Let $f=\sum_{n,m \geq 0}^{\infty}[a_{nm}]p^ny^m \in \mathbb Z_p[[y]]$, where $a_{nm} \in \mathbb F_p$ and $[\cdot]$ means the Teichmüller lifting. Define $I(f)=\sum_{n \geq 0}[a_{nn}]p^nt^n \in \mathbb Z_p [[t]]$.

If $f$ is a rational function of $y$, must $I(f)$ lie in a finite extension of $\mathbb Q_p(t)$?

Motivation: the equal characteristic case is true (and the generalization to several variable case is also true in positive characterestic), for example see "Intégration sur un cycle évanescent" by Deligne, which gives a beautiful geometric proof.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.