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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 ...
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.