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The Cartier operator is defined for smooth varieties over finite fields. If $C$ is an algebraic curver over $\mathbb{Z}_p$. Let $C_p$ the reduction of $C$ to positive characteristic. My question is if we can lift the cartier operator from $C_p$ to $C$. If it is posible, in modular curves, this lift has relation with the Hecke operator?.

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