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. 



Reference to the conjecture about injectivity of Abel-Jacobi map
**Question 1:** [Height pairing between algebraic cycles](https://books.google.nl/books?id=Wa16CwAAQBAJ&pg=PA18), by A.A. Beilinson (1987), with reference to an independent work by S. Bloch, [Algebraic cycles and values of L-functions II](https://projecteuclid.org/euclid.dmj/1077304437) (1985).


