# Questions tagged [complex-multiplication]

The complex-multiplication tag has no usage guidance.

101
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### 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 ...

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### Effective Hecke Equidistribution

In 1918 and 1920 Hecke introduced his L-functions attached to his Grössencharakteren (Hecke characters) and proved they are equidistributed in a sense to made precise momentarily. One can identify ...

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### Eisenstein $E_2$ at imaginary quadratic arguments

In the paper On Epstein's zeta-function, Chowla and Selberg give a formula for evaluating the Dedekind eta function
$$\eta (\tau)=e^{\pi i\tau/12}\prod_{n=1}^\infty (1-e^{2\pi i n\tau}),\quad \Im\tau\...

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### 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(...

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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 ...

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### What is Weber's mistake about Hilbert's 12th problem?

Today, We call the Kronecker's Jugendtraum Hilbert's 12th problem. But, Hilbert's interpretation of the "Jugendtraum" was not that intended by Kronecker.
And Weber missed his chance to ...

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### 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?...

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### Elliptic curve with CM and image of Galois representation in normalizer of nonsplit Cartan

I am trying to understand the following. Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication given by the ring of integers $\mathcal{O}_K$. We are given a fixed rational prime $p$ ...

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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 $R_F$....

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1
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### Why does $[I](P)＝0$ ($P\in E$) imply $[\psi(I)](P)＝0$ ? ($\psi$ is Hecke character of elliptic curve)

Let $K$ be a imaginary quadratic field, $R_K$ be ring of integers of $K$, and $E/K$ be elliptic curve which has CM over $K$.
Let $\psi_E$ be Hecke (Grössencharakter) character of $E/K$.
Let fix prime ...

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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 ...

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319
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### Primes of bad reduction for CM elliptic curves

$\DeclareMathOperator\Norm{Norm}$Suppose $E/\mathbb{Q}(j(E))$ is a CM elliptic curve and $d$ is a non-square. Let $E_d$ denote the twist of $E$ by $\mathbb{Q}(j(E))(\sqrt{d})$. I know if $d$ is ...

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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 \...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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### 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-...

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### 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....

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### $\mu=0$ 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 $...

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### 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 ...

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### 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
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1
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### 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 ...

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### 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 ...

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### 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 ...

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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 ...

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### 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) ...

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3
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### 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(\...

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1
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### 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 ...

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0
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### 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{...

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### 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,...

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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 ...

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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, ...

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### 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 ...

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### 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, ...

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### 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. ...

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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.
...

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### 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 ...

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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 ...

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### 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....

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### 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 \...

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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}...

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### 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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...