PDFLINK |

# Some Arithmetic Properties of Complex Local Systems

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, 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 weak) glued to the -cells -cell yield the loops on which we take the free group and the relations come from the finitely many , glued to the loops. -cells

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 on the Riemann sphere -points 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 the maximal abelian quotient -modules, 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

and induces

a representation which is *continuous* for the profinite topology on both sides. Recall that the profinite completion

However, Toledo in Tol93 constructed a smooth connected complex projective variety *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*

The invariants

Here the notation

More generally the finite generation of *Betti moduli space* of *character variety* of *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 *étale fundamental group* *scheme*

This profinite group is defined by its representations in finite sets. A representation of

We denote by

the composite morphism. Here

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

As the kernel of the projection *loc. cit.* implies that the specialization homomorphism

induces an isomorphism on the image of *good*, that is smooth, as well as the stratification of the boundary divisor if *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

The factorization defines the irreducible *arithmetic properties* of the initial

On the other hand,

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

We make the following assumption

**Assumption ** For a given rank

It is simple to describe

Note the condition on

If

Consequently, those finitely many *come from geometry*, that is there is a smooth projective morphism

A different way of thinking of finiteness is using Kronecker’s analytic criterion Esn23: the set of the rank *unitary* (i.e., lying in *integral* (i.e., lying in *unitarity and integrality*.

We first observe that *rigid* if we preserve the

of

holds.

A classical example where *While by super-rigidity they all are integral* (i.e., the image of the representations lie in *we do not know whether they come from geometry.*

Another example is provided by connected smooth projective complex varieties

## 5. Non-abelian Hodge Theory

We first assume that

There is a real analytic isomorphism

Simpson defines on Higgs bundles the algebraic *loc. cit.* that

We summarize this section: *The assumption *.

We cannot expect more as already on Shimura varieties of real rank

## 6. Arithmeticity

Again we fix