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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.

According to Wikipedia:

Complex multiplication is the theory of elliptic curves $E$ that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties $A$ having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at the identity element of $A$ is a direct sum of one-dimensional modules). Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.

It is useful in the study of class field theory.

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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. I'm not familiar with complex multiplication. I…
sherry
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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 $g$-dimensional abelian varieties over $K$ by…
Ashvin Swaminathan
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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 $z \in \mathbb{H}$, $f$ takes an algebraic value as…
Jesse
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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 $$y^2=x^3+g_2(\Lambda)x+g_2(\Lambda) \ \ \ \text{ where} \ \ \ \Lambda=\alpha\mathbb{Z}[i]$$ i.e. has…
Pulcinella
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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 $4 x^2 + 2 x y + 7 y^2 - z^3 \neq \pm 2 m^3, \; \pm 32 m^3$ when $m$ has certain prime…
Will Jagy
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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 like to study the arithmetical part related to…
Paramanand Singh
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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 references are Silverman's advanced topics on elliptic…
BlackAdder
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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 j-invariant 1, which in particular is an algebraic…
user3131035
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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 (-x,iy)$ make sense in the first place if $E$ is…
rollover
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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 Frobenius map $x\mapsto x^q$ is an endomorphism…
rogerl
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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.4.4, we have an injective ring morphism…
Alphonse
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Values of Grössencharacter attached to CM elliptic curve

Let $E$ be an elliptic curve defined over a number field $L$, having CM by by the ring of integers $\mathcal{O}_K$ for $K$ quadratic imaginary. If $K \subseteq L$, then (as constructed in Silverman's Advanced Topics in the Arithmetic of Elliptic…
Jupp
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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 Grossencharacter over $K$. My question is, if I take $E$ to be…
user119481
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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 $\mathbb C/\Lambda$ and $\mathbb C/O_K$ have the same area. The…
user8268
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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 field $H$ of $K$. Let it be given such a model. Let…
Bruno Joyal
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