# Conference "Non-archimedean analytic Geometry: Theory and Practice"

## August 24 - 28, 2015, Papeete, French Polynesia

Conference talks: titles and abstracts

• Omid Amini (CNRS, École Normale Supérieure de Paris) : Limit linear series and distribution of Weierstrass points
I will report on recent progress in constructing a general framework for the study of degenerations of linear series on degenerating families of smooth proper 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 of a theorem of Mumford and Neeman: let X be a smooth proper curve over a non-Archimedean 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.

• Francesco Baldassarri (Università degli Studi di Padova) : A p-adically entire function with integral values on Qp and additive characters of perfectoid fields.
Abstract and slides in pdf.

• Vladimir Berkovich (Weizmann Institute of Science) : Complex analytic vanishing cycles for formal schemes

• Marc Chapuis (Université Pierre et Marie Curie) : 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 similar statement 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 can be adapted to the difficulties that appear when one substitutes closed to open discs, or annuli to discs.

• Laura DeMarco (Northwestern University) : 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 P^1 (and an associated Berkovich space). As the height functions define dynamical "bifurcation measures" on the base variety, 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 P^1.
Slides of the talk and a movie.

• Antoine Ducros (Université Pierre et Marie Curie) : Reified valuation with infinitesimal elements and skeleta of Berkovich spaces
(joint work with Amaury Thuillier)
Let S_n be the standard skeleton of the Berkovich space A^n (over some fixed non-archimedean field k), let X be a k-affinoid space and let f: X---->A^n be an analytic morphism with zero-dimensional fibers. I will roughly present a proof of the fact that f^{-1}(S_n) 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 associated to X and some of its tameness properties which themselves rely on slightly involved commutative algebra (excellent schemes, regular morphisms, unibranch schemes...).

• Kazuhiro Fujiwara (Nagoya University) : 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 a nice class of adic rings for this purpose. Moreover I hope to discuss the expected global properties, in particular proper dominant descent.

• Fumiharu Kato (Kumamoto University) : Zariski Main Theorem for henselian rigid spaces
Abstract: 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 in algebraic geometry.
Slides of the talk.

• Klaus Künnemann (Universität Regensburg) : 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.
Slides of the talk.

• Emmanuel Lepage (Université Pierre et Marie Curie) : Tempered fundamental group of curves
The tempered fundamental group of a non-archimedean analytic curves classifies 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-archimedean field of mixed characteristics from its tempered fundamental group.

• Ruochuan Liu (Beijing Int. Center of Mathematical Research) : 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 Qp, the proetale cohomology groups of Qp-local systems are all finite dimensional Qp-vector spaces. This is joint work with Kedlaya.

• Florent Martin (Universität Regensburg) : 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 of a 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 each connected 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 generic fibers.

• Kentaro Mitsui (Kobe University) : 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 reduction of the abelian variety. To show the existence, we introduce and study minimal models of torsors under quasi-projective smooth group schemes.
Slides of the talk.

• Ambrus Pal (Imperial College London) : 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 (phi,nabla) modules over the 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 finiteness for 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's functor, and hence complete existing results and conjectures on independence and monodromy-weight. This is joint work with Chris Lazda.

• Jérôme Poineau (Université de Caen) : 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 properties of 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 as counterparts of the foundational theorems of Kiehl and Tate in the usual p-adic setting.
Slides of the talk.

• Joe Rabinoff (Georgia Institute of Technology) : The tropical skeleton
Given a closed subscheme X of a toric variety Y_Delta, we define a canonical locally 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 of compactified 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 tropical skeleton coincides with a canonically defined skeleton in the sense of Berkovich, as well as the parameterizing complex of Helm--Katz. We also give applications to continuity of the section of the tropicalization map on the tropical multiplicity-1 locus.

• Bertrand Rémy (Université Claude Bernard Lyon 1) : Automorphisms of Drinfeld half-spaces over a finite fields
This is joint work with Amaury Thuillier and Annette Werner.
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 of this result uses analytic geometry in the sense of Berkovich over the finite field equipped with the trivial valuation. We also take into account extensions of the base field.

• Juan Rivera-Letelier (Pontificia Universidad Católica de Chile) : On the equidistribution of points of small toric height
On a toric variety, we study the asymptotic distribution of points that are small with respect to a given toric height. In this setting, the usual method to prove 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 for smooth 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 a joint work with Burgos, Philippon, and Sombra.

• Michael Temkin (Hebrew University of Jerusalem) : 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 of multiplicity n and its relation to Herbrand function and the ramification theory. If time permits we will also talk about the different function associated to a morphism.

• Chengyang Xu (Beijing Int. Center of Mathematical Research) : 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 K_X is semi-ample. In the second part, I will discuss examples of the topology of the dual complex of special types of varieties.
Slides of the talk.

• Kazuhiko Yamaki (Kyoto University) : 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 $\mathrm{codim} (X) = 1$ or $\dim (X) = 1$. In particular, we obtain the Bogomolov conjecture for curves in their Jacobian over function fields in full generality. In a very important step of the proof, the canonical measures over non-archimedean analytic spaces play crucial roles. We will explain how they are used.
Slides of the talk.

• Shou-Wu Zhang (Princeton University) : 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, then define p-adic modular forms, p-adic L-functions, p-adic period integrals, and finally state a p-adic Waldspurger formula.