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?

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