Antiholomorphic involutions of analytic families of abelian varieties

Author:
Allan Adler

Journal:
Trans. Amer. Math. Soc. **254** (1979), 69-94

MSC:
Primary 10D20; Secondary 14K22

MathSciNet review:
539908

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Abstract: In this paper, we investigate antiholomorphic involutions of Kuga-Satake analytic families of polarized abelian varieties *V*. A complete set of invariants of the Aut(*V*)-conjugacy classes of antiholomorphic involutions of *V* is obtained. These invariants are expressed as cohomological invariants of the arithmetic data defining *V*. In the last section, the fibre varieties of Kuga-Satake type belonging to totally indefinite quaternion division algebras over totally real fields are investigated in more detail, and the cohomological invariants are related to results of Steve Kudla. The group of holomorphic sections of *V* is computed for this case. It is also shown that in general the fibre structure of *V* is intrinsic.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0539908-9

Keywords:
Symmetric space,
antiholomorphic,
cohomology,
cohomology with nonabelian coefficients,
quaternion algebra,
totally indefinite,
algebraic number field,
totally real,
fibre variety,
maximal order,
ideal class group,
connection,
properly discontinuously

Article copyright:
© Copyright 1979
American Mathematical Society