On the Exceptional Sets of $p$-adic Transcendental Analytic Functions

In this paper, we study the exceptional sets $S_f$ of $p$-adic transcendental analytic functions $f$ with rational and algebraic coefficients. We establish a necessary condition for a subset $S\subseteq \bar{\mathbb{Q}}\cap B(0,\rho )$ to be the exceptional set of a $p$-adic transcendental analytic function with rational coefficients, demonstrating that, in general, the answer to Mahler’s Problem C over ${\mathbb{C}}_p$ is negative. However, we prove that if $S$ is closed under algebraic conjugation and contains 0, there exist uncountably many transcendental analytic functions $f\in {\mathbb{Q}}_{\rho }[[z]]$ such that $S_f=S$. Furthermore, if $\rho \ge 1$, $f$ can be taken in ${\mathbb{Z}}_{\rho }[[z]]$. Additionally, we demonstrate that any $S\subseteq \bar{\mathbb{Q}}\cap B(0,\rho )$ containing 0 can be the exceptional set of uncountably many transcendental analytic functions $f\in {\bar{\mathbb{Q}}}_{\rho }[[z]]$.