Isogeny Estimates for Abelian Varieties, and Finiteness Theorems

1993 ◽  
Vol 137 (3) ◽  
pp. 459 ◽  
Author(s):  
David Masser ◽  
Gisbert Wustholz
2018 ◽  
Vol 154 (8) ◽  
pp. 1571-1592 ◽  
Author(s):  
Martin Orr ◽  
Alexei N. Skorobogatov

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.


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

2001 ◽  
Vol 236 (1) ◽  
pp. 191-200 ◽  
Author(s):  
Shigeharu Takayama

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