In this paper, we prove the potentially abelian case of the finiteness conjecture of Fontaine-Mazur (Conjectures 2b and 2c of [4]), assuming the rationality of the representations.
Let $K$ be an algebraic number field of finite degree over the rational number field $\Q$ and $\GK = \Gal(\Kbar/K)$ its absolute Galois group. Let $p$ be a prime number, $\Qp$ the $p$-adic number field, and $d$ a positive integer. Then we prove:
Theorem. There exist only finitely many isomorphism classes of potentially abelian $\Qbar$-rational semisimple geometric representations $\rho : \GK \to \GLd(\Qpbar)$ with a fixed Hodge-Tate type and bounded inertial level.
Note that we do not really need the condition ``geometric'' because it follows from the $\Qbar$-rationality ([5], Th\'eor\`eme 2).
A similar result is proved (for {\it admissible systems} of $\ell$-adic representations) in [1], 4.5; they prove it as a corollary to their finiteness theorem on representations of the Weil group of $\Q$ into $\GLd(\C)$ with bounded conductor. Here we give a direct $p$-adic proof.
We are motivated also by a similar question in the mod $p$ case ([8], [9]).
After giving some preliminaries in \S 1, we prove the Theorem in \S 2.