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On Abelian Varieties

Published online by Cambridge University Press:  22 January 2016

Hisasi Morikawa
Hisasi Morikawa*
Affiliation:
Mathematical Institute, Nagoya University
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In a Bourbaki seminary note, La Théorie des Fonctions Thêta, A. Weil has discussed two fundamental theorems of the general theory of Theta functions. The first, due to H. Poincaré, was proved very skilfully in the note by means of harmonic integrals on a torus and the second, due to Frobenius, was treated by the systematic use of the notion of analytic structure.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 6 , October 1953 , pp. 151 - 170
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1953

References

1) Chow, W. L.: On the quotient variety of an abelian variety, Proc. Nat. Aca. Sci. Vol. 38, No. 12(1952)Google Scholar.

2) Siegel, C. L.: Additive Theorie der Zahikörper, Math. Ann. Bd. 88(1923)Google Scholar.

3)

4) We mean by a non-degenerate divisor X a divisor such that Xt—X∽O for only finite number of points t.

5) Z≡0 means that Zt—Z∽0 for all points t. We call such a divisor Z algebraically equivalent to zero.

6) See Lemma 10, N°35, §V, [V].

7) δAis the identical endomorphism of A.

8) See N°40 § V, [WJ].

9) Let ℬ0be the ring generated by Then ℬ0is an algebra of type S. Therefore ℬ0is semi-simple. This shows that there is a non-singular matrix with Iadic element F such that F—1M1(β)F is diagonal for every β∈ℬ0.

10) For see Definition in preceding Lemma 5.