Scholze on Torsion, Part III

Posted on June 22, 2013 by Persiflage

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. \(\ \)

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

    3. \(\ \)

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. \(\ \)

  2. 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.

    3. \(\ \)

  3. 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 \(\widetilde{I}\).

    4. \(\ \)

  4. 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 \(\psi\).

    5. \(\ \)

  5. 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.

    6. \(\ \)

  6. 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.

    7. \(\ \)

  7. 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!

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Scholze on Torsion, Part II

Posted on June 19, 2013 by Persiflage

This is a sequel to Part I.

Section V.1:
Today we will talk about Chapter V. We will start with Theorem V.1.4. This is basically a summary of the construction of Galois representations in the RACSDC case, which follows, for example, from work of Shin. We know a little bit more than this theorem states (namely, local-global compatibility).

Corollary V.1.7 is just the statement that the cohomology groups \(H^0(X,\mathcal{E}_{\mathrm{sub}})\) for the sub-canonical extensions of automorphic sheaves \(\mathcal{E}\) are computed by forms \(\pi\) whose transfer to \(\mathrm{Res}_{F/\mathbf{Q}}\mathrm{GL}(n)\) are RACSDC. The sub-canonical extension corresponds to imposing a vanishing condition at the cusps. For example, the sub-canonical extension of \(\omega^k\) on the modular curve is \(\omega^k(-\infty)\). There is a nice action of the Hecke algebra \(\mathbf{T}\) on this space, which is compatible with the associated Galois representations in all the expected ways (Satake parameters to Frobenius eigenvalues) at the unramified primes. So far, this is all classical (as of 2011).

Determinants: We will be using congruences to obtain Galois representations, but the information that gets glued is really the Hecke eigenvalues. So one wants a convenient way to pass from one to the other. The classical approach with modular forms is to remember the “standard” Hecke operators \(T_x\) which correspond to the traces of Frobenius. Knowing the trace is enough to determine a two dimensional representation away from characteristic \(2\), if one has residual irreducibility. This is the theory of pseudo-representations. Naturally enough, for larger dimensional Galois representations, it helps to remember more than the trace, namely, the entire characteristic polynomial. The corresponding theory was worked out by Chenevier. Namely, given an \(n\)-dimensional representation \(\rho\) of the group \(G\) over a commutative ring \(A\), there is a map:

\(D: A[G] \rightarrow M_n(A) \rightarrow A\)

given by formally extending \(\rho\) in the obvious way and then composing with the determinant. For example, if \(n = 2\), then

\(T(g):=D([g] + [1]) – D([g]) – D([1]) = \mathrm{Tr}(\rho(g)).\)

Now the map \(D\) has to satisfy a bunch of formal properties due to the constraints of coming from an \(n\)-dimensional representation. Writing all these down gives the correct notion of Chenevier’s generalized “determinant.” (Original paper here.) For those who like pseudo-representations, note that when \(n=2\), one can define \(D\) using the formula:

\(D(g) = \displaystyle{\frac{T(g)^2 – T(g^2)}{2}},\)

where \(T\) is the trace. So for \(n = 2\) in characteristic greater than two, the notions are equivalent. And indeed Chenevier’s notion of determinants is the same as a pseudo-representation whenever \(n!\) is invertible, but is better behaved in small characteristics. Determinants satisfy the nice properties that pseudo-representations do, and that Galois representations sometimes don’t (but do in the residually absolutely irreducible case), namely:

  1. You can glue determinants: \(D \rightarrow A/I\) and \(D \rightarrow A/J\) which agree on \(A/(I + J)\) to get a determinant \(D \rightarrow A/(I \cap J).\)
  2. Given a formal variable \(X\), there is a natural determinant map \(D: A[X][G] \rightarrow A[X]\) such that \(D(1 – X g)\) is the characteristic polynomial of \(g\) if the determinant comes from an actual representation.

Here I follow Scholze in using \(\mathrm{Det}(I – X \cdot M)\) rather than \(\mathrm{Det}(M – I \cdot X)\) as the definition of a characteristic polynomial — this is just a bookkeeping issue (the dreaded arithmetic versus geometric Frobenius). Returning to automorphic forms from coherent cohomology, since \(H^0\) is torsion free, the module \(\mathbf{T}\) is flat over \(\mathbf{Z}\). Since the characteristic zero forms give rise to Galois representations coming from RACDSC forms, we naturally obtain a determinant map:

\(D: \mathbf{Z}_p[G_{F}] \rightarrow \mathbf{T}\)

such that \(D(1 – X \cdot \mathrm{Frob}_x)\) is exactly as one would expect. (This is Corollary V.1.11). Note that the ring \(\mathbf{T}_c\), which arises at this point, is just the inverse limit of the corresponding classical \(\mathbf{T}\) over all \(p\)-power levels; this is defined in Chapter IV which we shall talk about later.

Segue on Completed Cohomology: I have to recall here a few basics about completed cohomology (one reference is here.) I already know about completed cohomology (and so do many of my loyal readers) so I don’t really feel obliged to say too much about it, but since most of you have been sent here from Quomodocumque, I will cough up a few pointers. The basic definition (for any congruence arithmetic manifold corresponding to a group \(\mathbb{G}\)) is as follows:

\(\widetilde{H}^i(X,\mathbf{Z}/p^n \mathbf{Z}) := \lim_{K \rightarrow} H^i(X(K),\mathbf{Z}/p^n \mathbf{Z}).\)

Here the limit is over shrinking compact open subgroups \(K\) of \(\mathbb{G}(\mathbf{Z}_p)\). The tame level is fixed and can be included in the notation somewhere. One can also adorn the cohomology groups in the usual way, namely, by considering compactly supported cohomology. So what’s the point of completed cohomology? Apart from having a natural action of \(\mathbb{G}(\mathbf{Q}_p)\), which is always the type of group one wants to act on a candidate space for automorphic representations of any kind, a matter of experience and intuition suggested (to Matt and me) that it should be the “correct” space of automorphic forms modulo \(p^n\) when \(\mathbf{G}(\mathbf{R})\) does not have discrete series (and even when it does). One way to justify this is via the following four properties, the final one conjectural:

  1. The completed cohomology groups \(\widetilde{H}^i(X,\mathbf{Z}/p \mathbf{Z})\) are co-finitely generated over \(\Lambda = \mathbf{F}_p[[\mathbb{G}(\mathbf{Z}_p)]]\). This latter ring has nice properties, e.g. after shrinking the group \(\mathbb{G}(\mathbf{Z}_p)\) slightly to get a powerful torsion free pro-\(p\) group, \(\Lambda\) is a local Noetherian ring which is Auslander regular (see Lazard and also Venjakob.)

    2. \(\ \)

  2. The Pontryagin dual groups \(\widetilde{H}^i(X,\mathbf{Q}_p/\mathbf{Z}_p)^{\vee}\), which are finitely generated (by part one and Nakayama’s Lemma and the usual long exact sequences) are not torsion \(\Lambda = \mathbf{Z}_p[[\mathbb{G}(\mathbf{Z}_p)]]\)-modules if and only if one is in middle degree and the corresponding real group admits discrete series (see this paper).

    3. \(\ \)

  3. The completed homology groups satisfy a Poincaré duality spectral sequence. The completed cohomology groups are compatible with the Hochschild–Serre sequence from which one can recover classical cohomology groups.

    4. \(\ \)

  4. Given a torus bundle, or more generally a nilmanifold, the completed cohomology disappears outside degree zero.

    5. \(\ \)

  5. Conjecturally: for any reductive algebraic group there will be a dominating term \(\widetilde{H}^i(X)\) in degree \(i = q_0\) which will have co-dimension \(l_0\) as a \(\Lambda\)-module, where \(2 q_0 + l_0\) is the real dimension of \(X_G\), and the degrees \([q_0,q_0+1,\ldots,q_0 + l_0]\) are exactly the degrees in which tempered automorphic representations contribute to cuspidal cohomology. More directly, \(l_0\) for a semi-simple group is the rank of \(\mathbb{G}(\mathbf{R})\) minus the rank of the maximal compact. For example, \(l_0\) is equal to zero if and only if the real group admits discrete series. Hence this bullet point is a conjectural generalization of point (2). As an example, in the case of \(\mathrm{GL}_2\) over an imaginary quadratic field, the completed cohomology \(\widetilde{H}^1(\mathbf{F}_p)\) should have codimension exactly one.

    6. \(\ \)

(For the last three points I’ll refer you once again to this survey.)

Section V.2: The key starting point, as mentioned last time, is that one can relate the cohomology of the group we are interested in — \(\mathrm{Res}_{F/\mathbf{Q}}(\mathrm{GL}_n)\) — to the cohomology of Shimura varieties by realizing the first group as the Levi \(M\) inside a maximal parabolic \(P\) inside a group \(G\) corresponding to a Shimura variety. The first step is to compare the cohomology of what we are interested in (coming from the Levi \(M\)) to the cohomology of the boundary piece coming from the parabolic \(P\) inside \(G\) containing \(M\). This is pretty standard: what happens is that the resulting space \(X_P\) which actually occurs in the boundary of the Borel–Serre compactification \(X^{BS}_G\) of \(X_G\) is a torus bundle over \(X_M\). Well, not literally always a torus bundle, but rather a nilmanifold \(N\) coming from the unipotent part of \(P\). The nilmanifold fibres spread the cohomology around by a Künneth type formula like a Frenchman expectorating over-oaked California Chardonnay into a spittoon. (Usually this fibration arises as a quotient from a fibration with a contractible fibre, which means that the cohomology really is just the derived product of the cohomology of the base and the cohomology of \(N\), so it’s not really so bad.) One way to avoid this mess is by passing to completed cohomology. On the boundary this has the effect of collapsing all the torus like directions in the nilmanold, and obtaining a map from the completed cohomology of the arithmetic manifold corresponding to the Levi into the completed cohomology of the total space. Compare with equation (1.4) of this survey again.

Hecke Operators from \(M\) to \(G\). One thing we have to understand is how to compute the Hecke operators at unramified primes on the completed cohomology of the boundary of \(X_G\) in terms of the action of the Hecke operators on the original object of interest \(X_M\). Let us fix an unramified prime \(x\) which is prime to everything. To orient you, we are at the top of page 82 of Scholze. I’m going to be more prosaic in my notational choices and write \(T_G\), \(T_P\), and \(T_M\) for the local Hecke algebras at the prime \(x\) (Scholze does all the unramifed primes at once). Yes, I know this is an abuse of notation, because here the groups \(G\), the parabolic \(P\) and the Levi \(M\) are really the local versions at the prime \(x\). (You will cope.) There are natural maps:

\(T_G \rightarrow T_P \rightarrow T_M.\)

Let’s actually consider what these are in the case when \( M\) comes from \(\mathrm{GL}(2)\) over an imaginary quadratic field \(F\) in which \(x\) splits, and \(G\) comes from \(U(2,2)\) which also splits over \(F\). So locally at \(x\), the group \(G\) is just \(\mathrm{GL}(4)\), and \(M\) is the levi \(\mathrm{GL}(2) \times \mathrm{GL}(2)\), and \(P\) is what it obviously has to be. In this case, we have isomorphisms:

\(T_G \simeq \mathbf{Z}_p[X^{\pm}_1,\ldots,X^{\pm}_{4}]^{S_4}, \qquad T_P \simeq \mathbf{Z}_p[Y^{\pm}_1,Y^{\pm}_2]^{S_2} \times \mathbf{Z}_p[Z^{\pm}_1,Z^{\pm}_2]^{S_2}.\)

Perhaps we are required to adjoin \(\sqrt{x}\) to both sides in order to normalize this appropriately. Consider it done. Now the map \(T_G \rightarrow T_{M}\) is the one sending \((X_1,X_2,X_3,X_4) \mapsto (x Y_1,x Y_2,x^{-1} Z_1,x^{-1} Z_2)\). The choice here must be coming from the choice of \(M\) (for a fixed torus) corresponding to a choice of subgroup \(S_2 \times S_2\) of the Weyl group. One can write down analogous formulas for the inert and ramified primes. The corresponding maps of Satake parameters indicates that the if our original eigenclass has a Galois representation \(\rho\), then the Hecke eigenvalues of the class which has been pulled back is associated to \(\rho^{\vee} \oplus \rho^c\). (Edit: In the previous version I omitted the dual. Note that \(\rho^{\vee} \det(\rho) = \rho\) for \(n = 2\). End Edit) Now this statement seems to be somewhat in conflict with my previous post, where I claimed that the action of the Hecke algebra on the cohomology of \(U(2,2)\) corresponded to the Galois representation \(\rho \otimes \rho^c\). This is because of a subtlety which I think I can explain. Suppose you start from a classical modular form \(f\) and base change it to a Hilbert modular form \(f_E\) over a real quadratic extension. Then the corresponding map of Satake parameters is just the obvious one corresponding to the restriction of the Galois representation. In particular, if \(\alpha,\beta\) are the Satake parameters of a local unramified component \(\pi_x\) of \(f\), and if \(x\) splits in the quadratic field and \(y\) is a prime above \(x\), then \(\pi_y\) of \(f_E\) will have the same Satake parameters, and \(f_E\) will have the same Hecke eigenvalue for \(T_y\) that \(f\) has for \(T_x\). However, the actual Galois representation occurring inside the etale cohomology of the Hilbert modular surface is not the restriction of the Galois representation to \(E\), but rather the (four dimensional) tensor induction. This also reflects an important point: we will not be finding the desired Galois representation inside etale cohomology (which, apparently by an argument of Clozel and Harris, is impossible), but rather we will simply be “following the Hecke eigenclasses.” In this context, for example, cuspidal automorphic representations for \(U(2,2)\) contain all the information for the associated four-dimensional representations, but the ones occurring in cohomology are (tensor inductions!) of \(\wedge^2\). That is why in this post we see the Hecke eigenvalues as looking like the direct sum \(\rho^{\vee} \oplus \rho^c\), whereas the action on cohomology via Eichler-Shimura looked like \(\wedge^2 (\rho^{\vee} \oplus \rho^c)\), which contains \(\rho \otimes \rho^c\) up to twist.

The arguments on the lower half of page 82 are just related to the fact that the boundary of the compactification on \(X_G\) can have a number of components, and these components can have their own boundary, and so on. If one takes the case where \(X_G\) corresponds to \(U(2,2)\) over an imaginary quadratic field, then the only boundary components are (torus bundles over) Bianchi manifolds \(X_M\), and the only boundaries that they have are hyperbolic cusps. In particular, in this case, using remark (4) on completed cohomology above, the completed cohomology of the boundary \(\widetilde{H}^k(\partial X_G)\) (denoting \(X^{BS}_G \setminus X_G\) by \(\partial X_G\)), is given by

\(\widetilde{H}^k(\partial X_G,\mathbf{Z}/p^n \mathbf{Z}) = \mathrm{Ind}^{\mathbb{G}(\mathbf{Z}_p)}_{\mathbb{P}(\mathbf{Z}_p)} \left(\widetilde{H}^k(X_M,\mathbf{Z}/p^n \mathbf{Z})\right).\)

So we are interested in the Hecke action on the right hand side, which we have now transferred to the left hand side. (Of course, the local Hecke algebras combine by taking tensor powers to get the Hecke algebra at all unramified primes, which surjects onto the corresponding global Hecke algebras \(\mathbf{T}_G\) and \(\mathbf{T}_M\).) There is a natural long exact sequence of completed cohomology associated to a manifold with corners as follows:

\(\ldots \rightarrow \widetilde{H}^{k-1}(X_G,\mathbf{Z}/p^n \mathbf{Z}) \rightarrow \widetilde{H}^{k-1}(\partial X_G,\mathbf{Z}/p^n \mathbf{Z}) \rightarrow \widetilde{H}^{k}_c(X_G,\mathbf{Z}/p^n \mathbf{Z}) \rightarrow \widetilde{H}^k(X_G,\mathbf{Z}/p^n \mathbf{Z}) \rightarrow \ldots \)

So to get a Galois representation (or, to begin with, a determinant) on \(\widetilde{H}^{k-1}(\partial X_G)\), we can start by finding determinants for the two surrounding terms.

Special Case: Let’s continue discussion the special case where \(X_M\) is a Bianchi manifold, and \(X_G\) comes from \(U(2,2)\) which splits over the corresponding imaginary quadratic field. The key term of interest will be (for the Bianchi manifold) \(\widetilde{H}^1(X_M)\) or, equivalently, \(\widetilde{H}^1(\partial X_G)\). In fact, by Hochschild-Serre, the completed cohomology \(\widetilde{H}^1(X_M)\) captures all the interesting Hecke actions coming from torsion in Bianchi groups as long as one localizes away from the Eisenstein primes coming from the cusps. The cusps in the Borel–Serre compactification of the Bianchi group are elliptic curves with CM by the underlying imaginary quadratic field. The difference between the classical classes in \(H^1\) and \(\widetilde{H}^1\) proved themselves to be a real pain in my book with Akshay, because when one wants a numerical correspondence, one can’t ignore Eisenstein terms. Yet blessedly, in this context, we can localize away from them. Hence the key terms are those in the following boundary exact sequence:

\(\widetilde{H}^{1}(X_G) \rightarrow \widetilde{H}^1(\partial X_G) \rightarrow \widetilde{H}^{2}_c(X_G)\)

Let’s consider the first term. The group \(U(2,2)\) has real rank two. In particular, by super rigidity, any non co-compact lattice in \(U(2,2)\) will have the congruence subgroup property. It follows that \(\widetilde{H}^1(X_G)\) is trivial! The point is that if all the finite quotients of a lattice in \(U(2,2)\) come from congruence quotients, then pulling back over all such quotients kills everything. Actually, this is not strictly correct, because completed cohomology only pulls back over \(p\)-power quotients, and there may be cohomology coming from the tame level. However, it is easy to see (by Hochschild–Serre) that any such cohomology will be Eisenstein. In particular, after localizing at a non-Eisenstein (in the appropriate sense) ideal, we get an injection from \(\widetilde{H}^1(\partial X_G)\) to \(\widetilde{H}^{2}_c(X_G)\), and thus from Theorem IV.3.1, we obtain a determinant to the Hecke algebra of \(\widetilde{H}^1(X_M)\) (localized away from Eisenstein ideals) without any need to quotient out by an ideal with fixed zero power as in Corollary V.2.6. I don’t think this trick will really work in any other examples, however, since it’s very hard to say anything in general about \(~H^2\). (There is recent work on on stable completed co/homology here, but that will never be enough to give something useful in this context.)

General Case: The general case is now quite similar, except know to understand \(\widetilde{H}^k(\partial X_G)\) one needs to understand both boundary terms. There is also going to be some loss of information coming from the corresponding extension class. If one had determinants on \(\widetilde{H}^*(X_G)\) and \(\widetilde{H}^*_c(X_G)\), then one would immediately get Corollary V.2.6 with an ideal \(I\) with \(I^2 = 0\). However, Theorem IV.3.1 (which is being invoked here) only applies to \(\widetilde{H}^*_c(X_G)\). Now \(\widetilde{H}^*(X_G)\) is related to its compact cousin by a Poincaré duality spectral sequence, but this will once again spread out some terms and necessitate replacing \(I^2 = 0\) by some power involving the dimension. At any rate, while there is room for improvement in general, there is still the fundamental problem (mentioned in part zero!) of controlling whether this boundary cohomology is going forwards or backwards in the long exact sequence above (or worse, being mixed). I’m going to give some heuristics next time on what one expects should happen (short answer: after localizing at a nice maximal ideal, it should work out as well as the Bianchi case, but that will be hard to prove.)

Note that Scholze actually works with classical cohomology here, and then relates it back to completed cohomology using Hochschild-Serre on p.86. The point in either argument is that all the terms in the spectral sequence (on every page) are, by Theorem IV.3.1, modules for the Hecke ring \(\mathbf{T}_c\) which acts on coherent cohomology. Hence the limit terms have filtrations by a fixed bounded number of such objects.

Next time: I’ll say a little more about how one might expect the “simplification” in the Bianchi case above to apply more generally, and I’ll talk about the final section V.3 of chapter V, in which we extract the \(n\)-dimensional representations from our \(2n\)-dimensional determinants.

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Scholze on Torsion, Part I

Posted on June 16, 2013 by Persiflage

This is a sequel to this post, although as it turns out we still won’t actually get to anything substantial — or indeed anything beyond an introduction — in this post.

Let me begin with some overview. Suppose that \(X = \Gamma \backslash G/K\) is a locally symmetric space, where \(G\) is a semi-simple group which does not admit discrete series. To be concrete, suppose that \(G = \mathrm{SL}_2(\mathbf{C}),\) and that \(\Gamma\) is an congruence subgroup of level \(N\) in the Bianchi group \(\mathrm{SL}(\mathcal{O}_F)\) — recall here that \(F\) is an imaginary quadratic field. Since days of yore (Langlands, Clozel, Fontaine-Mazur, etc.), everyone has expected that there should exist a bijection between the following objects:

  1. Regular algebraic cuspidal autormophic representations \(\pi\) for \(\mathrm{GL}_2(\mathbf{A}_F)\) of level \(N\) and weight zero (= the same infinitesimal character as the trivial representation).
  2. Cuspidal (= non-boundary) cohomology classes \(H^1(\Gamma,\mathbf{R})\) which are eigenforms for the ring of Hecke operators \(\mathbf{T}\).
  3. Weakly compatible families of two-dimensional Galois representations of \(F\) which are irreducible of level \(N\) and Hodge-Tate weight \([0,1]\).
  4. Irreducible semi-stable \(p\)-adic Galois representations of weight \([0,1]\) and level (determined in the usual way) dividing \(N\).
  5. Abelian Varieties of \(\mathrm{GL}(2)\)-type over \(F\) which don’t have CM by \(F\).

The equivalence between \((1)\) and \((2)\) follows from Matsushima/Franke. The (conjectural) relationship between \((1)\) and \((3)\) is the problem of reciprocity in the Langlands programme. It consists of two directions; existence (i.e. constructing Galois representations from automorphic forms) and modularity (i.e. showing nice Galois representations are automorphic). Both of these directions are difficult. The passage from \((5) \Rightarrow (3) \Rightarrow (4)\) is easy, whereas \((4) \Rightarrow (5)\) is basically the Fontaine-Mazur conjecture (not so easy). Actually even that last statement is not quite correct: if one knows that \(V\) is pure of weight one with non-negative Hodge-Tate weights and arises up to twist in the cohomology of some smooth proper algebraic variety, then it should actually arise in cohomology without having to twist and hence come from an Abelian variety; proving this, however, is probably hard (as in Standard conjectures hard). Note that \((1) \Rightarrow (5)\) follows, in the case of classical modular forms, from a geometric construction of Shimura, but that idea doesn’t work here, and in fact this arrow is completely open and we shall say no more about it. In the particular case of imaginary quadratic fields, Harris, Soudry, and Taylor showed in 1992 that \((1) \Rightarrow (3)\) under certain favourable conditions. This case is slightly exceptional in this regard, since there exist functorial transfers of such \(\pi\) to \(\mathrm{GSp}(4)\) which do contribute to the \((\mathfrak{q},K)\)-cohomology of Shimura varieties and hence can be directly related to the coherent cohomology of Shimura varieties (although not directly to Betti cohomology, because the corresponding weights are not regular.) As readers of this blog know, only very recently, Harris-Lan-Taylor-Thorne established the same result for \(\mathrm{GL}_n\) over a CM field. (Small lie: not all the desired properties of the corresponding Galois representations, — i.e. local-global compatibility — have been established. I think Ila Varma is working this out for her thesis.)

It was observed early on that the cohomology groups of \(X\) are not, in general, torsion free. So what then does a torsion class represent? Computations by Grunewald, Helling, and Mennicke in an 1978 paper suggested that torsion classes (specifically, two-torsion with Hecke eigenvalues in \(\mathbf{F}_2\)) in these groups should be associated to \(\mathrm{GL}_2(\mathbf{F}_2) = S_3\) Galois representations of the field \(F\). Apparently there are even some unpublished notes from Grunewald in 1972 doing similar things, although I have only ever heard rumors of their existence (to be fair, I heard those rumours from Grunewald, so they’re probably pretty reliable). So very early on there were hints that a further story was going on between torsion classes and Galois representations that wasn’t immediately related to the (conjectural) story coming from automorphic forms. The first general and precise conjecture along these lines were formulated by Ash in his 1992 Duke paper, with further refinements by Ash–Sinnott, and Ash–Doud–Pollack. Unlike the previous speculations of Grunewald, these conjectures were precisely formulated and falsifiable, and in the spirit of Serre’s original conjecture. Moreover, Herzig did actually come along and falsify them, by finding a more natural prediction for the set of possible Serre weights which turned out to be different from the formulation of Ash et. al., and Herzig’s formulation subsequently proved (numerically!) to give the right answer in these cases (Edit: see comments, this is not quite correct). At any rate, for quite some time, we have expected that mod-p torsion classes give rise to Galois representations, and following the conjectures of Serre, Ash, and others, one can be quite precise about exactly the local properties the corresponding Galois representations should have at primes of bad reduction. What is perhaps more recent is the idea that, especially for groups \(G\) with no discrete series, that torsion is not merely a techincal nuisance, but rather is the source of “most” of the interesting Galois representations. In particular,

  • The phenomenon whereby Galois representations coming from the countably many classical automorophic forms are dense in a suitable universal deformation ring (Böckle, Gouvêa-Mazur, Chenevier) will be totally false when \(G\) does not have discrete series. On the other hand, the representations coming from torsion should cut out all of the universal deformation ring.
  • That in order to answer the most pressing questions concerning reciprocity (even in characteristic zero!) one needs to understand torsion classes.

    • \(\qquad\)

    For one take on this, I might suggest reading Section 1.1, Speculations on
    p-adic functorality, of this paper. For another interesting perspective, you should also read this, as well as the accompanying review.

    So let us assume then that studying torsion representations and associating them to Galois representations is an Important Goal. How do we construct them? An observation also going back a long way (I believe to Harder??) is the following. Even though \(G\) may not admit discrete series, there may exist a group \(H\) containing a parabolic \(P\) with \(G\) as a Levi. If \(H\) does admit discrete series, then there will exist a Shimura variety \(X_{H}\) whose Borel-Serre compactification will have at least one boundary component which is a torus bundle over \(X_{G}\), and as a result one obtains a map (with some mixing of degrees) \(H^*(X_{G},\mathbf{Z}) \rightarrow H^*(\overline{X}_{H},\mathbf{Z})\). Now one is theoretically in better shape, because this map should be compatible (in some sense) with Hecke operators, and the latter group has a chance to admit comparisons to étale cohomology groups which do come with Galois representations. There are three immediate problems:

    1. The compactification \(\overline{X}_{H}\) will be singular, except in the case of modular curves.
    2. Given a long exact sequence for (co)homology relative to a boundary divisor, it’s not clear whether the cohomology in the boundary ends up in \(H^{i-1}\) or \(H^{i}\).
    3. Just because a class \([c]\) has an interesting Hecke eigensystem doesn’t mean that that étale cohomology sees an interesting Galois representation.

    \( \ \)

    The third issue is a genuine problem. If one has a Hecke eigenclass \([c] \in H^*(X)\) in the etale cohomology of a Shimura variety, even in characteristic zero, then all one can deduce is that the corresponding Galois representation is annihilated by the corresponding characteristic polynomial of Frobenius. But this is not always enough to get the correct Galois representation! It’s probably worthwhile to consider two examples.

    First, a somewhat degenerate example. Let \(\mathbb{G} = \mathrm{GL}(1)/\mathbf{Q}\), \(G = \mathbf{R}^{\times}\), \(\mathbb{H} = \mathrm{SL}(2)/\mathbf{Q}\), and \(H = \mathrm{SL}_2(\mathbf{R})\). Now \(X_{G}\) is (for some level structure) a finite set of points, and \(X_{H}\) is a modular curve with cusps. The boundary map realizes \(X_{G}\) as a set of cusps in \(X_{H}\). The Hecke operators act on cusps by the degree map, i.e. \(T_p[c] = (1+p)[c]\). This coincides with the action of Hecke operators on \(H^0(X_H)\). So, in the etale cohomology group \(H^0(X_H)\) we have a Hecke eigenclass which we imagine (looking at the Hecke operators) to be associated to the Galois representation \(1 \oplus \chi\) where \(\chi\) is the cyclotomic character. Yet \(H^0(X_H)\) is one dimensional, and so we only see half of the Galois representation, namely, the trivial character. Now as it turns out, the other piece of the Galois representation can be seen in \(H^1(X_H)\), which is now mixed because \(X_H\) is not projective (so the cuspidal part has motivic weight one, and this other piece of the Eisenstein series has weight two). So even in this trivial case, we see that a Hecke eigenclass in etale cohomology may have a less interesting Galois representation than the Hecke eigenvalues might suggest. From the Eicher-Shimura relation, we do get that the (trivial) Galois representation which does occur is annihilated by the characteristic polynomial of Frobenius \((\sigma – 1)(\sigma – \chi(\sigma))\), indeed it is annihilated by the first factor.

    Second, let \(\mathbb{G} = \mathrm{GL}(2)/F\), \(G = \mathbf{GL}_2(\mathbf{C})\), \(\mathbb{H} = U(2,2)/\mathbf{Q}\), and \(H = U(2,2)\). Here \(\mathbb{H}\) is taken to split over \(F\). The cohomology of (a torus bundle over) the Bianchi group maps into the cohomology of \(U(2,2)\). The characteristic polynomials of the Hecke operators are, morally, the following. If \(\rho\) is the (conjectural) Galois representation associated to an eigenclass on the Bianchi group, then \(r = \wedge^2 (\rho \oplus \rho^c)\) is a six dimensional (reducible) representation which is a direct sum \(r = s \oplus \psi \oplus \psi^c\) for a four dimensional representation \(s = \rho \otimes \rho^c\) and a Grossencharacter \(\psi\) and its conjugate (which are related to the central character of the original form and its conjugate). Now the characteristic polynomials of Frobenius on this Galois representation are, by Eichler–Shimura, the characteristic polynomials of Hecke on the image of this cohomology class in the cohomology \(H^*(X_{H})\). Without assuming one has \(\rho\), one can phrase the above purely in terms of Satake parameters, but this way of saying it makes clearer what is going on, even though we don’t know yet that \(\rho\) actually exists. If one could find the Galois representation \(r\) (and in particular \(s\)) inside the etale cohomology of \(X_H\) one would (almost) be done, but instead, the classes which actually turn up in etale cohomology in these degrees are the reducible terms in \(r\) corresponding to the Grossencharacters rather than to the interesting representation \(s\) we are looking for. So as above, even in characteristic zero, one has the interesting Hecke eigenclass, but not the Galois representation.

    These examples suggest that to understand what is going on we first need to get a better understanding of Shimura varieties. Most of the recent history of understanding Shimura varieties (and the Galois representations associated to automorphic forms) has concentrated on the cohomology arising from cuspidal automorphic representations. In this classical setting, the automorphic representations have a classical avatar as global sections of certain coherent bundles on \(X_H\). (For example, classical modular forms of weight \(\ge 1\) are global sections of the line bundle \(\omega^{\otimes k}\).) If we want to restrict to cusp forms, we can also take the corresponding extension of these sheaves to minimal (or toroidal, doesn’t matter) compactifications which vanish appropriately at the boundary. If we denote these automorphic bundles by \(\mathcal{E}_{\mathrm{sub}}\), then another way of saying this is that the action of Hecke operators \(\mathbf{T}\) on

    \(\bigoplus_{\mathcal{E}} H^0(\overline{X}_{H},\mathcal{E}_{\mathrm{sub}})\)

    is now understood if \(X_H\) is, for example, a Shimura variety of unitary type over a totally real field. Even getting this far is a somewhat monumental task that required, amongst other things, Ngo’s work on the Fundamental Lemma, work of Kottwitz, Clozel, some large fraction of Jussieu, the work of Shin, and many more. In fact, as far as local-global compatibility goes, the ink is barely dry on the most recent work. Now we can at least state, in vague terms, the following:

    Theorem [Scholze, IV.3.1]: For (many) Shimura varieties \(X_H\), the action of \(\mathbf{T}\) on torsion classes in Betti cohomology factors through the action on coherent cusp forms in characteristic zero.

    Two examples: If \(X_H\) is the modular curve, then this says that the action of Hecke operators on \(H^1(X_H,\mathbf{Z}/p^n \mathbf{Z})\) can be realized by the action on classical modular cuspidal eigenforms modulo powers of \(p\). Given how we think about modular forms, this is almost tautological, because, by Eichler-Shimura, we can pass between cohomology classes and classical modular forms (in this case, we can even do this via the Hodge decomposition of \(H^1\)). However, there is a little wrinkle: we do see Eisenstein classes in Betti cohomology, and this theorem says that we can realize these as coming from cusp forms, so this result also implies that there exist cusp forms which are congruent to Eisenstein classes modulo \(p^n\). Since we are ultimately interested in classes coming from the boundary of some compactification, we don’t want to ignore this case. Still, it’s not so difficult to prove.

    If \(X_H\) comes from \(U(2,1)/\mathbf{Q}\) (so it is a arithmetic complex hyperbolic manifold of real dimension \(4\), also known as a Picard modular surface), then we can look at the group \(H_1(X_H,\mathbf{Z}_p)\). The characteristic zero classes here are known to correspond to endoscopic automorphic representations (and thus to not exist in the co-compact case) and are understood. However, unlike in the modular curve case, we no longer know that this group is torsion free, and in general, it may not be. So, a priori, all we know about the torsion classes and their Hecke operators is that there exists a Galois representation which is annihilated by the characteristic polynomial of \(T_p\), using Eichler-Shimura. These polynomials are all of fixed degree (three in this case), but that doesn’t give any lower bound on the dimension of this representation. This is an even more stark example of the well known phenomenon that Eichler-Shimura is pretty much useless for constructing Galois representations outside the case of dimension two where knowing both the trace and determinant tells you a lot. For example, suppose you have an irreducible representation \(V\) of a finite group \(G\) in characteristic zero such that all the elements of \(g\) have a minimal polynomial of degree at most \(d\): then you can’t a priori bound the dimension of \(V\)! As an example, the extra-special \(2\)-group of order \(2^{1 + 2n}\) has a representation of dimension \(2^n\) all of whose elements have images satisfying the degree two polynomials \(x^2 – 1 = 0\) or \(x^2 + 1 = 0\). So, before Scholze, we could not say anything about the dimensions of mod-\(p\) Galois representations arising from torsion in the first homology of \(U(2,1)\). However, using Scholze, we can now deduce that any such representation comes from a classical cusp form, and hence must (in this case) have dimension three!

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En Passant II

Posted on June 15, 2013 by Persiflage

Let’s party like it’s 1995! The Boston conference on Fermat produced a wonderful book, but now you can watch the original videos. Some first impressions: some of you used to have more hair (not naming names).

Forum of Mathematics Pi is, supposedly, a gold open access journal which is freely available to anyone. However, the designers of the journal decided to modify the model slightly to what I might call “access only to those unencumbered by taste,” by plastering a ridiculous sickly aquamarine running head across every page (click on the link here), making the entire paper an eyesore that is presumably impossible to print. Perhaps the point of this journal was to make it apparent that Elsevier does actually provide added value to their journals by not scribbling crayon markings all over the final product? Fortunately, the paper in question can also be downloaded here. (Hat tip to TG.)

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Scholze on Torsion 0

Posted on June 13, 2013 by Persiflage

This will be the first zeroth of a series of posts talking about Scholze’s recent preprint, available here. This is mathematics which will, no question, have more impact in number theory than any recent paper I can think of. The basic intent of this post is to commit to future posts in which I will discuss the details. I should remark that Scholze’s writing is pretty clear, so these posts will mainly be for my own benefit rather than yours.

Here are some of the specific points that I might cover:

Basics: The Hodge-Tate Period map, Perfectoid spaces, etc. To be honest, I will probably skip the details here to begin with, and discuss them only at points where they become fundamental for understanding.

Theorem IV.3.1: The action of Hecke on the completed cohomology groups \(\widetilde{H}^i(\mathbf{Z}/p^n \mathbf{Z})\) for Shimura varieties is detected by the action of Hecke on classical cuspidal automorphic forms. Although it may end up being no easier to consider, this result is already intersting in some quite degenerate cases. For example, this is new even for \(X = U(2,1)/\mathbf{Q}\) and \(i = 1\) (Gee and Emerton’s results, for example, are contingent on the relevant Galois representations being three dimensional — now one knows that they are!). A very similar example is the case of a compact inner form of \(U(2,1)\) (so called Rogawski lattices) or, more generally, the simple Shimura variety of Kottwitz-Harris-Taylor type. Can one show in those cases that \(\widetilde{H}^i\) vanishes outside degree zero and outside the middle dimension? A weaker question: can one compute the completed cohomology in degree one? Compare with the work of Pascal Boyer.

Local Global Compatibility: Suppose one is in the ordinary case. Then the HLTT approach (via congruences, discussed previously on this blog (here, here, and here) should allow one to establish some cases of local-global compatibility. At ramified primes \(\ell \ne p\), the HLTT approach should also work, especially if one is also willing to assume that the residual representation is absolutely irreducible (using base change arguments). What can one do in the torsion case?

The Nilpotent Ideal: Scholze ultimately constructs Galois representations over \(\mathbf{T}/I\) for an ideal \(I\) such that \(I^m = 0\). The necessity of this ideal arises from a spectral sequence argument. (The parameter \(m\) only depends on the degree of the field and \(n\).) The Calegari-Geraghty modularity lifting argument (in the minimal case) can still be made to apply even with the presence of this ideal if one is in the minimal case, but not in the non-minimal case which will require \(m = 1\) (the Taylor Ihara’s avoidance trick requires more precise control than the minimal case). Are there any circumstances (extra assumptions, etc.) in which one can prove that \(m =1\)?

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Bad Headlines, and Schoenberg Redux

Posted on June 11, 2013 by Persiflage

A bad headline can go a good way towards making an otherwise interesting article seem condescending or off point. Slate seems particularly inept (or adept, depending on the intention) in choosing titles, a characteristic which must be particularly irritating for authors. A case in point: this recent article by J. Bryan Lowder in which he discusses Schoenberg’s Pierrot Lunaire. This blog has previously linked to Pierrot Lunaire, so there’s no complaint here about the selection, nor indeed of the article itself. But the headline “A Schoenberg piece you will actually like” suggests something quite different, and presenting Schoenberg’s Pierrot under that headline is a little like recommending Finnegans Wake in a column entitled “some light reading for the Beach this summer.” (No doubt when JSE gets around to writing the definitive Slate piece on Grothendieck, we can expect the title to be “Rings, you’re doing it wrong.”) As far as user friendly Schoenberg goes, there’s an obvious choice, namely the Verklärte Nacht Op. 4 string sextet from his pre twelve-tone days. Curiously enough, even this was panned as ultra-modern in its time. One critic described the sound “as if the score of Tristan had been smeared while the ink was still wet”, which actually sounds pretty good to me.

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Swans

Posted on May 30, 2013 by Persiflage

Since I first saw it, I’ve always been very impressed by Maya Plisetskaya’s dying swan. But Charles Riley gives her a run for her money in his own original performance with Yo Yo Ma. Compare and constrast!

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Elsevier’s answer to public criticism:

Posted on May 28, 2013 by Persiflage

The following was sent to editors for an Elsevier journal; a copy of the email mysteriously fell into my hands, and I reproduce it here (in part):

Following discussions with the board and at Elsevier this year, we feel that a change in how you are compensated for your editorial work is needed for next year. We sincerely appreciate the time and expertise that you dedicate to the journal, and know that without it, the journal would not be what it is today. Beginning in 2013, we would like to offer all Associate Editors an honorarium per handled manuscript (be it a rejection or acceptance) of 60 USD.

I guess this is one way to prevent editorial board revolts!

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Understatement

Posted on May 27, 2013 by Persiflage

This supposition, the so-called Twin Prime Conjecture, is not necessarily obvious .

“He wasn’t a big name, and I get the impression that he wasn’t one of the leading analytical number theorists,” said Richard Taylor, a respected mathematician and a member of the journal’s editorial board.

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Bush, the Messiah (and Emma Kirkby)

Posted on May 22, 2013 by Persiflage

The title of this post is, in part, a public service message to use the Oxford comma. However, there is a thread (in my mind) linking the three titular subjects. The connection between George Bush and the Messiah is not an obvious one, I admit, but hear me out. When I think of Bush, I think of the phrase “either you’re with us, or your against us.” I then always associate this phrase with “If God be for us, who can be against us?.” Was Bush consciously echoing the King James Bible? To me, of course, the latter phrase does not recall the Bible but rather Handel’s Messiah. All of which is a roundabout way of saying that this is another music post, with (who else) but Emma Kirkby performing with Christopher Hogwood and the Academy of Ancient Music (I have a recording on CD by the same ensemble which sounds to have been made contemporaneously with the video). The vintage of the haircuts is more H.W. than G., however.

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