arXiv subjects
Publications and source records attributed to Pietro Piras.
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 $p$.
A field in which the (logarithmic) Weil height is bounded from below by a strictly positive constant is said to have the Bogomolov property (property (B)). Given a normalized eigenform $f\in S_k(\Gamma_0(N))$ Amoroso and Terracini proved (B) for the field "cut out" by the adelic representation associated to $f$ under some assumptions on $f$, generalizing the earlier work of Habegger on elliptic curves. In this paper we extend this result to the case of normalized eigenforms with nontrivial nebentypus character. We also introduce the notion of ADZ field, inspired by earlier work of Amoroso, David and Zannier, exhibiting a class of fields in which property (B) is preserved under (arbitrary) composition.