Non-Archimedean
Analytic Geometry:
Theory & Practice
welcome booklet

Agence Nationale
de la Recherche

Université de la
Polynésie française

European
Research
Council

Institut
Universitairede France@Dear conference participant,Welcome to French Polynesia.
French Polynesia, or Polynésie française, is an overseas country of the French
Republic. It comprises about 130 islands partitioned in five archipelagos. The
largest and most populous island, Tahiti, is part of the Îles du Vent group of theÎles de la Société archipelago.
Polynesia has been visited by European explorers since the sixteenth century,with French settlers arriving in the early nineteenth century. France gave French
Polynesia the status of an overseas country in 1946; essentially, Polynesians votein nation-wide French legislative and executive elections, as well as to elect a local
government and assembly. Those local bodies have more leeway than those incharge of similar administrative divisions in mainland France, although matterssuch as justice and defense remain under the authority of the nation.
Tahiti is an extinct volcano, which explains its mountainous center andsurrounding lagoon. It is home to 180, 000 people, which comprises about two
thirds of the population of French Polynesia. The capital of French Polynesia isthe city of Papeete where the conference is held.
Most people in Tahiti speak French, although Tahitian is the mother tongueof many. Other Polynesian languages are widely spoken in the surroundingarchipelagos. Common Tahitian words and expressions are also frequently used
in French; during your stay you will likely hear most if not all of the following
ones:’ia ora na
maeva
nana
māuruuru
vahine
tane
fenua

hello
welcomegoodbyethank youwoman
manthe land3

Practical Information
Location
The conference will be held at the Muriavai Hall of the Maison de la Culture
in downtown Papeete. The participants will stay at the Royal Tahitien hotel
located in a tropical garden on the beach of a lagoon.
A free shuttle service will operate from the hotel to the conference centerand back according to the following schedule:

MondayTuesday
ThursdayFridayHotel to conference
08:00
07:40
07:40
07:40

Conference to hotel
18:30
18:30
21:15
18:30

The Royal Tahitien hotel does not provide a complimentary breakfast. However, conference days will begin with a coﬀee break during which Viennoiseries
and fruit juice will be served. Lunches (free of charge for participants) will beheld at the Manava restaurant on Monday and Tuesday and at the La Romana
restaurant on Thursday and Friday. Both restaurants are a ten-minutes walk from
the Maison de la Culture.

4

Events
On Wednesday an excursion will take us to the inner parts of the island up the
Papenoo valley. We will have lunch at the Relais de la Maroto with a wonderfulview of the surrounding mountains. The departure is scheduled at 08:15 fromthe Royal Tahitien hotel, we should be back by 18:00. Please dress comfortably
for this trip.
Later on the same day, the shuttle will pick us up at the Royal Tahitien at
18:40 and drive us to the conference dinner held at the InterContinental Hotel.
This will be an occasion to discover the Marquesas Islands dances performed bya group of warriors. The buﬀet will feature Polynesian and Marquesas specialties.
The shuttle bus back to the Royal Tahitien will depart from the InterContinental
hotel at 21:30.
If you come as a couple and your partner wishes to participate to the excursion
and/or the conference dinner, please let us know about it early on Monday sothat we are able to make necessary arrangements. Note that this is will imply
additional charges.
On Thursday evening we plan to dine together at the roulottes of the Vaietesquare and have a beer at “Les 3 Brasseurs”.

5Monday
August 24
8:00
8:30
9:00
9:30

coﬀee
coﬀee & registration

Florent Martin

opening remarks
Laura DeMarco

10:00
10:30

Tuesday
August 25

coﬀee
Emmanuel Lepage

Juan Rivera-Letelier

11:00

Jérôme Poineau

11:30
12:00
12:30
13:00

lunch

lunch

Francesco Baldassarri

Kazuhiro Fujiwara

coﬀee

coﬀee

Ruochuan Liu

Fumiharu Kato

Ambrus Pal

Marc Chapuis

13:30
14:00
14:30
15:00
15:30
16:00
16:30
17:00
17:30

Wednesday
August 26

free

ThursdayAugust 27

FridayAugust 28

coﬀee

coﬀee

Antoine Ducros

Klaus Künnemann

coﬀee

coﬀee

Joe Rabinoﬀ

Omid Amini

Chengyang Xu

Kazuhiko Yamaki

lunch

lunch

Michael TemkinBertrand Rémy

coﬀee

coﬀee

Shou-Wu Zhang

Vladimir Berkovich

Kentaro MitsuiTalk Abstracts
Monday 09:30, Laura DeMarco
Variation of canonical height, illustrated
Around 1990, Joe Silverman wrote a series of three articles on the variation of
canonical height in families of elliptic curves. I will discuss connections between
these results and dynamical systems on P1 (and an associated Berkovich space).
As the height functions define dynamical ”bifurcation measures” on the basevariety, I will show illustrations of these measure densities. In new work with
Dragos Ghioca, we exploit these ideas to study rationality of canonical heights
for maps of P1 .

Monday 10:30, Juan Rivera-Letelier
On the equidistribution of points of small toric height
On a toric variety, we study the asymptotic distribution of points that aresmall with respect to a given toric height. In this setting, the usual method toprove equidistribution, introduced by Ullmo, Szpiro, Zhang, can only be applied
to Weil heights. In fact, this method requires the height function to satisfy
Zhang’s inequality with an equality, and recently Burgos, Philippon, and Sombra
showed that this only holds for Weil heights. Nevertheless, we show that forsmooth toric heights the equidistribution of points of small height does hold,and as a consequence we derive the Bogomolov property for these heights. We
also give a host of examples of non-smooth heights for which equidistribution
fails. We also give the first example of a height for which the Bogomolov property
fails.
This is joint work with Burgos, Philippon, and Sombra.
8

Monday 14:30, Francesco Baldassarri
A p-adically entire function with integral values on Q p and additive characters of perfectoid fields
We give an essentially self-contained proof of the fact that a certain p-adic
power seriesΨ = Ψ p (T ) ∈ T + T p−1 Z[[T p−1 ]],which trivializes the addition law of the formal group of Witt p-covectors is
p-adically entire and assumes values in Z p all over Q p . We also carefully examine
its valuation and Newton polygons. For any perfectoid field extension (K, k) of(Q p , k p ) contained in (C p , k p ), and any pseudo-uniformizer $ = $ (i ) i ≥0 of
K [ , we consider the element
π = π($) :=X$ (i ) p i +

i ≥0

X
i <0

$ (0)

 p −ipi ∈ K .

We use the isomorphism between the Witt and the Cartier (hyperexponential)
group over Z( p) , which we extend to their p-divisible closures, and the propertiesof Ψ p , to show that the map x 7→ exp(πx ), a priori only defined for v p (x ) >1
p−1 − v p (π), extends to a continuous additive character
Ψ$ : Q p → 1 + K ◦◦ .
A similar character for the cyclotomic p-extension of Q p appears in Colmez’
work. I will also give the numerical computation of the first coeﬃcients of Ψ p ,for small p.

9Monday 16:00, Ruochuan Liu
Finiteness of cohomology of relative (φ, Γ )-modules
We will show finiteness of the higher direct images of geometric (φ, Γ )-modules
for proper smooth morphisms of smooth rigid analytic varieties. As a consequence, for proper smooth rigid analytic varieties over finite extensions of Q p ,the proetale cohomology groups of Q p -local systems are all finite dimensional
Q p -vector spaces.
This is joint work with Kedlaya.

Monday 17:00, Ambrus Pal
Rigid cohomology over Laurent series fields
It is possible to refine Berthelot’s rigid cohomology over Laurent series fields
of characteristic p to a cohomology theory taking values in (φ, ∇)-modules overthe bounded Robba ring, not just over the larger Amice ring, asfurnished by
Berthelot’s original theory. One gets the refined theory by replacing Berthelot’s
notion of a rigid frame with a suitably modified version, which however requires
the usage of adic spaces instead of rigid analytic spaces. Nevertheless the arguments of the subject all carry over without much diﬃculty to prove the finitenessfor this cohomology theory, and a comparison theorem with Berthelot’s original
construction. As a consequence we can attach Weil–Deligne representations
to the rigid cohomology of varieties over Laurent series fields using Marmora’sfunctor, and hence complete existing results and conjectures on independence
and monodromy-weight. This is joint work with Chris Lazda.

10

Tuesday 08:30, Florent Martin
Two connectedness results on Berkovich spaces
We prove two topological results for Berkovich k-analytic spaces. First, when
k is discretely valued, we give a new proof and generalize a result of Siegfried
Bosch about the connectedness of tubes over a closed point in the reduction ofa strictly k-aﬃnoid space. Secondly, we prove that a subanalytic set has finitely
many connected components, and when k is discretely valued, we show that eachconnected component is itself subanalytic. We use as a main tool a result of A.J.
de Jong relating special formal schemes and bounded functions on their genericfibers.

Tuesday 10:00, Emmanuel Lepage
Tempered fundamental group of curves
The tempered fundamental group of a non-archimedean analytic curvesclassifies a category of covers encompassing at the same time finite étale covers
and topological covers of the Berkovich analytic space. In this talk, we will discuss
what can be recovered of a curve over an algebraically closed non-archimedeanfield of mixed characteristics from its tempered fundamental group.

11

Tuesday 11:00, Jérôme Poineau
Theorems A and B for Berkovich spaces over Z
Although Berkovich spaces usually appear in a non-archimedean setting,their general definition actually allows arbitrary Banach rings as base rings, e.g.
Z endowed with the usual absolute value. Over the latter, Berkovich spaces look
like fibrations that contain complex analytic spaces as well as p-adic analytic
spaces for every prime number p. After recalling the main known propertiesof those spaces, we will show that relative disks of arbitrary dimension satisfy
the conclusions of Cartan’s theorems A (global generation) and B (vanishing of
higher cohomology) for coherent sheaves. Those results can be considered ascounterparts of the foundational theorems of Kiehl and Tate in the usual p-adic
setting.

Tuesday 14:30, Kazuhiro Fujiwara
Proper dominant descent in rigid geometry
In this talk some basic and foundational aspects of rigid geometry is discussed.
If one starts with formal schemes, to make the functoriality work, it is necessary
to introduce non-noetherian adic rings in the theory. I will show that there is anice class of adic rings for this purpose. Moreover I hope to discuss the expected
global properties, in particular proper dominant descent.

12

Tuesday 16:00, Fumiharu Kato
Zariski Main Theorem for henselian rigid space
This is a joint-work with Shuji Saito (Tokyo Institute of Technology). We
announce an analogue of Zariski Main Theorem in henselian rigid geometry,describing the sketch of the proof, and discuss a few possible applications inalgebraic geometry.

Tuesday 17:00, Marc Chapuis
Tamely ramified forms of discs and annuli
Using Michael Temkin’s theory of graduated reduction, Antoine Ducros has
shown that any tamely ramified form of an open polydisc is trivial. Later, Tobias
Schmidt proved that the result holds for closed discs. In fact we have a similarstatement both for polydiscs, either closed or open, and for (one-dimensional)
annuli, either closed or open in one or the other of their edges. We shall recall
the main steps of Ducros’ proof which we follow and show broadly how they canbe adapted to the diﬃculties that appear when one substitutes closed to open
discs, or annuli to discs.

13

Thursday 08:30, Antoine Ducros
Reified valuation with infinitesimal elements and skeleta of Berkovich
spacesLet Sn be the standard skeleton of the Berkovich space An (over some fixed
non-archimedean field k), let X be a k-aﬃnoid space and let f : X → An be an
analytic morphism with zero-dimensional fibers. I will roughly present a proofof the fact that f −1 (Sn ) is piecewise monomial, i.e. locally looks like a compact
subset of (R+ )n defined by finitely many non-strict monomial inequalities. As
a key tool it uses a compact, totally disconnected space of valuations associatedto X and some of its tameness properties which themselves rely on slightly involved commutative algebra (excellent schemes, regular morphisms, unibranch
schemes…).
This is joint work with Amaury Thuillier.

Thursday 10:00, Joe Rabinoﬀ
The tropical skeleton
Given a closed subscheme X of a toric variety Y∆ , we define a canonicallocally closed subset STrop(X ) of X an , called the tropical skeleton, as the set of
Shilov boundary points of the fibers of the tropicalization map. This is a kind ofcompactified c -skeleton in the sense of Ducros. Perhaps surprisingly, STrop(X )is not always closed; we will discuss equidimensionality conditions under which
a limit of a sequence of points of STrop(X ) is contained in STrop(X ). We give
applications to so-called schön subvarieties of a torus, in which case the tropicalskeleton coincides with a canonically defined skeleton in the sense of Berkovich,as well as the parameterizing complex of Helm–Katz. We also give applications tocontinuity of the section of the tropicalization map on the tropical multiplicity-1
locus.

14

Thursday 11:00, Chengyang Xu
Skeleton and dual complex
In the first part of this talk, I will discuss a construction of the essential
skeleton from the dual complex of a relative minimal model, for a variety X
over K = k((t )) such that KX is semi-ample. In the second part, I will discuss
examples of the topology of the dual complex of special types of varieties.

Thursday 14:30, Michael Temkin
Wild coverings of Berkovich curves
I will describe the structure of finite morphisms between smooth Berkovich
curves. The special accent will be on the description of the loci of points ofmultiplicity n and its relation to Herbrand function and the ramification theory.
If time permits we will also talk about the diﬀerent function associated to amorphism.

15

Thursday 16:00, Shou-Wu Zhang
A p-adic Waldspurger formula
In this talk, I will explain a p-adic Waldspurger formula proved by Bertolini–
Darmon–Prasanna under the Heegner condition, and in full generality later by
Liu–Zhang–Zhang. I will start with a classical Waldspurger formula on complex
modular forms and a Gross–Zagier formula on rational modular forms, thendefine p-adic modular forms, p-adic L-functions, p-adic period integrals, andfinally state a p-adic Waldspurger formula.

Thursday 17:00, Kentaro Mitsui
Closed points on torsors under abelian varieties
We show the existence of a separable closed point of small degree on any
torsor under an abelian variety over a complete discrete valuation field under
mild assumptions on the residue field of the valuation ring and the reductionof the abelian variety. To show the existence, we introduce and study minimalmodels of torsors under quasi-projective smooth group schemes.

16

Friday 08:30, Klaus Künnemann
Metrics and delta-forms in non-archimedean analytic geometry
We report on joint work with Walter Gubler from Regensburg. We consider
metrics on line bundles over the non-archimedean analytification of an algebraic
variety. Extending work by Chambert-Loir and Ducros we introduce delta-forms,
discuss their basic properties, and describe their use in the construction of first
Chern forms, Monge–Ampère measures, and local heights.

Friday 10:00, Omid Amini
Limit linear series and distribution of Weierstrass points
I will report on recent progress in constructing a general framework forthe study of degenerations of linear series on degenerating families of smoothproper curves over a field of characteristic zero, using in particular tools from
non-Archimedean geometry, generalizing the Eisenbud–Harris theory of limit
linear series from the eighties to any semistable curve.I will then discuss an application to the problem of understanding the limiting behavior of Weierstrass
points on such families. This leads to thefollowing non-Archimedean version ofa theorem of Mumford and Neeman: let X be a smooth proper curve over a nonArchimedean field of residue characteristic zero, and L an ample line bundle on
X . The Weierstrass points of powers of L are equidistributed according to the
Zhang measure on the dual graph of a semistable model.
The talk is partially based on joint works with M. Baker and E. Esteves.

17Friday 11:00, Kazuhiko Yamaki
Recent progress in the geometric Bogomolov conjecture
Let A be an abelian variety over a function field and let X be a closed subvariety of A. The geometric Bogomolov conjecture claims that if X has a dense subset
of small points, then X should be a special subvariety, where a special subvariety
is a sum of a torsion subvariety and a subvariety defined over the constant field.
In a recent work, we prove that the above conjecture holds under the assumption of codim(X ) = 1 or dim(X ) = 1. In particular, we obtain the Bogomolov
conjecture for curves in their Jacobian over function fields in full generality. In avery important step of the proof, the canonical measures over non-archimedean
analytic spaces play crucial roles. We will explain how they are used.

Friday 14:30, Bertrand Rémy
Automorphisms of Drinfeld half-spaces over a finite fields
We show that the automorphism group of Drinfeld’s half-space over a finite
field is the projective linear group of the underlying vector space. The proof ofthis result uses analytic geometry in the sense of Berkovich over the finite fieldequipped with the trivial valuation. We also take into account extensions of thebase field.
This is joint work with Amaury Thuillier and Annette Werner.

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Friday 16:00, Vladimir Berkovich
Complex analytic vanishing cycles for formal schemes
Let K be a non-Archimedean field with nontrivial discrete valuation whose∼ring of integers K ◦ contains C and C → Ke , and let Π be the subgroup of Galois group G of K which is the preimage of Z under the canonical isomorphism∼ b
G → Z.
I will describe a construction that associates to every special formal
scheme X over K ◦ an exact functor Λ· 7→ RΨηh (Λ·X ) from the derived categoryηof Π-modules to the derived category of abelian Π-sheaves on the complex analytification X hs of the closed fiber X s of X. This functor possesses many of theproperties established in my previous work for the similar functor on complexesof discrete torsion G -modules and extends the latter. Furthermore, given a morphism of complex analytic germs (X , x ) → (C, 0), a scheme Y of finite typeover OX ,x , a subscheme Z ⊂ Y ⊗OX ,x C, and a homomorphism OC,0 → K ◦ defined by a generator of the maximal ideal of K ◦ , the above functor associatedto the formal completion YÒ/Z coincides with the restriction of the corresponding complex analytic vanishing cycles functor (from SGA7), associated to theanalytification Y h of Y , to Z h . The construction allows one to define, forevery compact strictly K -analytic space X , integral “étale” cohomology groups
ca . These are finitely generated abelian groups providedbK
H q (X , Z) of X = X ⊗with a quasi-unipotent action of Π and functorial in X , and they give rise to theétale l -adic cohomology groups of X and, if X is rig-smooth, to the de Rham
cohomology groups of X .

19

Acknowledgments
We are grateful to the organizing committee, scientific committee, and oursponsors who dedicated time and resources to make this conference possible.

Organizing Committee
Gaetan Bisson

Université de la Polynésie française

French Polynesia

Jean Chaumine

Université de la Polynésie française

French Polynesia

Philippe Lebacque

Université de Besançon

FranceRoger Oyono

Université de la Polynésie française

French Polynesia

Scientific Committee
Antoine Ducros

Université Paris VI

FranceJohannes Nicaise

Katholieke Universiteit Leuven

BelgiumJérôme Poineau

Université de Caen

FranceAlexey Zykin

University of French Polynesia

French Polynesia

Sponsors
Agence Nationale de la Recherche

Université de la Polynésie française

European Research Council

Institut Universitaire de France
21

ParticipantsAlexey Zykin
Ambrus Pal
Antoine Ducros
Bertrand Rémy
Chengyang Xu
Emmanuel Lepage
Éric Férard
Florent Martin
Francesco Baldassarri
Fumiharu KatoGaëtan Bisson
Jean Chaumine
Jérôme Poineau
Joe Rabinoﬀ
Juan Rivera-Letelier
Kazuhiko Yamaki
Kazuhiro Fujiwara
Kentaro Mitsui
Klaus Künnemann
Laura DeMarco
Marc Chapuis
Mattias Jonsson
Michael Temkin
Mihnea Popa
Omid Amini
Roger Oyono
Ruochuan Liu
Shou-Wu Zhang
Shu Kawaguchi
Vladimir Berkovich
Ye LuoUniversité de la Polynésie française
Imperial College London
Université Pierre et Marie Curie
Université Claude Bernard Lyon 1
Beijing Center of Math. Research
Université Pierre et Marie Curie
Université de la Polynésie française
Universität Regensburg
Università degli Studi di Padova
Kumamoto University
Université de la Polynésie française
Université de la Polynésie française
Université de Caen
Georgia Institute of Technology
Pontificia Universidad Católica
Kyoto University
Nagoya University
Kobe University
Universität Regensburg
Northwestern University
Université Pierre et Marie Curie
University of Michigan
Hebrew University of Jerusalem
Northwestern University
École Normale Supérieure
Université de la Polynésie française
Beijing Center of Math. Research
Princeton University
Doshisha University
Weizmann Institute of Science
Rice University

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