On potentially abelian geometric representations (Q1430280)

Let \(K\) be a number field, and fix a natural number \(d\) and a prime number \(p\). The author proves that there are only finitely many isomorphism classes of potentially abelian geometric Galois-representations \(G_K\rightarrow \text{GL}_d(\overline{{\mathbb Q}_p})\) with fixed Hodge-Tate-type and bounded inertial level. This proves the potentially abelian case of a more general conjecture of \textit{J.-M. Fontaine} and \textit{B. Mazur} in [Coates, John (ed.) et al., Elliptic curves, modular forms, \& Fermat's last theorem; Cambridge, MA: International Press. Ser. Number Theory. 1, 41--78 (1995; Zbl 0839.14011)].