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1

You can always map $(T,h)$ to the trivial Shimura datum and then this into the Siegel one. I assume this isn't what you want however. May I therefore modify the question to ask whether a zero dimensional variety can be embedded in a Siegel variety. I.e. You want to know if all zero dimensional Shimura varieties are of Hodge type. I think this is certainly ...

1

If I am not mistaken, the claim that $\mathrm{Lie}(u')$ is not an inclusion of a direct summand is erroneous. Here is why. Let $G'$ be the abelian $R'$-scheme with generic fiber $F'_K/E'_K$. Then according to Raynaud's Theorem A.1 of the 1996 Compositio paper of Abbes and Ullmo the sequence $$E' \rightarrow F' \rightarrow G'$$ of abelian $R'$-schemes ...

4

Here are two (almost three) comments that you might find useful. I don't know the answer to your question in general unfortunately. First, Autissier proved that for abelian varieties over $\overline{\mathbb Q}$ with good reduction, the Faltings height of $A$ equals the theta height of $A$ plus some contribution at infinity. See the main theorem on p. 1 of ...

5

It suffices to prove the vector bundle property for all cohomological degrees when the base is an artin local ring $R$ (by considerations with direct limits to pass to the noetherian case and then using the standard base change formalism). We can make a flat local extension $R \rightarrow R'$ to an artin local ring with algebraically closed residue field, ...

11

Another proof that $L = \,\overline{\bf \!Q\!}\,$: Clearly $L$ is contained in $\,\overline{\bf \!Q\!}\,$, so we need only show $L$ contains every algebraic number $x \notin \bf Q$. Let $P(X)$ be the minimal polynomial of $x$. If $\deg P$ is odd, then the class of $((x,0)) - (\infty)$ is a $2$-torsion point on the Jacobian of the elliptic or hyperelliptic ...

11

If $\lambda\in\overline{\mathbb{Q}}$, the elliptic curve $$E_\lambda\colon y^2=x(x-1)(x-\lambda)$$ has $(\lambda,0)$ as $2$-torsion point and is defined over (a subfield of) $L=\mathbb{Q}(\lambda)$. Its Weil restriction $A_\lambda:=\operatorname{Res}_{L/\mathbb{Q}}(E_\lambda)$ is an abelian variety defined over $\mathbb{Q}$ and shares the same points of ...

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