Now we define an elliptic curve as "a smooth projective curve of genus one with a specified base point". (A little question by the way: Is the requirement "with a specified base point" necessary? If we do not say that, what's going to happen?)  

We know that the Jacobian of an elliptic curve is itself, moreover, it's also an abelian variety of dimension 1. So an elliptic curve is an abelian variety of dimension 1.  

But how to prove that an abelian variety of dimension 1 is indeed an ellptic curve? Can anyone give me some guidance?  

Remark: the question is rised from <https://math.mit.edu/classes/18.783/2015/LectureNotes2.pdf>



Why an elliptic curve can be defined as an abelian variety of dimension 1?
Abelian varieties are smooth projective, and if they are of dimension $1$ then they are curves.


Check that the tangent bundle of an abelian variety is trivial by differentiating the multiplication map. For curves, this implies genus $1$.


Then the identity gives a base point.


