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

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Hardware ADC Sample rate calculations

We have an ADC hardware which claims to provide samples at a rate of 3.6 GSPS (Giga samples per second). The interface multiplexes the data out of 2 parallel ports (the earlier sample of the 2 samples ...
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Big trouble multiplying 2 matrix

I'm having a big trouble when I have to multiply 2 matrix. I think I have a problem with my calculator (HP 50g) because I get a correct answer but not the one my professor has. For example, I have to ...
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The product of multiple univariate Gaussians

What is the final result of $I=\mathcal{N}_{x}(\mu_1,v_1)\,\mathcal{N}_{x}(\mu_2,v_2)\ldots\,\mathcal{N}_{x}(\mu_n,v_n)=\frac{1}{\sqrt{2\pi\,v_1} } e^{ -\frac{(x-\mu_1)^2}{2v_1} } ...
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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 ...
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413 views

What does the multiplication of standard deviation of two variables gives?

If we need to find the correlation between two variables it is given by the formula - co variance of two variables divided by the multiplication of Standard deviation of the two variables. My ...
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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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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{ ...
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437 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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Math and Taxes - Items versus Sum Total differs

First, The question is a bit confusing as I am not really sure how to word this problem as a question. The math problem I encountered which is a bit of an anomaly is this : Suppose you are ...
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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
143 views

Product of all complex roots of z^n=a+bi?

How can one prove that the product of all the roots of a complex equation is the same as one root to the power of equation? e.x. $z^n=a+bi$ has $n$ roots (from de Moivre's formula), prove that their ...
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1answer
114 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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173 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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1answer
75 views

Is Multiplication A System?

I don't understand how to identify the properties of a system. What possible properties could a system have? Are a certain number of properties required in order to be classified as a system? The ...
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687 views

Recursive FFT java implementation

Given below is my java program for FFT. For the input {0,2,3,-1} its returns a false output in complex point representation. ...
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203 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 ...