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Asvin
Asvin
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Let $R$ be the ring of integers in a (complete) algebraic closure of $\mathbb Q_p$ with maximal ideal $\mathfrak p$. Suppose I have an Abelian surface $\mathcal A/R$ such that over every $R/\mathfrak p^n$, there exist elliptic curves $E_n, E_n'$ over $R$ such thatwith $\mathcal A$ is isogenous to $E_n\times E_n'$ over $R/\mathfrak p^n$.

Does this imply that $\mathcal A$ splits over $R$? Note that I do not require any compatibility conditions between the elliptic curves or the isogenies over varying $n$.

Let $R$ be the ring of integers in a (complete) algebraic closure of $\mathbb Q_p$ with maximal ideal $\mathfrak p$. Suppose I have an Abelian surface $\mathcal A/R$ such that over every $R/\mathfrak p^n$, there exist elliptic curves $E_n, E_n'$ over $R$ such that $\mathcal A$ is isogenous to $E_n\times E_n'$ over $R/\mathfrak p^n$.

Does this imply that $\mathcal A$ splits over $R$? Note that I do not require any compatibility conditions between the elliptic curves or the isogenies over varying $n$.

Let $R$ be the ring of integers in a (complete) algebraic closure of $\mathbb Q_p$ with maximal ideal $\mathfrak p$. Suppose I have an Abelian surface $\mathcal A/R$ such that over every $R/\mathfrak p^n$, there exist elliptic curves $E_n, E_n'$ over $R$ with $\mathcal A$ isogenous to $E_n\times E_n'$ over $R/\mathfrak p^n$.

Does this imply that $\mathcal A$ splits over $R$? Note that I do not require any compatibility conditions between the elliptic curves or the isogenies over varying $n$.

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Asvin
Asvin
  • 7.7k
  • 2
  • 21
  • 52

Lifting a splitting of an Abelian variety to characteristic 0

Let $R$ be the ring of integers in a (complete) algebraic closure of $\mathbb Q_p$ with maximal ideal $\mathfrak p$. Suppose I have an Abelian surface $\mathcal A/R$ such that over every $R/\mathfrak p^n$, there exist elliptic curves $E_n, E_n'$ over $R$ such that $\mathcal A$ is isogenous to $E_n\times E_n'$ over $R/\mathfrak p^n$.

Does this imply that $\mathcal A$ splits over $R$? Note that I do not require any compatibility conditions between the elliptic curves or the isogenies over varying $n$.