Non-abelian reciprocity laws and higher Brauer-Manin obstructions
J. P. Pridham
TL;DR
This paper reinterprets Kim's non-abelian reciprocity maps as obstruction towers in etale homotopy types, extending the theory to include new obstructions like the Brauer--Manin, with applications to Shimura varieties and modular curves.
Contribution
It introduces a new perspective on Kim's reciprocity maps as obstruction towers, removing previous technical constraints and extending the framework to broader classes of varieties.
Findings
Obstruction towers can recover the Brauer--Manin locus.
Non-trivial reciprocity maps exist for Shimura varieties.
Obstructions relate to Galois cohomology of modular forms.
Abstract
We reinterpret Kim's non-abelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of etale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer--Manin obstruction, allowing us to determine when Kim's maps recover the Brauer-Manin locus. A tower based on relative completions yields non-trivial reciprocity maps even for Shimura varieties; for the stacky modular curve, these take values in Galois cohomology of modular forms, and give obstructions to an adelic elliptic curve with global Tate module underlying a global elliptic curve.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Non-abelian reciprocity laws and higher Brauer–Manin obstructions
J. P. Pridham
Abstract.
We reinterpret Kim’s non-abelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of étale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer–Manin obstruction, allowing us to determine when Kim’s maps recover the Brauer–Manin locus. A tower based on relative completions yields non-trivial reciprocity maps even for Shimura varieties; for the stacky modular curve, these take values in Galois cohomology of modular forms, and give obstructions to an adélic elliptic curve with global Tate module underlying a global elliptic curve.
The author was supported during this research by the Engineering and Physical Sciences Research Council [grant number EP/I004130/2].
Introduction
In [Kim], Minhyong Kim introduced a sequence of non-abelian reciprocity maps on the adélic points of a variety over a number field equipped with a global point and satisfying certain cohomological conditions, with the global points contained within the kernel of all the maps. When , this sequence just consists of a single map, the Artin reciprocity law
[TABLE]
from the finite idèles of to the abelianisation of its Galois group, with the property that .
In this paper, we give a topological construction of the non-abelian reciprocity maps, based on homotopical obstruction theory. These are defined under more general hypotheses than those of [Kim]. In particular, we do not need to assume existence of a global point in order to define the maps, so our reciprocity laws can be used to test the Hasse principle. For arbitrary varieties, the reciprocity maps exist as a tower of spaces over , with the cohomological conditions of [Kim] sufficing to ensure that the maps in the tower are injective.
Kim’s non-abelian reciprocity laws are based on the lower central series of the geometric fundamental group, but other variants are possible with our approach. One variant produces a tower starting with the Brauer–Manin obstruction, allowing us to compare it with Kim’s reciprocity laws. Another variant is based on relative completions, allowing us to study varieties whose geometric fundamental groups are perfect or nearly so.
For instance, the geometric fundamental group of the moduli stack of elliptic curves is the profinite completion of . This has finite abelianisation, so trivial pro-unipotent completion, which means the unipotent reciprocity maps of [Kim] are identically zero. However, the Malcev completion of relative to (resp. ) is a pro-unipotent extension of (resp. ) by a pro-unipotent group freely generated by duals of spaces of weight (resp. level ) modular forms. Elements in Galois cohomology of these tensors then give non-trivial obstructions to an adélic elliptic curve with global Tate module underlying a global elliptic curve.
Our point of view is that the reciprocity maps of [Kim] are obstruction towers in étale homotopy theory. The constructions of [AM, Fri2] associate a pro-simplicial set to any locally Noetherian simplicial scheme . When is smooth and quasi-projective over a field , with separable closure , [Fri2, Theorem 11.5] shows that for , the geometric homotopy type is the homotopy fibre of over , because the space is contractible. Moreover, is equivalent to the profinite completion of the homotopy type of the complex manifold , for any embedding , so is a whenever is so.
We are interested in the simplicial set
[TABLE]
i.e. the mapping space (or function complex) of pro-simplicial sets over (cf. Definition 2.1). The space is a , equivalent to the nerve of the Galois group . Since morphisms of schemes give rise to morphisms of étale homotopy types, there is then a natural map
[TABLE]
When is a (such as any hyperbolic curve, surface of general type, or abelian variety) over , we have (ignoring issues with basepoints)
[TABLE]
For smooth varieties , will always be of strictly negative weights, so , and we have
[TABLE]
a discrete set of points. This non-abelian cohomology set is the main focus of [Kim], and for hyperbolic curves , Grothendieck’s section conjecture amounts to the prediction that the morphism
[TABLE]
is an equivalence.
In this paper, we construct the reciprocity maps using obstruction theory analogous to [Bou]. The idea is to identify towers of quotients of over for which there exist non-abelian spectral sequences converging to , where . The crucial property making these spectral sequences special is that they incorporate fibre sequences
[TABLE]
giving obstructions to lifting homotopy classes of maps.
We can also take more general spaces as the source, considering a profinite homotopy type associated to the adèle ring , for a (possibly infinite) non-empty set of finite places. Reciprocity maps then arise in non-abelian spectral sequences converging to the homotopy groups of
[TABLE]
and he spaces in the spectral sequence are compactly supported cohomology groups , which can be rewritten as duals of Galois cohomology groups by Poitou–Tate duality. Defining the tower in terms of the lower central series of the geometric fundamental group recovers Kim’s reciprocity maps [Kim]. Subtler towers based on relative completions give rise to reciprocity laws in more general situations.
Explicitly, for a modular curve we can consider the set of adélic points for which the Tate module of the associated elliptic curve lifts to a -representation . We then construct a sequence of subsets (glossing over subtleties related to potential higher automorphisms for now)
[TABLE]
containing . These are defined inductively by , for reciprocity maps
[TABLE]
where the -vector spaces are given by homogeneous factors of a Lie algebra generated by
[TABLE]
for irreducible -representations ; via Eichler–Shimura, the groups can be interpreted as -adic realisations of motivic modular forms of weight and level . If we instead assume that the Tate modules lift to -representations for all primes , then we have a similar sequence, but with now defined in terms of modular forms of all levels. In this case, [HV] shows that whenever there is an adélic elliptic curve compatible with the representations , there must exist a rational elliptic curve giving rise to them, but our obstructions should measure the difference between these elliptic curves.
We can even incorporate higher homotopical information in constructing reciprocity laws for Deligne–Mumford stacks , by looking at completions of étale homotopy types instead of their fundamental groups. The first obstruction map in the spectral sequence is then just the Brauer–Manin obstruction when we take the base of the tower to be , with refinements for (pro-)étale covers given by the subtler obstruction towers.
The structure of the paper is as follows. Section 1 lays the topological foundations for constructing reciprocity laws, developing generalisations of Bousfield’s obstruction theory [Bou]. The most general statement is Proposition 1.5, giving obstruction spaces for homotopy limits of abelian extensions of simplicial groupoids.
Section 2 then applies this theory to give towers of obstructions to the existence of global points over a number field. The first such tower we consider is Example 2.5. Writing , , and ,
with the closure of , this gives a non-abelian spectral sequence
[TABLE]
encoding Ellenberg’s obstructions. There is a unipotent generalisation Example 2.12, and further refinements for relative completion. Notably, Examples 2.16 and 2.17 give obstructions, in terms of modular forms, to lifting a -representation to an elliptic curve over with Tate module .
In Section 3, this approach is refined to consider the difference between the obstruction towers for and , yielding reciprocity laws in terms of Poitou–Tate duality. The main examples of resulting spectral sequences appear in §3.2.2, with Examples 3.16 and 3.18 recovering and generalising Kim’s non-abelian reciprocity laws [Kim], while reciprocity laws for the stacky modular curve appear in Example 3.19, giving obstructions to an adélic elliptic curve being defined over when its Tate module is known to be a -representation.
Constructions in terms of higher homotopy types are then given in Example 3.21, with §3.2.3 showing how the spectral sequences for higher homotopy types start with the Brauer–Manin obstruction (or a pro-étale generalisation) as the first stage in the tower. Proposition 3.26 gives a sufficient condition for Kim’s non-abelian reciprocity laws to recover the Brauer–Manin set. In §3.3, we then discuss more concrete ways to construct the reciprocity laws, with a fairly explicit description of the first obstruction for modular curves, and a discussion of the relation between higher Brauer–Manin obstructions and Massey products.
Appendix A contains the technicalities needed to work with higher étale homotopical invariants of adèle rings, giving a morphism from to the homotopy type governing restricted products of local cohomology groups.
Readers unfamiliar with abstract homotopy theory are advised to skip §1 entirely, starting with §3.3 for an overview before reading the examples in §§2, 3. We should warn at this stage that none of the examples exhibits explicit classes in Galois cohomology on which to evaluate the obstructions, but the weights of the Galois representations involved suggests they must exist in great generality.
I would like to thank Minhyong Kim for many helpful discussions, Felipe Voloch for alerting me to relevant references, Ambrus Pál for a helpful observation, and the anonymous referee for catching several errors and suggesting improvements.
Notation
We will write for isomorphism and for weak equivalence. Let denote the category of simplicial sets with the Kan model structure, and the category of bisimplicial sets. We denote mapping spaces in model categories by ; in the case of simplicial model categories, these simplicial sets are just given by derived functors of the simplicially enriched bifunctor, and in general they are given by the function complexes of [Hov, §5.4].
We fix a number field , and a (possibly infinite) non-empty set of finite places of . Then denotes the Galois group of , and its quotient is the Galois group of the maximal extension of unramified outside . We write for the adèle ring
[TABLE]
Contents
1. Obstruction theory from abelian extensions
Given a fibration of spaces with fibre , there is a long exact sequence
[TABLE]
of homotopy groups and sets, where the final map need not be surjective (and at this stage we are being deliberately vague about basepoints).
Our primary goal in this section is to look for cases where this sequence extends one stage further, giving an obstruction map from to some pointed set such that the fibre over the basepoint is the image of . This will happen if there is some space and a map in the homotopy category of spaces, with the homotopy fibre over a point , and in this case above is automatically the loop space .
An obvious example of this phenomenon is when is a principal -bundle over for a topological group , so arises as the homotopy fibre of a map . We then have a long exact sequence
[TABLE]
noting that .
In this form, this statement is telling us nothing new, since is automatically surjective in such cases. However the characterisation of as a homotopy fibre also passes to homotopy limits of such diagrams. Given a small category , together with -diagrams and in simplicial sets and simplicial groups, and a principal -bundle over , we can characterise as the homotopy fibre of a map in the homotopy category, and then
[TABLE]
is a homotopy fibre sequence, so gives rise to a long exact sequence of homotopy groups and sets of the desired form; this is essentially the content of Corollary 1.11 below.
1.1. Central and abelian extensions of simplicial groups
1.1.1. Central extensions
We now look at principal fibrations in the category of groups. First observe that an internal group object in the category of groups is an abelian group by the Eckmann–Hilton argument, with multiplication being a group homomorphism.
An -space in groups is then a group equipped with a group homomorphism such that the diagram
[TABLE]
commutes. In other words, , for the group homomorphism to the centre of given by . The -action is faithful if is injective, and then is a principal -space over .
Applying the nerve functor, we have a simplicial abelian group (the group homomorphism inducing a multiplication , and for every principal -space in groups over , we get a principal -fibration over .
Definition 1.1**.**
Define to be the right adjoint to Illusie’s total functor given by . Explicitly,
[TABLE]
with operations
[TABLE]
Given a simplicial diagram of groupoids, the nerve is a bisimplicial set, and we write , noting that this agrees with the definition of [GJ, §V.7] when has constant objects.
Note that the loop space of is weakly equivalent to , so in particular , with .
In [CR], it is established that the canonical natural transformation
[TABLE]
from the diagonal is a weak equivalence for all . Thus is a model for the homotopy colimit
[TABLE]
and in particular a model for .
Proposition 1.2**.**
Given a surjection of simplicial groups with central kernel , there is a simplicial set weakly equivalent to and a map with fibre , which is also the homotopy fibre. Moreover, the space and weak equivalence can be chosen functorially.
Proof.
Writing , the statement is essentially the well-known result ([GJ, Theorem V.3.9]) that classifies principal fibrations. The reasoning above applied to simplicial groups gives us a bisimplicial abelian group and a principal -fibration over . Applying the codiagonal functor then gives us a simplicial abelian group and a principal -fibration over . The map then just comes by taking the homotopy quotient of by the action of .
Explicitly, we set , for the simplicial groupoid with objects and morphisms given by acting on the right. Applying twice to the map of groupoids in groups gives the weak equivalence , since and the fibre is contractible. Similarly, the Kan fibration comes from the map of groupoids in groups. ∎
1.1.2. Abelian extensions
More generally, given a group , a group object in the comma category of groups over is of the form , for an abelian group equipped with an -action.
Then a -space in groups over consists of a group and a surjection together with an associative action (all maps being group homomorphisms). Equivalently, for the group above, we have a group homomorphism over , hence a -equivariant map .
The condition for to be a principal -space is then just that the map be an isomorphism. In other words, a pair is the same as an abelian group equipped with an -action together with a surjective group homomorphism with kernel .
Given such a , we can take the nerve, giving a surjective fibration of simplicial sets with fibre over the unique vertex of . The simplicial set is a group object in simplicial sets over , and is a principal -bundle.
Proposition 1.3**.**
Take a surjection of simplicial groups with abelian kernel . Then there exists a fibration
[TABLE]
for which the projection is a weak equivalence, with
[TABLE]
Moreover, the space and weak equivalence can be chosen functorially.
Proof.
We adapt the proof of Proposition 1.2. Set , for the simplicial groupoid with objects and morphisms . Applying twice to the map of groupoids in groups gives the weak equivalence , since and the fibre is contractible. Similarly, the Kan fibration comes from the map of groupoids in groups. ∎
1.1.3. Groupoids
The constructions above generalise to groupoids, and we will not concern ourselves with the full generality of internal groups in groupoids. We just observe that any abelian group is a fortiori an internal group in groupoids with one object, and that for any groupoid , an -representation in abelian groups has associated groupoid , which is a group object in groupoids over .
Definition 1.4**.**
Say that a morphism is an abelian extension if it is an isomorphism on objects, surjective on morphisms, and the groups are abelian for all objects of .
Thus for any abelian extension of groupoids with kernel , we get a surjective fibration of simplicial sets, and the fibre over is just . Moreover, is a group object in simplicial sets over , and is a principal -bundle.
Proposition 1.5**.**
Given an abelian extension of simplicial groupoids with abelian kernel , there is fibration such that the projection is a weak equivalence, with
[TABLE]
Moreover, the space and weak equivalence can be chosen functorially.
Proof.
The proof of Proposition 1.3 carries over, replacing groupoids in groups with groupoids in groupoids. ∎
1.2. Passage to homotopy limits
For a small category , we have a limit functor from -diagrams of simplicial sets to simplicial sets. Recall from [GJ, §VIII.2] or [Hir, Ch. 18] that is the right-derived functor of ; in other words, it is the universal functor under preserving weak equivalences.
Definition 1.6**.**
Given a small category and simplicial group-valued functors , we say that a natural transformation is a central (resp. abelian) extension if the maps are so, for all .
Proposition 1.7**.**
Given a central extension of -diagrams with kernel , there is a morphism in the homotopy category of simplicial sets with homotopy fibre over the distinguished point .
Proof.
We just apply the derived functor to the diagrams from Proposition 1.2. ∎
Note that when , the simplex category, this recovers a fairly general case of Bousfield’s obstruction maps from [Bou].
Corollary 1.8**.**
In the scenario of Proposition 1.7, there is a sequence
[TABLE]
of sets, exact in the sense that the fibre of over [math] is the image of .
Moreover, there is a group action of on whose orbits are precisely the fibres of .
For any , with , the homotopy fibre of over is weakly equivalent to , and the sequence above extends to a long exact sequence
[TABLE]
Proof.
This is just the long exact sequence of a fibration ([GJ, Lemma I.7.3]) applied to , and noting that
[TABLE]
so
[TABLE]
for all . ∎
Remark 1.9*.*
Were it not for the final term, Corollary 1.8 would just be the long exact sequence of homotopy for the map . The essential purpose of all our effort so far has thus been to incorporate the extra term , giving an obstruction to lifting connected components.
Proposition 1.10**.**
Given an abelian extension of -diagrams with kernel , there is a morphism in the homotopy category of simplicial sets over with a homotopy pullback diagram
[TABLE]
In particular, if the adjoint action of on factors through some quotient , then for any , we have a fibration sequence
[TABLE]
on homotopy fibres over .
Proof.
We just apply the derived functor to the diagrams from Proposition 1.3. ∎
Now, given an -diagram , write .
Corollary 1.11**.**
In the scenario of Proposition 1.10, an element lies in the image of
[TABLE]
if and only if , where denotes the homotopy fibre of over .
Moreover, for each such there is a transitive group action of on the fibre of .
For any with , the homotopy fibre of over is weakly equivalent to , and the sequence above extends to a long exact sequence
[TABLE]
Proof.
The proof of Corollary 1.8 carries over. ∎
2. Towers of Diophantine obstructions
Recall that we are fixing a number field , and a (possibly infinite) non-empty set of finite places of . When consists of all finite places, we have a weak equivalence ; in general, [Čes, Appendix A, Equation (1)] combines with [Fri2, Corollary 6.5] to show that the homotopy fibre of the surjective map becomes contractible on derived pro- completion, i.e. pro- completion (cf. [AM, Theorems 3.4 and 4.3]) for the set of integer primes all of whose -prime factors lie in .
Given any profinite group and a pro-surjection (such as when is the arithmetic fundamental group of an -scheme), we have a fibration of pro-simplicial sets in the model structure of [Isa].
Thus for any pro-simplicial set over , we may consider the mapping space
[TABLE]
for the same model structure; when , this is the space of sections of .
Explicitly, [Isa, Proposition 10.9] allows us to describe mapping spaces of pro-simplicial sets in terms of the Edwards–Hastings strict model structure [EH], reducing to the following description.
Definition 2.1**.**
For pro-simplicial sets , such that each has finitely many non-zero homotopy groups, we may define the simplicial set in terms of mapping spaces of simplicial sets as the homotopy limit
[TABLE]
For general pro-simplicial sets , , we may define as the homotopy limit
[TABLE]
where denotes a Postnikov tower.
For a diagram of pro-simplicial sets, the relative mapping space is the homotopy fibre of over .
2.1. Abelian extensions
Assume that we have abelian extension of profinite groups with kernel , such that the conjugation action of on factors through some quotient of . When working with nilpotent completions of geometric fundamental groups, we may take , but for relative completions (as needed for modular curves), will be larger.
Writing , we have:
Proposition 2.2**.**
In the scenario above, and for any pro-simplicial set over , there is a natural fibration sequence
[TABLE]
of mapping spaces, the fibre being taken over the zero map .
Proof.
The idea behind this statement is that the extension defines an element of , which we can write as a morphism in the homotopy category of simplicial profinite groups over . The proof of [Pri3, Proposition 1.19] adapts to any Artinian category, and in particular to finite groups, allowing us to regard simplicial profinite groups as pro-objects in the category of (bounded) finite simplicial groups. We can then recover as a homotopy fibre product
[TABLE]
leading to the fibration sequence above.
More formally, we write as a filtered limit of finite quotient groups, inducing compatible expressions , and with .
The mapping spaces are given by
[TABLE]
so we apply Proposition 1.10 to the abelian extension
[TABLE]
of -diagrams in groups, and then take homotopy fibres over the canonical basepoint of . ∎
We think of the base of the fibration as an obstruction space; via the description of Definition 2.1, its homotopy groups are given by equivariant cohomology groups
[TABLE]
so we have an exact sequence
[TABLE]
In particular, the obstruction to lifting a homotopy class of maps to lies in , and the ambiguity in this lift is given by an action of on the fibres.
Remark 2.3*.*
Given an abelian extension of pro-simplicial groups with kernel , such that the conjugation action of on factors through some quotient of , there is a natural fibration sequence
[TABLE]
of mapping spaces, for any pro-space over .
Example 2.4*.*
In order to understand the first obstruction map explicitly, consider the case when is reduced and connected, so and an element of is a conjugacy class of pro-group homomorphisms over . Here, is a pro-group with generators and relations for . Since is surjective, we may lift to a morphism of pro-sets. The obstruction then measures the failure of to be a group homomorphism, in the form of the -cocycle
[TABLE]
2.2. Nilpotent obstruction towers
We can of course iterate the construction of Remark 2.3, by considering towers of surjections whose kernels are abelian -representations. The motivating examples are given by the quotients of by the lower central series of , and by their pro- completions relative to for .
Writing for the kernel of and , we then have an exact couple
[TABLE]
similar to that in [GJ, §VI.2], but with the extra final terms . Here, the connecting homomorphism is of homological degree , so we have
[TABLE]
This induces a non-abelian spectral sequence
[TABLE]
of groups and sets, where the terms are only defined for , and the indexing convention follows [GJ, §VI.2], with . Unlike the fringed Bousfield–Kan spectral sequence of [GJ, §VI.2], we have terms ensuring that we can recover the images of
[TABLE]
from our spectral sequence.
Explicitly, writing
[TABLE]
there are long exact sequences
[TABLE]
(as in [GJ, Lemma VI.2.8], but with extra final terms ).
The first page just corresponds to the exact sequences
[TABLE]
Example 2.5* (Nilpotent completion of ).*
If is a scheme over , and , with some geometric point , then the simplest examples are given by taking lower central series
[TABLE]
where for a profinite group we define inductively to be the closure of , with .
Thus , and taking , we get the non-abelian spectral sequence
[TABLE]
of groups and sets, where we write . If lies over a point in , then is just the semi-direct product of and the pro-nilpotent completion of .
Since points in map to elements in , this spectral sequence gives obstructions to the existence of such rational points. The same constructions work when is a Deligne–Mumford stack instead of a scheme, in which case we have a morphism from the groupoid to the fundamental groupoid .
The maps are just Ellenberg’s obstructions, which can be described in terms of Massey products as in Wickelgren’s thesis [Wic1].
Another variant is given by taking a smooth scheme over admitting a smooth relative compactification, and setting , with some geometric point . For a prime all of whose -prime factors lie in , we can consider the relative pro- completion (or more generally pro-nilpotent pro- for a set of such primes) of over in the sense of [HM2], which will have the effect of replacing with -torsion groups — the corresponding maps are described in [Wic2].
Alternatively, if we replaced with the étale homotopy type of an -scheme , we would instead obtain topological obstructions to the existence of a map over .
Example 2.6* (Relative completion of — descent obstructions).*
When the geometric fundamental group of is perfect, its nilpotent completion is trivial, so the construction of Example 2.5 gives no information. However, we can remedy this by taking the completion relative to a larger group than . We may take any quotient of bigger than , then write , and set .
This gives a non-abelian spectral sequence
[TABLE]
of groups and sets. Here, the Galois action on depends on the relevant section .
When is a finite extension of , each section as above gives a finite étale group scheme over with , and hence having étale homotopy type . Even when is not a finite extension of , we can write it as a filtered limit of such finite extensions, with each section giving a pro-(finite étale) group scheme over . Maps then correspond to -torsors , and we may substitute in the spectral sequence above.
Example 2.7* (Relative completion of ).*
As a special case of Example 2.6, take a congruence subgroup ; we may then form a stacky modular curve over a number field . The geometric fundamental group is the profinite completion of , so a point gives an isomorphism . The Tate module of the universal elliptic curve over gives rise to a -local system of rank on , and hence a map
[TABLE]
(for any choice of basepoint).
Since the local system has determinant , this induces a map
[TABLE]
and we may then take the relative pro-nilpotent completion over the image, or the relative pro- completion over the image in . Since the maps are pro- extensions, completion relative to gives the same limit from a different tower.
For , with , the spectral sequence resulting from the pro-nilpotent tower relative to is
[TABLE]
where .
The spectral sequence relative to instead has
[TABLE]
for .
2.3. Unipotent extensions
We now look to consider towers of unipotent extensions of Lie groups over , where all -prime factors of lie in of our set of places of .
Definition 2.8**.**
Say that a simplicial group is bounded if its Dold–Kan normalisation (given by ) is so.
Lemma 2.9**.**
If is a bounded simplicial unipotent algebraic group over , equipped with a continuous action of a profinite group , then is the filtered colimit of its bounded simplicial profinite -equivariant subgroups.
Proof.
This is a slight generalisation of [Pri6, Lemmas LABEL:weiln-admissprop and LABEL:weiln-admissiblelattice], which address the case where the -action is semisimple. Standard arguments give a -equivariant bounded simplicial -submodule of the Lie algebra of , with of finite rank and . The closure of under monomial operations in the Campbell–Baker–Hausdorff product is still bounded and of finite rank, as is nilpotent, and the groups realise as a filtered colimit of the required form. ∎
Corollary 2.10**.**
Take an affine algebraic group over and a surjection of simplicial affine group schemes, with bounded unipotent. Then for any Zariski-dense profinite group , the simplicial topological group
[TABLE]
is a filtered colimit of those simplicial profinite subgroups which are bounded nilpotent extensions of .
Proof.
Since is the fibre of , it suffices to prove this for reductive. As in [Pri1], the simplicial unipotent extension then admits a section (i.e. a Levi decomposition), unique up to conjugation by ; this gives an isomorphism . Since is Zariski dense in the reductive group , its action is semisimple so we may appeal to Lemma 2.9, writing
[TABLE]
for bounded -equivariant simplicial profinite subgroups of . ∎
The nerve is then an ind-pro-simplicial set, and defining mapping spaces for these by the usual convention
[TABLE]
for profinite, we may apply Proposition 2.2 to unipotent extensions, by passing to filtered colimits:
Proposition 2.11**.**
Take a unipotent extension of algebraic groups over with commutative kernel , such that the conjugation action of on factors through some quotient of . Then for any Zariski-dense map with profinite, and for any pro-simplicial set over , there is a natural fibration sequence
[TABLE]
of mapping spaces, the fibre being taken over the zero map .
Example 2.12* (Unipotent completion of ).*
If is a smooth scheme over admitting a smooth relative compactification, and , with some geometric point , then we may use Proposition 2.11 to give a variant of Example 2.5. For simplicity, assume that we have a point under (if not, we can recover analogues of the constructions below by taking a -equivariant set , then consider the -equivariant surjection from the groupoid to the contractible groupoid on objects ).
Now, [Fri1] shows that has equivalent derived pro- completion to the homotopy fibre of . Since the section splits the long exact sequence of homotopy groups, this gives an isomorphism between the relative pro- completion of over and the semi-direct product . We then consider the lower central series
[TABLE]
of the pro-unipotent Malcev completion .
Thus , and taking we get a non-abelian spectral sequence
[TABLE]
of groups and sets, where we write .
Although this gives weaker obstructions than Example 2.5, the obstruction spaces are easier to calculate. The vector spaces are the graded pieces of a pro-nilpotent Lie algebra with generators and relations non-canonically isomorphic to . Since points in map to elements in , this spectral sequence gives obstructions to the existence of such rational points.
2.4. Pro-unipotent extensions
Relative Malcev completion was introduced by Hain in [Hai3] for discrete groups, and as in [Pri2] we consider the natural generalisation to profinite groups as follows:
Definition 2.13**.**
Given a topological group , a reductive pro-algebraic group over , and a Zariski-dense continuous representation , Define the Malcev completion to be the universal diagram
[TABLE]
with a pro-unipotent extension, and the composition equal to .
When the representation is clear from the context, we will write .
Remark 2.14*.*
The pro-unipotent radical is then given by for a pro-(finite-dimensional nilpotent) Lie algebra . For the ring of algebraic functions on over , equipped with its left -action, the abelianisation of is dual to the continuous cohomology , and there is a presentation of with relations dual to . In particular, if , then there are canonical isomorphisms
[TABLE]
where for the Lie operad . Explicitly, when is finite-dimensional, is the subspace of the free Lie algebra on generators consisting of homogeneous terms of bracket length .
Also note that if is a discrete group and its profinite completion, then for any representation of , the map is necessarily an isomorphism.
Examples 2.15* ().*
Our main motivating example is to take and its profinite completion , with (regarded as a group scheme over ) and the natural map.
Since the ring of functions is given by , for the irreducible -representation of dimension over and the underlying vector space, we have
[TABLE]
Thus , and Eichler–Shimura gives a description of in terms of the decomposition of into modular forms and cusp forms of weight and level .
Our groups of interest are We may think of the spaces as -adic realisations of motives of modular forms, as in [Del]. These -vector spaces admit -actions via the interpretation as summands of , interpreting as the -fold product of the universal elliptic curve over the moduli stack of elliptic curves (the Tate twists arise because we wish to regard as a Tate module rather than its dual).
More generally, we can take to be a congruence subgroup of , giving a similar expression involving modular forms of higher levels, but with relations coming from whenever it is non-zero.
Alternatively, we can look at the relative Malcev completion of the canonical morphism
[TABLE]
where we regard the profinite group as an affine group scheme over . Then we still have , and Leray–Serre gives
[TABLE]
so
[TABLE]
giving generators of in terms of modular and cusp forms of all weights and levels.
We can also just look at the relative Malcev completion of the canonical morphism , again regarding as an affine group scheme over . We then have
[TABLE]
with the corresponding vanishing, giving generators for in terms of modular and cusp forms of weight and all levels.
For our purposes, Proposition 2.2 is now not quite general enough, as our group schemes might not be of finite type. Consider an affine group scheme over and a surjection of simplicial affine group schemes, with bounded pro-unipotent, together with a Zariski-dense profinite group . We can then canonically write the morphism as a filtered limit of unipotent extensions of affine algebraic groups, with Corollary 2.10 giving that is an ind-profinite group, so is naturally a pro-ind-profinite group.
The nerve is then a pro-ind-pro-simplicial set, and defining mapping spaces for these by the usual convention
[TABLE]
for profinite and ind-profinite, Proposition 2.11 extends verbatim to pro-unipotent extensions .
Example 2.16* (Modular forms of level ).*
If is the stacky modular curve over , and , then the Tate module gives a surjective homomorphism whose relative pro- completion (or equivalently that over ) is the same as that of . In particular, there is a natural action of on the relative Malcev completion , and we may consider the pro-unipotent extension
[TABLE]
setting
[TABLE]
As in Example 2.7, for the representation given by the Tate motive , we have , so a section of the projection is equivalent to giving a -representation of rank over , with determinant .
For the universal elliptic curve , we have the Tate module , a lisse -sheaf of rank on , giving a -action on by identifying it with , for the structure morphism .
Write
[TABLE]
Adapting Example 2.5, the pro-unipotent generalisation of Proposition 2.11 then combines with Examples 2.15 to give a non-abelian spectral sequence
[TABLE]
where the map is given by . Note that is mixed of weights (cusp forms and their conjugates) and (Eisenstein series), and that is pure of weight . Thus is mixed of weights and , so is of strictly positive weights, and .
Now set ; thus consists of representations whose determinant is the Tate motive, conjugation by giving equivalences, so consists of elements of commuting with the action of on . Since for , we then have exact sequences
[TABLE]
with a map . Here, is the nerve of the groupoid of maps , so is the set of isomorphism classes of elliptic curves over , and the group of automorphisms of the elliptic curve over ; the higher homotopy groups all vanish.
In other words, given a -representation of rank over , with determinant , these sequences give a tower of obstructions to lifting to an elliptic curve over with Tate module , and characterise the ambiguity of the lift at each stage. As in Examples 2.15, there is an entirely similar treatment for profinite completions of congruence subgroups , replacing with the modular curve .
Example 2.17* (Modular forms of all levels).*
Again taking to be the stacky modular curve over a number field and , we may consider the pro-unipotent extension
[TABLE]
setting
[TABLE]
Choose a section of the projection ; this is equivalent to giving a -representation of rank over , with determinant . Write
[TABLE]
As in Example 2.16, we then have a non-abelian spectral sequence
[TABLE]
where the map is given by . Since
[TABLE]
is mixed of weights (cusp forms of all levels and their conjugates) and (Eisenstein series of all levels), so is of strictly positive weights, and .
Now set ; thus consists of representations whose determinant is the Tate motive, conjugation by giving equivalences. Since for , we then have exact sequences
[TABLE]
with a map .
Example 2.18* (Modular forms of weight ).*
Again taking to be the stacky modular curve over , and , we may consider the pro-unipotent extension
[TABLE]
setting
[TABLE]
As in Example 2.18, choose a -representation of rank over , with determinant . Write
[TABLE]
thus is related to weight modular forms; as a Galois representation it is mixed of weights and . We then have a non-abelian spectral sequence
[TABLE]
where the map is given by .
Set ; since for , we then have exact sequences
[TABLE]
with a map .
Example 2.19* (Étale fundamental groups).*
For any smooth Deligne–Mumford stack over admitting a smooth relative compactification, with , we can generalise the examples above by considering any -equivariant Zariski-dense representation to a pro-reductive affine group scheme over . If there is no rational basepoint, we can instead take a -equivariant set of basepoints, then consider the -equivariant surjection from the groupoid to the contractible groupoid on objects , with relative Malcev completions as in [Pri1, §LABEL:htpy-malcev]).
We may then set
[TABLE]
with a quotient of described as in Remark 2.14.
For any section of the projection , we then have a non-abelian spectral sequence
[TABLE]
where the map is given by .
Example 2.20* (Étale homotopy types).*
We may refine the previous example by considering étale homotopy types in place of fundamental groups. Take a locally Noetherian simplicial scheme , and a geometric point . We can then form the étale topological type as defined in [Fri2, Definition 4.4]. In particular, we can apply this to a simplicial scheme resolving a locally Noetherian algebraic stack (by [Pri4, Theorem 4.7], such resolutions exist even for higher Artin stacks, and the description of [Pri4, Theorems 4.10 and Remark 4.11] ensures that the choice does not affect the homotopy type).
Note that is the set of geometric points of (with some bound imposed on the cardinalities of the associated fields). Consider the reduced pro-simplicial set given by setting to consist of -simplices with fixed vertex . We may then apply the simplicial loop groupoid functor of [DK] to get a pro-simplicial groupoid , and restricting to the vertex gives a pro-simplicial groupoid with .
If is defined over , with each admitting a smooth relative compactification, set . Now fix a Zariski-dense representation to a pro-reductive pro-algebraic group , and let be the Zariski closure of , and set . We now need to consider fibre sequences, because does not explicitly act on our model for . If the -representation is an extension of -representations for all -representations , then [Pri6, Theorem LABEL:weiln-lfibrations] combines with [Fri1] to give a fibre sequence
[TABLE]
of pro-algebraic homotopy types over . By [Čes, Appendix A, Equation (1)]), and have isomorphic cohomology for -torsion coefficients, so and we have a long exact sequence
[TABLE]
of pro-algebraic homotopy groups; in particular we will have an exact sequence of completed fundamental groups whenever , i.e. if is -good relative to in the sense of [Pri6, Definition LABEL:weiln-relgood2] and [Pri7, §LABEL:heid-relgoodsn].
We may then set to be the simplicial topological group given by the homotopy fibre product
[TABLE]
where ; in particular, . Note that since is equipped with a map from , there is a canonical morphism
[TABLE]
in the homotopy category of pro-ind-pro-simplicial sets.
We will then have a non-abelian spectral sequence
[TABLE]
with
[TABLE]
where is dual to (associating with an ind-lisse sheaf via and Remark 2.14), for now the cofree graded Lie coalgebra, and . Beware that the terms depend on an element of to determine the Galois action on .
The filtration corresponds to the good truncation filtration on , but there are variants for other filtrations, replacing with the page of the associated spectral sequence. For the case of the weight filtration on a quasi-projective variety, with representations tamely ramified around the divisor, see [Pri6, Corollary LABEL:weiln-htpyleray].
Note that taking path components of simplicial groups gives morphisms
[TABLE]
for the groups of Example 2.19. When (such as for the stacky modular curve), the filtration is just equivalent to the lower central series filtration of Example 2.19, so the morphisms are weak equivalences. For general , the towers will be different, but whenever the higher relative Malcev homotopy groups of vanish, the towers will converge to the same limit.
Remark 2.21*.*
To recover Example 2.16 from Example 2.20, we take to be the Zariski closure of the image of the representation
[TABLE]
given by combining the monodromy representation on with the pullbacks of the -representations . Then the Zariski closure of the image of is just , and the quotient is the Zariski closure of the representation . The conditions of [Pri6, Theorem LABEL:weiln-lfibrations] are then satisfied by construction.
Remark 2.22* (Algebraic monoids and weighted completion).*
A variant of Example 2.20 is given by taking a Zariski-dense representation to a pro-reductive pro-algebraic monoid , with the Zariski closure of , and . Then the theory of relative Malcev completion still works to give a homotopy fibre sequence
[TABLE]
of simplicial pro-algebraic monoids, and we may restrict to invertible elements ( etc.) and proceed as before.
For Example 2.16, that would mean adapting Remark 2.21 by taking to be the Zariski closure of the image of the representation
[TABLE]
The group would still be , and the obstruction spaces would be the same, but this gives a smaller sequence deriving them by ignoring data from irrelevant representations.
As noted in [Pri7, Remark LABEL:heid-monoidrk], the weighted completions of [HM1] relative to a pro-reductive group with central cocharacter can also be regarded as completions relative to a monoid, namely . This has the effect of excluding some, but not all, irrelevant representations in Example 2.16 (analogous to the distinction between effective motives and motives of non-negative weight).
Again, weighted completions generate the same obstructions as unweighted completion. In particular, for a relative curve in characteristic [math] and a generic point of , we may consider the fibre sequence as in [Hai1]. If we let be the Zariski closure of the natural representation from to (or to , or to any algebraic monoid in between), then we have a fibre sequence
[TABLE]
The associated obstruction and lifting data of the same type as the unipotent obstructions we encountered in Example 2.12, with relative completion in this case just providing an alternative description. These obstructions (and particularly the second stage of the tower) are the main technical ingredient of [Hai1].
3. Non-abelian reciprocity laws as obstruction maps
3.1. Adélic mapping spaces and compact supports
Definition 3.1**.**
In the category of pro-simplicial sets, we set
[TABLE]
where , and is the inertia subgroup; beware that both the coproduct and and the limit are taken in the category of pro-simplicial sets.
Note that there is a natural map .
Definition 3.2**.**
Given a finite abelian group equipped with a continuous -action, define
[TABLE]
where denotes the continuous cohomology complex and ranges over all finite subsets of containing the places at which the action on is ramified.
Definition 3.3**.**
Given a continuous profinite -representation , define
[TABLE]
where the range over the finite Galois-equivariant quotients of .
Similarly, given a continuous discrete torsion -representation , define
[TABLE]
where the range over the finite Galois-equivariant subgroups of .
Definition 3.4**.**
Given a continuous -representation in finite-dimensional vector spaces over , define
[TABLE]
where the range over the filtered direct system of all profinite subrepresentations of .
Given a continuous -representation in profinite-dimensional vector spaces over , define
[TABLE]
Note that for any -equivariant lattice in a finite-dimensional -representation over , the system of profinite subrepresentations is cofinal, so
[TABLE]
Remark 3.5*.*
Given a finite abelian group equipped with a continuous -action, observe that for all ,
[TABLE]
so .
The convention of Definition 3.4 ensures that this equivalence extends to profinite groups or (pro-)finite-dimensional -vector spaces (regarding as a pro-simplicial set or a (pro-)ind-pro-simplicial set).
Beware that is not necessarily the same as the étale homotopy type . However, there is a map from the former to the profinite completion of the latter (see Corollary A.5); on the level of fundamental groups this is just the observation that a finite lisse étale sheaf on is only ramified at for finitely many places in .
We may now adapt all the examples from §2 to consider adélic points instead of rational points. In particular:
Example 3.6* (Nilpotent completion of ).*
Using the pro-simplicial set , we may adapt Example 2.5. If is a Deligne–Mumford stack over , and , with some geometric point , again consider the lower central series
[TABLE]
where we write , . Thus , and taking in the tower of §2.2, we get the non-abelian spectral sequence
[TABLE]
of groups and sets, where we write .
The reasoning above (without recourse to Corollary A.5) gives a morphism of groupoids from to the fundamental groupoid , so the spectral sequence gives obstructions to the existence of such adélic points.
A variant of this construction is given by taking smooth over admitting a smooth relative compactification. For , we can then take to be the relative pro- completion of over (with our convention that all -prime factors of lie in ), giving a non-abelian spectral sequence
[TABLE]
with a morphism .
Example 3.7* (Unipotent completion of ).*
For unipotent adélic obstructions, we can adapt Example 2.12, taking a smooth scheme over admitting a smooth relative compactification, with and a geometric point. Assume that we have a point under (if not, there are analogous statements using a -equivariant set of basepoints instead), and consider the lower central series
[TABLE]
of the pro-unipotent Malcev completion .
Thus , and taking , we get a non-abelian spectral sequence
[TABLE]
of groups and sets, where we write . As in Example 3.6, there is a natural morphism of groupoids, but the obstruction spaces are easier to calculate in this setting.
Example 3.8* (Modular forms of level ).*
As in Example 2.16, let be the stacky modular curve, take , and consider the resulting pro-unipotent extension
[TABLE]
then set
[TABLE]
Using Example 2.7, a lift of the homomorphism is equivalent to giving -representations of rank over for , with determinant , such that for each , there are only finitely many with ramified. Write for the system .
As in Example 2.7, write . The pro-unipotent generalisation of Proposition 2.11 then combines with Examples 2.15 to give a non-abelian spectral sequence
[TABLE]
where the map is given by . Note that as is of non-zero weights.
Now set ; thus consists of sets as above, conjugation by giving equivalences, so consists of elements of commuting with the actions of the on . Since for , we then have exact sequences
[TABLE]
with a map . Here, is the set of isomorphism classes of elliptic curves over , and the group of automorphisms of the elliptic curve over .
In other words, given a system of rank local Galois representations over as above, these sequences give a tower of obstructions to lifting to an elliptic curve over with Tate module , and characterise the ambiguity of the lift at each stage. As in Examples 2.15, there is an entirely similar treatment for profinite completions of congruence subgroups , replacing with the modular curve .
Example 3.9* (Étale homotopy types).*
We now consider étale homotopy types in place of fundamental groups, as in Example 2.20. Take a smooth Deligne–Mumford stack over admitting a smooth relative compactification, and set . For a geometric point and a Zariski-dense representation to a pro-reductive pro-algebraic group , let be the Zariski closure of , and set .
We then look at the pro-simplicial group associated to the étale topological type . If the -representation is an extension of -representations for all -representations , then we may again set to be the simplicial topological group given by the homotopy fibre product
[TABLE]
where . Note that since is equipped with a map from , Corollary A.5 gives a canonical morphism
[TABLE]
in the homotopy category of pro-ind-pro-simplicial sets.
We then have a non-abelian spectral sequence
[TABLE]
with
[TABLE]
where is dual to .
3.2. Reciprocity laws
The idea behind non-abelian reciprocity laws is to compare the towers of obstructions for rational and adélic points, giving a relative obstruction tower for rational points over adélic points.
Definition 3.10**.**
Given a continuous -representation , we set
[TABLE]
where can be any of the types of representation considered in Definitions 3.2–3.4.
3.2.1. Abelian Poitou–Tate duality
Definition 3.11**.**
Define a contravariant functor on the category of abelian groups by
[TABLE]
Definition 3.12**.**
Define a contravariant functor on the category of continuous -representations in locally compact topological torsion abelian groups (in the sense of [HS2]) by
[TABLE]
Note that preserves the subcategory of finite representations, and interchanges profinite and discrete representations.
Lemma 3.13**.**
If is a finite set of finite places containing all primes dividing , and a continuous pro- -representation, then there is a canonical equivalence
[TABLE]
If is a continuous -representation in finite-dimensional vector spaces over , then we also have
[TABLE]
Proof.
The first statement is the formulation of Poitou–Tate duality given in [Lim], refining a homological isomorphism from [Nek]. For the second statement, take a -equivariant lattice , and then (writing ),
[TABLE]
the last isomorphism following because has finite rank, being finite. The result now follows because . ∎
Lemma 3.14**.**
If is a possibly infinite set of finite places, and a continuous -representation in profinite abelian groups whose order is a unit outside , then there is a canonical equivalence
[TABLE]
following the continuous cohomology conventions of Definition 3.3.
Proof.
When is finite, this is essentially the Poitou–Tate duality of [Mil, 1.4.10]. In general, writing for finite, we have
[TABLE]
∎
Remark 3.15*.*
If we wanted to extend Lemma 3.14 to more general coefficients, we would have to pass to a larger category than the category of locally compact topological torsion groups. The category precisely consists of the Tate objects over the category of finite abelian groups in the sense of [BGW]. Since and are functors from finite groups to complexes of Tate objects, their natural extension to coefficients in will take values in complexes of -Tate objects over finite abelian groups (or equivalently Tate objects over ), and Poitou–Tate duality will extend formally to that category.
3.2.2. Non-abelian reciprocity laws
We may now adapt all the examples from §2 to obtain obstructions to adélic points being rational points, with terms in the spectral sequence given by Galois cohomology with compact supports. Since the coefficients we consider have negative weights, the lower cohomology groups with compact supports tend to be small; when they vanish, the obstruction towers have no ambiguity in the lift at each stage.
Example 3.16* (Nilpotent completion of ).*
If is a Deligne–Mumford stack over , and , with some geometric point , then as in Examples 2.5 and 3.6 we may consider the lower central series
[TABLE]
where we write , and for the closure of . Write .
We then define the tower by the homotopy fibre products
[TABLE]
defined using the morphism from §3.1.
Taking homotopy fibres of the fibration sequences in §2.2, we then get a non-abelian spectral sequence
[TABLE]
of groups and sets, where we write . This comes from the exact couple
[TABLE]
with of cohomological degree .
As in §2, we have a map , so the spectral sequence gives obstructions to an adélic point being rational. When is a scheme (or algebraic space), and for .
By Lemma 3.14, is isomorphic to . Thus elements of give obstructions to lifting points in to , and the ambiguities of the lifts at each stage are dual to the groups , which are often finite for weight reasons as in [Jan]. The higher homotopy groups are necessarily [math], by vanishing of .
Remark 3.17*.*
Since is contained in the centre of , it seems that the spectral sequence in Example 3.16 can alternatively be obtained as an inverse limit of the non-abelian Poitou–Tate exact sequence of [Sti, Theorem 168].
Example 3.18* (Unipotent completion of ).*
Examples 2.12 and 3.7 adapt along the lines of Example 3.16. Take a smooth Deligne–Mumford over admitting a smooth relative compactification, with and a geometric point. Assume that we have a point under (if not, there are analogous statements using a -equivariant set of basepoints instead).
Now set
[TABLE]
and
[TABLE]
to give a non-abelian spectral sequence
[TABLE]
of groups and sets, where we write .
Lemma 3.14 shows that is isomorphic to . Since is smooth, the space is a pro-finite-dimensional Galois -representation of negative weights, so the local monodromy weight conjectures (as in the Poitou–Tate dual form of [Jan, Conjecture 6.3]) would imply for , with the exact couple yielding the spectral sequence then degenerating to exact sequences
[TABLE]
so the tower becomes a sequence of subsets.
Example 3.19* (Modular forms of level ).*
As in Examples 2.16 and 3.8, let be the stacky modular curve, take , and set
[TABLE]
where is here regarded as an algebraic group over .
We now write
[TABLE]
Since , the space consists of pairs with an adélic point and a -representation of rank over with determinant , together with an isomorphism of -representations.
Writing , Proposition 2.11 and Examples 2.15 then give a non-abelian spectral sequence
[TABLE]
In other words, given a global Galois representation and, for each , a local elliptic curve lifting each underlying -representation, with constraints on ramification, these sequences give a tower of obstructions to lifting to an elliptic curve over with Tate module and localisations ; the sequences also characterise the ambiguity of the lift at each stage.
As in Example 2.15, the group consists of modular forms and cusp forms of weight and level . Thus is a Galois -representation of weights , so it follows that is a pro-finite-dimensional Galois -representation of weights . As in Example 3.18, the local monodromy weight conjectures would cause the exact couple yielding the spectral sequence to degenerate to the exact sequences
[TABLE]
equipped with a map .
As in Examples 2.15, there is an entirely similar treatment for congruence subgroups , replacing with the modular curve . If we instead started from a representation over , relative Malcev completion of over as in Example 2.17 would give rise to reciprocity laws associated to modular forms of all levels. Meanwhile, relative Malcev completion of over as in Example 2.18 gives rise to reciprocity laws associated to weight modular forms of all levels.
Remark 3.20*.*
We may write
[TABLE]
As in Example 2.16, we may then consider the sheaf of relative Tate modules on , with , for the structure map . Applying Poitou–Tate duality in the form of Lemma 3.14 to this -lattice then gives
[TABLE]
providing an expression for as a summand of . As we will see in Example 3.24, the case is a part of the Brauer–Manin obstruction, divisible elements in cohomology giving rise to obstructions.
By [HV], for cofinite and , non-emptiness of implies non-emptiness of . The variants of Example 3.19 for relative Malcev completions of over or over should then help to identify .
Example 3.21* (Relative Malcev étale homotopy types).*
As in Examples 2.20 and 3.9, we may consider étale homotopy types in place of fundamental groups. Take a smooth Deligne–Mumford stack over admitting a smooth relative compactification, a geometric point and a Zariski-dense representation to a pro-reductive pro-algebraic group , let be the Zariski closure of , and set .
Now set to be the simplicial topological group given by the homotopy fibre product
[TABLE]
where .
The formula of Example 3.19 then gives a tower of spaces and an associated non-abelian spectral sequence
[TABLE]
with
[TABLE]
where is dual to and
[TABLE]
Remark 3.22*.*
As in [Pri6, Theorem LABEL:weiln-laff], Lafforgue’s theorem and Esnault–Kerz ([Laf, Theorem VII.6 and Corollary VII.8] and [EK]) imply that the ind-lisse sheaf on is pure of weight [math]. If is smooth and proper, [Pri6, Corollary LABEL:weiln-wgtexistspin] then implies that the group in Example 3.21 is pure of weight .
The obstruction spaces for étale homotopy sections are given in the spectral sequence by the terms . Assuming that is of geometric origin, the local monodromy weight conjectures (as in [Jan, Conjecture 6.3]) would imply that the groups vanish, so the only non-trivial contributions to come from
[TABLE]
as in Example 3.18, and from
[TABLE]
The latter group can only be only non-zero for , when contains copies of the Tate motive, in which case the reciprocity map is detecting the Brauer–Manin obstruction of a pro-étale covering whose geometric fibres are -torsors as in Example 3.27 below. These copies of the Tate motive then generate a large contribution to the term, producing an ambiguity in the lift much larger than the new obstruction, meaning the map would then be far from injective.
3.2.3. Brauer–Manin obstructions
We now look at Example 3.21 and analogous completions of étale homotopy types, giving rise to obstruction towers refining the non-abelian reciprocity laws by incorporating higher homotopical information. A common feature is that the first obstruction map in the tower is just the Brauer–Manin obstruction, or related (pro-)étale refinements in the case of relative completion. Because the higher obstructions induce non-abelian reciprocity laws, they will be non-trivial in any case where the higher reciprocity maps of [Kim] are non-zero on the relevant Brauer–Manin set.
If is a -equivariant -form for the ring of functions on the reductive group featuring in Example 3.21, then we may use Poitou–Tate duality to rewrite the term as
[TABLE]
when (unipotent completion of the geometric fibre), we have , and the first obstruction map is the rationalised Brauer–Manin obstruction
[TABLE]
Remark 3.23*.*
We may write as cohomology of a complex defined in terms of the Lie operad and the complexes for . In particular,
[TABLE]
where acts by switching the factors in . For , the expression in Remark 3.20 for modular curves generalises whenever , but usually there are extra factors reflecting the difference between reduced and non-reduced cohomology.
Taking nilpotent completion instead of unipotent completion gives the following:
Example 3.24* (Étale homotopy types and the Brauer–Manin obstruction).*
Take a smooth Deligne–Mumford stack over admitting a smooth relative compactification, and a geometric point . Applying relative pro- completion over levelwise (cf. [Pri6, §LABEL:weiln-profinitesn]) to the pro-simplicial group of Example 2.20 gives a pro-(finite simplicial group) as in the proof of Proposition 2.2; up to homotopy, this is independent of the choices made, by [Pri6, Proposition LABEL:weiln-Lcohoweak]. When is the set of all primes (corresponding to ), note that is just the profinite completion of .
We now refine Example 3.16 by considering relative pro-nilpotent completions of the whole profinite homotopy type instead of the fundamental group. For completions relative to , we set and
[TABLE]
which is a pro-(finite simplicial group).
We then construct a tower of homotopy fibre products
[TABLE]
defined using the morphism from §3.1 and Corollary A.5.
This gives a non-abelian spectral sequence
[TABLE]
where we regard the simplicial abelian groups as chain complexes.
Nielsen–Schreier implies that the simplicial group is given levelwise by profinite completions of free groups, so the term is given by which is just the reduced homology complex of with coefficients, where the product runs over those primes which are units in . Poitou–Tate duality in the form of Lemma 3.14 applied to the complexes thus gives
[TABLE]
where runs over all primes which are units in and we follow the usual convention for continuous cohomology, regarding as the ind-sheaf .
If we set to be the -torsion cohomological Brauer group, then the first obstruction map is thus the map
[TABLE]
induced by the natural map which is just the Brauer–Manin obstruction of [Man] when .
Writing for the kernel of , we thus have
[TABLE]
for the tower above, and the later pages of the spectral sequence give obstructions to lifting further up the tower. Beware, however, that when , the lifts are not unique at each stage; in particular if a point lies in the kernel of , we have a -torsor of possible choices on which to apply the secondary obstruction.
When is an algebraic space rather than a stack, we have , and may simply write for the image of .
Remark 3.25*.*
Because the simplicial pro-group of Example 3.24 is given levelwise by pro- completions of free groups, the Magnus embedding (applied to profinite groups as in [Wic1]) gives an isomorphism , where is the free Lie algebra functor, graded by bracket length, and the profinite completion of , applied levelwise to the simplicial abelian group. These functors are homotopy invariant when applied to chain complexes of projective modules via the Dold–Kan correspondence, but are not easy to calculate; they give the terms arising in the unstable Adams spectral sequence.
Over , the functor corresponds via the Dold–Kan correspondence to the free Lie algebra functor on chain complexes. Thus the spaces are much simpler to describe in terms of free Lie algebras, but they correspond to the obstructions for the unipotent completion of Example 3.21 (with ).
We are now in a position to compare Kim’s non-abelian reciprocity laws with the Brauer–Manin obstruction. Restricting to a single prime would give a similar statement for the -torsion part of the Brauer–Manin obstruction.
Proposition 3.26**.**
If the natural maps
[TABLE]
is surjective for all primes which are units in , then the image of the map from Example 3.16 is contained in the Brauer–Manin set .
Proof.
Take a free pro-simplicial resolution of , and observe that the cofibrancy of ensures that the natural map lifts to a map , unique up to homotopy.
Since a point of incorporates the datum of a -valued Galois representation, the composite map
[TABLE]
is necessarily [math]. The kernel of the middle map is the -torsion Brauer–Manin set as in Example 3.24, and via Poitou–Tate duality we can rewrite the final map as
[TABLE]
where denotes the reduced cohomology complex.
It suffices to show that this map is injective, or equivalently that its dual is surjective. This will follow from the Leray spectral sequences provided the maps
[TABLE]
are isomorphisms for and surjective for . The first condition is automatic and the second is our hypothesis. ∎
Considering the relative merits of the higher Brauer–Manin obstructions of Example 3.24 and the non-abelian reciprocity laws of Example 3.16, the latter generally avoid ambiguity of lifts to the higher stages of the tower, but converge more slowly.
Example 3.27* (Étale Brauer–Manin obstructions).*
While Example 3.24 considered completions of the étale homotopy type relative to , it also makes sense to consider completions with respect to larger quotients of over (i.e. relative pro- quotients of over ). We can write , and set .
As before, we define a tower by
[TABLE]
note that points in now include the data of sections of , because . The reasoning of Example 3.16 again gives a non-abelian spectral sequence
[TABLE]
of groups and sets. The terms depend on the section of induced by the relevant element of , the Galois action then coming from the natural -action on .
As in Example 2.6, each section above gives a pro-(finite étale -torsion) group scheme over with having étale homotopy type , and maps correspond to -torsors . The first obstruction map in the spectral sequence above is the disjoint union, over inner automorphism classes of sections , of the Brauer–Manin obstructions
[TABLE]
of the (defined as derived limits unless is finite), so we have
[TABLE]
(when is an algebraic space, we can drop the ’s). When , combining these for all finite extensions of will thus give Skorobogatov’s étale Brauer–Manin obstruction [Sko].
For smooth proper varieties, the space of adélic points is compact, and by Tychonoff’s theorem the inverse limit of non-empty compact spaces is non-empty, so considering pro-étale covers in this way will just recover the étale Brauer–Manin obstruction in this case.
When , the universal case to consider would take , with the spectral sequence then detecting exclusively higher homotopical information, and being a universal cover of . For this choice of , we may therefore set
[TABLE]
(again, we can drop the ’s when is an algebraic space).
Since has cohomological dimension , the higher homotopy groups never contribute to the obstruction spaces for in the non-abelian spectral sequence above. For the universal case , we have , and (the Hurewicz map for being an isomorphism). Thus for , meaning all higher obstructions vanish and
[TABLE]
Moreover the sequence is increasingly connected, so . Together, these phenomena imply that vanishing of the pro-étale Brauer–Manin obstruction alone implies the existence of a compatible section of the map of profinite étale homotopy types when is geometrically connected. This is not nearly as impressive as it might seem, since the construction of the pro-étale Brauer–Manin obstruction assumes a compatible section of .
Remark 3.28* (Relation to Harpaz–Schlank).*
Our spaces in this section are closely related to those of [HS1], which (after including Archimedean places) considers spaces broadly of the form
[TABLE]
as well as variants , and . In our terms, corresponds to replacing with above; the others are given by taking Postnikov towers.
Rather than imposing smoothness hypotheses and appealing to [Fri1] as we have done, [HS1] constructs a -equivariant homotopy type , and effectively works with the homotopy quotient in place of above. In [HS1, Theorem 11.1], the étale Brauer set is shown to correspond to the set , which is a somewhat stronger statement than our final observation in Example 3.27.
The main new ingredient in our constructions and comparisons is that by modelling profinite homotopy types as simplicial profinite groups and groupoids following [Pri6, §LABEL:weiln-profinitesn] and [Pri3, Proposition 1.19], we are able to work systematically with much more general towers than the Postnikov tower.
3.3. Alternative characterisations of the reciprocity laws
We now give a more pedestrian interpretation of the obstruction maps from §1, and show how this can give rise to a more explicit description of the first obstruction map in cases of interest. This first obstruction map seems to be well-known to experts, but we are not aware of a reference.
3.3.1. Cohomological obstruction classes
Extensions of a group by an abelian -representation are classified by
[TABLE]
by which we mean continuous cohomology when considering extensions of topological groups.
Given a group homomorphism , the obstruction to lifting to a homomorphism is then given by
[TABLE]
If , then the difference between two choices for is a derivation, so the set of choices is a torsor for the group
[TABLE]
Taking to be suitable quotients of the arithmetic fundamental group of a scheme over , the Diophantine obstruction maps on spaces of sections
[TABLE]
of §2 are all of this form. The adélic obstruction maps of §3.1 are a slight variant coming from looking at restricted products
[TABLE]
The reciprocity maps associated to an -point in §3.2 then effectively look at the difference between these obstructions, yielding an obstruction in via the exact sequence
[TABLE]
In general, this is not very easy to work with, but when the extension splits, so , the adélic point defines a derivation in , with associated abelian obstruction to lifting the adélic point to a rational point.
Example 3.29*.*
In nilpotent or unipotent settings such as Example 3.18, the first stage in the tower is a split extension
[TABLE]
Then an -point defines a class in whose image in is the first unipotent obstruction to being a rational point.
Example 3.30*.*
Relative Malcev completions as in Example 3.19 are a little more complicated. For the stacky modular curve, take , giving rise to a -representation of dimension over . We then set set , and
[TABLE]
with , where we are writing for the ring of algebraic functions on the scheme over .
Now, is an extension of by , so is given by a class in , where we may regard as an algebraic group. Since is reductive, the Leray–Serre spectral sequence then gives
[TABLE]
which vanishes because .
We therefore have a split extension . (For more general relative Malcev completions, a similar conclusion will still hold by combining Leray–Serre with the splitting of the extension .)
Thus an adélic elliptic curve defines a class in , whose image in is the first obstruction to being defined over with Tate module .
3.3.2. The first obstruction for modular curves
We now give an explicit description of the abelian obstruction of Example 3.30, seeking elliptic curves with given Tate module.
On the modular curve , the Tate module of the universal elliptic curve gives a lisse -sheaf of rank , and we write . On pulling back to , the sheaves correspond to the irreducible representations of , and we consider the Galois representations . For each , the adjunction defines a class
[TABLE]
Now take an adélic point , and assume that there is a -representation with and an isomorphism which is -equivariant for all . A necessary condition for to lie in compatibly with is that the class lies in the image of . Following the conventions of §3.2.1 to replace the product with a suitable restricted product, we get an obstruction
[TABLE]
Combining these gives a map
[TABLE]
which is the first reciprocity map associated to the relative completion of in Example 3.19, via the isomorphism . We may then use Poitou–Tate duality as in Example 3.24 to rewrite the target of the map as
[TABLE]
adapting Example 3.27, this can be recovered from the Brauer–Manin obstruction of an inverse system of finite étale covers of , which in this case correspond to twisted level structures associated to the -representations .
Remark 3.31*.*
An intermediate step in the construction above associates to each elliptic curve over a class in
[TABLE]
The corresponding construction for complex elliptic curves and mixed Hodge structures is given in [Hai2, Remark 13.3] (evaluating the section at the point ). The extension arises geometrically as the relative cohomology group .
3.3.3. Higher Brauer–Main obstructions via cochain algebras
The unipotent obstructions which we have considered were formulated in terms of morphisms of simplicial pro-unipotent groups, so could be thought of as a form of Quillen homotopy type [Qui]. An equivalent alternative formulation would be to look at morphisms of Sullivan homotopy types [Sul], which are just algebras of cochains.
Taking a Deligne–Mumford stack over and writing , the cochain complex carries a natural cup product, and is in fact naturally quasi-isomorphic to a commutative differential graded algebra over . Equivalently this means that carries the structure of a unital -algebra (or strongly homotopy commutative algebra): it has a symmetric bilinear multiplication , which is associative up to a homotopy , and there is a hierarchy of higher homotopies formulated in terms of the Lie operad. In the case, Example 3.21 looks at the morphism
[TABLE]
defined by an adélic point, and studies obstructions to lifting it to a -morphism which is equivariant for the global Galois group , rather than just the pro-groupoid
[TABLE]
formed from local Galois groups.
- (1)
The first reciprocity law seeks just to lift this as a morphism of complexes, fixing , so the first obstruction lies in
[TABLE]
this is just the rational Brauer–Manin obstruction. 2. (2)
The secondary obstruction of §3.2.3 depends on a choice of -equivariant chain map, together with a homotopy of -representations making compatible with our chosen adélic point. Such a lift exists whenever the rational -torsion Brauer–Manin obstruction vanishes, and we now need to look at whether it respects the cup product. We thus ask whether the diagram
[TABLE]
commutes, up to a homotopy , in the derived category of -representations, with a further -equivariant homotopy between and the homotopy providing the known -equivariant commutativity of . The resulting obstruction lies in
[TABLE]
but this restricts to the finer obstruction described in Remark 3.23 when we take symmetry and the unit into account. 3. (3)
The third obstruction is more complicated, measuring obstructions to choosing the next component of a -morphism. If we choose a model of which is strictly (graded-)commutative, this means we seek a map satisfying
[TABLE]
which must vanish on the unit and on shuffle products. The right-hand side and associated -equivariant homotopy in terms of give rise to an obstruction class in
[TABLE]
which is closely related to Massey triple products 4. (4)
Explicit descriptions for the higher obstructions follow from the formulae for -morphisms as in [LV, §§10.2.2, 13.1.13] (take the expression for -morphisms in [LV, Proposition 10.2.12] and replace with by taking invariants under shuffle permutations). These are related to higher Massey products.
To express Example 3.21 in these terms beyond the case, we may reformulate via [Pri1, Proposition 3.15 and Corollary 4.41] to seek -equivariant morphisms
[TABLE]
for a pro-reductive algebraic groupoid over and a Zariski dense Galois-equivariant homomorphism with a Galois-equivariant set of basepoints . The descriptions above adapt, with the sheaf (regarded as a -representation via the left and right actions) replacing .
Remark 3.32*.*
If we wished to construct obstructions in the nilpotent, rather than unipotent setting, we should seek Galois-equivariant morphisms of cosimplicial commutative rings. The first obstruction is just Brauer–Manin, but the torsion in the higher obstructions is very difficult to describe, as discussed in Remark 3.25.
Remark 3.33*.*
The description in terms of cochain algebras will readily adapt to more general cohomology theories with cup product. For instance, a motivic analogue of §2 would be given by seeking -morphisms of cohomological -motives, assuming existence of a suitable -structure enriching the cup product on motivic cohomology. The obstruction tower just depends on a filtration on the -operad, whereas a Postnikov-type filtration in terms of motivic homotopy groups [Pri5, §LABEL:HHtannaka2-motsn] would require a suitable -structure. This approach could also be used to construct motivic obstructions to adélic points being global, along the lines of this section, but it is not obvious what the motivic analogue of Poitou–Tate duality should be.
Appendix A Pro-finite homotopy types for adèles
Definition A.1**.**
Write for the category consisting of simplicial groupoids for which
- (1)
The simplicial set of is constant and finite; 2. (2)
each is finite; 3. (3)
the group is trivial for all but finitely many .
Note that the second condition is equivalent to saying that the map to the -coskeleton is an isomorphism for sufficiently large .
Lemma A.2**.**
The functor from to simplicial profinite groupoids given by is an equivalence of categories; moreover, we may restrict to inverse systems in which all morphisms are surjective.
Proof.
Since is an Artinian category, the proofs of [Pri3, Proposition LABEL:ddt1-cSp] (which dealt with Artinian local rings rather than finite groupoids) and of [Pri6, Lemma LABEL:weiln-levelwiseproL] carry over to this generality. ∎
Definition A.3**.**
Given a simplicial scheme , define to be the global sections functor from simplicial étale presheaves on to simplicial sets. Write for its right-derived functor with respect to the model structure for étale hypersheaves. Explicitly,
[TABLE]
where runs over simplicial étale hypercovers of .
Given an inverse system , set
[TABLE]
Lemma A.4**.**
There is a canonical morphism
[TABLE]
in , functorial in simplicial profinite groupoids .
Proof.
Because is quasi-compact, the category of quasi-compact hypercovers of is left filtering in the category of all hypercovers, by the argument of [Fri2, Proposition 7.1]. Thus for all simplicial presheaves ,
[TABLE]
is an equivalence, where is the category of simplicial hypercovers and the full subcategory of simplicial hypercovers with each quasi-compact.
Given a simplicial presheaf for which the map is an isomorphism, the map
[TABLE]
is an equivalence, where consists of quasi-compact hypercovers which are truncated in the sense that for some (in fact suffices for the case in hand).
Given a quasi-compact hypercover , write for its pullback along . Thus each is the spectrum of a finite product of finite field extensions of . Because is of finite type over , it is defined over for some finite set of primes. For , it then follows that is the spectrum of a finite product of finite unramified field extensions of . When the hypercover is -truncated, we can set , and then see that
[TABLE]
where consists of quasi-compact hypercovers built from unramified field extensions.
Writing
[TABLE]
we then get a map
[TABLE]
Returning to the statement of the lemma, since both functors send filtered inverse limits to homotopy limits, Lemma A.2 allows us to restrict to the case where . Thus the map is an isomorphism for some , so and satisfies the conditions for above. Then we have
[TABLE]
Now, we can rewrite the right-hand side as
[TABLE]
Since and this is weakly equivalent to
[TABLE]
which is just , as required. ∎
Corollary A.5**.**
There is a canonical morphism
[TABLE]
in the homotopy category of pro-simplicial sets, where denotes profinite completion, and the étale topological type as in [Fri2, Definition 4.4].
Proof.
Since simplicial profinite groupoids model profinite homotopy types by [Pri6, Proposition LABEL:weiln-cohochar], it suffices to show that we have natural morphisms
[TABLE]
for simplicial profinite groupoids , and this is precisely the content of Lemma A.4. ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[AM] M. Artin and B. Mazur. Etale homotopy . Lecture Notes in Mathematics, No. 100. Springer-Verlag, Berlin, 1969.
- 2[BGW] Oliver Braunling, Michael Groechenig, and Jesse Wolfson. Tate objects in exact categories. Mosc. Math. J. , 16(3):433–504, 2016. With an appendix by Jan Šťovíček and Jan Trlifaj.
- 3[Bou] A. K. Bousfield. Homotopy spectral sequences and obstructions. Israel J. Math. , 66(1-3):54–104, 1989.
- 4[Čes] K. Česnavičius. Poitou-Tate without restrictions on the order. Math. Res. Lett. , 22(6):1621–1666, 2015.
- 5[CR] A. M. Cegarra and Josué Remedios. The relationship between the diagonal and the bar constructions on a bisimplicial set. Topology Appl. , 153(1):21–51, 2005.
- 6[Del] Pierre Deligne. Formes modulaires et représentations l 𝑙 l -adiques. In Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363 , volume 175 of Lecture Notes in Math. , pages Exp. No. 355, 139–172. Springer, Berlin, 1971.
- 7[DK] W. G. Dwyer and D. M. Kan. Homotopy theory and simplicial groupoids. Nederl. Akad. Wetensch. Indag. Math. , 46(4):379–385, 1984.
- 8[EH] David A. Edwards and Harold M. Hastings. Čech and Steenrod homotopy theories with applications to geometric topology . Lecture Notes in Mathematics, Vol. 542. Springer-Verlag, Berlin-New York, 1976.