## 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.

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### 2 Responses to A public service announcement concerning Fontaine-Mazur for GL(1)

1. vytas says:

Is it known that say for 2-dimensional abs. irreducible representation of $G_\mathbb{Q, S}$ rational implies that it is geometric?

• Surely not known — but possibly not a very natural question?