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Questions tagged [complex-multiplication]

The theory of elliptic curves with large endomorphism rings. For questions on multiplication of complex numbers, use (complex-numbers) instead.

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CM Abelian surface with prescribed CM field

The number field $E=\mathbb Q(\sqrt{\sqrt{2}-3})$ is a CM-field since it is a totally imaginary extension of the totally real field $\mathbb Q(\sqrt 2)$. Is there a way to construct an abelian ...
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Abelian extension over imaginary quadratic field

Notation: For a finite abelian extension $F / K$, let $\mathfrak{f}_{F / K} \subset \mathcal{O}_K$ denotes its conductor such that $F$ is contained in the ray class field $K(\mathfrak{f})$. In ...
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six times is approx one times? huh?

So I'm discussing the number of atoms in the universe with a nice chap, but we can't seem to agree. Google told me the answer to be 10^82. He says it's 6x10^82. He says the six isn't relevant with a ...
Canis Fortunatus's user avatar
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How to construct an elliptic curve with complex multiplication and class number not 1?

There is a theorem that guarantees that when the j invariant of the elliptic curve E is an algebraic integer, the elliptic curve E has a complex multiplication. But how to construct an elliptic curve ...
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What is the result of (271698268*271698267)/2?

I need the answer in non-scientific form, to precision. The problem is I'm getting different results from different methods. Method Output Output (plain) Python ...
Dr-Bracket's user avatar
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Why this endomorphism over an elliptic curve $E$ acts as a multiplication-by-$m$ map?

Let $q \equiv 1 \pmod{4}$ be prime, let $E/\mathbb{F}_q : y^2 = x^3 + ax$ be an elliptic curve and let $i$ be a number satisfying $i^2 = -1$. Then, the map $\phi : E \to E$ such that $\phi(x,y) = (-x, ...
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Determine the numbers a and b that check the equality $\sqrt{\overline{aba}}=(a+b-1)\cdot \sqrt{a+b}$

Determine the numbers a and b that check the equality $\sqrt{\overline{aba}}=(a+b-1)\cdot \sqrt{a+b}$ MY IDEAS I thought of decomposing $\overline{aba}$ as $101\cdot a + 10\cdot b$ Then i thought that ...
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What is the reduction of the Weil pairing?

This is a question about the proof of II.4.4 in Silverman's Advanced Topics in the Arithmetic of Elliptic Cruves. In the proof, the author claims that there is an equality: $$\widetilde{e_E(x,y)} = e_{...
stillconfused's user avatar
2 votes
1 answer
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Why $[K:\Bbb{Q}]\leq2\dim A$ holds for CM abelian variety?

Let $A$ be an abelian variety. Let $K$ be a CM field. ${\rm End}^0(A):={\rm End}(A)\otimes_{\Bbb{Z}}\Bbb{Q} $. $A$ is said to have CM by $K$ if only if ${\rm End}^0(A)$ contains $K$. Then, why $[K:\...
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Complex multiplication on elliptic curves by $\sqrt{-11}$

I was trying to find the definition of CM by $\sqrt{-11}$, or at least a way to calculate it or to calculate a general CM. I can find a lot of examples on the LMFDB without any mention to how ...
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Definition of 'Abelian variety has CM by $K$'

Let $K$ be a number field, let $L⊆K$ be CM field. $A/K$ be an Abelian variety over $K$. $A/K$ is said to have CM by $L$ if there is embedding $L⊆End_K(A) \otimes \Bbb{Q}$. But if we adapt this ...
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Let $E/\Bbb{Q}:y^2=x^3+x$ has CM over $ \Bbb{Q}( \sqrt{-1})$. How can I prove a reduction of isogeny $[1+2\sqrt{-1}]$ is inseparable?

Let $E/\Bbb{Q}:y^2=x^3+x$ has CM by $ \Bbb{Q}( \sqrt{-1})$. How can I prove a reduction of isogeny $[1+2\sqrt{-1}] \pmod{(1+2\sqrt{-1}}$) is inseparable ? I know one way to judge this. Examing ...
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Let $E/ \Bbb{C} $ be an elliptic curve which has CM over $\Bbb{Q}( \sqrt{-5})$. Then, why $j(E)$ is real number?

Let $E/ \Bbb{C} $ be an elliptic curve which has CM over $\Bbb{Q}( \sqrt{-5})$. Then, why $j(E)$ is real number? If theory of complex multiplication is well known, we can explicitly calculate $j$ ...
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Tensor product vs ordinary product

I’m having a hard time understanding why can’t we see tensors as simple products on the base field. See, if i select two subfields of a field $(\mathbb{F}; +; *)$ in such a way that they form a finite ...
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Why $E/AutE$ is isomorphic to $\Bbb{P}^1$?

Let $E$ be an elliptic curve which has complex multiplication over number field $K$. Why $E/AutE$ is isomorphic to $\Bbb{P}^1$ ? I tried to use first isomorphism theorem. Let $E→\Bbb{P}^1$ be $(x,y)→x$...
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How convert Division to its multiplication equivalent [duplicate]

Question I have a division that is x / 2 and I know that is equivalent to x ⋅ 0.5 or $ \frac{x}{2}\; = x \cdot 0.5 $ What is ...
Federico Baù's user avatar
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Is $K(E_{tors})/K$ always infinite degree extension

This question is related to complex multiplication and Kronecker's young dream.   Let $K$ be a imaginary quadratic number field and let $E/\Bbb{C}$ be an elliptic curve which has complex ...
Poitou-Tate's user avatar
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Mistaken in Silverman's AAEC Theorem 4.3 ? Where am I mistaken?

According to Silverman's ''Advanced topics in the arithmetic of elliptic curves'', p122, theorem 4.3, Theorem4.3: Let $E_1,E_2,・・・,E_h$ be complete set of representatives for $ell(R_K)$(elliptic ...
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Complex multiplication over p adic fields

Elliptic curve $E$ over a field $K$ is said to have complex multiplication if $E$ has endomorphism except for multiplication by $n$($n∈\Bbb{Z}$) maps. We usually consider $K$ as $ \Bbb{C}$.In this ...
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Simplifying following integral related to Fourier transforms

I'm working on some operations relating to Fourier transforms. I would like to neatly combine the multiplication of these two integrals, preferrably grouping Y(w)'s or y(t)'s together somehow, y(t) ...
Kaan Kesgin's user avatar
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Explicit description of maps $\phi_\alpha\in End(E)$ for a CM elliptic curve

Let $E$ be an elliptic curve with complex multiplication. When $n\in \mathbb{Z}\hookrightarrow End(E)$, we can define its associated map as $\phi_n: P\mapsto P+..._{n \text{ times}}+P$. However when $\...
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Field over which CM endomorphism is defined

Let $E$ be an elliptic curve with coefficients over some number field $K$. Is it true that if $E$ has complex multiplication by $\mathbb{Q}[\sqrt{-D}]$, then any endomorphism $\phi: E \rightarrow E$ ...
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Lang's proof concerning ray class fields of imaginary quadratic number fields

In Lang's book Elliptic Functions, he shows how to generate the ray class field with conductor $N$ of an imaginary quadratic number field $k$ using the $j$-invariant of an elliptic curve $A/\mathbb{C}$...
mxian's user avatar
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Reductions of elliptic curves over number fields (implementation on Sage)

I have found two elliptic curves over a number field whose reductions are isomorphic to supersingular elliptic curves (used Deuring Lifting Theorem). How can I check which reduction is isomorphic to ...
Andy's user avatar
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Fast Multiplication of matrix and vector

Multiplication of the DFT matrix and any vector can be implemented by FFT. I'm interesting about other fast Multiplication. Suppose there are three matrix of size $N\times N$, $F$ , $Q$ and $P$, where ...
double lee's user avatar
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CM points on Shimura curves

I am trying to understand CM points on Shimura curves and I got confused. Before I get the point that I got confused and stuck I need to introduce some notations. $F$: a number field, $\mathbb{A}_f$: ...
user794509's user avatar
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Equivalent definitions of CM abelian varieties

I am reading Milne's notes on CM (page 27) https://www.jmilne.org/math/CourseNotes/CM.pdf He defined a CM abelian variety $A$ to satisfy $$2\dim A=[\text{End}^{0}(A):\mathbb{Q}]_\text{red}.$$ Then ...
finiteness's user avatar
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Definition of "CM-field"

(Sorry for my bad English. ) In Wikipedia, "CM-field" is defined as follows; A number field $K$ is a CM-field if it is a quadratic extension $K/F$ where the base field $F$ is totally real ...
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$\mathbb{Q}(\sqrt{p^*})$ is contained in the ring class field of conductor $p$

Let $K$ be an imaginary quadratic field, $p$ a prime of $\mathbb{Q}$ and $H_p$ the ring class field of $K$ of conductor $p$, i.e. the abelian extension of $K$ with Galois group isomorphic to the class ...
Fraz's user avatar
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Shimura reciprocity law

Let $X_s$ be the compact modular curve of level $\Gamma_0(N)\cap\Gamma_1(p^s)$, with $N\in\mathbb{N}$ and $p$ prime, $(N,p)=1$. Then noncuspidal points on $X_s$ correspond to triples $(E,\frak{n},\pi)$...
Fraz's user avatar
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Constructing a Hyperelliptic Curve with given characteristic polynomial

According to this lecture, how can a hyperelliptic curve (of genus $2$) be constructed in the following example? Let $C: f(x)=y^2$ (where $f(x)$ is of degree $5$) be the curve we want to construct ...
J. Linne's user avatar
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5 votes
3 answers
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Reduction of CM Elliptic Curves

I'm working on Exercise 2.30 of Silverman's Advanced Topics of Elliptic Curves: Suppose that $E/L$ is an elliptic curve with CM by an imaginary quadratic field $K$. Suppose that $L$ does not contain $...
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1 answer
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Example of complex multiplication for elliptic curve

In Mathematics of Isogeny Based Cryptography by De Feo, he mentions the following example: It seems I haven't understood something important about complex multiplication. How does $ (x,y) \mapsto (-...
rollover's user avatar
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Definitions of CM abelian varieties

I was reading through a presentation by Oort (https://www2.math.upenn.edu/~chai/UPenn2013-beamer.pdf) and noticed something which disturbed me: he defines (slide 38) a simple CM abelian variety to be ...
SdV's user avatar
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2 votes
1 answer
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Is the ring of integers of any imaginary quadratic field equal to $\mathrm{End}(E)$ for some $E / \Bbb Q$?

Let $K / \Bbb Q$ be an imaginary quadratic field, and let $O_K$ be its ring of integers. Is there an elliptic curve $E / \Bbb Q$ such that its ring of integers $\mathrm{End}_{\overline{ \Bbb{Q}}}(E)$ ...
Alphonse's user avatar
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1 answer
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Calculate missing vector based on the angle I'm supposed to get [closed]

Let's say I have a vector (-1,2,-3) and I want to combine this with an other vector, so the angle of these 2 vectors would come out to 30°. How would I go about calculating this? My professor showed ...
gamer42069's user avatar
2 votes
1 answer
61 views

When multiplication is defined on objects other than the real numbers, is there an attempt to define this operation in terms of addition?

I do not understand "abstract" multiplication. If for integers it is repeated addition (in n*k, n times where n is an integer), then what could it even mean for something as simple as ...
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Elliptic Curves with Complex Multiplication

So I've been reading about elliptic curves with CM recently. I am aware of the following theorem: Let $E/\mathbb{C}$ be an elliptic curve and let $\Lambda=\mathbb{Z}\oplus\mathbb{Z} \tau$ the ...
Nom's user avatar
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0 answers
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good reduction for CM elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve with Complex Multiplication by the ring of integers of an imaginary quadratic field $K$. Let $p$ be an odd prime of good supersingular reduction. We know by ...
debanjana's user avatar
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Finding the Hecke grossencharachter of a CM newform

The definition of "newform with CM" I'm interested in is given on p. 415 of this paper. You start by specifying your imaginary quadratic field $K = \mathbb{Q}(\sqrt{D})$ for $D < 0$, an ...
Freddie's user avatar
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Quaternionic and octonionic analogues of the Basel problem

It is a well-known fact that $$\sum_{0\neq n\in\mathbb{Z}} \frac{1}{n^k} = r_k (2\pi)^k$$ for any integer $k>1$, where $r_k$ are rational numbers which can be given explicitly in terms of Bernoulli ...
pregunton's user avatar
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2 votes
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modular forms with complex multiplication

I would like the definition of a modular form with complex multiplication and if possible a reference. Thank you !
Homieomorphism's user avatar
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1 answer
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Lattices which have complex multiplication by $\sqrt{-3}$

I’m reading David Cox’s “Primes of the form $x^2+ny^2$”. Right after Corollary 10.20 he says: First, consider all lattices which have complex multiplication by $\sqrt{-3}$. This means that we are ...
Nah's user avatar
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6 votes
1 answer
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Endomorphism rings of elliptic curves over finite fields

I understand that any elliptic curve $E$ defined over a finite field $\mathbb{F}_q$ has an endomorphism ring $End_{\overline{\mathbb{F}}_q}(E)$ that is strictly larger than $\mathbb{Z}$, since the ...
rogerl's user avatar
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Find the value of $k$ if $AB=BA$

$A=\begin{bmatrix} 2 & 6\\ 3 & 5 \end{bmatrix}$ $b=\begin{bmatrix} 1 & 0\\ 0 & k \end{bmatrix}$ Both are $2×2$ matrices. I have multiplied it but that's where I get stuck.
Muhammad Ali's user avatar
6 votes
1 answer
326 views

Endomorphism ring of $y^2=x^3-x$ over $\Bbb F_p$

Consider the elliptic curve $E$ defined by $y^2 = x^3-x$ over $\Bbb Q$. Let $p \equiv 3 \pmod 4$ be a prime, and $E_p$ be the reduction of $E$ modulo $p$. By Silverman "Advanced topics...", prop. II....
Alphonse's user avatar
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Design a matrix to reflect a signal

I have a signal $R$, can be thought of a vector of say 10,000 samples (created by taking magnitude of complex values). I have two more signals $T_1$ and $T_2$ (exact copy) each with 10,000 samples, ...
Nusrat's user avatar
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1 vote
1 answer
209 views

Clarification regarding Silverman's proof of the description of Hilbert class field of a quadratic imaginary field

I was reading the proof of the following fact from Silverman's Advanced Topics in the Arithmetic of Elliptic Curves (p. 122). The Hilbert class field of a quadratic imaginary field $K$ with ring of ...
standard reduction's user avatar
3 votes
0 answers
513 views

Why convolution in time domain is multiplication in frequency domain?

I am trying to understand intuitively why convolution is multiplication in frequency domain. I started at the mathematical derivation of this, but didn't understand what is happening intuitively. This ...
Omibuddyy's user avatar
1 vote
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194 views

Why is complex multiplication of an elliptic curve called complex multiplication?

Let $E$ be an elliptic curve over a field $k$. Let $\text{End}_k(E)$ denote the endomorphism ring of $E$, i.e., $$\text{End}_k(E) = \{\text{base point preserving morphism} \ f:E \to E\}.$$ Since ...
Yuhang Chen's user avatar