By L. D. Olson
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This implies 32 GENERAL THEOREMS ON ABELIAN VARIETIES [II, § 2] that all the functions of L (c - b ) have a zero at P . Since c can be selected rational over E, this space of functions has a basis defined over E, and we thus get a contradiction. \J = ! Pi' From the hypothesis deg a = 0, we see immediately that l(a) = 0 or 1. \J) = 1. \J) = O. ):> is rational over that field. \J and q have the same smallest field of rationality containing K . Since the transcendence degree of this field over K is equal to g, and since q has degree g, we conclude that q is of the desired type, thereby proving our lemma.
Furthermore P'u is a point of B, and is even a generic point of B over k because the kernel of P'c> is finite. This shows that B is defined over k, and concludes our proof. We shall end this section with another application of the hypothesis v(nc» -=I=- 0, namely Poincare's theorem of complete + 28 GENERAL THEOREMS ON ABELIAN VARIETIES [II, § 1] reducibility. As for the preceding theorem, we shall not use this result until Chapter V. THEOREM 6. Let A be an abelian variety, B an abelian sub- variety of A.
Proof: Let ;. : A ~ A / B be the canonical homomorphism of A onto the factor group, which we denote by H. The idea of the proof is to define a cross section, or an approximate cross section of H into A. Let u be a generic point of A over k, and put ;'(u) = v. Let W be the variety ;'-l(V). It is the locus of u over k(v). 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.