By Serge Lang (auth.)
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Then W = Bu is the translation B + u of B by u. It is a homogeneous space for B. If W has a rational point over k(v), then we can define the above mentioned section, and the theorem would be proved. In general, we are going to use an approximate section. More precisely, let P be a point of W which is rational over a separable algebraic extension of k(v), and let Pi be its conjugates over k(v). We can write Pi = Qi + u with Qi € B. Take the sum on A. We get ! Pi = ! Qi + d· u. This shows that! Qi is a point of B, rational over k(u).
We summarize the preceding construction as follows. THEOREM 8. Let C be a complete non-singular curve 01 genus g > O. Then there exists an abelian variety J 01 dimension g. and a rational map I : C --+ J such that il K is a lield 01 delinition lor I. • Pfl are independent generic points 01 Cover K. ,/(P i ) is a generic point 01 J over K and K(u) =cK(P 1 • .... ' Conversely. every generic point 01 J can be expressed in this manner. and the Pi are uniquely determined. up to a permutation. Finally.
We observe immediately that the homomorphism g* in (ii) is c', with a homomorphism g', then unique. If we have g = g'l g' = g*. Indeed, let u be a generic point of A and PI' . , P n independent generic points of V such that u = 2/(P;) . We have g(P i ) = g*/(P i ) + c = g'/(P i ) + c'. Taking the sum, we find + g*(u) = g'(u) +b for some constant bE B . Since g* and g' are homomorphisms, we have g*(O) = g'(O) = 0, and hence b = 0, and g' = g*. We shall say that g* is the homomorphism induced by g.