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This is a bit embarassing, but what are good examples of the lift of Frobenius of abelian varieties? In explicit terms? If I consider an ordinary abelian variety $A/k$, then by Serre-Tate theory there exists a canonical lift of $A$ to $W(k)$. However I find myself unable to write down a concrete example, or rather I'm stuck with the following computation: Consider $y^2=x(x-1)(x+2)$. This is ordinary for $p=5$. Then we have a lift of the abelian variety to $W(\mathbb{F}_5)=\mathbb{Z}_5$ which affine locally is given by $\mathbb{Z}_5[x,y]/(y^2-x(x-1)(x+2))$. Then to show that $$x\mapsto x^5, \quad y\mapsto y^5$$ is the Frobenius lift, I need to show that $$y^{10}-x^5(x^5-1)(x^5+2)\in (y^2-x(x-1)(x+2)).$$ However, that is not true by pluging-in $x=2,y=3$, then $$2^{10}-3^5(3^5-1)(3^5+2)$$ is not divisible by $(2^2-3(3-1)(3+2))$. So this can't be the lift of Frobenius. However, I don't know how to find it..... Any other explicit examples of lift of Frobenius would also be appreciated. I'd just like to have some examples on hand to be able to "play" with them.

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  • $\begingroup$ You mean divisible over the integers, or divisible modulo $5$? Does that fix things? $\endgroup$ Apr 12, 2021 at 22:32
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    $\begingroup$ You elliptic curve is isomorphic to $y^2 = x^3 + x$, so we can choose the lift with the same equation lmfdb.org/EllipticCurve/Q/64/a/4, unlike the lift you took this has CM by $\mathbf Z[i]$ so the lift of Frobenius should be one of the elements $1 + 2i$ in this ring. I can't see a quick way to calculate the formula for this isogeny, but there should be one that lifts the map $x\mapsto x^5 ,y\mapsto y^5$, note that this may have other terms whose coefficients are divisible by 5. $\endgroup$ Apr 13, 2021 at 1:36

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