Some Results on Abelian Varieties

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The theory of divisors on an abelian variety over the field of complex numbers has been much developed by making use of the theory of theta functions. But, in the case of an abstract abelian variety, many problems are still left open. In the present paper we shall study some properties of non-degenerate divisors. First the theorem of Riemann-Roch will be stated as follows : Let X be a positive non-degenerate divisor on an abelian variety A of dimension n. Then the dimension l(X) of the complete linear system |X| is equal to (X^<(n)>)/n! and also equal to (-1)^<n+1>x_A(X), where (X^<(n)>) means the n-fold intersection number of X and x_A(X) means the virtual arithmetic genus of X. In the next place let A and B be isogenous abelian varieties and let λ be a homomorphism from A onto B. If Y is a divisor on B, then two matrices E_l(λ^<-1>(Y)) and E_l(Y) are combined by the relation E_l(λ^-(Y))=^tM_l(λ)ヅ_l(Y)ネ_l(λ), where l is a prime number different from the characteristic of our geometry. This suggests to us that, if Y is positive non-degenerate, then l(λ^<-1>(Y)) is equal to v(λ)レ(Y). Actually a proof was given under an additional assumption in a recent paper [4] by Morikawa. In §2 we shall show that this additional assumption can be omitted. The above equality plays an important role in the algebraic treatment of the theorem of Frobenius, and we shall discuss it in a forthcoming paper. Lastly the existence theorem of a basic polar divisor (in the sense of numerical equivalence) on a polarized abelian variety will be proved. I wish to express here my hearty thanks to Professors S. Koizumi and T. Matsusaka for their kind advices.
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Some Results on Abelian Varieties

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