Reciprocity legislation of varied types play a significant function in quantity idea. within the least difficult case, one obtains a clear formula via roots of cohesion, that are specified values of exponential features. an analogous conception may be constructed for detailed values of elliptic or elliptic modular features, and is termed advanced multiplication of such services. In 1900 Hilbert proposed the generalization of those because the 12th of his recognized difficulties. during this booklet, Goro Shimura offers the main entire generalizations of this sort by way of declaring a number of reciprocity legislation when it comes to abelian kinds, theta capabilities, and modular features of a number of variables, together with Siegel modular functions.

This topic is heavily hooked up with the zeta functionality of an abelian sort, that's additionally lined as a first-rate subject within the e-book. The 3rd subject explored by means of Shimura is many of the algebraic family members 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 number of the subject matters mentioned during this ebook haven't been lined earlier than. particularly, this can be the 1st booklet during which the subjects of assorted algebraic kinfolk one of the sessions of abelian integrals, in addition to the exact values of theta and Siegel modular features, are handled extensively.

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