By Serge Lang (auth.)
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Extra info for Abelian Varieties
This shows that B(pm) is defined over k, and hence that k(u(pm») is a regular extension of k. One sees immediately that K(u) is a purely inseparable extension of k(u(pm»). Let p' be its degree. Then the cycle p' . (tt) is rational over k(u
Proof: Since the corollary is birational with respect to the variety, we may assume that it is a product of straight lines. Using Theorem 3, we may therefore assume that V is of dimension 1, and is either the affine line viewed as a group variety under addition, or the multiplicative group. According to Theorem 4, there exist two constants a, b such that f(x + y) = f(x) + f(y)+a and f(xy) = f(x) + fey) + b, where x, yare two independent generic points of V, and x + y, xy denote the addition on the affine line and the product, respectively.
Let (P, a) be a point of the intersection Tn (P X A). Since W is complete, there exists a point (P, Q, a) in r projecting on (P, a) . Since W is non-singular, the map I is defined at (P, Q), and hence takes on the constant value of I on P X W. This shows that the point a must be equal to this constant, and that our intersection consists of one point. According to the dimension theorem, we have dim T < dim V, and consequently dim T = dim V . If (M, N) is a generic point of V X W over a suitable field k, and z = I(M, N) then (M , z) is a generic point of T, and z must be algebraic over k (M).
Abelian Varieties by Serge Lang (auth.)