Survey on rigid local systems and related
arithmetic questions
Hélène Esnault, joint with Michael Groechenig

Regensburg, July 7th, 2020

Hélène Esnault, joint with Michael Groechenig

Rigid

Notation• X sm conn quasi-proj var /C of dim dHélène Esnault, joint with Michael Groechenig

Rigid

Notation• X sm conn quasi-proj var /C of dim d• XR model /R ⊂ C of ft /Z (i.e.: XR /R flat and XC = X )Hélène Esnault, joint with Michael Groechenig

Rigid

Notation• X sm conn quasi-proj var /C of dim d• XR model /R ⊂ C of ft /Z (i.e.: XR /R flat and XC = X )• integrable connection (E , ∇), ∇ : E → Ω1X ⊗OX E : E vb, ∇additive, lin / ground ring, Leibniz rule, ∇ ◦ ∇ = 0Hélène Esnault, joint with Michael Groechenig

Rigid

Notation• X sm conn quasi-proj var /C of dim d• XR model /R ⊂ C of ft /Z (i.e.: XR /R flat and XC = X )• integrable connection (E , ∇), ∇ : E → Ω1X ⊗OX E : E vb, ∇additive, lin / ground ring, Leibniz rule, ∇ ◦ ∇ = 0• Higgs bundle (V , θ), θ : V → Ω1X ⊗OX V : V vb, θ OX -lin,θ ◦ θ = 0;Hélène Esnault, joint with Michael GroechenigRigid

Notation• X sm conn quasi-proj var /C of dim d• XR model /R ⊂ C of ft /Z (i.e.: XR /R flat and XC = X )• integrable connection (E , ∇), ∇ : E → Ω1X ⊗OX E : E vb, ∇additive, lin / ground ring, Leibniz rule, ∇ ◦ ∇ = 0• Higgs bundle (V , θ), θ : V → Ω1X ⊗OX V : V vb, θ OX -lin,θ ◦ θ = 0;
• X ,→ X̄ good comp, D = ∪i Di = X̄ \ X , r ∈ N>0 ,Ki ⊂ GLr (C) quasi-unip conj cl /K ⊂ C number field, L a rk 1
local system /C with finite monodromy /K ; (L, ∇),Riemann-Hilbert-Simpson
LdR = (L, ∇), LDol = (L, 0).
Hélène Esnault, joint with Michael GroechenigRigid

Moduli (Simpson; CR Langer)• ∃ coarse quasi-proj moduli sp MB /OK [1/N] with: MB (C) = {isocl of irred rk r C loc syst V, det(V) ∼= L and local monodromies/Di in Ki }. There is an étale µr -gerbe M → MB where M is a algebraic stack, so MB is étally fine.

Hélène Esnault, joint with Michael GroechenigRigid

Moduli (Simpson; CR Langer)• ∃ coarse quasi-proj moduli sp MB /OK [1/N] with: MB (C) = {isocl of irred rk r C loc syst V, det(V) ∼= L and local monodromies/Di in Ki }. There is an étale µr -gerbe M → MB where M is a algebraic stack, so MB is étally fine.
• X proj: ∃ coarse quasi-proj moduli sp MdR /R with:MdR (k) = {iso cl of stable for int conn rk r integrable connections(E , ∇), det(E , ∇) = LdR , Hilbert pol = Hilbert pol of ⊕r OX }.
char. 0 (E , ∇) always ss and is stable iff it is irr.

Hélène Esnault, joint with Michael GroechenigRigidIn

Moduli (Simpson; CR Langer)• ∃ coarse quasi-proj moduli sp MB /OK [1/N] with: MB (C) = {isocl of irred rk r C loc syst V, det(V) ∼= L and local monodromies/Di in Ki }. There is an étale µr -gerbe M → MB where M is a algebraic stack, so MB is étally fine.
• X proj: ∃ coarse quasi-proj moduli sp MdR /R with:
MdR (k) = {iso cl of stable for int conn rk r integrable connections(E , ∇), det(E , ∇) = LdR , Hilbert pol = Hilbert pol of ⊕r OX }.
char. 0 (E , ∇) always ss and is stable iff it is irr.

• X proj: ∃ coarse quasi-proj moduli sp MDol /R with:
MDol (k) = {iso cl of stable for Higgs bdls rk r Higgs bundles (V , θ),det(V , θ) = LDol , Hilbert pol = Hilbert pol of ⊕r OX }.

Hélène Esnault, joint with Michael Groechenig

RigidIn

Geometric local systems
ExampleMain ex: V (resp. (E , ∇)) geometric i.e. ∃f : Y → U sm proj,
U ⊂ X dense, so V (resp.(E , ∇)) subq of R i f∗ C, i ∈ N>0 (resp.
R i f∗ Ω•Y /U ) ( ! summand by Deligne’s ss thm). e.g. f finiteétale, i = 0.)Hélène Esnault, joint with Michael Groechenig

Rigid

Some properties of geometric local systems
Property (Betti: integrality)
V geometric ⇒ V defined /OL , L number field ⊃ K , i.e. V
integral.

Hélène Esnault, joint with Michael GroechenigRigid

Some properties of geometric local systems
Property (Betti: integrality)
V geometric ⇒ V defined /OL , L number field ⊃ K , i.e. V
integral.
Property (dR: crystalline)(E , ∇) geometric ⇒ ∀Zq → Spec(R), R some ring of defn of(X , (E , ∇)), Spec(Fq ) ∈ Spec(R) cl pt of good reduction, then(E , ∇)Qq on X̂Qq is an isocrystal with Frobenius structure. We sayfor short: (E , ∇) is crystalline.

Hélène Esnault, joint with Michael Groechenig

RigidRiemann-Hilbert-Simpson correspondence /CMB (C)S◦RH ∼=R−an

RH ∼=C−an%y/ MdR (C)
S∼
=R−an

MDol (C)Hélène Esnault, joint with Michael Groechenig

RigidRiemann-Hilbert-Simpson correspondence /C
MB (C)S◦RH ∼=R−an

RH ∼=C−an%/ MdR (C)S∼
=R−anyMDol (C)In particular 0-diml comp called rigid:
MBrig (C)S◦RH ∼=R−an

RH ∼=C−an/ M rig (C)dR%yS∼=R−anrigMDol(C)Hélène Esnault, joint with Michael Groechenig

Rigid

Elementary properties of rigid local systems• ∃ finitely many (⇐ quasi-proj. MB )Hélène Esnault, joint with Michael Groechenig

Rigid

Elementary properties of rigid local systems• ∃ finitely many (⇐ quasi-proj. MB )• If σ ∈ Aut(C), then V ⇔ V σ rigid (⇐ base change MB /C)

Hélène Esnault, joint with Michael Groechenig

Rigid

Elementary properties of rigid local systems• ∃ finitely many (⇐ quasi-proj. MB )• If σ ∈ Aut(C), then V ⇔ V σ rigid (⇐ base change MB /C)• MBrig ⊂ MB defined /OL [1/D], L ⊂ C number field ⊃ K (⇐ elsecould deform)Hélène Esnault, joint with Michael GroechenigRigid

Elementary properties of rigid local systems• ∃ finitely many (⇐ quasi-proj. MB )• If σ ∈ Aut(C), then V ⇔ V σ rigid (⇐ base change MB /C)• MBrig ⊂ MB defined /OL [1/D], L ⊂ C number field ⊃ K (⇐ else
could deform)• ∀λ ∈ Spec(OL [1/D]), V ∈ MBrig is étale, i.e. factors throughπ top (X (C)) → π top\(X (C)) = π ét (X ) = π ét (X).
111Hélène Esnault, joint with Michael Groechenig

CRigid1Frac(R)Katz’ and Simpson’s theorems; Simpson’s conjecture
Theorem(Katz): dim(X ) = 1 ⇒ MBrig geometric;(Simpson): MBrig factors through π1ét (XF ), F ⊃ Frac(R) finite, i.e.
rigid ⇒ arithmetic.

Hélène Esnault, joint with Michael Groechenig

Rigid

Katz’ and Simpson’s theorems; Simpson’s conjecture
Theorem(Katz): dim(X ) = 1 ⇒ MBrig geometric;(Simpson): MBrig factors through π1ét (XF ), F ⊃ Frac(R) finite, i.e.
rigid ⇒ arithmetic.
Conjecture (Simpson)
rigid ⇒ geometricHélène Esnault, joint with Michael Groechenig

Rigid

Katz’ and Simpson’s theorems; Simpson’s conjecture
Theorem(Katz): dim(X ) = 1 ⇒ MBrig geometric;(Simpson): MBrig factors through π1ét (XF ), F ⊃ Frac(R) finite, i.e.
rigid ⇒ arithmetic.
Conjecture (Simpson)
rigid ⇒ geometric
Conjecture (subconjectures)
V rigid ⇒ V integral(E , ∇) rigid ⇒ (E , ∇) crystallineHélène Esnault, joint with Michael Groechenig

Rigid

Main theorem (E-Groechenig)
Theorem
V coh. rigid ⇒ V integral(E , ∇) rigid ⇒ (E , ∇) crystalline

Hélène Esnault, joint with Michael Groechenig

Rigid

Main theorem (E-Groechenig)
Theorem
V coh. rigid ⇒ V integral(E , ∇) rigid ⇒ (E , ∇) crystalline
Remark• coh. rigid: smooth isolated point, i.e.
T[V] MB = H 1 (X̄ , j!∗ End 0 (V)) = 0.
• Katz: dim(X ) = 1: rigid ⇒ coh. rigid.
• So far: not a single example of a rigid V which is not coh. rigid.

Hélène Esnault, joint with Michael Groechenig

Rigid

Corollary 1
Corollary ( ⇐ Thm + L.Lafforgue-Abe)(E , ∇) rigid ⇒ ∀C ,→ X curve c.i. of smooth ample divisors(E , ∇)|ĈQ geometric.
q

Hélène Esnault, joint with Michael GroechenigRigid

Corollary 1
Corollary ( ⇐ Thm + L.Lafforgue-Abe)(E , ∇) rigid ⇒ ∀C ,→ X curve c.i. of smooth ample divisors(E , ∇)|ĈQ geometric.
qRemarkL. Lafforgue: irred arithmetic `-adic local syst with fin det aregeometric. Abe: companions ` ! isocrystals with Frobenius
structure. Big problem: extend geometricity on curves togeometricity on X and then geometricity on X over a finite field togeometricity on X over C.

Hélène Esnault, joint with Michael Groechenig

Rigid

Corollary 2
Corollary (⇐ method of proof)(E , ∇) rigid with p-curvature 0 for all Spec(Fq ) ∈ dense open ofSpec(R) ⇒ unitary monodromy.

Hélène Esnault, joint with Michael Groechenig

Rigid

Corollary 2
Corollary (⇐ method of proof)(E , ∇) rigid with p-curvature 0 for all Spec(Fq ) ∈ dense open ofSpec(R) ⇒ unitary monodromy.
RemarkSo p-curvature conjecture for coh rigid connections ⇐ [(E , ∇)rigid with p-curvature 0 for all Spec(Fq ) ∈ dense open of Spec(R)⇒ (E , ∇)σ rigid with p-curvature 0 for all Spec(Fq ) ∈ dense openof Spec(R) ∀σ ∈ Aut(C), (E , ∇)σ ↔ V σ ].

Hélène Esnault, joint with Michael GroechenigRigid

Corollary 3
Corollary (⇐ method of proof)
Deligne’s conjectural companion correspondence for V arithm`-adic irred with fin det on XF̄p such thatH 1 (X̄F̄p , j!∗ End 0 (V)) = 0including the crystalline components: ∀ι : Q̄` → Q̄`0 including`0 = p, ∃ V ι with the same local char. pol.

Hélène Esnault, joint with Michael GroechenigRigid

Corollary 3
Corollary (⇐ method of proof)
Deligne’s conjectural companion correspondence for V arithm`-adic irred with fin det on XF̄p such that
H 1 (X̄F̄p , j!∗ End 0 (V)) = 0including the crystalline components: ∀ι : Q̄` → Q̄`0 including`0 = p, ∃ V ι with the same local char. pol.
RemarkRecall Deligne’s conjecture is known on curves (L. Lafforgue, Abe),and ``0 6= p on X smooth in any dim (Drinfeld), and withDrinfeld’s method p` (E-Abe, Kedlaya).
Hélène Esnault, joint with Michael Groechenig

RigidOn proof of the integrality theorem
V coh. rigid, λ ∈ Spec(OL [1/D]), L number fieldVλ `-adic on
X /C thus on XF̄p , p >> 0 prime to the order of L, r , the order ofthe res. representation.

Hélène Esnault, joint with Michael Groechenig

RigidOn proof of the integrality theorem
V coh. rigid, λ ∈ Spec(OL [1/D]), L number fieldVλ `-adic on
X /C thus on XF̄p , p >> 0 prime to the order of L, r , the order ofthe res. representation.
Vλ arithmetic (Simpson)companions Vλ0 . As

H 1 (X̄F̄p , j!∗ End 0 (Vλ0 )) ⊂ ⊕i H i (XF̄p , End 0 (Vλ0 ))is the pure weight 1 part it is recognised on the L-function thus=0
V 0 coh. rigid on X /C.

Hélène Esnault, joint with Michael GroechenigRigid

On proof of the integrality theorem
V coh. rigid, λ ∈ Spec(OL [1/D]), L number fieldVλ `-adic onX /C thus on XF̄p , p >> 0 prime to the order of L, r , the order ofthe res. representation.
Vλ arithmetic (Simpson)companions Vλ0 . As

H 1 (X̄F̄p , j!∗ End 0 (Vλ0 )) ⊂ ⊕i H i (XF̄p , End 0 (Vλ0 ))is the pure weight 1 part it is recognised on the L-function thus=0
V 0 coh. rigid on X /C.
VV 0 bijection.

Hélène Esnault, joint with Michael Groechenig

RigidOn proof of crystalline theorem: crystalline partrigMdR⊂ MdR has certain mult. µ ⇒ ∀Spec(Fq ) → Spec(R),p >> 0, µ << p.

Hélène Esnault, joint with Michael Groechenig

RigidOn proof of crystalline theorem: crystalline part
rigMdR⊂ MdR has certain mult. µ ⇒ ∀Spec(Fq ) → Spec(R),p >> 0, µ << p.

X / perfeckt field k: p-curv ∈ H 0 (X , End(E ) ⊗ F ∗ Ω1X 0 )∇ ⇒χdR = pth root of char. pol. of p-curv: MdR → A with χ−1dR (0) =conn. with nilp. p-curv.

Hélène Esnault, joint with Michael Groechenig

RigidOn proof of crystalline theorem: crystalline partrig
MdR⊂ MdR has certain mult. µ ⇒ ∀Spec(Fq ) → Spec(R),p >> 0, µ << p.

X / perfeckt field k: p-curv ∈ H 0 (X , End(E ) ⊗ F ∗ Ω1X 0 )∇ ⇒χdR = pth root of char. pol. of p-curv: MdR → A with χ−1dR (0) =conn. with nilp. p-curv.
classical fact: (E , ∇)Qq crystal ⇔ p-curv (E , ∇)Fq nilpotent.

Hélène Esnault, joint with Michael Groechenig

RigidOn proof of crystalline theorem: crystalline partrig
MdR
⊂ MdR has certain mult. µ ⇒ ∀Spec(Fq ) → Spec(R),p >> 0, µ << p.

X / perfeckt field k: p-curv ∈ H 0 (X , End(E ) ⊗ F ∗ Ω1X 0 )∇ ⇒χdR = pth root of char. pol. of p-curv: MdR → A with χ−1dR (0) =conn. with nilp. p-curv.
classical fact: (E , ∇)Qq crystal ⇔ p-curv (E , ∇)Fq nilpotent.
[(existence (E , ∇)Zq /p2 + χdR ((E , ∇)Fq ) 6= 0) ⇒ (E , ∇)Fq deforms
to order (p − 1)] ⇐ Ogus-Vologodsky.

Hélène Esnault, joint with Michael Groechenig

Rigid