textit{S. Bloch} and \textit{K. Kato} [in: ``The Grothendieck Festschrift'', Vol. I, Prog. Math. 86, 333-400 (1990; Zbl 0768.14001)]. The author replaces for even integers \(i=2m\) the Galois module \(W= H^{2m}(\overline{X},\mathbb Q_p(m))\) by the image \(V\) of \(CH^m(X_L) \otimes \mathbb Q_p\) under the étale cycle map \[ cl_L: CH^m(X_L) \rightarrow H^{2m}(\overline{X},\mathbb Q_p(m)) \] for \(L/K\) a sufficiently large finite Galois extension. The main result shows that for any \(r \geq 1\) the order of vanishing of the associated Artin \(L\)-function \(L(V,s)\) at \(s=1-r\) is equal to the dimension of the \(\mathbb Q_p\)-vectorspace \( H^1_f(K,V(r))\). Furthermore, the author shows that there are elements in \(H^{2m+1}_{\mathcal M}(X,\mathbb Q(m+r))_{\mathbb Z}\), independent of \(p\), which map to generators of \(H^1_f(K,V(r))\) under the \(p\)-adic regulator map \(r_p: H^{2m+1}_{\mathcal M}(X,\mathbb Q(m+r))_{\mathbb Z} \rightarrow H^1_f(K,V(r))\). Hence, if the cycle map \(cl_{\overline{K}}\) is surjective, e.g., if \(m = 0\) or \(m = \text{dim} X\) or \(X\) is a Calabi-Yau \(3\)-fold and \(0 \leq m \leq 3\), then the conjecture of Bloch-Kato is true and the \(p\)-adic regulator map is surjective.