Postdoc hiring season will be upon us soon! I have two excellent graduate students who will be applying for academic jobs soon, Chengyang Bao and Andreea Iorga. I have mentioned Chengyang’s first project before here and an introduction to the results in Andreea’s thesis is here. Today I wanted to talk about Chengyang’s thesis.
Fix a local mod-p representation, say
\( \overline{\rho}: G_{\mathbf{Q}_p}
\rightarrow \mathrm{GL}_2(\mathbf{F}_p)\)
given on inertia by \(\omega_2 \oplus \omega^p_2\). Associated to this residual representation is a Kisin deformation ring \(R\) corresponding to fixed determinant crystalline lifts of weights \([0,k-1]\), for some fixed positive integer \(k \equiv 2 \bmod (p-1)\). The special fibres \(R/p\) of these rings have dimension one, and so, if one denotes their maximal ideal by \(\mathfrak{m})\), then the Hilbert series
\(H_k(x) = \sum \dim(\mathfrak{m}^k/\mathfrak{m}^{k+1}) X^k\)
has the form
\(H_k(x) = \displaystyle{\frac{P_k(x)}{1 – x}}\)
where \(P_k(x)\) is a polynomial. The Hilbert-Samuel multiplicities of these rings are given by the Breuil-Mezard conjecture (also proved by Kisin). These numbers are explicitly given by the values \(P_k(1)\).
It seems quite surprising that understanding the seemingly simple number \(P_k(1)\) is so intimately linked to the proof of the Fontaine-Mazur conjecture. At the same time, we know very little about these rings \(R\) (or their special fibres) when \(k\) is large.
In the example above, the unrestricted (fixed determinant) local deformation ring \(R^{\mathrm{loc}}\) is is formally smooth of dimension three over \(\mathbf{Z}_p\). Although the rings \(R/p\) only have dimension one, one expects that for larger and larger \(k\) they start to “fill out” the unrestricted deformation ring. It is natural to wonder: how fast does this happen?
More explicitly, the Hilbert-Samuel series of the special fibre of the unrestricted deformation ring with fixed determinant is
\(\displaystyle{\frac{1}{(1-x)^3}} = 1 + 3 x + 6 x^2 + 10 x^3 + \ldots \)
So one can ask: what weight does one have to go to to see all three dimensions of the tangent space? How far does one have to go to see all of \(R^{\mathrm{loc}}/(p,\mathfrak{m}^n)\)?
This was the thesis problem of Chengyang Bao, which grew out of (in part) questions arising during her work here. In this particular case, it seems that one has to go to weight \(k = p^2 + 1\) to see the entire tangent space. Actually, an even more basic question (which also came up here), is whether there exists surjective map
\(R_{k+p-1}/p \rightarrow R_k/p,\)
this seems very tricky and is still open (but Chengyang’s work stongly suggests that it is true).
One of the difficulties in this project is that close to nothing was known about the rings \(R\) for \(k\) anything larger than \(k = 2p\) or so (although there has certainly been quite a bit of work understanding the link between \(a_p\) and the residual representation, including work of Buzzard-Gee, Sandra Rozensztajn, and many others using p-adic Langlands for larger \(k\)).
Chengyang’s approach was, perhaps surprisingly, to use global methods. The basic summary of the Taylor-Wiles method as formulated by Kisin is that via patching one finds that a patched Hecke ring may be identified with a power series ring over \(R\). By reverse engineering this, if one finds a residual representation with sufficiently nice global properties, one can use explicit Taylor-Wiles primes to get arbitarily close approximations to the Kisin deformation ring \(R\). One of the tricks here is to be able to do this in a way where one can work efficiently after fixing a residual representation and then increasing the weight.
By doing these computations, Chengyang generated lots of explicit data about these rings \(R\) from which one can start making conjectures. I said before how the Breuil-Mezard conjecture amounts to predicting the value of \(P_k(x)\) at \(x = 1\). Chengyang has, at least in this particular case, been able to formulate an exact conjectural answer for the *entire* polynomial \(P_k(x)\). As a consequence, one can read off from this the answer of how large a weight one has to go to see all the directions in \(R^{\mathrm{loc}}/\mathfrak{m^n}\). I mentioned before that for \(n=2\) the answer is \(p^2+1\). and my guess was that the answer in general might be of order \(p^n\). But somehow the conjectural answer (up to constants which depend on \(p\)) turns out to be of order \(O(n^2)\), which is honestly completely different from anything that I would have guessed. I think of this conjecture as a new “horizonal Breuil-Mezard conjecture.” But really, it’s only half a conjecture; the hope is that one can understand and interpret Chengyang’s conjecture on the \(\mathrm{GL}_2(\mathbf{Q}_p)\)-side, and working this out is an exciting problem.
At the same time, there are lots of other things one can start to guess from looking at these explicit rings. Chengyang has a precise conjecture which says when the rings \(R\) in this setting are complete intersections or Gorenstein, and it also seems that they are always Cohen-Macaulay.
Even though we “know” \(p\)-adic Langlands for \(\mathrm{GL}_2(\mathbf{Q}_p)\) much better than in any other situation, there seems to be a real opportunity here to tease out many more precise and explicit conjectures from Chengyang’s work, and really to discover new phenomena which have hitherto never been noticed because computations of these rings has been so limited. (Another basic question: how many components does the generic fibre of \(R\) have in terms of \(k\)?).
I recommend that anyone interested in more details read Chenyang’s research statement!