Polarized Varieties, Fields of Moduli and Generalized Kummer Varieties of Polarized Abelian Varieties

1958 ◽  
Vol 80 (1) ◽  
pp. 45 ◽  
Author(s):  
T. Matsusaka
Keyword(s):  
Abelian Varieties ◽  
Fields Of Moduli

Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

2014 ◽  
Vol 66 (5) ◽  
pp. 1167-1200 ◽  
Author(s):  
Victor Rotger ◽  
Carlos de Vera-Piquero
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Quadratic Field ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Main Idea ◽  
Galois Representation ◽  
Rational Points ◽  
Shimura Curves ◽  
Fields Of Moduli

AbstractThe purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

scholarly journals The coincidence of fields of moduli for non-hyperelliptic curves and for their jacobian varieties

1981 ◽  
Vol 82 ◽  
pp. 57-82 ◽  
Author(s):  
Tsutomu Sekiguchi
Keyword(s):  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Jacobian Varieties ◽  
Axiomatic Treatment ◽  
Fields Of Moduli

The notion of fields of moduli introduced first by Matsusaka [8] has been developed by Shimura [12] exclusively in the area of polarized abelian varieties. Later Koizumi [7] gave an axiomatic treatment for the notion.

scholarly journals The Fields of Moduli for Polarized Abelian Varieties and for Curves

1972 ◽  
Vol 48 ◽  
pp. 37-55 ◽  
Author(s):  
Shoji Koizumi
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Essential Role ◽  
Positive Characteristic ◽  
Abelian Varieties ◽  
Complex Multiplications ◽  
Definition Of ◽  
Fields Of Moduli

In the study of moduli of polarized abelian varieties and of curves as well as in the theory of complex multiplications, the notion of fields of moduli for structures plays an essential role. This notion was first introduced by Matsusaka [7] for polarized varieties with some pleasing properties and later was given a more comprehensible treatment by Shimura [10] in the case of polarized abelian varieties or polarized abelian varieties with some further structures. Both authors discussed fields of moduli not only in algebraic geometry of characteristic zero but also in that of positive characteristic, but in the latter case the definition of fields of moduli seems somewhat artificial and there have been no essential applications of them so far.

scholarly journals Classification of nonrigid families of abelian varieties

1993 ◽  
Vol 45 (2) ◽  
pp. 159-189
Author(s):  
Masa-Hiko Saitō
Keyword(s):  
Abelian Varieties
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