Overview

Abelian varieties are projective and connected group varieties. They are of fundamental importance to the study of all projective smooth varieties over fields. In the late 1960s, Serre and Tate developed a deformation theory of abelian varieties over fields of positive characteristic, and introduced several ways to identify the generic ones, which are called ordinary abelian varieties. This monograph provides a comprehensive generalisation of the Serre and Tate theory of ordinary abelian varieties and their deformation spaces. This generalisation deals with abelian varieties equipped with additional structures. The additional structures can be not only a classical action of a semi-simple algebra and a polarisation, but can also be, more generally, the data given by some “crystalline Hodge cycles” (i.e., a crystalline version of a Hodge cycle in the sense of motives). Compared to Serre–Tate ordinary theory, new phenomena appear in the generalised setting. The generalised theory is presented both in abstract contexts of F-crystals endowed with reductive groups and minuscule Hodge co-characters, and in geometric contexts provided by good moduli spaces of abelian varieties endowed with additional strictures, which are called integral canonical models of Shimura varieties of Hodge type. This monograph studies the generalised notions of ordinariness from multiple points of view, such as Newton polygons, canonical lifts, formal Lie groups, complex multiplications, etc. Researchers and graduate students working in related areas, such as arithmetic algebraic geometry and number theory, will find this monograph truly valuable due to its comprehensive nature and general setting, which provide key new tools for tackling numerous other problems pertaining to all integral canonical models of Shimura varieties of Hodge type.

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9781571462770

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Adrian Vasiu, Binghamton University, New York, USA

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