Let $C$ be a curve defined over $\mathbb{Q}$, of positive genus. Let $J$ denote its Jacobian. I would like to ask a couple of basic (I presume) questions:
0) Why is $J$ an algebraic variety?
1) For a number field $K$, is a $K$-rational point on $C$ mapped by the Abel-Jacobi map to a $K$-rational point on $J$?
2) When $C$ is an elliptic curve defined by the equation $y^2 = x^3 + ax + b$, is its Jacobian isomorphic to $C$ as algebraic varieties/is it birationally equivalent to $C$?
Thank you.