Questions tagged [complex-multiplication]
The complex-multiplication tag has no usage guidance.
37
questions with no upvoted or accepted answers
14
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answers
285
views
What's the dimension of the space of CM cusp forms?
I would guess that the following is very well known, but I don't know the answer and I couldn't find anything with some googling.
Let $\Gamma \subset \mathrm{SL}(2,\mathbf Z)$ be a congruence ...
12
votes
0
answers
759
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, ...
12
votes
0
answers
1k
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Are there analogues of Beilinson's conjectures for motives with coefficients?
There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...
11
votes
0
answers
359
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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 ...
11
votes
0
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324
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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.
...
8
votes
0
answers
420
views
Twists of CM modular forms
Let $K$ be an imaginary quadratic field of class number 1 and $E$ an elliptic curve over $\mathbf{Q}$ with CM by $\mathcal{O}_K$. Let $\psi$ be the Groessencharacter of $K$ attached to $E$, and
$$ g_{...
7
votes
0
answers
482
views
Structure of elliptic curve $y^2 = x^3 - x$ over $\mathbb{F}_p$ with $p=(a+i)(a-i)$
I hope this question is good enough for this network.
I am trying to compute the group structure as the title says of $E:y^2=x^3 - x$ over $\mathbb{F}_p$ with $p\equiv 1\bmod 8$ and $p-1$ a square, ...
6
votes
0
answers
389
views
Kisin module for CM elliptic curve
Let $E$ be a CM elliptic curve with CM by the field $K$ and assume that $p$ is ramified in $K$ so that $\pi^2 = p \in \mathcal{O}_K$. In particular, then $E$ has supersingular reduction at $p$ and by ...
5
votes
0
answers
241
views
$\mu=0$ for CM Elliptic curves?
Let $E$ be an elliptic curve defined over $F$ with CM by $\mathcal{O}_K$ where $K$ is an imaginary quadratic field. We may assume the $F$ contains $K$ and also contains the $p$-division points, where $...
5
votes
0
answers
248
views
Principally Polarized CM Abelian Variety
I am interested in considering examples of abelian varieties that are principally polarized with CM in dimension three. However, I am struggling to construct or find even a single instance.
In ...
5
votes
0
answers
201
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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 ...
4
votes
0
answers
249
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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 ...
4
votes
0
answers
139
views
About the main theorem of CM for elliptic curves
The classical main theorem of CM by Shimura states, among other things, that:
Consider a quadratic imaginary $K/\mathbb Q$ and an elliptic curve $E= \mathbb C / \Lambda$ with CM s.t. $End(E)\otimes \...
4
votes
0
answers
296
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 ...
4
votes
0
answers
359
views
Result of Deuring, intuitive way to see it's true/quickest way to prove?
There is the following result of Deuring that goes as follows:
Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary ...
4
votes
0
answers
246
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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 ...
4
votes
0
answers
361
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 ...
3
votes
0
answers
130
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
votes
0
answers
163
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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
199
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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 ...
3
votes
0
answers
137
views
Values of Grössencharacter attached to CM elliptic curve
I am trying a cross-post here, as my previous post on stackexchange was not as fruitful as I hoped. The link to the older post is: https://math.stackexchange.com/questions/3327269/values-of-...
3
votes
0
answers
95
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CM abelian surfaces (computed locally)
Let $K$ be a CM field such that $[K:\mathbb{Q}] = 4$ and let $K^+\subseteq K$ be the totally real subfield of $K$. For simplicity, assume that $K/\mathbb{Q}$ is Galois and suppose that $p\in \mathbb{Z}...
2
votes
0
answers
69
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Question on a certain reduced isogeny of CM elliptic curves
My question has to do with some hypotheses showing up in a Lemma of Joseph Silverman's Advanced Topics book. Here is some of the set up:
Let $K$ be an imaginary quadratic field and $E/H$ an elliptic ...
2
votes
0
answers
255
views
Eisenstein Series at CM points
Suppose that $L= \mathbb{Z}\tau + \mathbb{Z}$ is a lattice with CM. Consider the Eisenstein sum
$$ G_{2k}(L) = \sum_{(m,n)\neq (0,0)} \frac{1}{(m\tau+n)^{2k}}$$
where $k$ is a positive integer greater ...
2
votes
0
answers
135
views
Hasse invariant of abelian varieties with complex multiplication
Is there a good way to compute Hasse invariants of elliptic curves or higher dimensional Abelian varieties with complex multiplication?
For example, if $E$ is an elliptic curve with CM by an ...
2
votes
0
answers
229
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 ...
2
votes
0
answers
377
views
Number of CM lifting of an ordinary elliptic curve
Before asking my questions I will start with an example: There are two CM elliptic curves over $\mathbb{Q}$ with CM field $\mathbb{Q}(\sqrt{-7})$, whose $j$-invariants are $-3^3.5^3$ and $3^3. 5^3. 17^...
2
votes
0
answers
132
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Genus 2 hyperelliptic cryptography : typical discriminant and class number
As far as I know, there is no standard yet for cryptography based on the DLP over Jacobians of genus 2 curves. Yet, what can we say about the class number, and the discriminant of the complex ...
2
votes
0
answers
161
views
CM Hodge structures
There are several definitions of Hodge structures with complex multiplication. One of them is the following:
Definition. A Hodge structure $H$ is CM if it is polarizable and its Mumford-Tate group ...
1
vote
0
answers
45
views
Abelian surface with CM by a prescribed quartic field
Given a quartic field $K$, is it possible to exhibit an explicit abelian surface $A$ defined over a number field with CM by the field $K$?
For example, let's take the non-Galois CM-field $K=\mathbb Q(...
1
vote
0
answers
96
views
When a CM abelian variety has complex multiplication by $\mathcal{O}_E$?
I'm reading Milne's note, Complex Multiplication. There are many properties, such as Shimura-Taniyama Formula provided that $A$ is an abelian variety with complex multiplication by $\mathcal{O}_E$. So ...
1
vote
0
answers
91
views
Abelian group extensions
Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. Is there a way to see that $K(j(E), h(E[\mathfrak{p}]))/K$ is an Abelian extension for some $\mathfrak{...
1
vote
0
answers
163
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$F$-rational isogenies of CM Elliptic Curves
Let $F$ be a number field and $\mathcal{O}$ an order in an imaginary quadratic field $K$. Assume $K\subseteq F$. In Lang's Elliptic Functions, it is shown that over that there is a bijection between, ...
1
vote
0
answers
182
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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 ...
1
vote
0
answers
94
views
How quickly can we mutliply Cayley-Dickson hypercomplexes?
Assuming that all of the coordinates of two Cayley-Dickson Hypercomplex numbers are non-negative integers less than a prime $p$, how quickly can we multiply these numbers? I'm also interested in what ...
0
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0
answers
164
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Why Lubin Tate character acts on torsion points of CM elliptic curve implies the group of torsion points is infinite?
Let $F$ be quadratic imaginary field, and $R_F$ be its ring of integers.
Let $E /\Bbb{Q} $ be an elliptic curve which has CM by $F$. Suppose $E$ has good reduction at $P$,which is prime ideal of $R_F$....
0
votes
0
answers
120
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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 ...