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1

Yes. This is easy to see in terms of Galois cohomology. Let's work in the category $\mathcal{A} \otimes \mathbb{Q}$ of "abelian varieties up to isogeny". Then for an object $A$ of $\mathcal{A} \otimes \mathbb{Q}$, we have $$\operatorname{Aut}(A_{\overline{K}}) = \mathbb{Q}^\times = \left\langle - 1 \right\rangle \oplus \bigoplus_{p \textrm{ prime}} ...


3

Yes. The group of isogenies form a locally free module of rank $1$ over the endomorphism ring of $A$, hence are generated by a single isogeny of minimal degree $k$. So every isogeny is that isogeny composed with an endomorphism of $k$, so has degree $n^{2g} k$ for $n \in \mathbb Z$. Hence there are two isogenies with a given degree. The Galois action on the ...


1

I hesitate to answer my own question, but the answer below may be useful to some people. The Siegel cusp form $\chi_{10}$ of weight $10$ of genus $2$ has the following asymptotic behaviour as $z$ goes to $0$ $$ \chi_{10}(x,y,z) = \eta(x)^{24}\eta(y)^{24}(\pi z)^2+O(z^4), $$ where $x,y,z$ are the coordinates of the Siegel upper-half plane with $z$ off ...


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The action of $L$ on global 1-forms would give an embedding of $L$ into the algebra $M$ of $g$-by-$g$ matrices over $\mathbb{Q}$ (since we're in characteristic 0). But any maximal commutative $\mathbb{Q}$-subalgebra of $M$ is $g$-dimensional.



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