Questions tagged [complex-multiplication]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
0answers
74 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 ...
0
votes
0answers
88 views

$A$ is a CM abelian variety over $\mathbb{C}$, $\sigma \in$ Aut$(\mathbb{C})$. How to define $\sigma A$?

This question may be a little silly: $A$ is a CM abelian variety over $\mathbb{C}$, $\sigma \in$ Aut$(\mathbb{C})$. How to define $\sigma A$? In my opinion, $\sigma$ can only act on $A$ when it’s ...
1
vote
0answers
82 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 ...
4
votes
2answers
216 views

Proof in Schertz's Complex Multiplication

I'm reading through Complex Multiplication by Reinhard Schertz, and I'm stuck at Theorem 3.1.8. Let $\mathfrak{O}_t$ be the order of conductor $t$ in an imaginary quadratic field $K$. He defines ...
3
votes
0answers
163 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 ...
3
votes
0answers
100 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
1answer
109 views

Quadratic orders embedded in matrices

Let $\tau$ be a CM point and let $\mathcal O$ be the quadratic order corresponding to the lattice $[\tau,1]$, that is $$\mathcal O =\lbrace \lambda \in \mathbb C: \lambda[\tau,1]\subset[\tau,1]\rbrace....
4
votes
0answers
175 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 $...
2
votes
0answers
149 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 ...
1
vote
1answer
192 views

Local root numbers of the Hecke character associated with some specific CM elliptic curves, should they be some roots of unity?

TL;DR. Some local root numbers of the Hecke character associated with our specific CM elliptic curve by $\mathbf{Q}(i)$ seem to have value in $\mu_4$. But apparently our computation via Rohrlich's ...
21
votes
1answer
525 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 ...
12
votes
2answers
887 views

Complex Multiplication and algebraic integers

Let $q=e^{2\pi i\tau}$ and $$E_2(\tau) = 1 - 24 \sum_{n=1}^\infty\frac{nq^n}{1-q^n}$$ be the Eisenstein Series of weight $2$ and let $E_2^*(\tau) = E_2(\tau) - \frac{3}{\pi\cdot Im(\tau)}$ be the ...
4
votes
0answers
125 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 ...
10
votes
0answers
237 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 ...
1
vote
1answer
192 views

Confusion on supersingular reduction of elliptic curves with complex multiplication

Let $A/L$ be an elliptic curve, with complex multiplication by a quadratic imaginary field $K$. A theorem by Deuring ([13, paragraph 4], Theorem 12 on page 182 of Elliptic Functions by Serge Lang) ...
23
votes
3answers
2k views

Why are values of Eisenstein $E_2^*$ algebraic integers?

I'm looking for a proof that the following term is an algebraic integer whenever $\tau_N=\frac{N+\sqrt{-N}}{2}$ is a quadratic irrationality with class number $1$: $$A_N:=\sqrt{-N}\cdot\frac{E_2(\...
2
votes
1answer
107 views

Isogenies of degree 3 of elliptic curves with j-invariant 0

Let $E/\mathbb{Q}$ be an elliptic curve with $j(E)=0$. I.e., $E$ has Weierstrass equations $$ y^2 = x^3+ B$$ for some $B\in \mathbb{Q}$. $E$ has complex multiplication by $\mathcal{O}:= \mbox{ ring ...
1
vote
0answers
80 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{...
6
votes
1answer
528 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,...
4
votes
0answers
193 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 ...
1
vote
0answers
125 views

$F$-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
0answers
111 views

Example of the Main Theorem of Complex Multiplication [closed]

I am trying to understand the main theorem ov CM for elliptic curves. I work with the version stated in the second chapter of Silvermans "Advanced Topics in the Arithmetics of Elliptic Curves". I ...
6
votes
0answers
210 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, ...
8
votes
2answers
253 views

Number of points of elliptic curve over $\mathbb{F}_p$ with CM by $\sqrt{-2}$ when $p\equiv 1\bmod 8$

I was trying to calculate the number of points of the curve $E:y^2 = x^3 + 4x^2 + 2x$ over $\mathbb{F}_p$ for $p\equiv 1\bmod 8$ (In order to have $\sqrt{-2}\in\mathbb{F}_p$) but I did not succeed. ...
11
votes
0answers
312 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. ...
6
votes
1answer
302 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 ...
1
vote
0answers
177 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 ...
2
votes
1answer
184 views

CM Elliptic Curves and a result concerning ray class fields

Let $K$ be an imaginary quadratic field and suppose that $E/K$ has complex multiplication by $\mathcal{O}_K$. Let $\psi$ be the Hecke character associated with $E$ and $\mathfrak{f}$ its conductor (i....
3
votes
0answers
105 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 \...
3
votes
0answers
87 views

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}...
6
votes
0answers
307 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 ...
6
votes
1answer
455 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 ...
4
votes
0answers
177 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 ...
3
votes
1answer
242 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 ...
10
votes
1answer
460 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 ...
11
votes
2answers
522 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 ...
6
votes
1answer
672 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 ...
6
votes
1answer
371 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
votes
1answer
285 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 $\mathbb{Q}_{p}$ ...
1
vote
0answers
68 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 ...
3
votes
0answers
232 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 ...
2
votes
1answer
272 views

Reduction of Abelian Varieties with Complex Multiplication have Complex Multiplication

Let $A$ be an abelian variety of dimension $g$ over $C$ with complex multiplication by a CM field $K$ where $[K:Q] =2g$. By this I mean that End($A$) $\cong \mathcal{O}_K$. Then, $A$ has a model over ...
5
votes
2answers
401 views

Field cut out by a CM modular form is imaginary

Let $f=\sum_{n=1}^\infty a_nq^n$ be a newform of level $N$ and weight $k\ge 2$. Suppose that $f$ is a CM modular form in the sense of §3 of Ribet's paper Galois representations attached to eigenforms ...
2
votes
0answers
272 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^...
5
votes
1answer
651 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 ...
9
votes
1answer
419 views

Why there are only finitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves with CM by $\mathcal{O}$?

For someone who does not have a very extensive knowledge of number theory, what is a good intuitive explanation as to why there are only finitely many $\overline{\mathbb{Q}}$ isomorphism classes of ...
4
votes
0answers
345 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 ...
3
votes
1answer
314 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 $K$, ...
3
votes
1answer
239 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\...
5
votes
1answer
286 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 ...