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I think there are references for this question, but I didn't find it. We know that for a simple abelian variety $A/k$, the rign $\mathrm{End}^0 (A)$ is a division algebra. One use the fact that every homomorphism $\phi : A \rightarrow A$ is either an isogeny or $0$. To see this, one consider the identity connected component $G^0$ of $G := \mathrm{ker}(\phi)$ and one shows that $G^0_{\mathrm{red}}$ is an abelian subvariety of $A$. But I can't find a reference for the geometrical reducedness of $G^0_{\mathrm{red}}$ in the general case ($k$ not necessary being perfect). It would be appreciated for a reference or an explanation.

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You can find a proof here following hints of Raynaud. The main idea is that the prime-to-$p$ torsion of $G^0$ is dense in $G^0_\mathrm{red}$ and its Zariski closure (with reduced structure) is geometrically reduced.

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