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I wonder if there's any work that considers the algebracity/transcendentality of Coleman integral (over $\mathbb{Q}_{p}$). The reason I think about this is because, for hyperelliptic curves, there are some theories about such things over $\mathbb{C}$. For example, the period integrals like $$ \int_{1}^{\infty} \frac{dx}{\sqrt{x^{3} - x}} = \frac{\Gamma(1/4)^{2}}{2^{3/2}\pi^{1/2}} $$ is known to be transcendental by Siegel (see here). In case of Coleman integral, it seems that the exact value of the integral is not the most interest of people, but its nonzeroness is much more important due to its relation to the rational points on curves. There are some explicit examples in this thesis and also in other papers, and the exact values of them seems to be out of interest.

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    $\begingroup$ There is a relation between Coleman integrals and non-critical values of p-adic zeta functions; and p-adic zeta values are known to be irrational over $\mathbb{Q}$ in some cases. See e.g. this beautiful paper of Calegari: doi.org/10.1155/IMRN.2005.1235. This is vastly weaker than what you asked for, but I thought I'd post it in case it's of interest. $\endgroup$ Nov 14, 2021 at 10:29
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    $\begingroup$ Simple cases of Coleman integrals give the $p$-adic logarithms, $p$-adic elliptic logarithms, ... There is quite a bit of research on linear forms of such. In certain iterated integration, terms linked to $p$-adic height pairings appear. Apart from cm elliptic curves, the transcendence of these are unknown, but conjectured. Maybe you need to narrow down, which integrals you are interested in. $\endgroup$ Nov 14, 2021 at 12:17

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