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

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

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.

**[1]**M. Kuga,*Fibre varieties over a symmetric space whose fibres are abelian varieties*, Lecture Notes, Univ. of Chicago, Chicago, I11., 1964.**[2]**G. Shimura,*On analytic families of polarized abelian varieties and automorphic functions*, Ann. of Math.**78**(1963). MR**0156001 (27:5934)****[3]**-,*Introduction to the arithmetic theory of automorphic functions*, Publ. Math. Soc. Japan**11**(1971). MR**0314766 (47:3318)****[4]**S. Kudla,*On the*, Bull. Amer. Math. Soc.**R**-forms of certain algebraic varieties**81**(1975), 471-473. MR**0379505 (52:410)****[5]**H. Jaffee,*Real forms of hermitian symmetric spaces*, Bull. Amer. Math. Soc.**81**(1975), 456-458. MR**0412490 (54:613)****[6]**Shimizu,*On the zeta function of a quaternion algebra*, Ann. of Math.**196**.**[7]**G. Shimura,*On the real points of an arithmetic quotient of a bounded symmetric domain*, Math. Ann.**215**(1975), 135-164. Also, On abelian varieties with complex multiplication, Proc. London Math. Soc.**34**, 65-86. (Especially, pp. 79-82.) MR**0572971 (58:27992)****[8]**K.-Y. Shih,*Antiholomorphic automorphisms of arithmetic automorphic function fields*, Ann. of Math.**103**(1976), 81-102. MR**0466027 (57:5910)**

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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