## Scholze on Torsion, Part III

This is a continuation of Part I and Part II.

Before I continue along to section V.3, I want to discuss an approach to the problem of constructing Galois representations from the pre-Scholze days. Let’s continue with the same notation from last time, where $X_M$ is the symmetric space whose cohomology is of interest, and $X = X_G$ is the Shimura variety with Borel–Serre compactification $X^{BS}$ whose boundary contains (simplified assumption: is) a generalized torus bundle $X_P$ over $X_M$. If we localize at a “non-Eisenstein” ideal, then the completed cohomology groups $\widetilde{H}^n(X_M)$ should vanish outside a single degree $q_0$. For this discussion, let us define non-Eisenstein classes to be those which do not occur in degrees $< q_0$ in $H^*(X_M)$. By Hochschild-Serre, any cohomology class in lowest degree (after localization) always survives in the completed limit, so even if one doesn't assume the expected vanishing in higher degrees, the module $\widetilde{H}^{q_0}(X_M)$ will contain all the information about the classes in $H^{q_0}$ at classical level after localization. Hence, to obtain the desired Galois representations for these classes, one wants to prove:

1. The vanishing of $\widetilde{H}^{q_0-1}(X^{BS})$ after localization.
2. $\$

3. There are Galois representations (of the correct form) associated to classes in $\widetilde{H}^{q_0}_c(X^{BS})$.
4. $\$

The hope was that one could try to prove this via the following idealized argument. There is a spectral sequence:

$\mathrm{Ext}^i(\widetilde{H}^{BM}_j,\Lambda) \Rightarrow \widetilde{H}_{d-i-j},$

where $d = 2 \cdot \mathrm{dim}(X)=2n$ is the real dimension of the Shimura variety $X$. There is an identical sequence with the roles of completed homology and completed Borel-Moore homology reversed. Note that the completed homology groups are (Pontryagin dual) to the cohomology groups, which relates compactly supported cohomology to homology and cohomology to Borel-Moore homology. The non-commutative Ext groups in the spectral sequence vanish for any value of $i$ that is less than the co-dimension of the corresponding module. Recall from last time that $\widetilde{H}^*_j$ is torsion except for the middle degree $j = n$. Now suppose that one can show that the completed homology groups $\widetilde{H}^{*}_j$ have sufficiently large co-dimension outside the middle degree. Then from these bounds (and from trivial bounds on the cohomology of the boundary) the spectral sequence should degenerate, and one should have isomorphisms of the following form (after localization):

$\widetilde{H}_{n-i} = \mathrm{Ext}^i(\widetilde{H}^{BM}_n,\Lambda), \ i \le n, \quad \widetilde{H}_{n+i} = 0, \ i > 0, \quad \widetilde{H}^{BM}_n = \mathrm{Hom}(\widetilde{H}_n,\Lambda).$

(To recall, even though we are localizing at an ideal whose avatar on $H^*(X_M)$ is maximally non-Eisenstein, the corresponding ideal on $H^*(X_G)$ will be Eisenstein.) From these equalities, we see that to understand the action of the Hecke operators on completed cohomology, we are reduced to understanding the action on the completed cohomology in middle degree, which we know to be a module of positive rank and hence (even after localization) contain many cusp forms which are known to have interesting Galois representations. At the very least, this would prove the existence of the residual Galois representations associated to such a non-Eisenstein ideal $\mathfrak{m}$. The approach I am outlining here is the one in the (currently non-existent) paper that Matt and I had planned to write. Let’s suppose that one attempts to apply this approach in the Bianchi case. There’s no issue in defining Eisenstein classes here, since the classes that occur in $H^0(X_M)$ are easy to understand, and $q_0 = 1$. So the first step in the above program is to show that $\widetilde{H}^1(X^{BS})$ vanishes, at least if we pass to finite tame level. As we noted last time, this follows from the congruence subgroup property which is known because $U(2,2)$ has real rank two and the corresponding lattice in this group is (obviously) not co-compact. Here the Shimura variety has complex dimension four. So one only has to show that $\widetilde{H}_j$ is small for $j = 2$ and $j = 3$. In particular, one wants, explicitly, that:

$\mathrm{codim}(\widetilde{H}_2) > 4, \ \mathrm{codim}(\widetilde{H}_3) > 3$

The dimension of $\Lambda = \mathbf{Z}_p[[G]]$ is, for reference, $1 + \dim SL_4(\mathbf{Z}_p) = 16$. As noted previously, we know that these cohomology groups are torsion and so have co-dimension at least one. The proof of this result ultimately relied on facts concerning the growth of spaces of automorphic forms. However, it is impossible to determine anything further about the codimension by naïve automorphic considerations, because already $\Lambda/p$ has co-dimension one but no characteristic zero points. So, to prove this conjecture, one really needs to understand the torsion in the cohomology of Shimura varieties. This was where, basically, we were stuck. Note that even understanding $\widetilde{H}^1$ in this case took a powerful result. Understanding $\widetilde{H}^2$ is already much harder. As the real rank increases, it won’t be the case that such completed cohomology groups completely disappear, since there will exist not only trivial stable classes in characteristic zero, but also exotic torsion classes which will be related to K-theory and regulators (as can be seen here). One implication of our conjectures (as noted above) is that the completed cohomology groups vanish for Shimura varieties above the middle dimension. Scholze proves this! (IV.2.3). However, he doesn’t prove it by showing that the $\widetilde{H}_j$ are small for small $j$, and instead deduces a (weaker form) of such an estimate in reverse. I think it’s an interesting problem to understand $H^2(\Gamma,\mathbf{F}_p)$ for groups where the only characteristic zero classes are invariant under $G$, in both the stable and non-stable range. The first case I mentioned previously, and there is something in this direction (in the second case) in section 4.5 of this book.

Section V.3 OK, continuing on from last time, we now have a determinant of dimension $2n$ with image in $A_0/I = \mathbf{T}/I$ for some ideal $I$ with $I^{m} = 0$ for an integer $m$ which only depends on $\mathrm{dim}(X)$. The goal is now to extract an $n$-dimensional determinant, i.e., to recover $\rho$ from $\rho^{\vee} \oplus \rho^c$. Of course, the idea is not to do this from simply one class, but rather allowing twisting, so that we also know $r_{\psi} = \rho^{\vee} \mathrm{det}(\rho) \psi^{-1} \oplus \rho^c \psi^c$ for some Hecke character $\psi$. We may as well take $\psi$ to be a collection of characters of $\mathbf{Q}$, so that $\psi^c = \psi$.

Let’s first make some simplifying assumptions, namely, that the ideal $I = 0$, that we are in characteristic zero, and that the image of $r$ is through a finite group $G$, and the image of all the twists factors through the group $\Gamma:=G \times \mathbf{Z}$ where $\psi$ is a finite order character of the second factor, and $\psi^2 \ne 1$. We would like to imagine that there are equalities:

$r = W =^{?} U \oplus V, \quad r_{\psi} = W_{\psi} =^{?} U \psi \oplus V \psi^{-1}.$

Because the two factors of $\Gamma$ commute, it follows that $[\psi \otimes W_{\psi}] - [W]$
is a virtual character of $\Gamma$. Evaluating this character on the pairs $G \sim (g,1) \subset \Gamma$ defines a class function on $G$. Normalizing by $\psi^2(1) - 1 \ne 0$, this class function applied to $\mathrm{Frob}_{x}$ is the sum of the Satake parameters at $x$ corresponding to $U$, and we deduce that $[U]$, and hence also $[V]$, are virtual characters (with rational coefficients) of $G$. It now suffices to promote $[U]$ to an actual character. The virtual characters $[U]$ and $[V]$ tautologically promote to virtual characters of $\Gamma$ which decompose under the second factor into trivial representations. It follows that $[U \psi]$ and $[V \psi^{-1}]$ are (rational) sums of irreducible representations which decompose under the second factor as direct sums of the representation $\psi$ or $\psi^{-1}$. Assuming that $\psi \ne \psi^{-1}$, there can be no cancellation in $[U \psi] + [V \psi^{-1}]$, from which it follows that $[U]$ is already an actual character.

In general one has to modify this argument to work more integrally as well as to be compatible with the ideal $J$. As I told TG, “without having looking at this yet, it must essentially be trivial.” So, if you are like me, you can just ignore the following which took me a non-trivial number of hours to work out:

1. We take the characters $\psi$ to be characters of $\mathrm{Gal}(\mathbf{Q}(\zeta_{\ell^{\infty}})/\mathbf{Q})$ of $\ell$-power order, where $\ell$ is prime to two and $p$ and anything else inconvenient including the ramified primes. This auxiliary prime may vary.
2. $\$

3. Since we are going to allow $\psi$ to have order some arbitrarily large power of various primes, it is convenient to extend scalars to $A = A_0 \otimes W(\overline{\mathbf{F}}_p)$. Here $A_0$ is the Hecke algebra acting with coefficients modulo some fixed power of $p$. It’s useful to work with both rings, however, because whilst $A$ accepts characters of all orders, $A_0$ is literally a finite ring, which is convenient for finiteness arguments. We would like to show that the twisted determinants corresponding to $r_{\psi}$ have values in $A/I_{\psi}$ for the same $A$. This amounts to showing that, at the level of our original locally symmetric quotient $X_M$, we can twist by a sufficiently nice character $\psi$ and not change the Hecke algebra, except for extending scalars. This is straightforward, and is what is going on at the top of page 88.
4. $\$

5. If we have two determinants with a pair of corresponding ideals $I$ and $I_{\psi}$ with $I^m = I^m_{\psi} = 0$, then clearly $\widetilde{I} = I + I_{\psi}$ satisfies $\widetilde{I}^{2m-1} = 0$. So, at the cost of increasing the nilpotency, for any character $\psi$, we get two determinants with values in $A/\widetilde{I}$. Note that if $I$ and $I_{\psi}$ are both trivial, then so is $\widetilde{I}$.
6. $\$

7. We would also like the ideal $\widetilde{I}$ to be independent of $\psi$. Actually, we don’t need this, it will suffice to note that we can take $\widetilde{I} \cap A_0$ to be independent of $\psi$. Because $A_0$ is finite, there are only finitely many such ideals, and so we can take one that occurs for infinitely many primes $\ell$ and infinitely many of the corresponding characters $\psi$.
8. $\$

9. For any fixed character $\psi$, our determinant (which has twice the required dimension) will be defined on a finite quotient of

$\Gamma:=\mathrm{Gal}(L_{\infty}/F) = \mathrm{Gal}(L/F) \times \mathrm{Gal}(F_{\infty}/F),$

where $L/F$ is finite and $L_{\infty}, F_{\infty}$ are the pro-$\ell$ cyclotomic covers of $L, F$ respectively. This should hopefully look similar to our simplified problem in characteristic zero. We have two determinants $D$ and $D_{\psi}$ with the property that the characteristic polynomial of Frobenius $\mathrm{Frob}_x$ (which exists for determinants) is:

1. Of the form $P^{\vee}_x(X) P^c_x(X) \mod I$ for $D$.
2. Of the form $P^{\vee}_x(X/\chi(g)) P^c_x(X \chi(g)) \mod I_{\psi}$ for $D_{\psi}$

$\$

These polynomials $P^{\vee}_x(X)$ and $P^c_x(X)$ are what they obviously should be, namely, the polynomials with inverse roots given by the appropriate Satake parameters. (Or more accurately, with coefficients given by the appropriate Hecke operators.) Because these are determinants, these products are locally constant on the group $\Gamma$ because they are coming from honest Galois representations of rank $2n$. We would like to decompose these into products of two determinants of rank $n$. In the characteristic zero case, we took a character $\psi$ such that $\psi^2(1) - 1 \ne 0$ and used this as a fulcrum on which to tease out the representation $U$. Here we do something similar. A first step is to show that each of the four polynomials above is locally constant. We choose an element $1 \in \mathrm{Gal}(F_{\infty}/F)$ and a deep enough character $\chi$ so that $\chi^{2m}(1) -1 \ne 0$ for all $m = 1,\ldots,n$. We now find an open neighbourhood of $(G,1)$ where $D$ and $D_{\chi}$ are constant. Let $a(x)$ be the linear term of $P^{\vee}_x(X)$, and let $b(x)$ be the linear term of $P^c_x(X)$. Then we deduce that the following two terms are locally constant:

$a(x) + b(x), \quad a(x) \psi(x) + b(x) \psi^{-1}(x).$

So, because $\psi^2(x) -1 \ne 0$, we deduce that $a(x)$ and $b(x)$ are locally constant, and so $a(x) \mod \widetilde{I} \cap A_0$ is also locally constant. Given this, one proves that the quadratic terms are also locally constant in the same manner, and by induction one has the result for the entire polynomial. Thickening the open neighbourhood up, one proves the same result for the entire group $\Gamma$ minus the piece coming from $\mathbf{Z} \sim \mathrm{Gal}(F_{\infty}/F)$, which gives us Lemma V.3.4. Then by choosing a different auxiliary prime $\ell'$, one patches to get a well defined class function on $G$ in Lemma V.3.5.

10. $\$

11. So now we have a class function $D$ on the Galois group $\mathrm{Gal}(L/F)$ with values in characteristic polynomials (now of the right dimension!) in $A_0/I$ (dropping the tildes), and we want to promote it to a genuine determinant. Of course, over finite rings we can’t use the language of virtual characters. What Scholze does next is use the fact that we have such a decomposition for infinitely many different characters in order to glue enough of them together to obtain a determinant map

$D: A[G \times \mathbf{Z}] \rightarrow A[t]/I, \quad D(1 - X \gamma^k g) = P^{\vee}_g(X/t^k) P^c_g(X t^k),$

where $\gamma$ is a generator of $\mathbf{Z}$ and $I$ has the expected properties of nilpotence. This consists of Lemmas V.3.6 and V.3.7.

12. $\$

13. Now we are at Lemma V.3.8. Bugger it, this is taking a long time, and quite possibly nobody is interested in these specific details. Let me cut some corners and replace determinants by pseudo-representations. We deduce from the above that we are in the following situation: we have a degree $2n$ pseudo-representation:

$T: G \times \mathbf{Z} \rightarrow A[t], \qquad T(g,m) = a(g) t^m + b(g) t^{-m}.$

We want to deduce that $a(g)$ and $b(g)$ are both pseudo-representations of degree $n$. We are allowed to use the fact (which is obvious) that $a(g)$ and $b(g)$ are not pseudo-representations of degree strictly less than $n$. (Actually, is it obvious? It’s certainly obvious for $n = 2$ that $a(g)$ and $b(g)$ are not a character. So let’s assume $n =2$. Ah, I see by passing to the trivial element we can compute that $a(1) = b(1) = n$, so it is obvious.) Now, if we abstract slightly and drop any knowledge about $a(g)$ and $b(g)$ other than they are class functions, the best we can hope to prove is that $a(g)$ and $b(g)$ are both pseudo-representations of degrees $A$ and $B$ respectively, where $A+B=2n$. This is what we do. Since $T$ is a pseudo-representation of degree $2n$, we have the following identity:

$\displaystyle{\sum_{S_{2n+1}} (-1)^d T_{\sigma}(g_i,m_i) = 0.}$

In fact, this identity on class functions characterizes pseudo-representations of degree at most $2n$, the only other information coming from evaluating on the identity. Suppose we take the $m_i$ to be sufficiently generic integers so that all the sums $\sum \pm m_i$ are distinct. Now let us partition the $m_i$ into two sets $M_A, M_B$ of cardinality $A+1$ and $B$ respectively, where $A+B=2n$. Consider the coefficient of $t^C$ in the sum above, where we take

$C:= \sum_{M_A} m_i - \sum_{M_B} m_i$

The corresponding coefficient must vanish. Moreover, because of the way that the $m_i$ were chosen, we know exactly what terms can arise with this coefficient: explicitly, the terms in $M_A$ must come from $a(g)$, and the terms in $M_B$ must come from $b(g)$. Hence we recognize the coefficient to be (up to sign)

$\displaystyle{\left(\sum_{S_{A+1} \curvearrowright M_{A}} (-1)^d a_{\sigma}(g_i) \right) \left( \sum_{S_B \curvearrowright M_{B}} (-1)^d b_{\sigma}(g_i) \right)}.$

We deduce that, for any decomposition $A+B=n$, either $a(g)$ is a pseudo-representation of degree at most $A$, or $b(g)$ is a pseudo-representation of degree at most $B-1$. Taking $B$ to be the smallest integer for which $b(g)$ is a pseudo-representation, we deduce the result (such an integer exists because $b(g)$ is at least a degree $\le 2n$ pseudo-deformation). We are, mercifully, done. Looking at Scholze, I think this lemma (and even roughly the argument) is quite similar to the proof of Lemmas V.3.8-V.3.15 but this is much easier, at the cost of assuming that $p > n$.

It looks as though one can probably skip step 6 simply by choosing the value of $t \sim \psi(1)$ to generate a sufficiently generic extension of $A_0$ inside $A$, although I guess that’s how one does step 6 anyway.

Section V.4 is just a matter of putting things together. Next time: onto Chapter IV!