# Tagged Questions

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### the CM type of a CM abelian variety

Let $(A, F, i)$ be a CM abelian variety, by which I mean an abelian variety $A$ defined over $\overline{\mathbb{Q}}$, say of dimension $n$, a CM number field $K$ of degree $2n$ and an embedding $i: F ...

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### Canonical lifts from $\mathbb F_q$ and CM-theory

One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...

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### abelian varieties with the same CM type are isogenous

Does anybody have a reference for the following fact?
All abelian varieties with complex multiplication and same CM type are isogenous over $\overline{\mathbb{Q}}$?
Here abelian variety with ...

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### Hasse-Weil L-Functions of CM Abelian Varieties

In Shimura's paper "On the Zeta Function of an Abelian Variety With Complex Multiplication", in his terminology, the `one-dimensional part' of the zeta function is identified with a Hecke $L$-function ...

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### can all CM types be realized by Jacobians?

The question is kind of self contained, but let me develop a bit further.
Assume K is a CM field of degree $2g$, that is, a quadratic imaginary extension of a totally real field. A CM type of $K$ is ...

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### field of definition of abelian varieties with extra endormorphism

Let $A$ be a complex abelian variety such that $\mathrm{End}(A)$ is strictly bigger than $\mathbb{Z}$.
Question: Is is true that $A$ is defined over $\overline{\mathbb{Q}}$?
This is of course what ...

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### complex multiplication

For an abelian variety $A$, it is said to be have $complex \ multiplication$ if $\mathrm{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ contains a number filed $F$ of degree $2 \cdot \mathrm{dim} (A)$. ...

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### books (or notes) on complex multiplication

This would be a vague question, but I still want to ask here. Do you have any recommended book on complex multiplicaton. I know only 2 books: Shimura's book ...