There’s a rumour going around that results from transcendence theory are required to prove the Fontaine-Mazur conjecture for . This is not correct. In Serre’s book on -adic representations, he defines a -adic representation of a global Galois group to be rational if it is unramified outside finitely many primes and if the characteristic polynomials of actually all lie in some fixed number field rather than over . Certainly being rational is a consequence of occurring inside the etale cohomology of a smooth proper scheme , and one might be motivated to make a conjecture in the converse direction assuming that is absolutely irreducible. But being “rational” is just a rubbish definition (sorry Serre), a mere proxy for the correct notion of being potentially semistable at all primes dividing (“geometric,” given the other assumptions on ). And the implication
A character is Hodge-Tate is automorphic
doesn’t require any transcendence results at all. One can’t really blame Serre for not coming up with the Fontaine-Mazur conjecture in 1968. The reason for this confusion seems to be the proof of Theorem stated on III-20 of Serre’s book on abelian -adic representations (with the modifications noted in the updated version of Serre’s book), namely:
Theorem (Serre-Waldschmidt): If is an abelian representation of which is rational, then is locally algebraic.
This argument (even for the case when is a composite of quadratic fields, the case considered by Serre) requires some transcendence theory. But the implication is abelian and Hodge-Tate is locally algebraic (also proved in Serre) only uses Tate era p-adic Hodge theory. The other ingredients for Fontaine-Mazur are as follows: First, there is the classification of algebraic Hecke characters (due to Weil, I think). A key point here is that the algebraicity forces the unit group to be annihilated by some element in the integral group ring. However, the representation occurs in with dimension if is non-trivial, so this forces the existence of representations of on which , corresponding to CM subfields. The final step is the theory of CM abelian varieties. So although the result is non-trivial, you can be rest assured, gentle reader, that you are not secretly invoking subtle transcendence results every time you twist an automorphic Galois representation by a Hodge-Tate character and claim that the result is still automorphic.