# Mixed characteristic analogue of algebraicity of the diagonal of two-variable power series?

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 characteristic), for example see "Intégration sur un cycle évanescent" by Deligne, which gives a beautiful geometric proof.

• Interesting question. I checked that for $1/(1-[\alpha]x) + 1/(1-[\beta]x)$, the resulting diagonal is rational, as $a_{nn}$ is the $n$-th Witt digit of $[\alpha^n] + \beta^n]$, which is periodic in $n$. But for $-x/(1-x)^2$, the diagonal is $\sum [c_n] (px)^n$, where $c_n$ is the $n$-th Witt digit of $n$. I don't know if the sequence $c_n$ satisfies a linear recurrence, but if it works then I am optimistic about the general case. Apr 9, 2019 at 23:17