ARITHMETIC PROPERTIES OF LOCAL SYSTEMSPARK CITY, JULY 8-9-10 2024

HÉLÈNE ESNAULTAbstract. A building block of homotopy theory is the fundamental group of varieties, in topology and in arithmetic geometry. We know very little on it. One way toapproach it is via local systems, that is linear representations modulo conjugation.
In doing so we lose a lot of information. Still it yields obstructions for a finitely
presented group to be the fundamental group of a smooth complex quasi-projectivevariety. Among those there are the ones coming from geometry, that is the motivicones. Deep conjectures predict over various fields when local systems should butmotivic.

1. Lecture 1
1.1. Geometric groups. For X a smooth connected complex quasi-projective variety, we denote by π1 (X, x) its group of homotopy classes of continuous loops based
at x. Up to isomorphism, the group does not depend on the choice of x. At path γxy−1◦ π1 (X, y) ◦ γxy = π1 (X, x) and another pathfrom x to y induces an isomorphism γxy
is of the shape γxy ◦ γxx . We write π1 (X) for its isomorphism class.
X(C) has the homotpy type of a finite CW complex, so is π1 (X, x) finitely presented(fp). Any fp is the fundamental group of a finite CW complex (the 1-cells are thegenerators, the 2 cells the relations), vice-versa the fundamental group of a finite CW
complex is fp.
We say that a fp group π is geometric if there is a smooth connected complex
quasi-projective variety X such that π = π1 (X, x).
Problem 1 (General Problem). How to recognize the geometric groups among all fpgroups?
There is no general answer to this problem, which is of the same magnitude as theHodge conjecture in complex geometry, the Tate conjecture in arithmetic geometry
etc...we do not even have a conjectural answer (unlike for those ‘abelian’ cases). So
we look for obstructions for fp groups to be geometric.
1.2. Examples.
1.2.1. Algebraic Geometry. Any finite group is geometric (Serre [Ser58]): the group
acts freely on a complete intersection of very high dimension and Lefschetz theory
implies that those complete intersections as simply connected, i.e. π1 = {1}.
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1.2.2. So in particular the abelianization of any fp group Zb ⊕ Γ, with Γ finite(abelian) is geometric, as π1 is compatible with products and Zb = π1 (C×b ).
1.2.3. Abelian Harmonic Analysis. The Malčev completion of π1 (X, x) admits a mixed
Hodge structure.
1.3. Local Systems. This is a representation ρ : π1 (X, x) → GLr (C) modulo conjugacy by GLr (C) (so may be written ρ : π1 (X) → GLr (C)). We write Lρ for thisconjugacy class. Can be thought of as a topological fibration Lρ → X in Cr vectorspaces, with transition functions defined by ρ, or simply (Xx × Cr )/π1 (X, x) whereXx → X is the universal covering based at x and π1 (X, x) acts diagonally.
We can make this definition abstract by replacing π1 (X, x) by any fp group.
Since we are disarmed to study π1 (X, x) directly, we study local systems, from acategorial and / or geometric viewpoint.
Warning 1. The study of local systems reflects only a small ? part of π1 (X, x).
Indeed: As π1 (X, x) is fp, so in particular fg (finitely generated), a representation
ρ has values in GLr (A) for A ⊂ C a Z-algebra of finite type, so ρ = ρA ⊗A C.
But any such A possesses a Ov point, so A ,→ Ov , where Ov is an ℓ-adic ring, sowith finite residue field. But GLr (Ov ) is profinite, thus ρOv := ρA ⊗A Ov factors
through ρ̂Ov : π1\(X, x) → GLr (Ov ), where the profinite completion π1\(X, x) is by theRiemann existence theorem the étale fundamental group pf X (based at x), and ρ̂Ov
is continuous for the profinite topology.
But (Toledo [Tol93]): the kernel K := Ker π1 (X, x) → π1\(X, x) of the profinite
completion may be non-trivial.
Problem 2. We know nothing on K.
Similarly we define ℓ-adic local systems as continuous representations ρ̂ : π\1 (X) →GLr (Ov ) where Ov is an ℓ-adic ring. Absolute irreducibility means that ρ̂ ⊗Ov Q̄ℓ isirreducible.
1.4. Weakly integral fp groups.
Definition 1 ([dJE23]). A fp group π is weakly integral if whenever there is anirreducible representation ρ : π → GLr (C) with determinant of finite order dividingδ ∈ N≥1 , then for all prime numbers ℓ, there is an absolutely irreducibleρ̂v : π̂ → GLr (Ov )where Ov is some ℓ-adic ring, with determinant of order dividing δ ∈ N≥1 .
Theorem 2 (Main Group Theoretic Theorem, [dJE23]).
π geometric =⇒ π weakly integral.

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1.5. Obstruction.
Corollary 3. Being weakly integral is an obstruction for a fp π to be geometric.
1.6. Breuillard’s examples. Γw = ⟨a, b|w = 1⟩ with w = a2 ba−2 b−2 , δ = 1, r = 2.
Let



11 1j 0A= √B=0 j22 −1 1
where j = e2iπ/3 is a primitive cube root of unity. Then irreducible, conjugate overQ(j) (quadratic over Q) yield the only 2 irreducible SL2 (C) local systems. Howeverj1 j
AB = √2 −1 jso Tr(AB) = − √12 ∈/ Z̄2 so Γw is not weakly integral as it is not integral by ℓ = 2!
Here the second example which I haven’t checked: choose w = a2 ba−2 ba−4 b−3 andA, B replace by C, D:√
1a √ϕ33 √−1C=−D=32 10 a−1√
where ϕ = (1+ 5)/2 is the golden mean, and a is a solution to the quadratic equation
a2 + aϕ + 1 = 0 (so that a + a−1 = −ϕ = 2 cos( 4π)). Note that C has order 12 and5
D order 5. We can also check (say using Mathematica) that:C 2 DC −2 = D3 C 4 D−1 .
Furthermore we can compute (again with Mathematica) equations for the character
variety of Γw , and check that (C, D) is an isolated point of it, and that the subgroupit generates is indeed Zariski-dense (and thus irreducible). Finally we readily check
thatϕTr(CD) = √ ,3
which is non-integral at the prime ℓ = 3 (it is a root of the non-monic Q-irreducible
polynomial 9X 4 − 9X 2 + 1); in particular CD has infinite order. So it does ℓ = 3!
Problem 3. Can we find for each separate ℓ a Breuillard type example? (It isinteresting first to justify Breuillard’s two statements, thus for ℓ = 2 and for ℓ = 3,above!)
2. Lecture 2
2.1. Non-Abelian Hamonic Analysis; Simpson’s theory on X smooth connected projective over C. In [Sim92] Simpson constructed via GIT moduli spaces
MB (X, r), MdR (X, r), MDol (X, r) of rank r semi-simple local systems L, semi-simple
flat connections (E, ∇), semi-stable Higgs bundles (V, θ) with vanishing Chern classes.

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4MB (X, r) depends only on π1 (X), while MdR (X, r), MDol (X, r) depend on the analytic structure. He extended the complex analytic Riemann-Hilbert isomorphism
between MB (X, r) and MdR (X, r) to a real analytic isomorphism between MdR (X, r)
and MDol (X, r). The Hitchin map h : MDol (X, r) → AN , where
N = dimC H 0 (X, ⊕ri=1 Symi Ω1X )is proper, MB (X, r) is affine. It has tremendous consequences. e.g. if N = 0 thenthe 3 spaces are 0-dimensional. This is the case e.g. if X is a Shimura variety of
exceptional type (Margulis superrigidity).
These isomorphisms extend to the stable locus on all sides, which are then finemoduli. For B those points are the irreducible local systems, so dR they are simple
flat connections, for Dol there are stable Higgs bundles with vanishing Chern classes.
MDol (X, r) is endowed with the Higgs flow
Gm × MDol (X, r) → MDol (X, r), (t, (V, θ)) 7→ (V, t · θ).
Its fix points correspond via Simpson’s correspondence of C PVHS (complex polarizable variations of Hodge structure).
It yields an obstruction for a fp group π to be geometric: its (affine) character
variety MB (π, r) should be real analytically endowed with a proper map h to anaffine space.
The correspondence holds if we fix a order δ of the determinant as above. The
theory does not quite extend in this shape to the non-proper case.
2.2. Betti moduli with boundary and determinant conditions. Fix X ⊂ X̄a good compatification, with X̄ \ X = ∪Di a strict normal crossings divisor. We fix
δ so det(L)δ = I. We also fix the eigenvalues λij ∈ µ∞ ⊂ C× of the monodromies
at infinity (so they are quasi-unipotent with a fix set of eigenvalues). We could also
fix the Jordan decomposition type. We don’t in the sequel. The resulting Betti
moduli, which is denoted by MB (X, r; δ, λij ), is easily constructed: fix small loops γiaround the Di and complete with cj to span π1 (X, x). Then consider for any ringR matrices Γi , Cj ∈ GLr (R) with the relations: those coming from the relations ofπ1 (X, x) and det(Γi )δ = det(Cj )δ = I and symmetric functions of the eigenvaluesof the Γi describing (λi1 , . . . , λir ). It is an affine finite type variety defined over Z,
on which GLr acts by conjugacy. Then MBall (X, r; δ, λij ) is the GIT quotient. It isan affine finite type variety defined over Z. Its stable points over C correspond toirreducible local systems with the (δ, λij ) conditions. Its points over C correspondto semi-simple local systems with the (δ, λij ) conditions. We then consider the open
MB (X, r; δ, λij ) in MBall (X, r; δ, λij ) which is the complement in MBall (X, r; δ, λij ) of theclosure of the Zariski closure of the non-absolutely irreducible points in characteristic
0. We denote byϵ : MB (X, r; δ, λij ) → Spec(Z), ϵall : MBall (X, r; δ, λij ) → Spec(Z)ARITHMETIC PROPERTIES OF LOCAL SYSTEMSPARK CITY, JULY 8-9-10 2024

5the structure morphism.
Theorem 4 (Main Geometric Theorem, [dJE23]). ϵ dominant =⇒ for all prime
numbers ℓ, ϵall admits a Z̄ℓ -point hitting MB (X, r; δ, λij ) ⊗Z Q non-trivially.
In fact more is true, but under certain irreducibility assumptions, the stronger form
is equivalent to this one.
Corollary 5 (Diophantine Corollary: Rumely’s theorem [Rum86], see [MB89],
Théorème 1.6 for this version). If
1) MBall (X, r; δ, λij )(C) is non-empty and consists of irreducible local systems;
2) MBall (X, r; δ, λij ) and MBall (X, r; δ, λij ) ⊗Z C are irreducible;
then MBall (X, r; δ, λij ) possesses an integral (i.e. Z̄) point meeting MB (X, r; δ, λij )⊗O C
non-trivially.
Problem 4. The conditions 1) and 2) in the Diophantine Corollary are difficultto meet. We thank Peter Sarnak for his interest and questions on the diophantine
consequences of our Main Geometric Theorem. Initial dream with J. de Jong: prove or disprove thatall (X, r; δ) → Spec(Z) is pure in the sense of Gruson-Peskine (would be sufficient for Diophantine Applications).
MBImpossible in the present state of knowledge (the problem being the use of companions).

2.3. de Jong’s conjecture, [dJ01]. It predicts that an arithmetic GLr (kv [[t]]) representation, where kv is a finite char. ℓ field and X is defined over Fq of char. prime
to ℓ, is geometrically finite. From the conjecture, Johan deduces the following: letρ̄ : π1t (XF̄p ) → GLr (kv ) be an absolutely irreducible representation of the tame fundamental group (i.e. tame on all curves, i.e. the Galois cover defined by ρ̄ is such thatin restriction to all curves, the wild ramification index at punctures is trivial and theresidual field extension is separable), Def(ρ̄) be Mazur’s deformation’s space which
represents the functor assigning to a formal local W (Fv )-algebra R̂ with residue field
kv the set of equivalence classes of representations π1 (XF̄p ) → GLr (R̂) lifting ρ̄. It is aformal scheme formally of finite type over W (Fv ). We can also decorate the functor
with a determinant condition and conditions at infinity etc.
Q̄ℓ -points of Def(ρ̄) are precisely ℓ-adic local systems with residual local system
equals to ρ̄. The group π1t (XF̄p ) is topologically finitely generated and GLr (kv ) isfinite, thus there are only finitely many such ρ̄, thus a power m say of Frobenius Φfixes ρ̄ thus Def(ρ̄). Replacing Fq by Fqm , Φ acts Def(ρ̄) and we can study its fix
points. Those are precisely the ℓ-adic local systems which lift ρ̄ and are arithmetic,that is descend to X/Fq once we have assumed that X(Fq ) ̸= ∅ to represent π1 (X)as an extension of Gal(F̄p /Fq ) = Ẑ with π1 (XF̄p ).
Theorem 6 (deJong, [dJ01]). If Def(ρ̄)red is formally smooth, then Def(ρ̄)Φ
red →ΦnSpf(W (Fv ))) is finite flat. Moreover ∪n∈N≥1 Def(ρ̄)red ⊂ Def(ρ̄) is Zariski dense.
Similarly with decorations.

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62.4. de Jong’s conjecture implies that the Main Geometric Theorem implies the Main Group Theoretic Theorem. See [dJE23], Sections 3 and 4.
First let us remark that there is a big difference between the two theorems. In
the geometric one, we do have the loops γi at infinity while in the group theoretic
one, π is abstract, so there are no privileged γi . Should it be geometric, it means inparticular that in it, there are hidden some γi with Γi quasi-unipotent etc.
We use the notation MB (X, r; δ) dropping the λij conditions at infinity. This defines
a forgetful morphism MB (X, r; δ, λij ) ,→ MB (X, r; δ) which is a closed embedding.
Note again the left hand side sees the geometry as we need the boundary divisor todefine it, the right hand side depends solely on π1 (X). As a application of de Jong’s
conjecture proved by Gaitsgory [Gai07] in general (for p ≥ 3) one obtains:
Proposition 7 ([EK23]).
∪λij ∈µ∞ MB (X, r; δ, λij ) ⊂ MB (X, r; δ)is Zariski dense.
Thus if ϵ on MB (X, r; δ) is dominant, so is it on some MB (X, r; δ, λij ) and we canapply the geometric theorem.
3. Lecture 3
The goal of this Lecture is to sketch the proof of the Main Geometric Theorem, see[dJE23, Section 2].
3.1. Companions. We refer to [Esn23, Sections 7.1,7.2,7.3] for the motivation on
the complex side of the existence of companions, the meaning in rank one and some
remarks on geometricity.
3.2. Algebraic Geometry Facts. Set ϵ : M := MB (X, r; δ, λij ) → Spec(Z).
• M of finite type =⇒ ϵ generically smooth on Mred . Let z ∈ M be a closed point,of residue field Fv = Fℓm . It corresponds to an absolutely irreducibleρ̄ : π1 (X) → GLr (Fℓm ).
• For a good model of X, Grothendieck’s specialization homomorphism =⇒ factorization
ρ̄F̄pspρ̄ : π1 (X) −→ π1t (XF̄p ) −−→ GLr (Fℓm ).
• On XF̄p has Mazur’s deformation’s space Def(XF̄p , ρ̄F̄p ; δ, λij )cz ∼M= Def(XF̄p , ρ̄F̄p ; δ, λij )as W (F̄ℓm )-local formal schemes.
• De Jong’s conjecture: there is a ℓ-adic local system L on XFq with residual localsystem ρ̄F̄p .

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• Argument stemming from [EG18]: for any algebraic isomorphism σ : Q̄ℓ ∼= Q̄ℓ′construct the companion (which respects after Deligne the boundary conditions andthe determinant) and consider sp−1 of it. It yields the theorem for all ℓ ̸= p.
• Redo for a larger p.
References[dJ01][dJE23][EG18][Esn23][EK23][Gai07][MB89][Rum86][Ser58][Sim92][Tol93]de Jong, J.: A conjecture on arithmetic fundamental groups, Israel J. Math. 12 (2001),
61–84.
de Jong, J., Esnault, H.: Integrality of the Betti moduli space, Transactions of the AMS
377 (1) (2024),431–448.
Esnault, H., Groechenig, M.: Cohomologically rigid connections and integrality, Selecta
Mathematica 24 (5) (2018), 4279–4292.
Esnault, H.: Local Systems in Algebraic-Arithmetic Geometry, Springer Lecture Notes inMathematics 2337 (2023), Springer Verlag, i–vii + 94 pages.
Esnault, H., Kerz, M.: Local systems with quasi-unipotent monodromy at infinity aredense, Israel J. Math. 257 (2023) no1, 251–262.
Gaitsgory, D.: On de Jong’s conjecture, Israel J. Math. 157 (2007), 155–191.
Moret-Bailly, L.: Groupes de Picard et problèmes de Skolem, Ann. Sc. ÉNS 4o série 22(1989), 161–179.
Rumely, R.: Arithmetic over the ring of all algebraic integrers, J. reine angewandte
Mathematik 368 (1986), 127–133.
Serre, J.-P.: Sur la topologie des variétés algébriques en caractéristique p, Symposium
internacional de topologia algebraica International symposium on algebraic topology,
Universidad Nacional Autonoma de Mexico and UNESCO, Mexico (1958), 24–53.
Simpson, C.: Higgs bundles and local systems, Publ. math. I.H.É.S. 75 (1992), 5–95.
Toledo, D.: Projective varieties with non-residually finite fundamental group, Publ. math.
IHÉS 77 (1993), 103–119.

Freie Universität Berlin, Berlin, Germany; Harvard University, Cambridge, USA
Email address: esnault@math.fu-berlin.de