Faltings proved the following:
Fix integers $w, d \geqslant 0$, and fix a number field $K$ and a finite set $S$ of primes of $\mathcal{O}_K$. There are, up to conjugation, only finitely many semisimple Galois representations $\rho: G_K \rightarrow \mathrm{GL}_d\left(\mathbf{Q}_p\right)$ such that
(a) $\rho$ is unramified outside $S$, and
(b) $\rho$ is pure of weight $w$, i.e. for every prime $\mathfrak{p} \notin S$ all the eigenvalues of Frobenius at $\mathfrak{p}$ are algebraic integers, all of whose conjugates have complex absolute value $\left|\mathcal{O}_K / \mathfrak{p}\right|^{w / 2}$.
I wonder is there a good bound on the number of such representations?
