By Serge Lang (auth.)

ISBN-10: 0387908757

ISBN-13: 9780387908755

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**Extra resources for Abelian Varieties**

**Example text**

H(P g ). It is symmetric in P v ... , P g , and the point v = H(Pv ... , P g ) is rational over k(u). Hence there exists a rational map (J. U = v. Taking into account Theorem 4, and recalling that I(P') = h(P') = 0, we see that (J. is a homomorphism. This proves our theorem. Let 1 : V ~ A be a rational map of a variety into an abelian variety. Then 1 induces a homomorphism of the group of cycles on V into A as follows. We denote by Zr(V) the group of cycles of dimension r on V. Let a = ~ ni(xi ) be an element of Zo(V)· We put I(a) = ~ ni(t(x i )).

Semi-pure) if its function field over some field of definition is a purely transcendental extension (resp. is contained in a purely transcendental extension) of the constant field. COROLLARY. Every rational map of a pure, or semi-pure, variety into an abelian variety is constant. 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.

Hence there exists a rational map (J. U = v. Taking into account Theorem 4, and recalling that I(P') = h(P') = 0, we see that (J. is a homomorphism. This proves our theorem. Let 1 : V ~ A be a rational map of a variety into an abelian variety. Then 1 induces a homomorphism of the group of cycles on V into A as follows. We denote by Zr(V) the group of cycles of dimension r on V. Let a = ~ ni(xi ) be an element of Zo(V)· We put I(a) = ~ ni(t(x i )). It is an element of Zo(A). Taking the sum on A, we get a point 5(t(a)) = 5,(a).

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