Ah, I see that this gives an example of 2). Thanks. I am interested to know if there are examples where the function field is not a rational function field.

Thanks for your answer. I found the group law as follows: let t(a,b) be the tangent line through the point (a,b). Then t(a,b) = y-a^qx +b^q. Check that: t(a,b)^q + t(a,b) = (x_a)^{q+1} where x_a = x-a. This equation is just the same as that defining the Hermitian curve with t(a,b) replacing y and x_a replacing x. Now let T(c,d) be the tangent line (in terms of t(a,b) and x_a) to this curve at the point (c,d). Then T(c,d) = t(a,b) - c^qx_a + d^q and one checks that T(c,d) equals t(a+c,b+d+ca^q). So, from the points (a,b) and (c,d) we 'get' the point (a+c,b+d+ca^q). This gives a group.