## Jacobians of Curves

For those familiar with algebraic geometry and algebraic curves, one can prove Theorem 21.2.3 from an alternative point of view. There is a bijection between nonsingular geometrically irreducible projective curves over and function fields over (which we assume are finite separable extensions of such that ). Let be the curve corresponding to . The group is in bijection with the divisors of degree 0 on , a group typically denoted . The quotient of by principal divisors is denoted . The Jacobian of is an abelian variety over the finite field whose dimension is equal to the genus of . Moreover, assuming has an -rational point, the elements of are in natural bijection with the -rational points on . In particular, with these hypothesis, the class group of , which is isomorphic to , is in bijection with the group of -rational points on an algebraic variety over a finite field. This gives an alternative more complicated proof of finiteness of the degree 0 class group of a function field.

Without the degree 0 condition, the divisor class group won't be finite. It is an extension of  by a finite group.

where  is the greatest common divisor of the degrees of elements of , which is when has a rational point.

William Stein 2004-05-06