Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta\_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta\_8$ is a primitive 8-th root of unity. Let $\Lambda={{\mathbb{Z}}}[\zeta\_8]$ denote the ring of integers of $L$.



> 
> **Question 1.** Does there exist an abelian variety $A$ over ${{\mathbb{C}}}$ of dimension **two** with
>  ${\mathrm{End\,}} A\simeq \Lambda$ ?
> 
> 
> 


I expect the answer "no". Note that I want the endomorphism ring to be exactly $\Lambda$, not just contain $\Lambda$. In particular, my abelian variety should be simple.



> 
> **Question 2.** Does there exist an abelian variety $A$ over ${{\mathbb{C}}}$ of dimension **four** with
>  ${\mathrm{End\,}} A\simeq \Lambda$ ?
> 
> 
> 


I expect the answer "yes".



> 
> **Question 3.** Is it possible to find an abelian variety $A$ over ${{\mathbb{C}}}$ of dimension 2 or 4 with
>  ${\mathrm{End\,}} A\simeq \Lambda$ and such that $A$ is definable (without endomorphisms) over a small number field,
>  say over ${{\mathbb{Q}}}(i)$ or even over ${{\mathbb{Q}}}$?
> 
> 
> 



Abelian variety with prescribed endomorphism ring
The answer to your Question 3 is YES with the ground field $\mathbb{Q}$.


Here is a sketch of the proof. For each positive integer $q$ and a "parameter" $t$ (in char 0) consider the smooth projective model $C\_{q,t}$ of an affine curve $y^q=x^3-x-t$. Let $P\_{8,t}$ be the Prym variety of the double cover
$$C\_{8,t}\to C\_{4,t}, (x,y)\mapsto (x,y^2).$$
Then $P\_{8,t}$ is an abelian fourfold provided with an embedding 
$$\mathbb{Z}[\zeta\_8]\hookrightarrow End(P\_{8,t}).$$
One may deduce from Theorem 1.5 of [arXiv:math/0601072 [math.AG]](http://arxiv.org/abs/math/0601072) that if $t$ is a transcendental number then $P\_{8,t}$ does not contain positive-dimensional abelian subvarieties of CM type. It follows from Th. 1.1 of a paper by [Jiangwei Xue and Chia-Fu Yu, arXiv:1304.6202 [math.NT]](https://arxiv.org/abs/1304.6202) that for such $t$ the endomorphism ring $End(P\_{8,t})$ coincides with $\mathbb{Z}[\zeta\_8]$. Now, by [Masser's specialization theorem](http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1996__124_3/BSMF_1996__124_3_457_0/BSMF_1996__124_3_457_0.pdf)
one may choose a rational number $c$ such that
$End(P\_{8,c})$ equals $End(P\_{8,t})$
and therefore coincides with $\mathbb{Z}[\zeta\_8]$.


