# Notation for a canonical quotient of an abelian variety in positive characteristic

This may be a somewhat silly question, but there it goes.

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $A=A_{/k}$ be an ordinary abelian variety of dimension $g\geq1$. One knows that the $p$-torsion of $A$ is a product: $$A[p]=\hat A[p]\times T_p(A)\otimes(\Bbb Z_p/p\Bbb Z_p).$$ Here $\hat A[p]$ is the maximal connected subgroup, $T_p(A)\otimes(\Bbb Z_p/p\Bbb Z_p)$ is etale, both factors are subgroups of rank $p^g$ and they're Cartier dual of each other.

The completely inseparable isogeny obtained by quotient by $\hat A[p]$ is the relative Frobenius $F_k:A\rightarrow A^{(p)}$ where $A^{(p)}$ is the abelian variety obtained by twisting the $k$-structure of $A$ by the geometric Frobenius i.e. the automorphism $x\mapsto x^{1/p}$ of $k$.

The isogeny $A\rightarrow A^\prime$ obtained by quotient by $C=T_p(A)\otimes(\Bbb Z_p/p\Bbb Z_p)$ is etale and after an identification $A\simeq(A^\prime)^{(p)}$ is the Verschiebung $V_k:(A^\prime)^{(p)}\rightarrow A^\prime$ associated with $A^\prime$.

Is there a standard notation for the abelian variety that I denoted $A^\prime$? I have checked a few textbooks and lecture notes about abelian varieties and/or group schemes, but I seemed not to be able to find any.

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