The following question is related to the discussion based on the discussion in Elliptic curves by L. Washington.
Let $E$ denote an elliptic curve over a field $K$ and ${\overline{K}}$ denote the algebraic closure of $K$. It is well known that $E({\overline{K}})$ is an abelian group. Define $E[n]$ be the subgroup defined by $$ E[n] = \big\{ (x,y) \in E({\overline{K}}) ~:~ n (x,y) = \infty \big\} $$ where $\infty$ denote the trivial element (point at infinity) of $E({\overline{K}})$. The following is Theorem 3.2 (Page 79, Washington).
$\bf{Theorem:}$ If $n$ is a positive integer so that ${\mathrm{Char}}~K \nmid n$ or ${\mathrm{Char}}~K = 0$, then $E[n] = {\mathbb{Z}}_n \oplus {\mathbb{Z}}_n$.
If ${\mathrm{Char}}~K = p > 0$ and $p \mid n$, write $n = p^r n^{\prime}$ so that $p \nmid n^{\prime}$. Then $E[n] = {\mathbb{Z}}_{n^{\prime}} \oplus {\mathbb{Z}}_{n^{\prime}}$ or ${\mathbb{Z}}_n \oplus {\mathbb{Z}}_{n^{\prime}}$.
Most of the proof assumes that the endomorphism $[n] =$ multiplication by $n$ to a point is a non-zero endomorphism. Notice that $E[n]$ is the kernel of $[n]$ here.
$\bf{Question:}$ Is there a classification to when $[n]$ can be a zero endomorphism?
For example if $K = {\mathbb{Q}}$, a deep theorem of Mazur (Rational isogenies of prime degree, Invent. Math. 44 (1978) 129–162) proves that the torsion subgroup $E_{\mathrm{tor}}$ is one of the following:
${\mathbb{Z}}_m ~(m = 1, \dotsc, 10, 12)$ or ${\mathbb{Z}}_2 \oplus {\mathbb{Z}}_{\nu} ~(\nu = 1, 2, 3, 4)$.
$\bf{Additional Remark:}$ From Mordell-Weil theorem it follows that $E({\overline{K}})$ is finitely generated. Now write $E({\overline{K}}) = {\mathbb{Z}}^r \oplus E({\overline{K}})_{\mathrm{tor}}$, and call $r$ as the infinite rank of $E({\overline{K}})$. In the paper by Rose(1995), it was pointed out that $r \neq 1$ for $K = {\mathbb{Q}}$ (possibly a consequence of BSD, or something simpler??). This paper constructs infinitely many examples so that $E({\mathbb{Q}}) = 0$ (opposed to the full $E({\overline{\mathbb{Q}}})$).
So the question is, is $r \neq 0$ for algebraically closed field is expected?