Density of Arithmetic Representations of Function
FieldsHélène Esnault, joint with Moritz Kerz

IAS, Basic Notions Seminar, May 19, 2020

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Background and Motivation
X is a smooth connected projective variety of dimension d;defined over an algebraically closed field k;
F is a semi-simple `-adic (i.e. Q̄` -valued) local system, ` prime to chark(more generally a shifted semi-simple perverse sheaf but forget this generality for the lecture)Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Background and Motivation
X is a smooth connected projective variety of dimension d;defined over an algebraically closed field k;
F is a semi-simple `-adic (i.e. Q̄` -valued) local system, ` prime to chark(more generally a shifted semi-simple perverse sheaf but forget this generality for the lecture)

Conjecture
Hard Lefschetz holds: if η ∈ H 2 (X , Q` ) is the Chern class of an ample
line bundle, then the cup-product map ∪i η : H d−i (X , F) → H d+i (X , F)should be an isomorphism for all i ∈ N.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Hodge Theory and Weights
Why? k = C: Harmonic theory ⇒ Hard Lefschetz. For F = C: Hodge’s
proof, in general semi-simplicity ⇔ existence of an harmonic metric(Simpson).

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Hodge Theory and Weights
Why? k = C: Harmonic theory ⇒ Hard Lefschetz. For F = C: Hodge’s
proof, in general semi-simplicity ⇔ existence of an harmonic metric(Simpson).
Char. k = p > 0, may assume k = F̄p , so X = X0 ⊗k0 k where k0 = Fq .
Pure weights on H j (X , F), different for different j ⇒ (ultimately) HardLefschetz (a central theorem in Weil II).

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Hodge Theory and Weights
Why? k = C: Harmonic theory ⇒ Hard Lefschetz. For F = C: Hodge’s
proof, in general semi-simplicity ⇔ existence of an harmonic metric(Simpson).
Char. k = p > 0, may assume k = F̄p , so X = X0 ⊗k0 k where k0 = Fq .
Pure weights on H j (X , F), different for different j ⇒ (ultimately) Hard
Lefschetz (a central theorem in Weil II).
If F is punctually pure, Deligne’s theory yields pure weights on H j (X , F),different for different j.
If F is defined over finite extension k0 ⊂ k00 ⊂ k, i.e. F is arithmetic, theLanglands correspondence of Drinfeld-Lafforgue implies that F ispunctually pure.
Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

So what if we have no weights?
If F is not arithmetic, we do not have the theory of weights at disposal.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

So what if we have no weights?
If F is not arithmetic, we do not have the theory of weights at disposal.
However, if F has rank 1, we could develop a strategy, inspired by
Drinfeld’s proof [de Jong conjecture ⇒ Kashiwara conjecture], resting on
the following theorem.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Rank 1
G geometric fundamental group;
G ab,` its pro−` abelianization, assumed to be torsionfree;F finite field of char. `, W (F) be the ring of Witt vectors; W (F) ,→ Q̄` .

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Rank 1
G geometric fundamental group;
G ab,` its pro−` abelianization, assumed to be torsionfree;F finite field of char. `, W (F) be the ring of Witt vectors; W (F) ,→ Q̄` .
Then: Spf(W (F)[[G ab,` ]]) is a formal torus;
W (F)[[G ab,` ]] ⊗W (F) Q̄` is noetherian and Jacobson;
S := Spm(W (F)[[G ab,` ]])(Q̄` ) noetherian topological space;
S =set of iso. classes of rank 1 F with trivial residual representation;
Frobenius Φ ∈ Gal(k/k00 ) acts on G ab,` , thus on S;
narithmetic points: = ∪n∈N>0 S Φ = (class field theory ) torsion points.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

The Theorem in Rank 1 (E-K)
Theorem
A Zariski closed subset Z ⊂ S which is stabilized by Φn for some integern > 0 is the Zariski closure of its torsion points.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

The Theorem in Rank 1 (E-K)
Theorem
A Zariski closed subset Z ⊂ S which is stabilized by Φn for some integern > 0 is the Zariski closure of its torsion points.

Theorem ⇒ Hard-Lefschetz.
Z 0 = {bad points} ⊂ S is Zariski constructible (main point), Φ-invariant;
Theorem ⇒ if Z =Zariski closure 6= ∅ then Z 0 contains arithmetic points;
impossible by Deligne’s Hard Lefschetz.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Higher rank: the set Sρ̄ generalizing S
Set Up: X0 smooth geometrically irreducible over k0 = Fq ,→ k = F̄p ;F finite field of char. ` 6= p;
semi-simple residual representation ρ̄ : G → GLr (F).
Define Sρ̄ = set of iso. classes of rank r F with (semi-simple)
residual representation ρ̄ .

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Higher rank: the set Sρ̄ generalizing S
Set Up: X0 smooth geometrically irreducible over k0 = Fq ,→ k = F̄p ;F finite field of char. ` 6= p;
semi-simple residual representation ρ̄ : G → GLr (F).
Define Sρ̄ = set of iso. classes of rank r F with (semi-simple)
residual representation ρ̄ .

RemarkIn rank 1, ρ̄ was chosen to be trivial, but if ρ̄ is any character
G → GL1 (F) = F× , then Sρ̄ is isomorphic to S by translation with theTeichmüller lift of ρ̄, so the theory is the same. In higher rank however
there is no Teichmüller lift and the Sρ̄ for various ρ̄ are different.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Pseudo-deformations
Taylor-Wiles-. . . -Chenevier: C be the category of complete local
W (F)-algebras (A, mA ) such that W (F) → A/mA identifies F with theresidue field of A;
The functor of pseudo-deformations PDρ̄ : C → Sets of ρ̄ assigns to A theset of continuous r -dimensional A-valued determinants D : A[G ] → A suchthat D ⊗A F : F[G ] → F is the F-valued determinant induced by ρ̄.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Pseudo-deformations II• A Determinant D : A[G ] → A is simply a compatible collection ofDB : B[G ] → B, where B is an A-algebra;• uniquely determined by the r -coefficients of the (monic) ’characteristicpolynomial’ charg = DA[t] (t − g ), g ∈ G ;• any continuous ρ : G → GLr (O), O finite extension of W (F) in C, with(semi-simple) residual representation isomorphic to ρ̄ defines D(ρ) determinedby D(ρ)A[t] (t − g ) = det(t − ρ(g )), g ∈ G ;• if r ! ∈ W (F)× , then standard
determined by D|G and (a)Pdefinition: Dsignσ
invariance by conjugation (b) σ∈Σr +1 (−1)
D(gσ(1) · · · gσ(r +1) ) = 0;Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Sρ̄ as a noetherian topological spaceP

Chenevier: PDρ̄ is representable by (C 3 Rρ̄P , D Rρ̄ : Rρ̄P [G ] → Rρ̄P ), Rρ̄P
noetherian algebra topologically spanned by the coefficients of thechargi , gi ∈ G . If ρ̄ is absolutely irreducible, then Rρ̄P is Mazur’sP

deformation space of ρ̄ and D Rρ̄ is the determinant determined by thecharacteristic polynomial of the universal representation.

Hélène Esnault, joint with Moritz KerzDensity

IAS, Basic Notions Seminar, May 19, 2020

Sρ̄ as a noetherian topological spaceP

Chenevier: PDρ̄ is representable by (C 3 Rρ̄P , D Rρ̄ : Rρ̄P [G ] → Rρ̄P ), Rρ̄P
noetherian algebra topologically spanned by the coefficients of thechargi , gi ∈ G . If ρ̄ is absolutely irreducible, then Rρ̄P is Mazur’sP

deformation space of ρ̄ and D Rρ̄ is the determinant determined by thecharacteristic polynomial of the universal representation.
We deduce: (i) Rρ̄P ⊗W (F) Q̄` noetherian and Jacobson;(ii) Sρ̄ = Spm(Rρ̄P ⊗W (F) Q̄` ) noetherian topological space;(iii) there are closed embeddings: charg : Sρ̄ ,→ (Arm )ˆp wherep = (p1 , . . . , pm ) are the characteristic polynomials of ρ̄ on well chosen
g1 , . . . gm ∈ G .

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

The arithmetic points of Sρ̄(arithmetic) Frobenius Φ ∈ Gal(k/k0 ) lifts to π1ét (X0 );thus acts by conjugation on G (modulo conjugation by G ).
Assume ρ̄ is Φ-invariant.
So Φ induces a well-defined action on Sρ̄ , and is an homeomorphism for
the Zariski topology.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

The arithmetic points of Sρ̄(arithmetic) Frobenius Φ ∈ Gal(k/k0 ) lifts to π1ét (X0 );thus acts by conjugation on G (modulo conjugation by G ).
Assume ρ̄ is Φ-invariant.
So Φ induces a well-defined action on Sρ̄ , and is an homeomorphism for
the Zariski topology.
nThis defines the arithmetic points A = ∪n∈N>0 Sρ̄Φ ⊂ Sρ̄ .
So an arithmetic point is a semi-simple `-adic local system with (semi-simple)
residue local system ρ̄ which is defined over X0 ⊗k0 k00 where k0 ⊂ k00 ⊂ kis a finite extension.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

The conjecture (E-K)

Conjecture
Strong form: A Zariski closed subset Z ⊂ Sρ̄ invariant under Φn for some
integer n > 0 is the Zariski closure of its arithmetic points A ∩ Z ⊂ Z .
Weak form: Sρ̄ is the Zariski closure of its arithmetic points A.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

The theorems (E-K)
Theorem
1) Strong form true for r = 1, X0 projective (mentioned in the motivation)or more generally when Φ is pure on H 1 (X , Q` ) of weight 6= 0.
2) Weak form true for X0 curve, ρ̄ absolutely irreducible and ` > 2.
3) Strong form true for X0 = P1 \ {0, 1, ∞}, r = 2 and ρ̄ tame.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

The theorems (E-K)
Theorem
1) Strong form true for r = 1, X0 projective (mentioned in the motivation)or more generally when Φ is pure on H 1 (X , Q` ) of weight 6= 0.
2) Weak form true for X0 curve, ρ̄ absolutely irreducible and ` > 2.
3) Strong form true for X0 = P1 \ {0, 1, ∞}, r = 2 and ρ̄ tame.

Proposition (as a further motivation)1) Strong form on X0 = P1 \ {0, 1, ∞}, r ≥ 2 and ρ̄ tame ⇒ strong formin any dimension and rank.
2) Strong form in any dimension and rank ⇒ Hard Lefschetz.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Comments on the theorems
1) Thm 1) is on the arXiv (last fall). Geometric proof, uses weights andClass Field Theory.
2) Thm 2) (weak form on curves) proved using the Langlands program
over function fields, de Jong’s method and de Jong’s conjecture, proven byGaitsgory for ` > 2 using the geometric Langlands program.
3) Thm 3) (strong form in rank 2 on P1 \ {0, 1, ∞} for a tame ρ̄) has ageometric proof, using weights, and ultimately Thm 1). It does not usethe Langlands program.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Comments on the theorems
1) Thm 1) is on the arXiv (last fall). Geometric proof, uses weights andClass Field Theory.
2) Thm 2) (weak form on curves) proved using the Langlands program
over function fields, de Jong’s method and de Jong’s conjecture, proven byGaitsgory for ` > 2 using the geometric Langlands program.
3) Thm 3) (strong form in rank 2 on P1 \ {0, 1, ∞} for a tame ρ̄) has ageometric proof, using weights, and ultimately Thm 1). It does not use
the Langlands program.

Remark
Our proof of Thm 3) yields also a geometric proof of de Jong’s conjecturein rank 2 on P1 \ {0, 1, ∞} for a tame ρ̄ and any `, thus without
Langlands program.
Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Leitfaden of Proof of Thm 3)
1-st step: Grothendieck’s specialization’s theory: the tame quotient G t of
G is a topological quotient of the profinite completion F̂2 of the free groupin 2 letters; it is topologically spanned by g0 , g1 the images (after choosingétale paths from 0, resp. 1 to the base point of G ) of the generators of thetame local inertia at 0, resp. 1. Similarly one has g∞ with, for appropriate
choices, the relation g0 · g1 · g∞ = 1. Set g = (g0 , g1 , g∞ ).

Hélène Esnault, joint with Moritz KerzDensity

IAS, Basic Notions Seminar, May 19, 2020

Leitfaden of Proof of Thm 3)
1-st step: Grothendieck’s specialization’s theory: the tame quotient G t of
G is a topological quotient of the profinite completion F̂2 of the free groupin 2 letters; it is topologically spanned by g0 , g1 the images (after choosingétale paths from 0, resp. 1 to the base point of G ) of the generators of thetame local inertia at 0, resp. 1. Similarly one has g∞ with, for appropriate
choices, the relation g0 · g1 · g∞ = 1. Set g = (g0 , g1 , g∞ ).
2nd step: Elementary invariant theory: for pa = char(ρ̄(ga )), a = 0, 1, ∞,one has charg : Sρ̄ → (A2·3=6 )ˆp closed embedding.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Leitfaden of Proof of Thm 3) II
3rd step: Arithmetic Frobenius Φ: Φ(ga ) conjugate to gaq , thus can makecharg equivariant. It reduces the strong form of the conjecture to(A2·3=6 )ˆp .

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Leitfaden of Proof of Thm 3) II
3rd step: Arithmetic Frobenius Φ: Φ(ga ) conjugate to gaq , thus can makecharg equivariant. It reduces the strong form of the conjecture to(A2·3=6 )ˆp .
ˆ2·3=6 )ˆ formal torus which separates
4th and last step: π : (G2·3m ) → (Apthe roots of the 3 degree 2 polynomials, translated by the Teichmüller liftsof those, on which Φ acts by raising to the q-th power. There one appliesThm 1).

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Leitfaden of Proof of Thm 3) II
3rd step: Arithmetic Frobenius Φ: Φ(ga ) conjugate to gaq , thus can makecharg equivariant. It reduces the strong form of the conjecture to(A2·3=6 )ˆp .
ˆ2·3=6 )ˆ formal torus which separates
4th and last step: π : (G2·3m ) → (Apthe roots of the 3 degree 2 polynomials, translated by the Teichmüller liftsof those, on which Φ acts by raising to the q-th power. There one applies
Thm 1).

Comment: Still for ρ̄ tame, which in view of Prop 1) is the mostimportant case, our proof shows that it is enough to find suitable words inthe ga which embed Sρ̄ and on which we control the eigenvalues of Φ.
Even for r = 3 this is difficult.
Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Leitfaden of Proof of Prop 1)
1st step: Reduction to X0 a curve: ’Lefschetz theory à la Wiesend’ but
for pseudo-characters yields a closed embedding of Sρ̄ on X to Sρ̄ on agood curve.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Leitfaden of Proof of Prop 1)
1st step: Reduction to X0 a curve: ’Lefschetz theory à la Wiesend’ butfor pseudo-characters yields a closed embedding of Sρ̄ on X to Sρ̄ on agood curve.
2nd step: Reduction to X0 a curve and ρ̄ trivial: taking the Galois coverassociated to ρ̄, one has to show that the density property is true
downstairs if it is upstairs.

Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020

Leitfaden of Proof of Prop 1)
1st step: Reduction to X0 a curve: ’Lefschetz theory à la Wiesend’ butfor pseudo-characters yields a closed embedding of Sρ̄ on X to Sρ̄ on agood curve.
2nd step: Reduction to X0 a curve and ρ̄ trivial: taking the Galois coverassociated to ρ̄, one has to show that the density property is true
downstairs if it is upstairs.
3nd step: Choose a tame ’Belyi’ map: h : X̄0 → P1 , where X0 ,→ X̄0 isthe normal compactification (possible by Saı̈di-Sugiyama-Yasuda), so h∗ ρ̄,(which has higher rank), is tame. One has to show that the density
property is true upstairs if it is downstairs. (Annoyance: no induction ofpseudo-representations documented in the literature, so one has to do itby hand).
Hélène Esnault, joint with Moritz Kerz

Density

IAS, Basic Notions Seminar, May 19, 2020