Isogeny Estimates for Abelian Varieties, and Finiteness Theorems

1993 ◽  
Vol 137 (3) ◽  
pp. 459 ◽  
Author(s):  
David Masser ◽  
Gisbert Wustholz
Keyword(s):  
Abelian Varieties ◽  
Finiteness Theorems

scholarly journals Finiteness theorems for K3 surfaces and abelian varieties of CM type

2018 ◽  
Vol 154 (8) ◽  
pp. 1571-1592 ◽  
Author(s):  
Martin Orr ◽  
Alexei N. Skorobogatov
Keyword(s):  
Abelian Varieties ◽  
Algebraic Closure ◽  
Complex Multiplication ◽  
Uniform Boundedness ◽  
Invariant Subgroup ◽  
Smooth Projective Variety ◽  
Number Fields ◽  
K3 Surfaces ◽  
Isomorphism Classes ◽  
Finiteness Theorems

We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford–Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case.

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

Very ample line bundles on quasi-abelian varieties

2001 ◽  
Vol 236 (1) ◽  
pp. 191-200 ◽  
Author(s):  
Shigeharu Takayama
Keyword(s):  
Abelian Varieties ◽  
Line Bundles
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