In appendix 2 of Katz's "p-Adic properties of modular schemes and modular forms", he describes a certain "Frobenius" endomorphism on the de Rham cohomology of ordinary elliptic curves and I am interested in understanding how this relates to the absolute Frobenius. The construction is as follows:

Let $R$ be the ring of $p$-adic modular functions of level $n$ over $W(\mathbb{F}_q)$ (with $q=p^k$ and $q \equiv 1 \mod n$), $E/R$ the universal ordinary elliptic curve and $H \subset E$ its canonical subgroup. Let $E'=E/H$ and $\pi: E \to E'$ denote the projection, then this induces a $R$-morphism $\pi^*:H^1_{dR}(E'/R) \to H^1_{dR}(E/R)$. Let $\varphi$ be the unique hom $\varphi: R \to R$ such that $E'=E^{(\varphi)}$ and using this defines a $\varphi$-linear endomorphism of $H^{1}_{dR}(E/R)$ by $F(\varphi):=\pi^{*} \circ \varphi^{-1}$.

**My question is what is the relationship between $F(\varphi)$ and absolute frobenius?**

If I understand things correctly, for an elliptic curve $E/\mathbb{Z}_p$, one can write absolute Frobenius as $$H^1_{dR}(E/\mathbb{Z}_p)\to H^1_{dR}(E'/\mathbb{Z}_p)\to H^1_{dR}(E/\mathbb{Z}_p)$$ where the second map is given by $\pi^*$ (in this setting) and the first map is given by the identification with crystalline cohomology of the special fibre.

The "Frobenius" in Katz's appendix respects the Hodge filtration whilst I think absolute Frobenius can't. For example:

- In the above composite for the absolute Frobenius, the second map respects the filtration whilst the first map usually won't (two lifts of the same curve will have very different Hodge filtrations).
- By Serre, the Tate module of an ordinary elliptic curve over $\mathbb{Q}_p$ decomposes as a direct sum of two one dimensional representations if and only if it has CM. This appears to happen if Frobenius respects the Hodge filtration.