Linear independence of p-adic logarithms (analog of Baker's theorem)

Question

We have the following theorem of Baker:

Theorem 1. Let $\alpha_1, \ldots, \alpha_m \in \mathbb{C}$ be algebraic numbers $\neq 0, 1$ such that $\log \alpha_1, \ldots, \log \alpha_m$ are linearly independent over $\mathbb{Q}$. Then $\log \alpha_1, \ldots, \log \alpha_m$ are linearly independent over the algebraic numbers.

Here I read that this theorem has an analog in the $p$-adic setting, so I guess it holds something like:

Theorem 2 (?). Let $p$ be a prime number and $\alpha_1, \ldots, \alpha_m \in \{x \in \mathbb{C}_p : |x - 1|_p < p^{-1/(p-1)}\}$ be algebraic numbers such that $\log_p \alpha_1, \ldots, \log_p \alpha_m$ (where $\log_p$ is the $p$-adic logarithm) are linearly independent over $\mathbb{Q}$. Then $\log_p \alpha_1, \ldots, \log_p \alpha_m$ are linearly independent over the algebraic numbers.

My question is: Is Theorem 2 true? Where I can find a reference?

Thanks.

Answer 1

Yes. For linear independence, the result goes back to A. Brumer in connection with the Leopoldt conjecture. Brumer, A. "On the units of algebraic number fields", Mathematika 14 (1967) 121–124

For lower bounds on linear forms, there was an early version due to van der Poorten that I think had some problems and then a correct version was proved by Kunrui Yu. You should be able to find Yu's papers through Mathscinet. I am sure there are even more recent developments for linear forms.