I want to understand what the Jacobian variety is from an algebraic (or arithmetic?) perspective.
I want to know:
- What is the definition of the Jacobian?
- Widely know facts about it.
- Why the Jacobian of an elliptic curve is the curve itself.
- How to map points on a hyperelliptic curve to the Jacobian.
- As much as possible about the torsion subgroup and torsion 2-subgroup of the Jacobian of a hyperelliptic curve
- What is the connection between the fundamental unit in the corresponding ring and torsion points of the Jacobian?
- How to get sum of points on the Jacobian (maybe the Mumford algorithm).
I want to understand it as quickly as I can, so I want to read a clear monograph. It would be great if that monograph (book or some lecture notes or survey or something) requires minimal possible background.
Thanks for any references.
Best Answer
Over $\mathbb C$, there's also Mumford's beautiful monograph Curves and their Jacobians, The University of Michigan Press, Ann Arbor, Mich., 1975. But a monograph covering all of the topics in your wish list is going to be rather superficial. It's simply too big a subject. And I appreciate your wanting to "understand it as quickly as [you] can," but as someone far wiser than me once said, "there's no royal road to geometry."