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.

It's important to be clear that this map on $H^1_{\mathrm{dR}}$ overlies a highly non-trivial map on the base-ring $R$. You can imagine a case where $R$ is something like $\mathbf{Z}_p\langle X \rangle$ and $\phi$ is $X \mapsto X^p$ (this is a bit of an oversimplification, but it might even be literally true for a few small values of $n$ and $p$). So $\phi$ acts highly non-trivally on the space $\operatorname{Spec} R$; the majority of $\mathbf{Z}_p$- points of Spec R won't be stable under $\phi$, and hence don't give us elliptic curves over $\mathbf{Z}_p$ with $\phi$ preserving the Hodge filtration. This will only happen for a few very special points which are preserved by $\phi$ -- e.g. the Serre canonical liftings of ordinary points of the special fibre will have this property.

  • $\begingroup$ Thanks! That makes a lot of sense. Do you know if there is a useful description of what (non-trivial) map is required to get from Katz's construction to the absolute Frobenius? For example, does something like Gauss—Manin help undo the fact that $\varphi$ has moved us to a different position in the family, as you mentioned? $\endgroup$ Apr 1 at 14:04
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    $\begingroup$ You're essentially asking for a rigid-analytic Frobenius lifting on $E$ (or on some open affinoid in it). These can be made really rather concrete, have a look at the literature on Kedlaya's algorithm for computing Frobenius via rigid cohomology. $\endgroup$ Apr 1 at 15:12

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