Reciprocity legislation of varied types play a primary function in quantity thought. within the least difficult case, one obtains a clear formula through roots of harmony, that are precise values of exponential capabilities. an analogous thought could be built for designated values of elliptic or elliptic modular capabilities, and is named complicated multiplication of such capabilities. In 1900 Hilbert proposed the generalization of those because the 12th of his well-known difficulties. during this ebook, Goro Shimura presents the main finished generalizations of this sort through declaring a number of reciprocity legislation by way of abelian kinds, 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 type, that is additionally coated as a primary subject within the publication. The 3rd subject explored by way of Shimura is many of the algebraic kinfolk one of the sessions of abelian integrals. The research of such algebraicity is comparatively new, yet has attracted the curiosity of more and more many researchers. some of the themes mentioned during this booklet haven't been coated ahead of. particularly, this can be the 1st ebook within which the subjects of assorted algebraic kinfolk one of the classes of abelian integrals, in addition to the specified values of theta and Siegel modular services, are handled extensively.
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