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Given an abelian variety $A$ over a field $K$ and a field extension $F/K$, there is a natural inclusion of endomorphism rings $\End(A)\subseteq\mathrm{End}(A_F)$, where $A_F$ denotes the base change of $A$ to $F$.

If $L$ is an extension of $K$, the set of endomorphism rings $\mathrm{End}(A_F)$ over subextensions $F/K$ forms a lattice (under inclusion). When $L$ is chosen so that $\mathrm{End}(A_L)=\mathrm{End}(A_{\overline{K}})$, this lattice is as large as possible and known as the endomorphism lattice of $A.$ Tensoring with $K$ (or any extension of $K$) yields a corresponding lattice of endomorphism algebras $\mathrm{End}(A_F)\otimes K$.

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• Review status: reviewed
• Last edited by Christelle Vincent on 2019-05-07 14:46:15
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