# Abelian varieties with complex multiplication and modular by Goro Shimura

By Goro Shimura

Reciprocity legislation of assorted types play a relevant function in quantity concept. within the least difficult case, one obtains a clear formula by way of roots of cohesion, that are particular values of exponential features. an analogous thought should be built for specific values of elliptic or elliptic modular capabilities, and is termed advanced multiplication of such capabilities. In 1900 Hilbert proposed the generalization of those because the 12th of his recognized difficulties. during this e-book, Goro Shimura presents the main complete generalizations of this sort via pointing out a number of reciprocity legislation by way of abelian forms, theta features, and modular capabilities of numerous variables, together with Siegel modular services.

This topic is heavily attached with the zeta functionality of an abelian style, that's additionally lined as a chief subject matter 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. the various issues mentioned during this booklet haven't been coated prior to. particularly, this can be the 1st e-book during which the subjects of assorted algebraic kinfolk one of the classes of abelian integrals, in addition to the detailed values of theta and Siegel modular capabilities, are handled greatly.

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**Additional resources for Abelian varieties with complex multiplication and modular functions**

**Example text**

M/; Symm V _ / ,! m/ Symm V _ dimensional quotients of Symm V _ , which lies naturally in P via the Plücker embedding. V /-equivariant embedding (see [Mum66, Lect. 15]): jm W Hilbd ,! m/; Symm V _ / ,! P. ŒX Pr 7! 1/ and we denote by m WD ss;m Hilbs;m Â Hilbd d Â Hilbd the locus of points that are stable or semistable with respect to m , respectively. If ŒX Pr 2 Hilbs;m Pr 2 Hilbss;m Pr is m-Hilbert d (resp. ŒX d ), we say that ŒX stable (resp. semistable). 3(i)]. In particular, Hilbs;m are d and Hilbd constant for m 0.

U/ WD Y ! S . Property (c) above implies that forming wps is compatible with base-change. The last assertion is clear from the above geometric description of the contraction t u s W Xs ! X /s on each geometric point of u. 12 If u W X ! X / ! g. [Knu83]). The wp-stable reduction allows us to give a more explicit description of the quasiwp-stable curves. 13 A curve X is quasi-wp-stable (resp. quasi-p-stable, resp. quasistable) if and only if it can be obtained from a wp-stable (resp. p-stable, resp.

Semistable, strictly semistable). ŒX ss r closed inside Hilbd then we say that ŒX P is Hilbert polystable. j Let Chowd ,! ˝2 Symd V _ / (see [Mum66, Lect. 16]). 2 Hilbert-Mumford Numerical Criterion for m-Hilbert and Chow. . 47 the locus of points of Chowd that are, respectively, stable and semistable with respect to . 4]): Ch W Hilbd ! Chowd ŒX Pr 7! ŒX Pr /: We say that ŒX Pr 2 Hilbd is Chow stable (resp. ŒX Pr / 2 Chowsd (resp. Chowss d , Chowd n Chowd ). V /-orbit is closed inside Chowd .