A public service announcement concerning Fontaine-Mazur for GL(1)

There’s a rumour going around that results from transcendence theory are required to prove the Fontaine-Mazur conjecture for \(\mathrm{GL}(1)\). This is not correct. In Serre’s book on \(\ell\)-adic representations, he defines a \(p\)-adic representation \(V\) of a global Galois group \(G_F\) to be rational if it is unramified outside finitely many primes and if the characteristic polynomials of \(\mathrm{Frob}_{\lambda}\) actually all lie in some fixed number field \(E\) rather than over \(\mathbf{Q}_p\). Certainly being rational is a consequence of occurring inside the etale cohomology of a smooth proper scheme \(X\), and one might be motivated to make a conjecture in the converse direction assuming that \(V\) 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 \(p\) (“geometric,” given the other assumptions on \(V\)). And the implication

A character \(\chi: G_F \rightarrow \overline{\mathbf{Q}}_p\) is Hodge-Tate \(\Rightarrow \chi\) 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 \(\ell\)-adic representations (with the modifications noted in the updated version of Serre’s book), namely:

Theorem (Serre-Waldschmidt): If \(V\) is an abelian representation of \(G_F\) which is rational, then \(V\) is locally algebraic.

This argument (even for the case when \(F\) is a composite of quadratic fields, the case considered by Serre) requires some transcendence theory. But the implication \(V\) is abelian and Hodge-Tate \(\Rightarrow V\) 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 \(V\) occurs in \(\mathcal{O}^{\times}_F \otimes \mathbf{C}\) with dimension \(\dim(V|c = 1)\) if \(V\) is non-trivial, so this forces the existence of representations \(V\) of \(G\) on which \(c = – 1\), 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.