Finiteness of the global deformation ring over local deformation rings

(This post is the result of a conversation I had with Matt). Suppose that

\(\overline{\rho}: G_{F} \rightarrow \mathrm{GL}_n(\mathbf{F})\)

is a continuous mod-\(p\) absolutely irreducible Galois representation. For now, let’s assume that \(F/F^{+}\) is a CM field, and \(\overline{\rho}\) is essentially self-dual and odd. Associated to this representation is a global deformation ring \(R\) (of essentially self-dual representations) consisting of representations with no local restriction at primes dividing \(p\) and the condition of being unramified at primes away from \(p\). One also has a (collection of) local (unrestricted) deformation rings for the set of primes \(v|p\), combining to give a ring \(R^{\mathrm{loc}}\). Let us also assume that \(\overline{\rho}\) has suitably big image (for example, its restriction to \(F(\zeta_p)\) is adequate). Then we have:

Proposition: The map \(R^{\mathrm{loc}} \rightarrow R\) is finite.

(Matt and Vytas prove this in the modular (odd) case when \(n = 2\) and \(F = \mathbf{Q}\), although I’m not sure whether the paper exists yet [actually, I’m pretty sure it doesn’t]. Possibly if I was listening closer to Matt’s talk at Fields I might have remembered the argument, since I vaguely think it came up there, although possibly only briefly.)

Here one has to be a little careful defining deformation rings in the local case, of course (for those worried by such issues, simply choose suitable framings). To prove this, it suffices to prove the result after base change, so we may assume that \(\overline{\rho}\) is unramified at all primes, and completely trivial at all primes dividing \(p\). By Nakayama’s lemma, the problem above reduces to the following:

Proposition: Let \(F^{\mathrm{ur}}\) be the maximal extension of \(F\) unramified everywhere. Let \(\Gamma\) be the Galois group of \(F^{\mathrm{ur}}\) over \(F\). Then \(\Gamma\) does not admit a continuous essentially self-dual representation:

\(\Gamma \rightarrow \mathrm{GL}_n(A)\)

such that \(A\) is a complete local Notherian \(\mathbf{F}\)-algebra of positive dimension.

This is a special case of the generalization of the unramified Fontaine-Mazur conjecture due to Boston. Recall that the group \(\Gamma\) may be infinite (Golod-Shafarevich), but that Fontaine-Mazur predicts that the image of any such representation into any characteristic zero \(p\)-adic analytic group has finite image. Boston conjectured that the same finiteness would hold for homomorphisms of \(\Gamma\) into \(\mathrm{GL}_n(A)\) for rings like \(A = \mathbf{F}[[T]]\). It turns out that even though the Fontaine-Mazur conjecture is hard, when \(A\) has characteristic \(p\) the conjecture is amenable to modularity lifting theorems by comparison to a new deformation ring in regular weight.

The proof is as follows:

Step 1: Using lifting theorems (Theorem 4.3.1 from BLGGT), we may assume, after a finite base change, that \(\overline{\rho}\) is potentially ordinarily modular of level one for some regular weight \(w\).

Step 2: Using minimal modularity theorems in the ordinary case (Section 10 from Thorne’s Jussieu paper, or Theorem 2.2.2 of BLGGT, both using work of Geraghty), deduce that the minimal weight \(w\) ordinary deformation ring \(S\) is finite over \(W(\mathbf{F})\), and hence that \(S/p\) is finite over \(\mathbf{F}\). Strictly speaking, theorems of this kind are required to prove the previous result.

Step 3: Note that the minimal everywhere unramified deformations of \(\overline{\rho}\) (i.e., the deformations coming from \(\Gamma\)) of characteristic \(p\) are all ordinary of weight \(w\), because everything unramified is ordinary, and in characteristic \(p\) any two weights are the same. Hence \(R/p\) is a quotient of \(S/p\), from which it follows from the finiteness of \(S\) that \(R\) is also finite.

While I am using the latest modularity lifting theorems here, weaker versions for \(n=2\) with some local assumptions on \(\overline{\rho}\) follow from 90’s era technology (say Taylor’s Remarks on a conjecture of Fontaine and Mazur paper from 2000, or even earlier if one assumes residual modularity).

Via the usual argument, this result also applies to even Galois representations \(\overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{F})\) with large image. In particular, the unramified deformation rings in these cases will be finite over \(W(\mathbf{F})\), and there will be at most finitely many counter examples to the unramified Fontaine-Mazur conjecture in characteristic zero for a fixed residual representation. One can also apply it to many classes of higher dimensional non-self dual representations by taking irreducible summands of \(\rho \otimes \rho^{\vee}\). For example, one can take any representation of \(\mathbf{Q}\) whose image contains \(\mathrm{SL}_n(\mathbf{F}_p)\) if \(n\) is even, since then the associated \((n^2 – 1)\)-dimensional representation \(\mathrm{Ad}^0(\overline{\rho})\) restricted to an auxiliary CM field is irreducible, odd, self-dual, and adequate for large enough \(p\). Similar remarks apply to representations over an arbitrary field \(F\) with generic enough image by taking the tensor induction down to \(\mathbf{Q}\).

If one starts allowing ramification at auxiliary primes, things become a little harder. One fix is to build the auxiliary primes into the local deformation ring \(R^{\mathrm{loc}}\), although this might be considered cheating. The problem is that one cannot deduce (in general) that more general ordinary deformation rings \(S\) are finite in the non-minimal situation. Although perhaps one can get by with the Taylor trick in some contexts. One should be OK with \(\mathrm{GL}_2\) by Ihara’s Lemma.