Let $M$ be a closed hyperbolic 3-manifold. One can use hyperbolic structure on $M$ to define hyperbolic volume $Vol(M)$. Thanks to Mostow's rigidity theorem the volume depends only on the topology of $M.$ Volume Conjecture gives a way to compute this invariant starting with any triangulation, by looking at the asymptotical behavior of Turaev-Viro invariants. These invariants are denoted $TV_n(M,q)$ and depend on the root of unity $q=e^{\frac{2\pi i}{n}}.$ They can be computed very explicitly for any triangulated $3-$manifold and are based on so-called $6j-$symbols for quantum group $U_q(sl_2)$. Here is the exact statement: $$ \lim_{n\longrightarrow \infty} \frac{2\pi i}{n}log\left ( TV_n(M,q) \right)=Vol(M). $$

Hyperbolic volume is an invariant of motivic origin. Namely, one can associate to every hyperbolic 3-manifold a motivic cohomology class in $H_{\mathcal{M}}^{1}(F, \mathbb{Q}(2))$ for some number field $F$. Hyperbolic volume $Vol(M)$ is obtained by applying Borel's regulator to it. It will be interesting to formulate an analog of the Volume Conjecture in some other motivic realization.

**Question 1** One can look at other regulators, related to other cohomology theories, for instance $p-$adic regulator. So, probably, there exists a $p-$adic version of hyperbolic volume. Has it been studied?

**Question 2** Is it possible to formulate an analog of the volume conjecture in this setting? For instance, substituting representations of $U_q(sl_2)$ with some similarly behaving category in characteristic $p?$

**Question 3** If the answer to both questions is "yes", is it possible to come up with a formulation of the Volume conjecture in some context, where analytical difficulties of proving convergence are less severe?