The Essential Dimension of Congruence Covers
Benson Farb, Mark Kisin, and Jesse Wolfson

TL;DR
This paper computes the minimal number of parameters needed to describe congruence covers of moduli spaces of abelian varieties and curves, using deformation theory and essential dimension concepts.
Contribution
It introduces new methods to determine the essential dimension of congruence covers of moduli spaces, connecting deformation theory with algebraic geometry.
Findings
Computed the essential dimension of congruence covers of moduli spaces
Determined the $p$-dimension of these covers
Applied results to moduli of curves and hyperelliptic loci
Abstract
Consider the algebraic function that assigns to a general -dimensional abelian variety an -torsion point. A question first posed by Kronecker and Klein asks: What is the minimal such that, after a rational change of variables, the function can be written as an algebraic function of variables? Using techniques from the deformation theory of -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential -dimension of congruence covers of the moduli space of genus curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties.
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The Essential dimension of congruence covers
Benson Farb, Mark Kisin and Jesse Wolfson
Department of Mathematics, University of Chicago
Department of Mathematics, Harvard
Department of Mathematics, University of California-Irvine
Abstract.
Consider the algebraic function that assigns to a general -dimensional abelian variety an -torsion point. A question first posed by Kronecker and Klein asks: What is the minimal such that, after a rational change of variables, the function can be written as an algebraic function of variables?
Using techniques from the deformation theory of -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential -dimension of congruence covers of the moduli space of genus curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety is proper. As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, non-abelian covering of a proper algebraic variety.
The authors are partially supported by NSF grants DMS-1811772 (BF), DMS-1601054 (MK) and DMS-1811846 (JW)
Contents
1. Introduction
This article grew out of an attempt to answer questions first raised by Kronecker and Klein.111See [KleinLetter, p. 171], [Kronecker, p. 309] and [Burkhardt1, Burkhardt2, Burkhardt3], esp. the footnote to [Burkhardt2, Ch. 11] on p. 216, and the treatment in §51-55 that follows. Let be an algebraically closed field of characteristic and consider the algebraic function that assigns to a general -dimensional (principally polarized) abelian -variety an -torsion point.
Question 1**.**
Let . What is the minimum such that, after a rational change of variables, the function can be written as an algebraic function of variables?
We can rephrase Question 1 in more modern language, using the moduli space of principally polarized abelian varieties and the notion of essential dimension introduced by Buhler–Reichstein [BR]. Let denote the coarse moduli space of -dimensional, principally polarized abelian varieties over , and let be the coarse moduli space of pairs where is a principally polarized abelian variety of dimension and is an -torsion point on
For any finite, generically étale map of -schemes, , define the essential dimension , or if the map is implicit, to be the minimal for which, generically, is a pullback:
[TABLE]
of a finite map of -dimensional -varieties via a rational map . In this case we call a (rational) compression of Question 1 can be rephrased as asking for the value of
Following Klein [KleinIcos], we can ask a related question, where we allow certain accessory irrationalities; that is, we can ask for
[TABLE]
for some class of finite, generically étale maps .222Given maps and , we use the notations and interchangeably to denote the fiber product. For example, the essential -dimension is defined (see [ReiYo], Definition 6.3) as the minimum of where runs over finite, generically étale maps of -varieties of degree prime to . For any map one always has
[TABLE]
where denotes the composite of Galois closures of the connected components of (see Lemmas 2.2.2, 2.2.3, cf. [BR, Lemma 2.3]). In particular, for any class of maps the answer to the question above does not change if we replace by the coarse moduli space of pairs where is a principally polarized abelian variety of dimension and is a symplectic basis for the -torsion (Here and below we fix once and for all an isomorphism so that we may speak of a symplectic basis for )
In this paper we apply techniques from the deformation theory of -divisible groups and finite flat group schemes to compute the essential -dimension of congruence covers of certain locally symmetric varieties, such as .
Theorem 2**.**
Let , and let be any prime with . Then
[TABLE]
Theorem 2 thus answers Question 1: the minimal equals . We in fact prove a more general result, that for subvarieties of satisfying some mild technical hypotheses, More precisely, we prove the following (see Theorem 3.2.6 below).
Theorem 3**.**
Let be prime, and let be an integer prime to . Suppose that an algebraic closure of let be its ring of integers and let be its residue field. Let be a locally closed subscheme that is equidimensional and smooth over , and whose special fiber meets the ordinary locus Then
[TABLE]
We give three applications of Theorem 3. The first is an analogue of Theorem 2 for the coarse moduli space of smooth, proper, genus curves over For any integer consider the level congruence cover , where denotes the moduli space of pairs consisting of a smooth, proper curve of genus together with a symplectic basis for where is the Jacobian of Applying Theorem 3 and the Torelli theorem, we will deduce the following.
Corollary 4**.**
Let . Let be any prime with . Then
[TABLE]
As a second application of Theorem 3, we answer a question encountered by Burkhardt in his study of hyperelliptic functions (see in particular [Burkhardt2, Ch. 11]). Let denote the coarse moduli of smooth, hyperelliptic curves of genus over For any integer consider the level congruence cover , where denotes the moduli space of pairs consisting of a hyperelliptic curve together with a symplectic basis for Analogously to the case of , we prove the following.
Corollary 5**.**
Let . Let be any odd prime. Then
[TABLE]
The hypothesis that is odd in Corollary 5 is necessary; see 3.3.5 below.
Our third application of Theorem 3 generalizes Theorem 2 to many locally symmetric varieties. Recall that a locally symmetric variety is a variety whose complex points have the form where is a Hermitian symmetric domain and is an arithmetic lattice in the corresponding real semisimple Lie group (see 4.2.3 below). Attached to there is a semisimple algebraic group over with for a maximal compact subgroup. By a principal -level covering we mean that the definition of does not involve any congruences at and is the subgroup of elements that are trivial mod A sample of what we prove is the following (see Theorem 4.3.5 below for the most general statement).
Theorem 6**.**
With notation as just given, suppose that each irreducible factor of is the adjoint group of one of with or for some positive integer If is unramified at (a condition which holds for almost all ) then for any principal -level covering , we have
[TABLE]
In fact our results apply to any Hermitian symmetric domain of classical type, but in general they require a more involved condition on the -group giving rise to they always apply if splits over Note that these results include cases where the locally symmetric variety is proper. As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, non-abelian covering of a proper algebraic variety. The only prior result for unramified covers of proper varieties of which we are aware is due to Gabber [CT, Appendix], who proved that if is a collection of connected, unramified covers of elliptic curves , then under certain conditions, the cover has essential dimension at equal to .
When is proper, the use of fixed-point techniques is precluded by the fact that one cannot use “ramification at infinity”. This is in analogy with Margulis superrigidity for irreducible lattices in higher rank semisimple Lie groups, where, for nonuniform (equivalently noncompact ), unipotents in play a crucial role. When is uniform, it contains no unipotents, and new ideas were needed (and were provided only later, also by Margulis).
There are many examples of finite simple groups of Lie type for which our methods give a lower bound on the essential -dimension of a covering of locally symmetric varieties with that group.
Corollary 7**.**
Let be a classical, absolutely simple group over with Then there is a congruence -cover of locally symmetric varieties such that satisfies :
- •
If is a form of which is split if is odd, then
- •
If is then
- •
If is a split form of then
- •
If is a form of and is not of type then
**Historical Remarks. **
- (1)
The study of essential dimension originates in Hermite’s study of the quintic [Hermite], Kronecker’s response to this [Kronecker], and Klein’s “Resolvent Problem” (see [KleinNU, Lecture IX] and [KleinIcos] esp. Part II, Ch. 1.7, as well as [Klein87], [KleinLast], and more generally the papers [KleinCW, p. 255-506]; see also [Tschebotarow]). Following Buhler-Reichstein [BR], the last two decades have seen an array of computations, new methods, generalizations, and applications of this invariant (see [ReICM] or [MerSurv] for recent surveys), but computations to date have largely focused on a different set of problems than those we consider here. 2. (2)
In contrast to Kronecker, Klein also advocated for the consideration of accessory irrationalities as a “characteristic feature” of higher degree equations, and called upon his readers to “fathom the nature and significance of the necessary accessory irrationalities.” [KleinIcos, p. 174, Part II, Ch.1.7] While essential dimension at partially answers this call, a full answer requires the notion of resolvent degree , which asks for the minimum such that an algebraic function admits a formula333Recall that a formula for an algebraic function on a complex variety is a tower of finite field extensions , a choice of primitive element for for each , and an embedding of fields over . One can then write for a rational function, which is to say a “formula” for in terms of the using only algebraic functions of or fewer variables (see [FW]). For example, Klein’s icosahedral formula for the quintic [KleinIcos] shows that for any algebraic function with monodromy , while the essential p-dimension can equal 2. At present, we do not know of a single example which provably has .
**Idea of Proof. ** To prove Theorem 2 and its generalization to subvarieties, we use arithmetic techniques, and specifically Serre-Tate theory, which describes the deformation theory of an ordinary abelian variety in characteristic in terms of its -divisible group. Let be an integer coprime to and let denote the universal abelian scheme over (now considered over ). Let be its -torsion group scheme, and let denote the fiber of at Given a rational compression of (in characteristic [math]) onto a smaller-dimensional variety, we show that there is an ordinary mod point of and a tangent direction at such that the deformation of corresponding to is trivial. From this we deduce that the deformation of corresponding to is trivial, a contradiction.
One might view our method as an arithmetic analogue of the “fixed point method” in the theory of essential dimension (see [ReICM]), where the role of fixed points for a group action is now played by wild ramification at a prime. In the fixed point method one usually works over a field where the order of the group is invertible. In contrast, for us the presence of wild ramification plays an essential role.
Acknowledgements. We would like to thank Aaron Landesman and Zinovy Reichstein for detailed comments on an earlier version of this paper and for extensive helpful correspondence. We also thank Maxime Bergeron, Mike Fried, Eric Rains, Alexander Polishchuk and Burt Totaro, for various helpful comments. Finally we thank the referee for many useful comments.
2. Preliminary results
2.1. Finite étale maps
We begin with some general lemmas on finite étale maps.
** 2.1.1***.*
Let be a map of groups, and a finite set with an action of We say the action of on lifts to if there is an action of on , which induces the given action of on We call such a -action a lifting of the -action on We say that the action of on virtually lifts to if there is a finite index subgroup containing the image of such that the action of on lifts to
For a positive integer we denote by the -fold product equipped with the diagonal action of and by , the projections.
Lemma 2.1.2**.**
Let be a map of groups, and a finite set with an action of
- (1)
Suppose the action of on admits a lifting to an action of on and fix such a lifting. Then there exists a finite index subgroup containing the image of such that and have the same image in 2. (2)
Let be a -stable subset such that Then the action of on virtually lifts to if and only if the action of on virtually lifts to
Proof.
Let be the kernel of and the subgroup generated by and the image of Since is finite, and hence has finite index in and satisfies the conditions in (1).
For (2), suppose that the action of on virtually lifts to By (1), after replacing by a finite index subgroup containing the image of we may assume that the action of on lifts to and that and have the same image in Then is -stable, so the action of on lifts to Conversely, if the action of on virtually lifts to then, by (1), we may assume that the action of on lifts to and that and have the same image in In particular, any element of acts on via an element of Since this element of is uniquely determined. The uniqueness implies that factors through a homomorphism which lifts the action of on ∎
** 2.1.3***.*
Let be a scheme, and a finite étale cover. We will say that is Galois if any connected component of is Galois over its image. If is a geometric point of then corresponds to a finite set equipped with an action of Conversely, if is connected and is a finite set with an action of we denote by the corresponding finite étale cover of
In the next three lemmas we consider a map of schemes and a finite étale cover equipped with an isomorphism If and are connected, and again denotes the geometric point of induced by then such a exists if and only if the action of on lifts to
Lemma 2.1.4**.**
If is Galois, then there is an finite étale such that factors through and is Galois.
Proof.
We may assume that and are connected. We apply Lemma 2.1.2, (1), to and Let be the finite étale map corresponding to the -set Since this set has a fixed point, factors through By construction, the images of and in are equal. Denote this image by and for denote by the stabilizer. Since is Galois, is a normal subgroup which does not depend on which is implies that is Galois. ∎
** 2.1.5***.*
Let be a finite ring. By an -local system on we will mean an étale sheaf of -modules which is locally isomorphic to the constant -module for some Such an is representable by a finite étale map This is clear étale locally on and follows from étale descent in general. Isomorphism classes of -local systems are in bijective correspondence with conjugacy classes of representations
For a map of schemes, and an -local system on we will denote by the pullback of to
Lemma 2.1.6**.**
Suppose that corresponds to an -local system Then there is a finite étale such that factors through and represents an -local system with
Proof.
Choose as in the proof of Lemma 2.1.4, using the construction in Lemma 2.1.2, (1). Then the finite set naturally has the structure of a finite free -module on which acts -linearly. Since the images of and in are equal, acts on -linearly, which implies the statement of the lemma. ∎
** 2.1.7***.*
For any integer we denote by the kernel of This is a finite flat group scheme on and we denote by the same symbol its pullback to any scheme This pullback is étale if any only if is a -scheme.
Suppose is a -scheme. For any algebra we again denote by or the étale sheaf For any non-negative integer we write For negative we set equal to the -linear dual of .
Lemma 2.1.8**.**
Let be an integer, integers, and a finite -algebra. Suppose that is a -scheme and that corresponds to an -local system which is an extension of by for some positive integers
Then there is a finite étale map such that factors through and the cover represents an -local system which is an extension of by with as extensions.
Proof.
Replacing by , we may suppose that By Lemma 2.1.6, we may assume represents an -local system and as -local systems. Let denote the sub -local system corresponding to so that corresponds to an -submodule
Since the group acts trivially on after replacing by we may assume that is an extension of by and is an isomorphism of extensions. A similar argument, applied to shows that we may assume that is an extension of by with as extensions. ∎
2.2. Essential dimension
** 2.2.1***.*
Let be a field, a -scheme of finite type, and a finite étale cover. The essential dimension [BR, §2] of over is the smallest integer such that there exists a finite type -scheme of dimension a dense open subscheme and a map such that is the pullback of a finite étale covering over The essential -dimension is defined as the minimum of where runs over dominant, generically finite maps, which have degree prime to at all generic points of
Note that when is irreducible, the definition does not change if we consider only coverings with irreducible. Indeed, for any as above, one of the irreducible components of will have degree prime to over the generic point of
Suppose is connected. A Galois closure of is a union of Galois closures of the connected components of If are finite étale, Galois and connected, then a composite of the is a connected component of Up to isomorphism, this does not depend on the choice of connected component. If we drop the assumption that is connected, we define the Galois closure and composite by making these constructions over each connected component of
Lemma 2.2.2**.**
Let be finite étale cover, and a Galois closure for For any map of -schemes we have
[TABLE]
In particular, we have
[TABLE]
Proof.
We may assume that is connected. Let be a geometric point for and a -set with Suppose first that is connected. Let be the (normal) subgroup which fixes pointwise. If and is the stabilizer of then for a finite collection of elements Let denote the -orbit of Then the stabilizer in of any point of is so is a Galois closure of
Now we drop the assumption that is connected, and let denote the connected components of with corresponding to a subset on which acts transitively. The above construction gives subsets (we may assume without loss of generality that does not depend on ) with a Galois closure of If then is a Galois closure for
From the construction one sees that and satisfy the conditions in Lemma 2.1.2, (2) (note that these conditions do not depend on the choice of ). For any map of groups these conditions continue to hold if we instead view and as -sets.
Now let be a map of -schemes, a connected open subset, and a map of -schemes. Suppose admits a geometric point mapping to Applying the above remark with Lemma 2.1.2, (2) implies that the action of on virtually lifts to if and only the action of on virtually lifts to As in the proof of Lemma 2.1.4, this implies that arises by pullback from a cover of for some finite étale if and only if the same condition holds for Since was an arbitary geometric point of this implies ∎
Lemma 2.2.3**.**
Let be connected Galois coverings of -schemes, and a composite of the Then for any map of -schemes we have
[TABLE]
In particular, we have
[TABLE]
Proof.
Let be a geometric point for and let be a -set with Let Then corresponds to a transitive set which surjects onto each Viewing we see that satisfies the conditions of Lemma 2.1.2. The lemma now follows as in the proof of Lemma 2.2.2. ∎
Lemma 2.2.4**.**
Let be a finite ring, a field, an -local system on a connected -scheme equipped with a geometric point and the representation corresponding to Let be the covering of corresponding to
Then is the composite of Galois closures of the connected components of In particular,
[TABLE]
For any prime we also have
[TABLE]
Proof.
Consider the action of on corresponding to The connected components of correspond to the stabilizers for A Galois closure of such a component corresponds to
[TABLE]
Thus the composite of such Galois closures corresponds to
The lemma now follows from Lemmas 2.2.2 and 2.2.3.
∎
Lemma 2.2.5**.**
Let be algebraically closed fields. If is a finite étale covering of finite type -schemes then
[TABLE]
Proof.
Let be a dense open and a morphism with such that arises from a finite cover Then the finite cover and the isomorphism are all defined over some finitely generated -algebra Specializing by a map produces the required data for the covering ∎
** 2.2.6***.*
It will be convenient to make the following definition. Suppose that is a finite étale covering of finite type -schemes, where is a domain of characteristic Set where is any algebraically closed field containing and similarly for By Lemma 2.2.5 this does not depend on the choice of
Lemma 2.2.7**.**
Let be an algebraically closed field, and a finite Galois covering of connected, finite type -schemes with Galois group Let be a central, cyclic subgroup of order with Then for any prime we have
[TABLE]
with equality if
Proof.
To show after shrinking we may assume there is a map such that for a finite étale covering which may be assumed to be connected and Galois by Lemma 2.1.4. The Galois group of is necessarily equal to and we have
For the converse inequality when we may assume there is a map such that for a finite étale covering which we may again assume is connected and Galois with group The image, of under
[TABLE]
is the obstruction to lifting to a -covering. Here, for the final isomorphism, we are using that is algebraically closed of characteristic prime to Viewing as a Brauer class, we see that it has order dividing This implies that (after perhaps shrinking further) there is an étale covering of order dividing a power such that is trivial [FarbDennis, Lemma 4.17]. In particular, has order prime to Replacing by their pullbacks to we may assume that and that for some Galois covering with group
The difference between the -coverings and is measured by a class in After replacing by the -covering corresponding to this class, we may assume that this class is trivial, and so This shows that ∎
3. Essential dimension and moduli of abelian varieties
3.1. Ordinary finite flat group schemes
In this subsection, we fix a prime and we consider a complete discrete valuation ring of characteristic with perfect residue field of characteristic and a uniformizer
By a finite flat group scheme on a -scheme we will always mean a finite flat, commutative, group scheme on of -power order. A finite flat group scheme on is called ordinary if étale locally on it is an extension of a constant group scheme by a group scheme of the form for integers In this subsection we study the classification of these extensions.
** 3.1.1***.*
Now let be an affine -scheme, and set Let and consider the exact sequence of sheaves
[TABLE]
in the flat topology of Taking flat cohomology of this sequence and its restriction to we obtain a commutative diagram with exact rows
[TABLE]
The group classifies line bundles on Hence, if is local it vanishes, and this can be used to classify extensions of by as finite flat group schemes. We have
[TABLE]
Here and below, the group on the left denotes extensions as sheaves of -modules.
Similarly, we can classify extensions of by as -divisible groups. If is such an extension, then is an extension of by and we have
[TABLE]
If is complete and local with residue field then the right hand side may be identified with the subgroup of units which map to in Thus we have
[TABLE]
** 3.1.2***.*
For the rest of this subsection we assume that Then by [SGA2]*XI, Thm 3.13, and we have a commutative diagram
[TABLE]
** 3.1.3***.*
We call an element of syntomic if it arises from an element of or equivalently from a class in and we denote by the subgroup of syntomic elements.
Lemma 3.1.4**.**
A syntomic class in arises from a unique class in
Proof.
If is a -power in then it is a power in as is normal. Hence the map is injective, and the lemma follows from the description of ’s above. ∎
Lemma 3.1.5**.**
Let for some integer and
[TABLE]
a local flat map of complete local -algebras. Suppose that where and that is syntomic. Then is syntomic.
Proof.
Let be an element giving rise to Since is a unique factorization domain we may write with and Since is syntomic, we may write with and Comparing the images of both sides in the group of divisors on one sees that So arises from ∎
** 3.1.6***.*
Let be the maximal ideal of and its image in The natural map
[TABLE]
is a bijection; both sides are -vector spaces spanned by We denote by the composite
[TABLE]
Here we have used Lemma 3.1.4 to identify and
Lemma 3.1.7**.**
With the notation of Lemma 3.1.5, suppose that
[TABLE]
is a subset such that the -span of is Then is an isomorphism.
Proof.
By functoriality of the association we have Hence the -span of is contained in image of It follows that surjects onto Since and are complete local -algebras, this implies that which is a subring of surjects onto Hence is an isomorphism. ∎
3.2. Monodromy of -torsion in an abelian scheme
We now use the results of the previous section to obtain results about the essential dimension of covers of of the moduli space of abelian varieties.
** 3.2.1***.*
Recall that an abelian scheme over a -scheme is called ordinary if the group scheme is ordinary for all This is equivalent to requiring the condition for
Let be an algebraically closed field of characteristic and let be a complete discrete valuation ring with residue field so that Let be an abelian scheme over of dimension We assume that is ordinary. Since is algebraically closed, this implies that is isomorphic to
Consider the functor on the category of Artinian -algebras with residue field which attaches to the set of isomorphism classes of deformations of to an abelian scheme over Recall [Katz:Serre-Tate, §2] that is equivalent to the functor which attaches to the set of isomorphism classes of deformations of and that is pro-representable by a formally smooth -algebra of dimension called the universal deformation -algebra of
We denote by the universal (formal) abelian scheme over Note that although is only a formal scheme over the torsion group schemes are finite over and so can be regarded as genuine -schemes.
Since is ordinary the -divisible group is an extension of by Hence is an extension of by This extension class is given by a matrix of classes with
Lemma 3.2.2**.**
With the notation of §3.1, the elements span
Proof.
Consider the isomorphism
[TABLE]
introduced in §3.1. The universal extension of -divisible groups over gives rise to a matrix of elements which reduce to
Let be the -span of the images of the elements or equivalently, the elements and set Using the isomorphism one sees that is defined over If then there exists a surjective map which sends to zero. Specializing by this map induces the trivial deformation of (that is the split extension of by ) over This contradicts the fact that pro-represents Hence which proves the lemma. ∎
** 3.2.3***.*
Let be a quotient of which is formally smooth over That is, is isomorphic as a complete -algebra to As in §3.1, we set and
Lemma 3.2.4**.**
Let for some integer and let
[TABLE]
be a local flat map of complete local -algebras. Set Suppose that is algebraically closed, and that there exists an -local system on which is an extension of by such that as extensions of -local systems. Then is an isomorphism.
Proof.
Using the notation of 3.2.1, we have spans by Lemma 3.2.2. In particular, if we again denote by the restrictions of these classes to then spans
Now by Lemma 3.1.5 the extension classes defining are syntomic. So arises from an extension of by as finite flat group schemes over If we denote by the corresponding matrix of elements of then Lemma 3.1.4, together with the fact that implies that It follows that the elements span which implies that is an isomorphism by Lemma 3.1.7. ∎
** 3.2.5***.*
Fix an integer , a prime , and a positive integer coprime to Consider the ring where is a primitive root of Using the isomorphism for any -scheme and any principally polarized abelian scheme over the -torsion scheme is equipped with the (alternating) Weil pairing
[TABLE]
We denote by the -scheme which is the coarse moduli space of principally polarized abelian schemes of dimension equipped with a symplectic basis of When this is a fine moduli space which is smooth over For a -algebra we denote by the base change of to If no confusion is likely to result we sometimes denote this base change simply by
Suppose that and let be the universal abelian scheme. The -torsion subgroup is a finite flat group scheme over which is étale over Let be a point with residue field of characteristic and the corresponding abelian variety over The set of points such that is ordinary is an open subscheme . For any we denote by the image of for any coprime to and
We now denote by a perfect field of characteristic and a finite extension with ring of integers and uniformizer We assume that is equipped with a choice of primitive root of We remind the reader regarding the convention for the definition of ed and introduced in 2.2.6.
Theorem 3.2.6**.**
Let and let be any prime. Let and coprime to and let be a locally closed subscheme which is equidimensional and smooth over and whose special fiber, , meets the ordinary locus Then
[TABLE]
Proof.
It suffices to prove the theorem when is algebraically closed which we assume from now on. Moreover, since is an arbitrary finite extension of it is enough to show that We may replace by a component whose special fibre meets the ordinary locus, and assume that and are geometrically connected.
Suppose that Then there exists a dominant, generically finite map of degree prime to at the generic points of and a map to a finite type -scheme with such that arises as the pullback of a finite étale covering of We may assume that is the scheme-theoretic image of under Next, after replacing both and by dense affine opens, we may assume that both these schemes are affine corresponding to -algebras and respectively, and that is flat.
Let be the normalization of in Let be the generic point of and the primes of over Since the degree of over is prime to for some the ramification degree and the degree of the residue field extension are prime to In particular, the residue field extension is seperable. By Abhyankar’s Lemma, it follows that, after replacing by a finite extension, we may assume that for some and that is étale at Shrinking further if necessary, we may assume that there is an affine open such that has dense image, and is étale. In particular, is smooth over
Now choose a finitely generated -subalgebra such that This is possible as is finitely generated over Then extends to a map Let be an ideal of and the blow up of Denote by the proper transform of by this blow up. That is, is the closure of in By the Raynaud-Gruson flattening theorem, [RG, Thm 5.2.2], we can choose so that is flat. Since is normal, the map is an isomorphism over the generic points of Hence, after replacing by an affine open in and shrinking we may assume that is flat.
Shrinking further, we may assume that the special fiber maps to the ordinary locus of Now let and denote the -adic completions of and respectively, and set and 444Although it would in some sense be more natural to work with formal schemes here, we stay in the world of affine schemes, so as to be able to apply the results proved in §1, and to deal with generic fibers without resorting to -adic analytic spaces. Since is ordinary, there is a finite étale covering such that is an extension of by Hence by Lemmas 2.1.6 and 2.1.8, factors through a finite étale map such that is the pullback of an extension of by on As is normal, we may assume is normal.
Let be the normalization of in As is normal, we have
[TABLE]
As is normal, is flat over the generic points of Hence, there exists which is nowhere nilpotent on and such that is flat over the complement of the support of the ideal Now let be a lift of and let and the -adic completions 555 corresponds to a formal affine open in the formal scheme of and Let and Then is flat over by [EGA, IV, 11.3.10.1]. Moreover, since is flat, the generic points of map to generic points of So the image of is dense in and in particular is non-empty.
Now choose a point and denote by its image. We write and for the complete local rings at and Since the maps
[TABLE]
are formally étale, is naturally isomorphic to the complete local ring at the image of in Let be the universal deformation -algebra of the abelian scheme Then is naturally a quotient of The map satisfies the conditions of Lemma 3.2.4, (cf. [EGA, IV, 17.5.3]) and it follows that this map is an isomorphism. In particular, this implies that
[TABLE]
which contradicts our initial assumption. ∎
Corollary 3.2.7**.**
Let be an integer, any prime, and an integer coprime to Let be a locally closed subscheme which is equidimensional and smooth over and whose special fiber, , meets the ordinary locus
If we also assume the following condition: For any generic point and the spectrum of an algebraic closure of the abelian variety over has automorphism group equal to Then
[TABLE]
Proof.
If the corollary follows from Theorem 3.2.6 and Lemma 2.2.4.
Suppose or Let be an integer coprime to We may assume that is equipped with a primitive root of The map of -schemes is a covering with group Consider the maps
[TABLE]
Then again corresponds to a covering, and corresponds to a covering. When these two coverings coincide.
Let Our assumption on the automorphisms of implies that at the generic points of the map is étale. Thus, after replacing by a fibrewise dense open, we may assume that is smooth over The observations of the previous paragraph, Lemma 2.2.7 when Lemma 2.2.4 and Theorem 3.2.6 imply that we have
[TABLE]
∎
Corollary 3.2.9**.**
Let and a positive integer coprime to Consider the finite étale map of -schemes Then for any we have
[TABLE]
Proof.
A fortiori it suffices to consider the case when is prime, which is a special case of 3.2.7. ∎
3.3. Moduli spaces of curves
Using the Torelli theorem one can use Theorem 3.2.6 to deduce the essential -dimension of certain coverings of families of curves.
** 3.3.1***.*
Let and let denote the coarse moduli space of smooth, proper, genus curves. For any integer let denote the -scheme which is the coarse moduli space of pairs consisting of a proper smooth curve of genus together with a choice of symplectic basis for where denotes the Jacobian of For this is a fine moduli space which is smooth over [DeligneMumford].
Theorem 3.3.2**.**
Let , and let be any prime dividing Then
[TABLE]
Proof.
Let be an integer. There is a natural map of -schemes taking a curve to its Jacobian. For in the pairs and are isomorphic via on Thus is an involution of which is non-trivial, unless We denote by the quotient of by this involution. When the fixed points of in any fibre of over are contained in a proper closed subset. Thus is generically smooth over every point of
By [OortSteenbrink] 1.11, 2.7, 2.8, the map of -schemes induces a map which is injective, an immersion if and an immersion outside the hyperelliptic locus if
It suffices to prove the theorem with replaced by the prime factor Let be coprime to and let It is enough to show that If then as coverings of and if we have natural degree maps of coverings of
[TABLE]
Thus, using Lemma 2.2.7 when it suffices to show that
[TABLE]
Since for example by comparing the degrees of these coverings, and meets the ordinary locus in [FabervanderGeer, 2.3], the theorem now follows from Theorem 3.2.6. ∎
** 3.3.3***.*
We now prove the analogue of Theorem 3.3.2 for the moduli space of hyperelliptic curves. Let be a -scheme. Recall that a hyperelliptic curve over is a smooth proper curve of genus equipped with an involution such that has genus Let denote the coarse moduli space of genus hyperelliptic curves over It is classical (and not hard to see) that over one has
[TABLE]
where is the moduli space of genus [math] curves with ordered marked points, and is the symmetric group on letters.
For any integer let denote the -scheme which is the coarse moduli space of pairs consisting of a hyperelliptic curve together with a symplectic basis for . As above, for this is a fine moduli space which is smooth over
Theorem 3.3.4**.**
Let , and let be any odd prime dividing Then
[TABLE]
Proof.
There is a natural map of -schemes which is generically an injective immersion on every fibre over as a general hyperelliptic curve has only one non-trivial automorphism [Poonen], Thm 1. By the Torelli theorem and [OortSteenbrink] Cor. 3.2, the map is also generically an injective immersion on every fibre over and thus so is
Now the Jacobian of a hyperelliptic curve over the generic point of has automorphism group [Matsusaka] p. 790, and meets the ordinary locus of by [GlassPries], Theorem 1. Hence the theorem follows from Corollary 3.2.7. ∎
** 3.3.5***.*
We remark that when Theorem 3.3.4 extends to as this is a special case of Theorem 3.3.2. However, an extension to is not possible when To explain this, recall that for a finite group and a prime denotes the supremum of taken over all -covers of finite type -schemes, for any characteristic algebraically closed field (the definition being independent of ). The covering
[TABLE]
is a component of for , the cover is disconnected, and all components are isomorphic.666The monodromy of was computed by Jordan [Jo, p. 364, §498] to factor as , where denotes the spherical braid group. See also [Di, p. 125], or for a more recent treatment, see the case of [Mc, Theorem 5.2]. The connected components of the cover are in bijection with the cosets . The equivalence of the components follows from the monodromy computation. We conclude that
[TABLE]
where the second equality follows from the versality of for , and the third follows from [MR, Corollary 4.2].
The lower bound can actually be recovered using the techniques of this paper. The point is that although is not generically an immersion in characteristic one can show that the image of the map on tangent spaces at a generic point has dimension We are grateful to Aaron Landesman for showing us this calculation.
4. Essential dimension of congruence covers
4.1. Forms of reductive groups
In this subsection we prove a (presumably well known) lemma showing that for a reductive group over a number field one can always find a form with given specializations at finitely many places.
** 4.1.1***.*
Let be a number field and an adjoint, connected reductive group over We fix algebraic closures and of and respectively, for every finite place of as well as embeddings
Recall [SGA3, XXIV, Thm. 1.3] that the automorphism group scheme of is an extension
[TABLE]
where is a finite group scheme. If is split, then this extension is split and is a constant group scheme which can be identified with the group of automorphisms of the Dynkin diagram of
We will also make use of the notion of the fundamental group [Borovoi]. This is a finite abelian group equipped with a -action. As an étale sheaf on one has where is the simply connected cover of and is the order
Lemma 4.1.3**.**
Let be a split, adjoint connected reductive group over and a finite set of places of The natural map of pointed sets
[TABLE]
is surjective.
Proof.
Recall the following facts about the cohomology of reductive groups over global and local fields [Kottwitzelliptic]: Let be an adjoint connected reductive group over For any place of , there is a map
[TABLE]
which is an isomorphism if is finite. For any finite set of places of consider the composite map
[TABLE]
Then by [Kottwitzelliptic, §2.2] is in the image of if Applying this to for some finite place we see that
[TABLE]
is surjective.
Next we remark that the map
[TABLE]
is surjective. Indeed, since is split a class in is simply a conjugacy class of maps and similarly for the local classes, so this follows from [Cal, Prop. 3.2].
Now let and let be the image of By what we have seen above, there exists mapping to Since we are assuming is split, (4.1.2) is a split extension, so there is a mapping to Let be the twist of by Recall that this means that, if we choose a cocycle representing then there is an isomorphism over such for and we have We have a commutative diagram
[TABLE]
such that the vertical maps send and to the trivial classes in the bottom line. Thus it suffices to show that
[TABLE]
is surjective, which we saw above. ∎
4.2. Shimura varieties
In this subsection we apply the results of Section 3 to compute the essential dimension for congruence covers of Shimura varieties. This will be applied in the next subsection to give examples of congruence covers of locally symmetric varieties where our techniques give a lower bound on the essential dimension. Since our aim is to give lower bounds on essential dimension, it may seem odd that we work with the formalism of Shimura varieties, rather than the locally symmetric varieties which are their geometrically connected components. However, many of the results we need are in the literature only in the former language, and it would take more effort to make the (routine) translation.
** 4.2.1***.*
Recall [DeligneCorvallis, §1.2] that a Shimura datum is a pair consisting of a connected reductive group over and a conjugacy class of maps This data is required to satisfy certain properties which imply that the commutant of is a subgroup whose image in is maximal compact and is a Hermitian symmetric domain.
Let denote the adeles over and the finite adeles. Let be a compact open subgroup. The conditions on imply that for sufficiently small, the quotient
[TABLE]
has a natural structure of (the complex points of) an algebraic variety over a number field called the reflex field of which does not depend on We denote this algebraic variety by the same symbol,
Now let equipped with a perfect symplectic form Set and We denote by the conjugacy class of maps satisying the following two properties
- (1)
The action of the real Lie group on gives rise to a Hodge structure of type
[TABLE] 2. (2)
The pairing on is positive or negative definite.
Then is a Shimura datum called the Siegel datum, and has an interpretation as the moduli space of principally polarized abelian varieties with suitable level structure.
We say that is of Hodge type if there is a map of reductive groups over which induces Given any compact open subgroup there exists a such that the above maps induce an embedding of Shimura varieties [DeligneTravaux, Prop. 1.15]
[TABLE]
** 4.2.2***.*
Now fix a prime and suppose that is the generic fibre of a reductive group over If no confusion is likely to result we will sometimes write simply for We take to be of the form where and where denotes the finite adeles with trivial -component. Under these conditions, is unramified in and for any prime of has a canonical smooth model over [Kisin, Thm. 2.3.8], [KimMadapusi, Thm. 1], which we will denote by In particular, we may apply this to if we take with
Given and of Hodge type, we may always choose and with such that induces a map of smooth -schemes
[TABLE]
which is locally on the source an embedding [KisinPappas, 4.1.5], [Kisin, Prop. 2.3.5]. That is, if is a closed point, and then the complete local ring at is a quotient of the complete local ring at In particular, there is an open subscheme of whose special fibre is dense in such that the restriction of to this open subscheme is a locally closed embedding. Fix such choices. As in §3, we denote by the universal abelian scheme over . Then we have
Lemma 4.2.3**.**
Suppose that the reflex field admits a prime with residue field Then a Galois closure of the étale local system is given by a congruence cover , with monodromy group isomorphic to over every geometrically connected component of
Moreover,
[TABLE]
Proof.
Let and denote geometrically connected components of and respectively with The étale local system corresponds to a representation
[TABLE]
where as above, and has image : By the smooth base change theorem and the comparison between the algebraic étale and topological fundamental groups for varieties over it suffices to check that the composite of the maps of topological fundamental groups
[TABLE]
has image This sequence of maps may be identified with [DeligneCorvallis, 2.1.2, 2.0.13]
[TABLE]
where and are -congruence subgroups of and -respectively, for some finite set of finite places not containing In particular, the image of the composite map is
By Lemma 2.2.4 we have
[TABLE]
We now consider the map of integral models corresponding to the prime of the lemma. Since has residue field every component of the image meets the ordinary locus of by [Wort, Thm. 1.1]. Now using that for some with there is a surjective map and Theorem 3.2.6, we conclude
[TABLE]
∎
4.3. Congruence covers
It will be more convenient to state the results of this subsection in terms of locally symmetric varieties. These are geometrically connected components of the Shimura varieties discussed in the previous subsection.
** 4.3.1***.*
Let be a semisimple, almost simple group over We will assume that is of classical type, so that (the connected components of) its Dynkin diagram are of type or
Let be a maximal compact subgroup. We will assume that is a Hermitian symmetric domain. The group is a product of simple groups for in some index set We denote by (resp. ) the set of with non-compact (resp. compact). Then is a product of the irreducible Hermitian symmetric domains for where is maximal compact. We use Deligne’s notation [DeligneCorvallis] for the classification of these irreducible Hermitian symmetric domains. Since we are assuming is of classical type, for is of type or The group is either the adjoint group of in case of type of in the case of type of in case of type or and of an inner form of if is of type
Since is almost simple, for the are all of the same type, except possibly if is of type in which case it is possible that both factors of type and occur among the We will say that is of Hodge type if all the factors are of the same type and the following condition holds: is simply connected, unless the are of type in which case is a product of special orthogonal groups.
** 4.3.2***.*
The Dynkin diagram is equipped with a set of vertices which is described as follows (cf. [DeligneCorvallis] §1.2, 1.3). For is the centralizer of a rank compact torus which is the center of Thus there are two cocharacters which identify with this compact torus. These cocharacters are miniscule, and each corresponds to a vertex of The two vertices are distinct exactly when are not conjugate cocharacters. In this case, they are exchanged by the opposition involution of which also gives the action of complex conjugation on We set to be the union of all the vertices above. Thus is empty if and consists of one or two vertices if In the latter case it consists of two vertices if and only if is either the adjoint group of with or of with odd.
** 4.3.3***.*
Fix an embedding The Galois group acts on We consider a subset such that consists of one element for .
We call -admissible if splits over an unramified extension of and for some embedding and some choice of the action of leaves invariant. This definition may look slightly odd; it will be used to guarantee that the reflex field of a Shimura variety built out of has at least one prime where the Shimura variety has a non-empty ordinary locus.
** 4.3.4***.*
For any reductive group over a congruence subgroup is a group of the form for some compact open subgroup An arithmetic lattice is a finite index subgroup of a congruence subgroup. If is a map of reductive groups whose kernel is in the center of and is an arithmetic lattice then is an arithmetic lattice.
If is an arithmetic lattice which acts freely on then has a natural structure of algebraic variety over [DeligneCorvallis, §2]. For any arithmetic lattice there is a finite index subgroup which acts freely on For a Shimura datum the geometrically connected components of have the form where is the image of a congruence subgroup of this was already used in the proof of Lemma 4.2.3.
Now suppose that is almost simple and splits over an unramified extension of Then extends to a reductive group over Let and be compact open, with and Let and We call a covering of the form a principal -level covering.
Theorem 4.3.5**.**
Let be an almost simple group of Hodge type which is -admissible, and let Then for any principal -level covering. we have
[TABLE]
Proof.
We slightly abuse notation and write for Let be a subset of the form described above. This corresponds to a -conjugacy class of cocharacters which we denote by Then is a Shimura datum and its reflex field corresponds to the subgroup of which takes to itself [DeligneCorvallis, Prop. 2.3.6]. Since is -admissible, there is a choice of and a prime of with
Then one sees using [DeligneCorvallis, Prop. 2.3.10] that one can choose a Shimura datum of Hodge type with and adjoint Shimura datum and so that all primes of above split completely in In particular, any prime of has residue field We have verified that satisfies the hypotheses of Lemma 4.2.3. The theorem follows by restricting the map of that lemma to geometrically connected components. ∎
** 4.3.6***.*
We can make the condition of -admissibility of in Theorem 4.3.5 somewhat more explicit, if we assume that has no compact factors.
Corollary 4.3.7**.**
Let be an almost simple group which splits over an unramified extension of Suppose either that
- (1)
* splits over or* 2. (2)
The irreducible factors of are all isomorphic to the adjoint group of one of with or for some positive integer
Then is -admissible, and for any principal -level covering we have
[TABLE]
Proof.
If splits over then acts trivially on and so leaves any choice of stable. For (2), one checks using the classification of [DeligneCorvallis] that in each of these cases, and a vertex is stable by any automorphism of the connected component of containing It follows that leaves stable. ∎
** 4.3.8***.*
The above results give examples of coverings for which one can compute the essential -dimension. It may be of interest to compare these numbers with the essential -dimension of the corresponding group, which can in principle be computed using the Karpenko-Merkjurev theorem [KarMer].
We call a reductive group almost absolutely simple if is semisimple and is absolutely simple. (That is, it remains simple over an algebraic closure).We have the following result.
Proposition 4.3.9**.**
Let be a classical, almost absolutely simple group over with Then there exists a Hermitian symmetric domain attached to an adjoint -group and arithmetic lattices corresponding to a principal -covering, with such that
[TABLE]
satisfies
- •
If is a form of which is split if is odd, then
- •
If is then
- •
If is a split form of then
- •
If is a form of and is not of type then
Proof.
Let There is a unique (up to canonical isomorphism) connected reductive group over with
We (may) now assume that is one of the four types listed, and in each of these cases we define a semisimple Lie group over as follows. If is a form of we take to be if is even and if is odd. If is we take If is a form of we take to be the inner form of (the compact group) which gives rise to the Hermitian symmetric domain of type (cf. [DeligneCorvallis, 1.3.9,1.3.10]). If is a form of we take to be
In all cases and are forms of the same split group. Thus by Lemma 4.1.3 there exists a semisimple reductive group over which gives rise to and over and respectively. By assumption is of Hodge type, and we now check that it can be chosen to be -admissible. This is necessarily the case by Corollary 4.3.7, except when is odd and is a form of or is a form of In these cases, we are assuming that is a split form, so permutes the components of simply transitively. If is a form of or with odd, then contains two points in (the opposition involution is nontrivial on in these cases), and we can take to be a -orbit of any point
When is a form of with even, then the opposition involution is trivial on and hence so is the action of complex conjugation. The set meets each component of in one vertex. Let be the -orbit of any vertex in There is an inner form of over such that (Note that the Dynkin diagrams of inner forms are identified so this makes sense.) Explicitly, let be an automorphism which preserves the connected components of and such that (Such an is unique except if is of type ) Then is given by twisting by the cocycle where lifts Using the surjection (4.1.4), we see that there is an inner twisting of over which is isomorphic to over and to over as an inner twist. As is stable by is -adimissible. Thus, the proposition follows from Theorem 4.3.5 and the formulae for the dimensions of Hermitian symmetric domains. ∎
Corollary 4.3.10**.**
Let be a classical, absolutely simple group over with Then there is a congruence -cover of locally symmetric varieties such that satisfies :
- •
If is a form of which is split if is odd, then
- •
If is then
- •
If is a split form of then
- •
If is a form of and is not of type then
Proof.
This follows immediately from Proposition 4.3.9 and Lemma 2.2.7. ∎
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