Elliptic curves over Q(i) are potentially automorphic

This spectacular theorem was announced by Richard Taylor on Thursday, in a lecture at the joint meetings.  Taylor credited this result and others to Allen-Calegari-Caraiani-Gee-Helm-Le Hung-Newton-Scholze-Taylor-Thorne (!), as an outcome of the (not so) secret mini-conference which took place at the IAS this fall.  The key new input here is work in progress of Caraiani-Scholze on the cohomology of non-compact unitary Shimura varieties, which can be leveraged to check (at least in some cases) the most difficult hypothesis in the Calegari-Geraghty method: local-global compatibility at l=p for torsion classes.

The slides from my talk can be found here. Naturally I managed to say “diamond” a bunch of times.

 

Nowhere-vanishing sections of vector bundles

Let X/k be a proper variety over some field, and let \mathcal{E} be a vector bundle on X.  The functor of global sections of \mathcal{E}, i.e. the functor sending a scheme f:S \to \mathrm{Spec}\,k to the set H^0(S \times_{k} X, (f \times \mathrm{id})^{\ast} \mathcal{E}), is (representable by) a nice affine k-scheme, namely the scheme \mathcal{S}(\mathcal{E}) = \mathrm{Spec}(\mathrm{Sym}_k H^0(X,\mathcal{E})^{\vee}). Let \mathcal{S}(\mathcal{E})^{\times} \subset \mathcal{S}(\mathcal{E}) denote the subfunctor corresponding to nowhere-vanishing sections s \in H^0(S \times_k X, (f \times \mathrm{id})^{\ast} \mathcal{E}). We’d like this subfunctor to be representable by an open subscheme. How should we prove this?

Let p: \mathcal{S}(\mathcal{E}) \to \mathrm{Spec}\,k be the structure map. The identity map \mathcal{S}(\mathcal{E}) \to \mathcal{S}(\mathcal{E}) corresponds to a universal section s^{\mathrm{univ}} \in H^0(\mathcal{S}(\mathcal{E}) \times_k X, (p \times \mathrm{id})^{\ast}\mathcal{E}). Let Z\subset |\mathcal{S}(\mathcal{E}) \times_k X| denote the zero locus of s^{\mathrm{univ}}. This is a closed subset. But now we observe that the projection \pi: \mathcal{S}(\mathcal{E}) \times_k X \to \mathcal{S}(\mathcal{E}) is proper, hence universally closed, and so |\pi|(Z) is a closed subset of |\mathcal{S}(\mathcal{E})|.  One then checks directly that \mathcal{S}(\mathcal{E})^{\times} is the open subscheme corresponding to the open subset |\mathcal{S}(\mathcal{E})| \smallsetminus |\pi|(Z), so we win.

I guess this sort of thing is child’s play for an experienced algebraic geometer, and indeed it took Johan about 0.026 seconds to suggest that one should try to argue using the universal section.  I only cared about the above problem, though, as a toy model for the same question in the setting of a vector bundle \mathcal{E} over a relative Fargues-Fontaine curve \mathcal{X}_S. In this situation, \mathcal{S}(\mathcal{E}) is a diamond over S^\lozenge, cf. Theorem 22.5 here, but it turns out the above argument still works after some minor changes.

Three things I learned from colleagues this semester

1 (from Patrick Allen) Let F be a number field, and let \pi be a cohomological cuspidal automorphic representation of some \mathrm{GL}_n(\mathbf{A}_F).  Suppose that \rho_\pi : G_{F,S} \to \mathrm{GL}_n(\overline{\mathbf{Q}_p}) exists and satisfies local-global compatibility at all places, and that H^1_f(F, \mathrm{ad}\,\rho_\pi) = 0 as predicted by Bloch-Kato.  Then the following are equivalent:

a) H^2(G_{F,S}, \mathrm{ad}\,\rho_\pi) = 0, as predicted by Jannsen’s conjecture;

b) H^1(G_{F,S}, \mathrm{ad}\,\rho_\pi) has the right dimension;

c) The product of restriction maps H^1_f(F,\mathrm{ad}\,\rho_\pi(1))\to \prod_{v|p} H^1_f(F_v,\mathrm{ad}\,\rho_\pi(1)) is injective.

The equivalence of a) and b) follows from Tate’s global Euler characteristic formula, but their equivalence with c) was news to me.  The question of whether or not c) holds came up incidentally in my work with Jack on Venkatesh’s conjecture, so it was very pleasing to learn that it follows from Bloch-Kato + Jannsen.

2 (from Keerthi Madapusi Pera) If \mathbf{G} / \mathbf{Q}_p is semisimple and simply connected, and isotropic (i.e. contains some \mathbf{Q}_p-split torus), then \mathbf{G}(\mathbf{Q}_p) has no proper finite-index subgroups.

3 (from Stefan Patrikis) Let \pi be as in 1) again. There are two number fields naturally associated with \pi (besides F): the field \mathbf{Q}(\pi) generated by its Hecke eigenvalues, and the “reflex field” E\subseteq F of its cohomological weight.  Is there any chance that E is always a subfield of \mathbf{Q}(\pi)?, I asked SP.  Yes, said he.

Artin vanishing is false in rigid geometry

Sorry for the lack of blogging.  It’s been a busy semester.

Let k be an algebraically closed field, and let X be a d-dimensional affine variety over k.  According to a famous theorem of Artin (Corollaire XIV.3.5 in SGA 4 vol. 3), the etale cohomology groups H^i_{\mathrm{et}}(X,G) vanish for any i > d and any torsion abelian sheaf G on X_{\mathrm{et}}. This is a pretty useful result.

It’s natural to ask if there’s an analogous result in rigid geometry.  More precisely, fix a complete algebraically closed extension k / \mathbf{Q}_p and a d-dimensional affinoid rigid space X=\mathrm{Spa}(A,A^\circ) over k.  Is it true that H^i_{\mathrm{et}}(X,G) vanishes for (say) any i>d and any \mathbf{Z}/n\mathbf{Z}-sheaf G on X_{\mathrm{et}} for n prime to p?

I spent some time trying to prove this before realizing that it fails quite badly.  Indeed, there are already counterexamples in the case where X=\mathrm{Spa}(k \langle T_1,\dots,T_d \rangle,k^\circ \langle T_1, \dots, T_d \rangle) is the d-variable affinoid disk over k.  To make a counterexample in this case, let Y be the interior of the (closed, in the adic world) subset of X defined by the inequalities |T_i| < |p| for all i; more colloquially, Y is just the adic space associated to the open subdisk of (poly)radius 1/p. Let j: Y \to X be the natural inclusion.  I claim that G = j_! \mathbf{Z}/n\mathbf{Z} is then a counterexample.  This follows from the fact that H^i_{\mathrm{et}}(X,G) is naturally isomorphic to H^i_{\mathrm{et},c}(Y,\mathbf{Z}/n\mathbf{Z}), together with the nonvanishing of the latter group in degree i = 2d.

Note that although I formulated this in the language of adic spaces, the sheaf G is overconvergent, and so this example descends to the Berkovich world thanks to the material in Chapter 8 of Huber’s book.

It does seem possible, though, that Artin vanishing might hold in the rigid world if we restrict our attention to sheaves which are Zariski-constructible.  As some (very) weak evidence in this direction, I managed to check that H^2_{\mathrm{et}}(X,\mathbf{Z}/n \mathbf{Z}) vanishes for any one-dimensional affinoid rigid space X.  (This is presumably well-known to experts.)

Riemann-Roch sur la courbe

Let C/\mathbf{Q}_p be a complete algebraically closed extension, and let X = X_{C^\flat} be the Fargues-Fontaine curve associated with C^\flat.  If \mathcal{E} is any vector bundle on X, the cohomology groups H^i(X,\mathcal{E}) vanish for all i>1 and are naturally Banach-Colmez Spaces for i=0,1.  Recall that the latter things are roughly “finite-dimensional C-vector spaces up to finite-dimensional \mathbf{Q}_p-vector spaces”. By a hard and wonderful theorem of Colmez, these Spaces form an abelian category, and they have a well-defined Dimension valued in \mathbf{N} \times \mathbf{Z} which is (componentwise-) additive in short exact sequences.  The Dimension roughly records the C-dimension and the \mathbf{Q}_p-dimension, respectively.  Typical examples are H^0(X, \mathcal{O}(1)) = B_{\mathrm{crys}}^{+,\varphi = p}, which has Dimension (1,1), and H^1(X,\mathcal{O}(-1)) = C/\mathbf{Q}_p, which has Dimension (1,-1).

Here I want to record the following beautiful Riemann-Roch formula.

Theorem. If \mathcal{E} is any vector bundle on X, then \mathrm{Dim}\,H^0(X,\mathcal{E}) - \mathrm{Dim}\,H^1(X,\mathcal{E}) = (\mathrm{deg}(\mathcal{E}), \mathrm{rk}(\mathcal{E})).

One can prove this by induction on the rank of \mathcal{E}, reducing to line bundles; the latter were classified by Fargues-Fontaine, and one concludes by an explicit calculation in that case.  In particular, the proof doesn’t require the full classification of bundles.

So cool!