Let $E$ be an elliptic curve over a number field $k$. We define the affine curve $C := E \backslash \{p_1,...,p_n\}$ by removing a finite number of points from $E$. Here, I would like to declare that I am not sure if its matters whether or not these points are $k$-points.
Then $C$ is a hyperbolic curve over $k$, i.e., it is the complement of $n$ points in a projective algebraic curve of genus $g$, such that $2g-2+n>0$. By the paper The Brauer-Manin obstruction for integral points on curves by Harari and Voloch, the paragraph preceding Lemma 2.1 states that affine curves whose complement in $\mathbb{P}^1_k$ is a reduced divisor of degree $\geq 2$ can be embedded in their generalized Jacobian, which in this case is an algebraic torus. Such affine curves include genus zero hyperbolic curves, i.e., open subsets of $\mathbb{P}^1_k$ with at least three points removed.
Question. Morphisms of varieties into their generalized Jacobians can be found in many literatures, but apart from the abovementioned example, I can't seem to find any where these morphisms are in fact embeddings. Does anyone know of some useful references, at least for our particular case $C$?
In general, the generalized Jacobian of $C$ is a semi-abelian variety $S$. One would suspect that $S$ could be the extension of the Jacobian variety $J_E$ of $E$ by some torus dependent on $n$, the degree of the divisor $D :=E\backslash C$. The closest thing I've managed to gather is that the "classical" Albanese variety of $E$ (which in this case is $J_E$) is the maximal abelian quotient of $S$. The kernel of the morphism $S \rightarrow J_E$ is some torus $T$ who dimension is the rank of the subgroup of divisors of $E \times _k \bar{k}$, supported on $D \times _k \bar{k}$, which are algebraically equivalent to zero.
EDIT. Here are some background on Albanese schemes, I'm referring to the paper Duality of Albanese and Picard 1-motives by Ramachandran. Given a connected scheme $X$, we have a universal morphism $u_X:X \rightarrow A_X$ from $X$ into its Albanese scheme $A_X$. The neutral component $A^0_X$ is called the generalized Albanese variety defined by Serre. If $X$ is proper, then $A^0_X$ is an abelian scheme.
In the case where $X$ is a normal integral curve, there exists a unique normal projective curve $X'$ containing $X$ as an open dense subscheme. Let $D := X'\backslash X$. The semiabelian variety $A^0_X$ is the generalized Jacobian of $X'$ (defined by Rosenlicht) corresponding to modulus $D$.
