Abstracts
Arithmetic properties of rigid local systems
Hélène Esnault(joint work with Michael Groechenig)
1. Introduction
The aim of the lecture, which relies on [3], is to generalize to the quasi-projective
non-projective case the theorem described in the introduction and Theorem 5.4
of [2] stating that irreducible complex rigid local systems, while restricted to ap-adic formal scheme with good reduction, for p large, underlie the structure of a
Fontaine-Lafaille module and define p-adic local systems.
2. Assumption and preliminary geometric facts
Assumption 1 (Throughout). X smooth quasi-projective /C; all irreducible complex local systems L of rank r with unipotent monodromies at ∞ are strongly cohomologically rigid i.e. H 1 (X, End(L)) = 0. (Verified for Shimura varieties of realrank ≥ 2 by Margulis superrigidity.)
Facts 2 (Preliminary geometry). 1) For j : X ,→ X̄ a good compactification, hasinj

H 1 (X̄, j!∗ End(L)) −→ H 1 (X, End(L))
so (strongly cohomologically rigid) =⇒ (cohomologically rigid) (fixing unipotent
conjugacy classes at ∞ and the determinant) =⇒ (rigid) (fixing unipotent conjugacy classes at ∞ and the determinant).
2) E Deligne’s extension of (L ⊗C Oan , 1 ⊗ d) with E → Ω1X̄ (log ∞) ⊗OX̄ Ewith nilpotent residues. SoH 1 (X̄, Ω• (log ∞) ⊗ End(E, ∇)) = H 1 (X, End(L)) = 0.
Atiyah class computation + Hodge theory ([4, Appendix B])=⇒0 = ci (E) ∈ H 2i (X̄, Q), i ≥ 1.
3) (E, ∇) necessarily semi-stable as saturated sub (E 0 , ∇0 ) ⊂ (E, ∇) has also
nilpotent residues, so (E 0 , ∇0 ) ⊂ (E, ∇) locally split outside of a codimension 2subset Σ ⊂ ∞ ⊂ X̄ in X̄, and for j : X̄ \ Σ ,→ X̄, has (E 0 , ∇0 ) = j∗ j ∗ (E 0 , ∇0 ) ⊂(E, ∇) = j∗ j ∗ (E, ∇). Thus (E 0 , ∇0 ) is Deligne’s extension as it is determinedoutside of codimension 2. Thus 0 = ci (E 0 ) ∈ H 2i (X̄, Q), i ≥ 1.
4) Langer moduli ([9, Theorem 1.1]) MdR (r), MDol (r) of stable log-objects
to the Hilbert polynom P (E) = P (⊕r1 OX̄ ) are defined over some S smooth /Z,MdR (r)S MDol (r)S → S flat and XS , X̄S → S relative NCD and base change forH 1 (X̄S , Ω• (log ∞) ⊗ End(ES , ∇S )).
5) The characteristic polynomials of the residues ES → Ω1 (log ∞S )⊗ES at ∞S
are regular functions on MdR (r)S , so the nilpotent residues condition defines closed
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subs MdR(X)(r)S ⊂ MdR (r)S , which after shrinking are flat /S. MdR(X)(r)S (C)consists precisely of the Deligne’s extensions of the underlying (strongly cohomologically rigid) irreducible local systems. So it is 0-dimensional, say of cardinality◦N . By taking an étale cover of S, we may assume that MdR(X)(r)S consists ofN -sections.
6)

Theorem 3 ([1], Theorem 1.1). All L are integral. (Finiteness of {L} implies all
L defined over one OL , L number field).
7) Mochizuki ([10, Theorem 10.5]): any (E, ∇) with nilpotent residues deforms
real analytically to a polarized C-VHS, so rigidity implies all the (E, ∇) underliea polarized C-VHS. So with 6)
Claim 4. Assumption 1 =⇒ all L underlie a polarized Z̄-VHS.
8) Boundedness of possible Hodge filtrations.
Definition 5 (Good model). S/Z smooth, condition 5), plus: all (E, ∇, F il)
defined over S, F il locally split /S. So (grF il (E), KS) stable Higgs, locally free/S nilpotent. We assume also for any Spec(W (Fq )) → S, char Fq > 2r + 2.
Theorem 6 ([3], Theorem A.4). Assumption 1 =⇒ on a good model, for anyd̄ carries
ˆ W )/XSpec(W ) → S, with W = W (Fq ), the formal connection (ÊW , ∇W
the structure of a locally free Fontaine-Lafaille module.
(Standard definition of a Fontaine-Lafaille module right now irrelevant as weshall work with an equivalent definition).
3. Sketch of Proof of Theorem 6◦
s closed point of Spec(W ). Has MdR(r)s =: (dR)◦s consisting of N s-points,and (Dol)◦s defined as the set of stable rank r log Higgs bundles (V, θ) with theresidues of θ being nilpotent and Hilbert polynomial P (V ) = P (⊕r1 OX̄ ).

Claim 7. C −1 (Ogus-Vologodksy [11, Theorem 2.8]) : (Dol)◦s → (dR)◦s and isinjective (in particular (Dol)◦s is finite).
Proof. C −1 defined for p > r + 1 plus lift to W2 , preserves stability, total Chernclasses and nilpotency of the residues at ∞.
Claim 8. H 1 (X̄s , End(C −1 (V, θ))) = H 1 (X̄s , End(V, θ)) = 0.
Proof. C −1 defined for 2r rank objects, preserves cohomology, LHS = 0 by 4). By Claim 7, |(Dol)◦s | = M ≤ N = |(dR)◦s |. Let M 0 ≤ M be the number ofobjects in (Dol)◦s of the shape (V, θ) = (grF il (E), KS).
Corollary 9. Given (V, θ) ∈ (Dol)◦s there is at most one possible possible (E, ∇, F il)with (V, θ) = (grF il (E), KS).

Non-Archimedean Geometry and Applications

3Proof. Given F il on (E, ∇), then Rees (⊕i∈Z (F ili E)t−i , ∇t ) on X[t, t−1 ] has fibre(grF il E, KS) at t = 0 and (E, ∇) at t = ∞. Deformation of (grF il E, KS) from
Fq [t]/(tn ) to Fq [t]/(tn+1 ) is computed by H 1 (X̄s , End(V, θ)) = 0. As any (E, ∇) isendowed with at least one F il (by 7) and good model), has N ≤ M 0 .
Corollary 10.
1) M 0 = M = N ; C is bijective; the p-curvature of any(E, ∇) ∈ (dR)◦s is nilpotent;
2) any (E, ∇) ∈ (dR)◦s carries precisely one F il;
3) gr : (dR)◦s → (Dol)◦s , (E, ∇) 7→ (grF il E, KS) is well defined and bijective.
Proof. Ad 1): N = M 0 ≤ M ≤ N (first inequality from Corollay 9, last inequality
by Claim 7). Thus M 0 = M = N . C −1 sends nilpotent Higgs to nilpotent pcurvature dR.
Ad 2): any (E, ∇) ∈ (dR)◦s carries at least one F il by the good model and more
would imply M > N by Corollary 9. 3) follows.
Corollary 11. σ := C −1 ◦ gr is a permutation of (dR)◦s and has finite order f |N !.
Definition 12. The chain(E0 , ∇0 , F il0 , φ0 : C −1 (grF il0 E0 , KS) ∼= (E1 , ∇1 ),E1 , ∇1 , F il1 , . . . , Ef −1 , ∇f −1 , F ilf −1 , φf −1 : C −1 (grF ilf −1 Ef −1 , KS) ∼= (E0 , ∇0 ))is called a f -periodic Higgs-de Rham flow. ([7], [8]).
d̄ in what is
Proposition 13. 1) The f -periodic Higgs-de Rham flow lifts to X
Wd̄
still a f -’periodic Higgs-de Rham flow’ over XW .
ˆ W )K .
2) The operator σ becomes the Frobenius on the isocrystals (ÊW , ∇Here W = W (Fq ), K = Frac(W ) and recall that the p-curvatures of the modp-reduction are nilpotent so we have isocrystals with a Frobenius structure.
d̄ together with their Hodge
Proof. The (E, ∇) in (dR)◦s lift by definition to XWd̄ yielding some (V̂ , θ̂ ) so (V , θ ) whichfiltration. So gr is defined on XWWWK Kd̄−1
in addition are stable. Cis defined on XW by Ogus-Vologodsky. As the liftˆ W ) is uniquely determined by its reduction to s =⇒ f -periodicity.
(ÊW , ∇Remark 14. Claim 8 =⇒ (semi-continuity of coherent cohomology)◦H 1 (X̄K , End(VK , θK )) = 0 =⇒ (VK , θK ) ∈ MDol(r)S (K).
◦◦If we define MDol(X)(r)S ⊂ MdR (r)S , so (V, θ) ∈ MDol(C) if the residues at ∞ are
1nilpotent and H (X̄, End(V, θ)) = 0, and we assume by étally shrinking S in the◦good model that in addition MDol
(X)(r)S (S) consists of different (finitely many)
S-sections, we see that in fact we have N such and they all come from the Higgs-deRham flow. If we had a log-Simpson correspondence at the level of moduli /C, wewould know this and could shorten the argument.

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Proposition 15. Lan-Sheng-Zuo, Lan-Sheng-Yang-Zuo: 1) Fully faithful functor:(f -periodic Higgs-de Rham flow with nilpotent residues level ≤ p − 1) → (logFontaine-Lafaille modules with F robf -structure, with nilpotent residues level ≤p − 1).
2) Generalization of Fontaine-Lafaille-Faltings [5] Theorem 2.6* and p.43 i)−1
F robi (object): fully faithful functor (log-Fontaine-Lafaille modulesapplied to ⊕fi=0fwith F rob -structure, with nilpotent residues, level ≤ p − 1) → (crystalline localsystems on XK with values in GLr (Zpf )).
Remark 16. Crystalline here is defined by Fontaine on K, Faltings on ’small’opens defined by a ring R, étale over the Tate algebra of Gdm over W , then on
their rings by admissibility of a Bcrys (R), then gluing. Generalization by TanTong, and just a few days ago Du-Lu-Moon-Shimizu. All those concepts restrictedto Spec(W ) → XW yield the same definition, which is Fontaine’s one. Matti
Würthen told us that he can construct directly a prismatic F -crystal in the senseof Bhatt-Scholze out of a Fontaine-Lafaille module. Granted this in the log-version,
one could enhance a bit Theorem 6 toTheorem 17 (?). Assumption 1 =⇒ on a good model, for any Spec(W ) → S,d̄ \ ∞ ) carries the structure of a prismaticˆ W )/(Xthe formal connection (ÊW , ∇WW
F -crystal.
4. Étale theoremHas O(S) ⊂ C and choose W ⊂ C for Spec(W ) → S as in Theorem 6. This∼=defines K̄ ⊂ C with (Grothendieck) π1 (XC ) −→ π1 (XK̄ ). By Theorem 3, under
Assumption 1, each L is integral. A p-place p of OL , p =char(Fq ), defines Lp onXC . Keep the same letter Lp for the O¯Lp local system on XK̄ .
Theorem 18 ([3], Theorem A.21). Assumption 1 =⇒ Lp defined on XK̄ descends
to a crystalline local system on XK with values in GLr (Zpf ) for some f |N !.
d̄ [6,Proof. By the compatibitliy of Faltings p-adic Simpson correspondence on XWTheorem 5] with his Fontaine-Lafaille functor [5] loc.cit, calling πi , i = 1, . . . , N
the GLr (Zpf ) local systems on XK defined Proposition 15, where f is a l.c.m. of
the periods, which divides N !, the πi /XK̄ correspond to the Higgs bundles (V, θ)K̄
which are stable, so in particular πi /XK̄ is irreducible. Likewise
dimCp Hom(XCp , π1 , π2 ) ≤ dimCp Hom(XCp , (V1 , θ1 ), (V2 , θ2 )) = 0as Faltings functor from small p-adic local systems to Higgs bundles is faithful. So
the number of isomorphism classes of πi /XK̄ is the same as the one of πi /XK which
is N . But by the complex Riemann-Hilbert correspondence, there are precisely N
isomorphism classes of Lp .
Remark 19. Theorem 18 via Remark 16 is the way Pila-Ananth Shankar-Tsimerman
use our work for Shimura varieties of real rank ≥ 2 in their proof of the André-Oort
conjecture for those [12, Theorem 1.2].

Non-Archimedean Geometry and Applications

5References[1] H. Esnault, M. Groechenig, Cohomologically rigid connections and integrality, Selecta Mathematica 24 (5) (2018), 4279–4292.
[2] H. Esnault, M. Groechenig, Rigid connections and F-isocrystals, Acta Math. 225 1 (2020),103–158.
[3] H. Esnault, M. Groechenig, Frobenius structures and unipotent monodromy at infinity,
Appendix to [12], https://arxiv.org/abs/2109.08788[4] H. Esnault, E. Viehweg, Logarithmic de Rham complexes and vanishing theorems, Inventiones math. 86 (1986), 161–194.
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[8] G. Lan, M. Sheng, Y. Yang, K. Zuo, Uniformization of p-adic curves via Higgs-de Rhamflows, Journal für die reine und angewandte Mathematik (Crelles Journal) 747 (2019), 63–108.
[9] A. Langer, Semistable modules over Lie algebroids in positive characteristic, Doc. Math. 19(2014), 509–540.
[10] T. Mochizuki, Kobayashi-Hitchin correspondence for tame harmonic bundles and an application, Astéerisque textbf309 (2006), viii+117.
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Hautes Études Sci. 106 (2007), 1–138.
[12] J. Pila, Ananth Shankar, J. Tsimerman, Canonical Heights on Shimura Varieties and theAndré-Oort Conjecture, https://arxiv.org/abs/2109.08788

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