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Let $f=\sum_{n\geq 1} a_n q^n$ be a normalized eigenform which is supersingular and crystalline at a prime $p$ and let $V_f$ be the associated crystalline representation, then it follows from the work of Scholl and Faltings that the characteristic polynomial of the $\Phi$ operator on $D_{cris}(V_f^*)$ is $X^2-a_p X+p^{k-1}$. This is alluded to in "Explicit reduction modulo p of certain $2$-dimensional crystalline representations"- Buzzard and Gee.

I've been trying to find the location of this result by looking through some of Scholl's papers, in which paper can it be found?

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    $\begingroup$ My guess is that what you get from Scholl and Faltings is simply that the associated Galois representation is crystalline. What you want should follow from this and some comparison/compatibility results saying the the characteristic polynomial of the $\Phi$ operator is the same as the characteristic polynomial of $Frob_p$ acting on the l-adic representation for $l\neq p$ but I don't know a precise reference for this. $\endgroup$
    – naf
    Commented Dec 9, 2018 at 7:12
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    $\begingroup$ Did you check Scholl's "Motives for modular forms", Inventiones 1990? $\endgroup$
    – Laurent Berger
    Commented Dec 9, 2018 at 7:41
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    $\begingroup$ Kato in his Astérisque paper refers to Saito (Modular forms and p-adic Hodge theory, Inventiones 1997) for this result. $\endgroup$
    – François Brunault
    Commented Dec 9, 2018 at 8:38
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    $\begingroup$ In fact Saito determines the char poly of $\Phi$ without any assumption on $f $. $\endgroup$
    – François Brunault
    Commented Dec 9, 2018 at 8:56
  • $\begingroup$ @Laurent, I did take a cursory look at that paper of Scholl's. $\endgroup$
    – user130124
    Commented Dec 9, 2018 at 16:22

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