The Bogomolov property for $p$-supercuspidal eigenforms

We prove a lower bound on the Weil height, the so-called Bogomolov property, for the algebraic extensions of $\mathbb{Q}$ cut out by the adelic Galois representations attached to certain eigenforms whose local component at a prime $p$ is supercuspidal. To this end, we give a method for constructing metric inequalities over $p$-adic Lie extensions of fields over $\mathbb{Q}$ that are finitely ramified at $p$.