Rigid Local Systems: Arithmetic Properties
Hélène Esnault
Freie Universität Berlin

Centennial of the International Mathematical Union
September 27, 2021

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Acknowledgements
We thankthe committee for the kind invitation, Michael Groechenig for ouryearlong collaboration, him and Bertrand Rémy for their helpfulcomments on our slides.

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Rigid Local Systems (RLS)• X topological space, for us complex algebraic variety;dfn• irreducible local system (LS) = irreducibleρ : π1 (X , x) → GLr (C) up to gauge transformation;• i.e.: irreducible fibre bundle Vρ → X with locally constanttransition functions.

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Rigid Local Systems (RLS)• X topological space, for us complex algebraic variety;dfn• irreducible local system (LS) = irreducibleρ : π1 (X , x) → GLr (C) up to gauge transformation;• i.e.: irreducible fibre bundle Vρ → X with locally constanttransition functions.
• ρ rigid: deformation ρt : π1 (X , x) → GLr (C[[t]]) gauge
equivalent to ρ0 , i.e. ∃gt ∈ GLr (C[[t]]) with ρt = gt ρ0 gt−1 , i.e.
moduli point [Vρ ] isolated.
• May fix determinant and conjugacy classes of local monodromy
at ∞ for X not proper.

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Occurencedim (X ) = 1 Katz: ∃ RLS =⇒• X = P1 \ {finitely many points};• moduli points have no multiplicity;• all RLS come from geometry, i.e. ’like’ (summand of)QVρ,τ ∈P1 \{∞,τ1 ,...,τn } = Hc1 (Yτ ), Yτ ⊂ A2 : y N = (x − τ ) n1 (x − τi ).

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Occurencedim (X ) = 1 Katz: ∃ RLS =⇒• X = P1 \ {finitely many points};• moduli points have no multiplicity;• all RLS come from geometry, i.e. ’like’ (summand of)QVρ,τ ∈P1 \{∞,τ1 ,...,τn } = Hc1 (Yτ ), Yτ ⊂ A2 : y N = (x − τ ) n1 (x − τi ).
Shimura varieties of rank ≥ 2: Margulis superrigidity =⇒• all irreducible LS are rigid;• moduli points have no multiplicity;• but we do not know whether they come from geometry.

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Simpson’s Geometricity Conjecture
Simpson’s geometricity conjecture (1990)
all RLS come from geometry.

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Simpson’s Geometricity Conjecture
Simpson’s geometricity conjecture (1990)
all RLS come from geometry.
Petrov’s theorem (2020) inserts it in a vast arithmetic program
Scholze+Liu−Zhu[Fontaine-Mazur conj. ==========⇒ Simpson conj.]/number fields

p−adic fields=======⇒ /C.

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Simpson’s Geometricity Conjecture
Simpson’s geometricity conjecture (1990)
all RLS come from geometry.
Petrov’s theorem (2020) inserts it in a vast arithmetic program
Scholze+Liu−Zhu[Fontaine-Mazur conj. ==========⇒ Simpson conj.]/number fields

p−adic fields=======⇒ /C.

Geometricity conjecture inacessible, except in dim. 1 (Katz).
Instead study consequences.

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Consequences of Simpson’s geometricity conjecture
Integrality conjecture (Simpson 1990)
RLS are integral, i.e. ρ : π1 (X , x) → GLr (O), O number ring.
 Betti cohomology is defined over Z. 

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Consequences of Simpson’s geometricity conjecture
Integrality conjecture (Simpson 1990)
RLS are integral, i.e. ρ : π1 (X , x) → GLr (O), O number ring.
 Betti cohomology is defined over Z. 
Crystalline conjecture (E-Groechenig 2018)
connection corresponding via the Riemann-Hilbert correspondence
to a RLS /XQq for a.a. p isi) an isocrystal with a Frobenius structure;ii) underlying a p-adic crystalline representation (Fontaine).
 Gauß-Manin connections are so by Deligne. 

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

The theorems
Integrality true (E-Groechenig 2018)• for RLS with moduli point with no multiplicity;• when (in a given rank) all irreducible LS are RLS.

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

The theorems
Integrality true (E-Groechenig 2018)• for RLS with moduli point with no multiplicity;• when (in a given rank) all irreducible LS are RLS.
Crystallinity true (E-Groechenig 2018 and 2021)• when X proper: i);• when (in rank ≤ r ) all irreducible LS (of rank ≤ r ) are RLS withmoduli point with no multiplicity: i) and ii).

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

The theorems
Integrality true (E-Groechenig 2018)• for RLS with moduli point with no multiplicity;• when (in a given rank) all irreducible LS are RLS.
Crystallinity true (E-Groechenig 2018 and 2021)• when X proper: i);• when (in rank ≤ r ) all irreducible LS (of rank ≤ r ) are RLS withmoduli point with no multiplicity: i) and ii).
Remark• Not a single RLS known with moduli point with higher
multiplicity.
• Crystallinity applies for all proper or rank ≥ 2 Shimura varieties.

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Methods
Integrality
uses Langlands program (Deligne’s companions’ conjecture, provenby L. Lafforgue and Drinfeld).

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

Methods
Integrality
uses Langlands program (Deligne’s companions’ conjecture, provenby L. Lafforgue and Drinfeld).
Crystallinity
uses theory of Higgs-de Rham flows (Lan-Sheng-Zuo).

Hélène Esnault Freie Universität Berlin

Rigid Local Systems

One consequence of our crystallinity theorem
Pila-Ananth Shankar-Tsimerman (posted on Sept. 21 2021)
André-Oort conjecture for all Shimura varieties.
[i.e: Zariski closure of special points is special ⊂ Shimura variety.]Hélène Esnault Freie Universität Berlin

Rigid Local Systems

One consequence of our crystallinity theorem
Pila-Ananth Shankar-Tsimerman (posted on Sept. 21 2021)
André-Oort conjecture for all Shimura varieties.
[i.e: Zariski closure of special points is special ⊂ Shimura variety.]
Methoduses (crucially) crystallinity.

Hélène Esnault Freie Universität Berlin

Rigid Local Systems