Antiholomorphic involutions of analytic families of abelian varieties
Author:
Allan Adler
Journal:
Trans. Amer. Math. Soc. 254 (1979), 6994
MSC:
Primary 10D20; Secondary 14K22
MathSciNet review:
539908
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Abstract: In this paper, we investigate antiholomorphic involutions of KugaSatake 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 KugaSatake 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]
Goro
Shimura, On analytic families of polarized abelian varieties and
automorphic functions, Ann. of Math. (2) 78 (1963),
149–192. MR 0156001
(27 #5934)
 [3]
Goro
Shimura, Introduction to the arithmetic theory of automorphic
functions, Publications of the Mathematical Society of Japan, No. 11.
Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton,
N.J., 1971. Kan\cflex o Memorial Lectures, No. 1. MR 0314766
(47 #3318)
 [4]
Stephen
Kudla, On the 𝑅forms of certain
algebraic varieties, Bull. Amer. Math. Soc.
81 (1975),
471–473. MR 0379505
(52 #410), http://dx.doi.org/10.1090/S000299041975137888
 [5]
Harris
A. Jaffee, Real forms of hermitian symmetric
spaces, Bull. Amer. Math. Soc. 81 (1975), 456–458. MR 0412490
(54 #613), http://dx.doi.org/10.1090/S000299041975137839
 [6]
Shimizu, On the zeta function of a quaternion algebra, Ann. of Math. 196.
 [7]
Goro
Shimura, On the real points of an arithmetic quotient of a bounded
symmetric domain, Math. Ann. 215 (1975),
135–164. MR 0572971
(58 #27992)
 [8]
Kuangyen
Shih, Antiholomorphic automorphisms of arithmetic automorphic
function fields, Ann. of Math. (2) 103 (1976),
no. 1, 81–102. MR 0466027
(57 #5910)
 [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 Rforms of certain algebraic varieties, Bull. Amer. Math. Soc. 81 (1975), 471473. MR 0379505 (52:410)
 [5]
 H. Jaffee, Real forms of hermitian symmetric spaces, Bull. Amer. Math. Soc. 81 (1975), 456458. 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), 135164. Also, On abelian varieties with complex multiplication, Proc. London Math. Soc. 34, 6586. (Especially, pp. 7982.) MR 0572971 (58:27992)
 [8]
 K.Y. Shih, Antiholomorphic automorphisms of arithmetic automorphic function fields, Ann. of Math. 103 (1976), 81102. MR 0466027 (57:5910)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197905399089
PII:
S 00029947(1979)05399089
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
