# Questions tagged [complex-multiplication]

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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 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? The ...
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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 $p$ ...
151 views

### 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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### 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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### 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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