All Questions

Are there counterexamples to the following: Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism defined over $\...

Let $\tau$ in the upper half plane lie in an imaginary quadratic field $K$. Then is $K \subset \mathbb{Q}(\{j(g \tau) \ | \ g \in GL_2^+(\mathbb{Q}) \})$? (here $j$ is the modular $j$-function, and ...

Let $\mu$ be the roots of unity and $S$ be the image under the modular $j$-function of all imaginary quadratic $\tau$. Then what is $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?