4 problems

Let $K$ be a number field and let $A/K$ be a principally-polarizable abelian surface whose absolute endomorphism ring satisfies … Let $\pi_{\operatorname{split}}(A/K,z)$ count prim…

Let $p$ be the prime defining the quaternion algebra $B_p$, and let $\mathcal{O},\mathcal{O}'\subset B_p$ be maximal orders of different types. For an integer $b>0$, say that…

Let ${\bf R}$ be a Noetherian local ring, let $E$ be a local ${\bf R}$-module, and let ${\Lambda}=\operatorname{End}_{\bf R}(E)$. Suppose that $E$ is ${\Lambda}$-cyclic and has fin…

Let $X$ be an absolutely simple abelian variety over a number field $K$. Say that a prime of $K$ is a splitting prime for $X$ when the reduction of $X$ at that prime is not simple.…