Tangent bundle of Jacobian of a curve

Question

In An Introduction to the Theory of Special Divisors on Algebraic Curves, Griffiths states that

Obviously the tangent bundle of the Jacobian is trivial.

Why is this the case?

(Unfortunately, I don't have access to the entire book.)

Answer 1

The Jacobian of a curve is a group scheme, it acts on itself transitively (by translations), and the tangent bundle is equivariant. In general, there is an equivalence between the category of equivariant vector bundles on a homogeneous space of a group scheme and representations of the stabilizer of a point. In the case of the Jacobian the stabilizer is trivial, hence any its representation is trivial, hence any equivariant vector bundle is trivial. In particular, so is the tangent bundle.