All Questions
Tagged with complex-multiplication ag.algebraic-geometry
32 questions
4
votes
0
answers
226
views
The definition of complex multiplication on K3 surfaces
I am reading this paper on the complex multiplication of K3 surfaces. It seems that this is only defined for complex K3 surfaces, or K3 surfaces over number fields. Is there a more general defintion ...
- 141
3
votes
0
answers
136
views
Are there CM complete intersections of arbitrarily large degree and codimension?
For every $d, c$ does there exist a smooth complete intersection $X \subset \mathbb{P}^N$ of codimension $c$ and multidegrees $d_1, \dots, d_c \ge d$ such that the Mumford-Tate group of $X$ is abelian?...
- 3,625
3
votes
0
answers
174
views
Reference Request: CM Motives over Function Fields
Inspired by Schutt and Shioda's lovely "An interesting elliptic surface over an elliptic curve", I have been investigating the following surface:
$$
\mathcal{E} : y^2 = x^3 - 27ux - 54v \...
3
votes
0
answers
201
views
Endomorphisms of elliptic curves, resp formal groups
Let
$E$ be an elliptic curve over a number field $K$,
$\mathcal{E}^w$ a fixed Weierstrass model for $E$ over $R := \mathbf{Z}[a_1,\ldots, a_6]$,
$\mathcal{E}$ the Néron model of $\mathcal{E}$ over ...
user147445
2
votes
0
answers
246
views
field of definition of CM abelian varieties
When $A$ is a CM abelian variety of dimension $1$ (i.e., an elliptic curve), then we have a result that if it has CM by a maximal order then it has a model over a number field $F$ where $F$ is the ...
- 443
21
votes
1
answer
648
views
The valuation of j-functions vs number of isomorphisms for an elliptic curve
Gross and Zagier prove the following fantastic result in their paper "Singular Moduli":
Let $R$ be a discrete valuation ring over $\mathbb Z_p$ with uniformizer $\pi$ such that $k = R/\pi$ is ...
- 7,736
7
votes
1
answer
1k
views
Fields of Definition of Elliptic Curves
I am currently studying the theory of complex multiplication and I find myself confused by the language in a lot of the literature.
In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves,...
- 901
4
votes
0
answers
259
views
Galois cohomology of the Serre group in the proof of the fundamental theorem of CM
I am working through J.S. Milne's note on the fundamental theorem of complex multiplication over $\mathbb{Q}$. Let $E$ be a CM-field Galois over $\mathbb{Q}$, and $S^E$ the Serre group corresponding ...
- 570
11
votes
0
answers
324
views
Why is the CM-type preserved after base changing from char 0 to char p?
There is a transition in the theory of complex multiplication which seems to be glossed over in all expositions I can find. I would like to explicitly find a theorem that allows me to do this.
...
- 3,539
1
vote
0
answers
183
views
Can we choose $h(\mathcal{O})$ elliptic curves such that they have same trace and endomorphism ring $\mathcal{O}$ but distinct j-invariants?
Let $\mathbb{F}_q$ be a finite field of characteristic $p$ with $q=p^a$ elements. Let $E$ be an ordinary elliptic curve defined over $\mathbb{F}_q$. We know that $\text{End}(E)\otimes\mathbb{Q}$ is ...
- 139
6
votes
1
answer
522
views
Endomorphisms of elliptic curves with CM; can we have an order?
Let $E$ be an elliptic curve over $\mathbb C$ with CM by ring of integers $O_K$ of an imaginary quadratic number field $K$. Let $O$ be an order of $O_K$.
Is there a number field $L$ such that $E$ has ...
- 119
5
votes
0
answers
205
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 ...
- 14.1k
3
votes
1
answer
288
views
Elliptic curve with CM by $(1+\sqrt{-11}) /2$
Can someone explain to me on how to obtain the endomorphism for elliptic curve with CM by $(1+\sqrt{-11}) /2$?
Given the elliptic curve over $F_{p}$ as $y^2=x^3-13824/539 x + 27648/539 \dots$ how do ...
- 41
10
votes
1
answer
575
views
Does every Shimura variety contain a generic point defined over a number field?
This question is related to my previous question, to which I got a partial answer.
Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta_8$ is a primitive ...
- 14.1k
11
votes
2
answers
652
views
Abelian variety with prescribed endomorphism ring
Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta_8$ is a primitive 8-th root of unity. Let $\Lambda={{\mathbb{Z}}}[\zeta_8]$ denote the ring of ...
- 14.1k
7
votes
1
answer
1k
views
Ordinary abelian varieties over a finite field
Let $q$ be a power of a prime $p$. Deligne's paper "Variétés abéliennes ordinaires sur un corps fini" seems to describe an equivalence of categories between
ordinary abelian varieties over a finite ...
- 647
3
votes
1
answer
434
views
Tate modules of elliptic curves with complex multiplications
Let $E/K$ be an elliptic curve with complex multiplication
over an imaginary quadratic field $K$. Then, I heard that
it is well-known that the Tate module $V_{p}(E)$ over
$\mathbb{Q}_{p}$ ...
- 31
5
votes
1
answer
704
views
Remark 4.23.4 in Hartshorne
Crosspost from math.stackexchange, since it's quite possible I might not get a response there.
Remark 4.23.4 in Chapter IV of Hartshorne's Algebraic Geometry references a paper by Elkies that ...
user76210
4
votes
1
answer
725
views
isogeny clases of CM abelian varieties
Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, ...
- 43
3
votes
1
answer
317
views
CM abelian varieties over the rationals
Let $K$ be a number field and let $A$ be an abelian variety of dimension $g$ over $K$. Let $L$ be a CM field and suppose that $[L:{\bf Q}]=2g$. Suppose that there exists an embedding $\iota:L\...
- 3,076
5
votes
1
answer
366
views
notion of $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ representation with complex multiplication
In usual Hodge theory, there is the notion of Hodge structure $H$ with complex multiplication, that can be defined in several ways, i.e. asking that there exists a CM number field $E$ such that $\dim ...
- 51
2
votes
1
answer
234
views
Difference between Frobenii on Tate modules of special and generic fibre
Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$.
If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...
- 91
21
votes
1
answer
734
views
Why is the CM closure of $\mathbb{Q}$ the "ultimate" coefficient field for motives?
In a rough way, a category of motives over a field $k$ with coefficients in a field $K$ gives a universal cohomology theory with coefficients in $K$ for algebraic varieties defined over $k$. I had the ...
- 6,920
2
votes
1
answer
799
views
Canonical lifts from $\mathbb F_q$ and CM-theory
One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...
- 647
4
votes
2
answers
482
views
abelian varieties with the same CM type are isogenous
Does anybody have a reference for the following fact?
All abelian varieties with complex multiplication and same CM type are isogenous over $\overline{\mathbb{Q}}$?
Here abelian variety with ...
- 41
1
vote
2
answers
304
views
how to see CM types as functions on the Galois group?
Let $K$ be a CM field, that is, an imaginary quadratic extension of a totally real number field. Its degree $[K: \mathbb{Q}]$ is a en even number $2n$.
(1) For me a CM type is a subset $\Phi \subset ...
- 11
4
votes
0
answers
264
views
modular curves with complex multiplication
Is the complete list of values of $N$ for which the modular curve $X_0(N)$ has complex multiplication?
I guess the answer is no...
Are there at least some examples known? The only one which comes ...
- 41
6
votes
2
answers
479
views
Rational points and torsion points of CM elliptic curve
Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order i.e., $\mathrm{End}_K(E)\cong \mathcal{O}=\mathcal{O}_K$. Let $\mathfrak{p}$ be a prime of $K$ such that ...
- 601
5
votes
2
answers
295
views
can all CM types be realized by Jacobians?
The question is kind of self contained, but let me develop a bit further.
Assume K is a CM field of degree $2g$, that is, a quadratic imaginary extension of a totally real field. A CM type of $K$ is ...
- 53
4
votes
3
answers
2k
views
transcendence of periods of CM elliptic curves
Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ defined by a Weierstrass equation
$$
y^2=4x^3+g_2x+g_3.
$$ Then $H^1_{dR}(E/\overline{\mathbb{Q}})$ is spanned by the classes of the ...
- 49
0
votes
0
answers
124
views
field of definition of abelian varieties with extra endormorphism
Let $A$ be a complex abelian variety such that $\mathrm{End}(A)$ is strictly bigger than $\mathbb{Z}$.
Question: Is is true that $A$ is defined over $\overline{\mathbb{Q}}$?
This is of course what ...
- 1
11
votes
3
answers
1k
views
Motive of CM elliptic curve and modular forms
I am trying to get some insight into the Deligne/Scholl construction of the motive of a modular form. First of all I would like to understand the case of weight two, especially when there is complex ...
- 111