All Questions
4 questions
5
votes
1
answer
291
views
Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathcal{O}_K$?
For $K$ a number field, denote by $\mathcal{O}_K$ its ring of integers and by $H_K$ its Hilbert class field.
For which imaginary quadratic field $K$ does there exist an elliptic curve $E$, defined ...
- 1,479
11
votes
0
answers
382
views
What are the possible bad reductions for an abelian variety of dimension $g$ and a maximal endomorphism ring?
Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added ...
- 13.8k
5
votes
0
answers
206
views
Real field of definition of an abelian variety of CM-type?
Question 0. Can a field of definitions (without automorphisms) of an (almost arbitrary) abelian variety of CM-type, originally defined over ${\mathbb{C}}$,
be chosen to be a totally real number ...
- 14.2k
8
votes
1
answer
823
views
Does complex multiplication for higher dimensional abelian varieties give some generalization of class field theory?
I am currently learning some aspects of the theory of complex multiplication for elliptic curves, and the relationship with class field theory.
As I understand it, there is a very special class of ...
- 3,058