Suppose $M$ is an object in the abelian category of mixed Tate motives over $\mathbb{Q}$, and it is an extension of $\mathbb{Q}(0)$ by $\mathbb{Q}(1)$
\begin{equation}
0 \rightarrow \mathbb{Q}(1) \rightarrow M \rightarrow \mathbb{Q}(0) \rightarrow 0.
\end{equation}
Suppose the Hodge realisation of $M$ is the one associated to $\log u, u \in \mathbb{Q}^\*$, i.e. $u$ is a nonzero rational number. Suppose $p$ is an unramified prime of $M$, and in the $p$-adic realisation of $M$, what is the matrix associated to the geometric Frobenius?


The matrix must be of the form
\begin{pmatrix}
1, ~~~0 \\
\*, 1/p
\end{pmatrix}
but is the unknown $\*$ in the matrix just the $p$-adic value of $\log u$, i.e. the $p$-adic logarithm valued at $u$.


Remark, I do not understand the $p$-adic realisations of the mixed Tate motives very well, so the statement of this question might not be very rigorous. References about the $p$-adic realisations of mixed Tate motives are welcomed.



$p$-adic realisation of Kummer motive and Frobenius matrix
The matrix entry $\*$ is $\log(u^{1-p})/p$ (where $\log$ is the $p$-adic logarithm). This can be obtained from the description of the inverse of Frobenius given in sections 2.9-2.10 of [1]. You can find more information about the crystalline realization of mixed Tate motives in section 4 of [2] (the Kummer motive shows up in Example 3).


[1] *Deligne, P.*, Le groupe fondamental de la droite projective moins trois points. , Galois groups over $\mathbb{Q}$, Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, 79-297 (1989). [ZBL0742.14022](https://zbmath.org/?q=an:0742.14022).


[2] *Yamashita, Go*, [**Bounds for the dimensions of $p$-adic multiple $L$-value spaces**](http://www.emis.de/journals/DMJDMV/vol-suslin/yamashita.html), Doc. Math. Extra Vol., Andrei A. Suslin’s Sixtieth Birthday, 687-723 (2010). [ZBL1251.11049](https://zbmath.org/?q=an:1251.11049).


