Suppose $k$ is a number field, and $\sigma:k \rightarrow \mathbb{C}$ is an embedding. Then there is the (generalised) Abel-Jacobi map \begin{equation} \text{CH}^j(X)_0 \rightarrow \frac{H^{2j-1}((X \times_{\sigma}\mathbb{C})(\mathbb{C}),\mathbb{C})}{H^{2j-1}((X \times_{\sigma}\mathbb{C})(\mathbb{C}),\mathbb{Q}(j))+F^j}=\text{Ext}_{\text{MHS}_{\mathbb{Q}}}(\mathbb{Q}(0),H^{2j-1}_{\sigma}(X)(j)) \end{equation} where $\text{CH}^j(X)_0$ consists of cycle classes which are homologous to zero, and $H^{2j-1}_{\sigma}(X)$ means the pure Hodge structure on $H^{2j-1}((X \times_{\sigma}\mathbb{C})(\mathbb{C}),\mathbb{Q})$.

Question 1: anyone who could point out precise references to the conjecture that the Abel-Jacobi map is injective up to torsions?

Question 2: Suppose the conjecture is true, does this mean that \begin{equation} \text{dim}_{\mathbb{Q}}\,\text{CH}^j(X)_0 \otimes_{\mathbb{Z}} \mathbb{Q} \, \leq \text{dim}\,\text{Ext}_{\text{MHS}_{\mathbb{Q}}}(\mathbb{Q}(0),H^{2j-1}_{\sigma}(X)(j)) \end{equation} here $\text{Ext}_{\text{MHS}_{\mathbb{Q}}}(\mathbb{Q}(0),H^{2j-1}_{\sigma}(X)(j))$ is a complex vector space quotient by a rational lattice, which is like a torus. E.g. let's say the dimension of $\mathbb{C}/\mathbb{Q}^2$ is one.

I am not sure whether I have state question 2 correctly, and it would be great if someone could explain it a little bit about this conjecture and something similar to question 2 rigorously.


Question 1: Height pairing between algebraic cycles, by A.A. Beilinson (1987), with reference to an independent work by S. Bloch, Algebraic cycles and values of L-functions II (1985).


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