Reciprocity legislation of varied varieties play a significant position in quantity idea. within the simplest case, one obtains a clear formula by way of roots of cohesion, that are designated values of exponential features. an analogous conception could be built for designated values of elliptic or elliptic modular services, and is named complicated multiplication of such services. In 1900 Hilbert proposed the generalization of those because the 12th of his recognized difficulties. during this booklet, Goro Shimura presents the main finished generalizations of this sort by way of declaring numerous reciprocity legislation when it comes to abelian forms, theta features, and modular features of numerous variables, together with Siegel modular functions.
This topic is heavily hooked up with the zeta functionality of an abelian style, that's additionally coated as a prime topic within the publication. The 3rd subject explored by way of Shimura is a number of the algebraic family one of the classes of abelian integrals. The research of such algebraicity is comparatively new, yet has attracted the curiosity of more and more many researchers. a few of the themes mentioned during this ebook haven't been lined sooner than. particularly, this is often the 1st publication during which the subjects of assorted algebraic family members one of the classes of abelian integrals, in addition to the distinctive values of theta and Siegel modular services, are handled extensively.
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