## New answers tagged abelian-varieties

3

Albert's classification works/is good enough for (algebraically closed) fields of characteristic zero. For the complete list of all possibilities for a given $g$ (including the case of prime characteristic) see a survey of Frans Oort:
``Endomorphism algebras of abelian varieties". Algebraic geometry and commutative algebra, Vol. II, 469–502, Kinokuniya, ...

3

the Rosati form is defined in definition 2.18 of Sato-Tate distributions and Galois endomorphism modules in genus 2

3

Recently I was wondering about generalizations of Beyli's theorem to higher dimensions and did some googling. As this issue is only discussed briefly in David Roberts' comment, I thought I contribute what references I found hoping someone might find it useful:
There is one direction of research which looks for actions of ...

0

I will answer the first formulation of the question: As ulrich said, the answer is that the codimension is $g-1$, given by $\mathcal{A}_{g-1}\times X(1)$ mapping quasi-finitely to $\mathcal{A}_g$.
To see this, let us work in Siegel space $\mathbb{H}_g$ with $Sp_{2g}(\mathbb{R})$ acting in the usual way. Having an extra automorphism is equivalent to being a ...

1

Let me call $m$ the symmetric biIinear form associated to $M$. In your first question you mean "all quadratic forms whose associated bilinear form is $m$"; then the answer is yes. There is indeed a canonical, simply transitive action of $V$ on the set of such quadratic forms; for $v\in V$, the form $v\cdot q$ is given by $x\mapsto q(x)+m(v,x)$. That does ...

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