Some Arithmetic Properties of Complex Local Systems

Hélène Esnault

Communicated by Notices Associate Editor Han-Bom Moon

Le niveau uniforme du varech sur toutes les roches marquait la ligne de flottaison de la marée pleine et de la mer étale.

(The uniform level of kelp on all the rocks marked the waterline of full tide and slack [étale] sea.)

—Victor Hugo, Les travailleurs de la mer, 1866, p. 257

1. Introduction

A group is said to be finitely generated if it is spanned by finitely many letters, that is, if it is the quotient of a free group on finitely many letters. It is said to be finitely presented if the kernel of such a quotient is itself finitely generated. This does not depend on the choice of generation chosen. For example the trivial group is surely finitely presented as the quotient of the free group in generator by itself (!). The following finitely presented group shall play a role in the note:

There are groups which are finitely generated but not finitely presented, see the interesting MathOverflow elementary discussion on the topic (https://tinyurl.com/3cavr69a).

The finitely presented groups appear naturally in many branches of mathematics. The fundamental group of a topological space based at a point is defined to be the group of homotopy classes of loops centered at . A group is finitely presented if and only if it is the fundamental group of a connected finite -complex based at a point . This is essentially by definition of a (closure-finite, weak) complex which is a topological space defined by an increasing sequence of topological subspaces, each one obtained by gluing cells of growing dimension to the previous one. So the -cells glued to the -cell yield the loops on which we take the free group , and the relations come from the finitely many -cells glued to the loops.

If is a smooth connected quasi-projective complex variety, its complex points form a topological manifold which has the homotopy type of a connected finite -complex .

The difference between and its complex points is subtle, and crucial for the note. If is projective for example, when we say we mean the set of defining homogeneous polynomials in finitely many variables with coefficients in . This collection of polynomials is called a scheme. On the other hand, only finitely many of those polynomials are necessary to describe them all (this is the Noetherian property of the ring of polynomials over a field), so in fact there is a ring of finite type over which contains all the coefficients of those finitely many polynomials. We write to remember , to remember . We can then take any maximal ideal in . The residue field is finite, say , and has characteristic . Then we write for the scheme defined by this collection of polynomials where the coefficients are taken modulo . Fixing an algebraic closure , and thinking of the polynomials as having coefficients in we write etc.

When we say , we mean the complex solutions of the defining polynomials. (Of course there is the similar notion , etc.)

The notion of a quasi-projective complex variety is easily understood on its complex points . They have to be of the shape where both and are projective varieties.

We do not know how to characterize the fundamental groups , where , among all possible . In this small text, we use the following terminology:

2. Classical Obstructions: Topology and Hodge Theory

A classical example comes from the uniformization theory of complex curves: any free group on letters, where is a natural number, is the fundamental group of the complement of -points on the Riemann sphere . This is because we understand exactly if has dimension , that is if is a Riemann surface. The simplest possible example is the Riemann sphere . Then as any loop centered at can be retracted to a point, see Figure 1.

The same holds true on . The first interesting example is . Then , and where is the circle turning around the origin , see Figure 2.

More generally, if a smooth compactification of has genus , topologically is a donut with holes. Then is spanned by elements with one relation . If consists of points, is spanned by elements with one relation . The literature is full of beautiful colored pictures visualizing this classical computation.

Beyond Riemann surfaces, that is, for of dimension , our understanding is very limited.

The in the in the previous example is more general: by the fundamental structure theorem on finitely generated -modules, the maximal abelian quotient , that is, the abelianization of , is isomorphic to a direct sum of for some natural number and of a finite abelian group .

Any abelian finitely presented group comes from geometry: Serre’s classical construction Ser58 realizes any finite group as the fundamental group of the quotient of a complete intersection of large degree in the projective space of large dimension, while the fundamental group of is equal to , see Figure 2. As the fundamental group of a product variety is the product of the fundamental groups of the factors (Künneth formula), we can take and there is no obstruction for to be the abelianization of the fundamental group of a smooth connected quasi-projective complex variety. If we require to be projective, then Hodge theory, more precisely, Hodge duality implies that is even. This is the only obstruction as we can then take instead, where is any elliptic curve, so is a donut with one hole, so , see Figure 3.

In the same vein, but much deeper is the fact that the pronilpotent completion of (also called Malčev completion) is endowed with a mixed Hodge structure. While so far we commented the topological structure of , Hodge theory studies in addition the analysis stemming from the complex structure, and the more refined properties, packaged in the notion of Kähler geometry and harmonic theory, which come from the property that is defined algebraically by complex polynomials. A modern way (due to Beilinson) to think of it is to identify the Malčev completion with the cohomology of an (infinite) simplicial complex scheme and to apply the classical Hodge theory on its truncations. We do not elaborate further.

3. Profinite Completion: The Étale Fundamental Group

Thus the difficulty lies in the kernel of the group to its abelianization. To study it, we first introduce the classical notion:

Why modulo conjugacy? A path from to defines an isomorphism . This isomorphism is not unique, any other path from to differs from this one by left multiplication by a loop centered at , which thus conjugates the isomorphism by . Thus not fixing the base point forces us to consider representations modulo conjugacy.

As is finitely presented, thus in particular finitely generated, factors through where is a ring of finite type. Any such can be embedded into the ring of -adic integers for some prime number , say . (For example if , has to be the natural pro--completion for any choice of . If we take in a transcendental element over and send to it, etc. The main point is that the field of fractions of , that is the field of -adic numbers, has infinite transcendence degree over ). Thus the datum of is equivalent to the one of whose range is profinite. In particular, factors through the profinite completion

and induces

a representation which is continuous for the profinite topology on both sides. Recall that the profinite completion of an abstract group is the projective limit over all finite quotients . It inherits the profinite topology compatible with the group structure for which a basis of open neighborhoods of is defined to be the inverse image of by one of those projections.

However, Toledo in Tol93 constructed a smooth connected complex projective variety with the property that is not injective. It answered a problem posed by Serre. It is an important fact which in particular implies that the study of complex local systems ignores . This leads us in two different directions.

The invariants and of the abelianization are seen on the complex abelian algebraic group

Here the notation means the set endowed with the (abelian) multiplicative group structure.

More generally the finite generation of enables one to define a “moduli” (parameter) space of all its irreducible local systems in a given rank . It is called the Betti moduli space of of irreducible local systems in rank or the character variety of of irreducible local systems in rank . It is a complex quasi-projective scheme of finite type. Its study is the content of Simpson’s non-abelian Hodge theory developed in Sim92. It is an analytical theory relying on harmonic theory, as is classical Hodge theory.

The second direction relies on the profinite completion homomorphism . By the Riemann existence theorem, a finite topological covering is the complexification of a finite étale cover. Thus is identified with the étale fundamental group of the scheme defined over , based at the complex point , as defined by Grothendieck in Gro71:

This profinite group is defined by its representations in finite sets. A representation of in finite sets is “the same” (in the categorial sense) as a pointed (above ) finite étale cover of .

We denote by

the composite morphism. Here is an algebraic closure of .

The notion of a complex local system (Definition 2) generalizes naturally:

As the kernel of the projection is a pro--group (that is all its finite quotients have order of power of ), Grothendieck’s specialization theory in loc. cit. implies that the specialization homomorphism

induces an isomorphism on the image of for larger than the order of . Here is a reduction of as explained in the introduction, and is good, that is smooth, as well as the stratification of the boundary divisor if is not projective. The upper script refers to the tame quotient of in case was not projective. We do not detail with precision the tameness concept, for which we refer to KS10. This roughly works as follows. Representations in finite sets of the étale fundamental group which factor through the tame quotient have base change properties “as if” were proper. We can contract the fundamental group of over a -adic ring with residue field to the one over in the way we do topologically in order to identify the topological fundamental group of a tubular neighborhood of a compact manifold to the one of the compact manifold. The natural identification of with where is an algebraic closure of the field of fractions of (this is called base change property) enables us to define . Grothendieck computes that induces an isomorphism on all finite quotients of and of order prime to .

The factorization defines the irreducible -adic local system on from which comes. This leads us to study in order to derive arithmetic properties of the initial . We can remark that again we know extremely little on the kernel of and that the study of complex local systems ignores them as well, for a chosen and large as before.

On the other hand, is defined over a field of finite type over , thus with a huge Galois group, and is defined over a finite field of characteristic , with a very small Galois group isomorphic to , the profinite completion of , topologically spanned by the Frobenius of . Nonetheless, we shall see that this small Galois group yields nontrivial information.

Our goal now is twofold. First we shall illustrate how to go back and forth between the Hodge theory side and the arithmetic side on a particular example. This by far does not cover the whole deepness of the theory, but we hope that it gives some taste on how it functions. Then we shall mention on the way and at the end more general theorems to the effect that deep arithmetic properties stemming from the Langlands program, notably the “integrality” illustrated on this particular example, enable one to find a new obstruction for the finitely presented group to come from geometry.

4. An Example to Study

Let be a smooth connected quasi-projective complex variety. If is not projective, we fix a smooth projective compactification so that the divisor at infinity is a strict normal crossings divisor (so its irreducible components are smooth and meet transversally). For each we fix roots of unity , possibly with multiplicity. They uniquely determine a conjugacy class of a semi-simple matrix of finite order. The normal subgroup spanned by the conjugacy classes of small loops around the components is identified with the kernel of the surjection . We fix an extra natural number .

We make the following assumption

Assumption : For a given rank , there are at most finitely many irreducible rank complex local systems on such that the determinant of has order dividing , and, if is not projective, such that the semi-simplification of falls in .

It is simple to describe : in the Betti moduli space we have the subscheme defined by the conditions . The condition means precisely that is -dimensional. Equivalently consists of finitely many points, or is empty.

Note the condition on depends only on so could be expressed on the character variety, not however the condition on . For this we have to know which in come from the boundary divisor, so we need the geometry.

If , we drop the condition on the determinant, and assume for simplicity that is projective. So the assumption becomes that there are finitely many irreducible rank complex local systems on . This then forces to be , so to be finite.

Consequently, those finitely many of rank have finite monodromy (i.e., is finite). This implies that the come from geometry, that is there is a smooth projective morphism where is a Zariski dense open in (in our case ), such that restricted to is a subquotient of the local system coming from the representation of in for some (in our case is finite étale and ).

A different way of thinking of finiteness is using Kronecker’s analytic criterion Esn23: the set of the rank local systems is invariant under the action of the automorphisms of acting on . Finiteness of the monodromy is then equivalent to the monodromy being unitary (i.e., lying in ) and being integral (i.e., lying in ). We now discuss the generalization of these two properties: unitarity and integrality.

We first observe that implies that the irreducible rank complex local systems are rigid if we preserve the conditions. As the terminology says, it means that we can not “deform” nontrivially the local system . Precisely it says that a formal deformation

of with the same conditions does not move , that is there is a such that in the relation

holds.

A classical example where is fulfilled is provided by Shimura varieties of real rank . Margulis super-rigidity Mar91 implies that all complex local systems are semisimple and all irreducible ones are rigid. While by super-rigidity they all are integral (i.e., the image of the representations lie in up to conjugacy), we do not know whether they come from geometry.

Another example is provided by connected smooth projective complex varieties with the property that all symmetric differential forms, except the functions, are trivial. In this case, non-abelian Hodge theory implies is fulfilled. Indeed, the Betti moduli space of semisimple rank complex local systems is affine, while the moduli space of semistable Higgs bundles with vanishing Chern classes (which we discuss below) admits a projective morphism to the so-called Hitchin base. The latter consists of one point under our assumption. As by a deep theorem of Simpson Sim92, both spaces are real analytically isomorphic, they are both affine and compact, thus are -dimensional. It is proven in BKT13, using Hodge theory, the period domain and birational geometry, that all the have then finite monodromy. This yields a positive answer to a conjecture I had formulated. As the proof uses Hodge theory, it is analytic. As of today, there is no arithmetic proof of the theorem.

5. Non-abelian Hodge Theory

We first assume that is projective. We discuss a little more the notion of Higgs bundles mentioned above. Simpson in Sim92 constructs the moduli space of stable Higgs bundles with vanishing Chern classes, where is a vector bundle of rank , is a -linear operator fulfilling the integrality condition , such that has finite order dividing . (The integrality notion here is for the Higgs field , and is not related to the integrality of a linear representation mentioned in Section 4). The stability condition is defined on the pairs , that is one tests it on Higgs subbundles. The finite order of implies that the underlying Higgs field of is equal to , so . The moduli space is a complex scheme of finite type. It has several features.

There is a real analytic isomorphism . So implies that consists of finitely many points.

Simpson defines on Higgs bundles the algebraic -action which assigns to for . It preserves stability near and semistability in general. Thus under the assumption , the -action stabilizes pointwise. Simpson proves in loc. cit. that -fixed points correspond to polarized complex variations of Hodge structure (PCVHS). Mochizuki in Moc06 generalized this part of Simpson’s theory to the smooth quasi-projective case so the conclusion remains valid in general.

We summarize this section: The assumption implies that the irreducible of rank , with determinant of order diving and semisimplification of falling in , underlie a PCVHS. This property is the analog of the unitary property in rank .

We cannot expect more as already on Shimura varieties of real rank , not all local systems are unitary. If they all were, as they are integral, they would have finite monodromy. This is not the case.

6. Arithmeticity

Again we fix . Once we obtain the finitely many local systems