All Questions
10 questions
5
votes
1
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291
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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
6
votes
1
answer
556
views
Analogue of j-invariant for CM fields
For any imaginary quadratic field $F$, the Hilbert class field $H$ is generated by the $j$-invariant of any elliptic curve with complex multiplication (CM) by $\mathcal O$, the ring of algebraic ...
- 528
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
4
votes
0
answers
308
views
Complex multiplication and ray class fields
This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...
- 41
13
votes
1
answer
1k
views
How did Takagi prove Kronecker's Jugendtraum for Q(i)?
In Noah Snyder's historical undergraduate thesis on Artin L-Functions, it mentions that Takagi proved Kronecker's Jugendtraum in the case of Q(i) in his doctoral thesis. Since I don't know how to get ...
- 15.4k
12
votes
0
answers
767
views
What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$
This is related to my first MO question and Kevin Buzzard's conjecture at
Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $
In December 2010 my question appeared in the M.A.A. Monthly, ...
- 25.7k
22
votes
1
answer
2k
views
Can one prove complex multiplication without assuming CFT?
The Kronecker-Weber Theorem, stating that any abelian extension of $\mathbb Q$ is contained in a cyclotomic extension, is a fairly easy consequence of Artin reciprocity in class field theory (one just ...
- 15.4k
4
votes
0
answers
370
views
Formal non-CM in local fields
An elliptic curve $E$ with complex multiplication by an imaginary quadratic field $F$ has $\ell$-adic Galois representations that essentially encode the class field theory of $F$ - in other words, the ...
- 15.4k
8
votes
4
answers
3k
views
Class Field Theory for Imaginary Quadratic Fields
Let $K$ be a quadratic imaginary field, and $E$ an elliptic curve whose endomorphism ring is isomorphic to the full ring of integers of $K$. Let $j$ be its $j$-invariant, and $c$ an integral ideal of $...
- 2,827
27
votes
5
answers
3k
views
A problem of Shimura and its relation to class field theory
In Chapter II.10 of The Map of My Life, Goro Shimura mentions a certain problem:
The second topic concerns a polynomial $F(x)$ with integer coefficients. Take
$$
F(x) = x^3 + x^2 - 2x - 1,
$$
...
- 1,764