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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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Question about multiplication/divisions of logarithms

While solving the following logarithm question: $\frac{\log_3 135}{\log_{15} 3}\ - \frac{\log_3 5}{\log_{405} 3}\\$ I came to the point where I have to multiply two equal log: $\...
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Multiplication by an element inducing the identity on a quotient

I am studying the chapter on the associated Grössencharacter of a CM ellipic curve in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves (II.9.) and have a question concerning a specific ...
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1answer
53 views

Methods for solving Elliptic curve over Q taking advantage of Complex Multiplication

In "An Introduction to the Theory of Numbers" by Hardy and Wright, they tantalizingly introduce a bunch of properties of elliptic curves, including the possibility of having Complex Multiplication, ...
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Complex operator $i$ and Exponents

I am trying to understand the complex numbers and exponents. I came across this question. I wonder how to explain the difference between $${2\cdot i} \text{ and } 2^i$$ as $i=\sqrt{-1}$ edit: ...
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How to prove the integrality of this Eisenstein Series?

Cohen and Strömberg included in their book Modular Forms: A Classical Approach the chapter "A Brief Introduction to Complex Multiplication" (pp. 199-203). In this chapter (p. 202) we find Proposition ...
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Problem in complex number multiplication [duplicate]

I want to know $\sqrt{-m}\sqrt{-n}=$? I tried in the following ways: Way 1:$$\sqrt{-m}\sqrt{-n}=\sqrt{(-m)(-n)}=\sqrt{mn}.$$ Way 2:$$\sqrt{-m}\sqrt{-n}=\sqrt{m}i\sqrt{n}i=\sqrt{mn}i^2=-\sqrt{mn}$$ ...
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1answer
52 views

Endomorphism ring of elliptic curves over $\Bbb Q$ vs over $\Bbb C$

Let $E$ be an elliptic curve over $\Bbb Q$. What is the relation between $End(E)$ and $End(E_{\Bbb C})$ ? We clearly have an inclusion $End(E) \subset End(E_{\Bbb C})$ : given $f :E\to E$, we can ...
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Matrices Multiplication [duplicate]

What is your favorite or own discovered way of doing matrices multiplication? For example: How can you multiply 3 x 3 matrices in an easy way?
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For which CM points $\tau$ is $\gamma(\tau)$ also a CM point?

Let $\tau$ be a CM point in the upper half plane $\mathcal{H}$ - that is, an element of $\mathcal{O}_K$ for an imaginary quadratic extension $K/\mathbb{Q}$ that lies in $\mathcal{H}$ after choosing an ...
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499 views

An explicit equation for an elliptic curve with CM?

The elliptic curve $$y^2=x^3+x$$ has complex multiplication by $i$ (the action of $i$ is $y\to iy$ and $x\to -x$), and any such has equation $$y^2=x^3+g_2(\Lambda)x+g_2(\Lambda) \ \ \ \text{ where} \ \...
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Is the standard complex multiplication the only one that makes $\Bbb R^2$ a field?

If $(a_1,a_2)*(b_1,b_2)=(a_1b_1-a_2b_2,a_1b_2+a_2b_1)$ then $\Bbb R^2$ is a field with pointwise addition and $\ast$ Is there another multiplication $\ast$' that has this property?
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Why does the multiplication in a division algebra depends on every component?

In a division algebra A over $\mathbb{R}$ we have this multiplication (A isomorphic to $\mathbb{R}^{n}$) $$\mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}^{n}:(x,y)\mapsto y=x\cdot y$$ where every ...
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CM elliptic curves and isogeny

Q: Show that any two CM elliptic curves with the same endomorphism algebra (say $K$) are isogenous. I was thinking of the following: Let $E_1\simeq \mathbb{C}/L_1$ and $E_2\simeq \mathbb{C}/L_2$, ...
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237 views

What is the Grossencharacter of this CM curve?

Let $E$ be an elliptic curve with complex multiplication by $\mathcal{O}$, the ring of integers of some imaginary quadratic field $K$. Then, the CM theory says that $E$ is related to a ...
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2answers
943 views

Finding an elliptic curve with CM by $\mathbb{Z}[\sqrt{-17}]$

I have the imaginary quadratic field $K= \mathbb{Q}(\sqrt{-17})$ with $\mathcal{O}_K = \mathbb{Z}[\sqrt{-17}]$. Now I want to have the $j$-Invariant of an elliptc curve $E$ with complex ...
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How does the Frobenius work on the Torsionpoints of an ellitptic curve with CM

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 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, that is, $p=(a+i)(a-i)$ over $\mathbb{Z}[i]$. I just ...
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Curves on reducible CM-abelian surfaces.

Let $E$ be an elliptic curve with complex multiplication. Consider an abelian surface $A = E \times E$. What can be said about curves on $A$? I guess they must be rather specific. I'm not even sure ...
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Finding Number Of Terms In a Given Number [closed]

Finding The Number Of Digits In Any Given (x)^n Type Of Number ? Eg-1997^8 ,462^31,697^5
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Index of Galois Representation of N-torsion for CM Elliptic Curves

I am currently working on a problem and I have no idea how to start. Following the proof of theorem 2.3 (pg.112-113) in Silverman's Book "Advanced Topics in the Arithmetic of Elliptic Curves". Let $...
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Relation between Discriminant of an Order and J invariant

I am reading this paper (Section 4) here they want to construct an elliptic curve over $\mathbb{F}_q$ whose order has discriminant $'x'$. They do that in two steps : Firstly they calculate the j-...
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How many elliptic curves have complex multiplication?

Let $K$ be a number field. Suppose we order elliptic curves over $K$ by naive height. What is the natural density of elliptic curves without complex multiplication? More generally, suppose we order $...
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1-1 correspondence of class group of an order '$\mathcal{O}$' and elliptic curves having complex multiplication by $\mathcal{O}$

I came across these two results Let $\mathcal{O}$ be an order in an imaginary quadratic field.There is a 1−1 correspondence between the ideal class group $C(\mathcal{O})$ and the homo-thety classes ...
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On Schoof's proof of deterministically finding $\sqrt{x} \bmod p$ when $p \neq 1 \bmod 16$

I am reading Schoof's paper in which he gave a polynomial time algorithm for counting points on an Elliptic curve over Finite field there he gave as an application an algorithm for deterministically ...
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287 views

How to test if a given elliptic curve has complex multiplication

Is there a general, reasonably easy to understand, algorithm for testing whether an elliptic curve has CM? For example, consider the curve $y^2=x^3+\frac{27}{1727}x+\frac{54}{1727}$ This has j-...
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1answer
357 views

Lattices in $\mathbb C$ as modules of the ring of integers in an imaginary quadratic field

Let $K$ be an imaginary quadratic number field and let $O_K\subset K$ be the ring of algebraic integers in $K$. Let us call a lattice $\Lambda\subset\mathbb C$ normalized if the tori $\mathbb C/\...
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2answers
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Quadratic twist of Elliptic curves with complex multiplication

Suppose $E/\mathbb{Q}$ is an elliptic curve that has complex multiplication by $\mathcal{O}_K$, where $K=\mathbb{Q}(\sqrt{D})$, for $D<0$ and squarefree. In "The main conjectures of Iwasawa ...
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632 views

Endomorphism Ring of an Elliptic Curve over Finite Field

Let $~E:y^2=x^3+x~$ be an elliptic curve over finite field $\mathbb{F}_{5},$ I compute the trace of Frobenius is $2$($E/\mathbb{F}_{5}$ obvious is ordinary). (By the theory of CM, I know (when $E$ ...
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When Frobenius map equal to multiplication-by-m map

Here is a homework, the result brought me some trouble. Let $p = 7$, and consider the finite field ${ \mathbb{F}}_{p^{2}}$ , which we may represent explicitly as $${ \mathbb{F}}_{p^{2}}\simeq {\...
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an Example of Elliptic Curve over finite field has no CM

I have known this property (from Silverman's The Arithmetic of Elliptic Curves): Let $\operatorname{char}(K)=p>0,$ and let $E/K$ be an elliptic curve with $j(E)~ \overline{\in}~ \overline{ \mathbb{...
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654 views

Where does a CM elliptic curve have bad reduction?

Let $d>1$ be square-free, and $K=\mathbf Q(\sqrt{-d})$. Choose an embedding of $K$ in $\mathbf C$, and let $E = \mathbf C/\mathcal O_K$. It is known that $E$ admits a model over the Hilbert class ...
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why say complex multiplication of elliptic curves is beautiful

David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science. Just as the title asked. ...
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Complex Multiplication of $y^2=x^3+B$

I would like to find out what the complex endomorphism for the class of elliptic curves given by $$y^2=x^3+B$$ looks like. I know that for the class of elliptic curves $$y^2=x^3+Ax,$$ the complex ...
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1answer
212 views

complex multiplication in elliptic curves

The following question is in my homework: How many complex elliptic curves (up to isomorphism) have complex multiplication by the ring $\mathbb{Z}[\frac{1+\sqrt{D}}{2}]$ of discriminant $D=-71$ and ...
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264 views

Galois Properties of the Values of Modular Forms

Let $f \in S_0(N)$ be a normalized Hecke eigenform. It is well known that its coefficients are algebraic integers, and $f^\sigma$ lies in $S_0(N)$ for $\sigma \in G_{\mathbb{Q}}$. At CM points $z \in \...
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Translation of Weber's Lehrbuch der Algebra vol 1, 2, 3

I have been trying to study elliptic functions and theta function for quite some time and have already got the hang of the classical theory (Jacobi/Ramanujan) based on real analysis, and now would ...
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Hecke Characters

My question today concerns Hecke characters or Größencharaktere. I'm doing a project on complex multiplication of elliptic curves and need some help understanding Hecke characters. My main ...
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317 views

What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$

EDIT: a selection of material relevant to this problem is available at INHOMOGENEOUS In December 2010 my question appeared in the M.A.A. Monthly, show that $4 x^2 + 2 x y + 7 y^2 - z^3 \neq \pm 2 m^...