You may want to read [Milne, Arithmetic Duality Theorems], Chapter I, Appendix A http://jmilne.org/math/Books/ADTnot.pdf and also [Neukirch, Schmidt, Wingberg], Cohomology of Number Fields, Chapter X, §1.
What makes it easier? The existence of the Jacobian and the known structure of its rational points, that everything is a curve over a field, so the theory is more geometric. Also the Zeta function being rational simplifies things.