## Periodic Reporting for period 4 - TOSSIBERG (Theory of Stein Spaces in Berkovich Geometry)

Reporting period: 2020-01-01 to 2020-12-31

Complex Stein spaces may be thought of as analytic analogues of the affine schemes of algebraic geometry. They may be characterized in several manners: using convergence of holomorphic functions, topological properties or potential-theoretic properties, for instance. Especially useful for applications is the fact that their coherent cohomology vanishes. Despite the crucial importance of this theory in complex analytic geometry, its p-adic counterpart has hardly been sketched.

In the setting of Berkovich geometry (one among the several notions of p-adic geometry), recent developments have enabled to get a fine understanding of the topology of the spaces (work of Berkovich and Hrushovski-Loeser) and to define the basic tools of potential theory (work of Baker-Rumely, Thuillier, Boucksom-Favre-Jonsson and Chambert-Loir-Ducros). The conditions for a comprehensive study of p-adic Stein spaces are now met; this will be our first goal. The theory will then be used to investigate envelopes of holomorphy and meromorphy. As an application, we plan to derive rationality criteria for power series over function fields.

The second part of the project is devoted to the theory of Stein spaces for Berkovich spaces over rings of integers of number fields (where all the places appear on an equal footing). Those spaces have hardly been studied and only a very small part of the usual analytic machinery is available in this setting. Here, our main goal will consist in proving the basic and fundamental fact that relative polydisks are Stein spaces (in the cohomological sense). This will allow a deeper investigation of rings of convergent arithmetic power series (i.e. with integral coefficients) and will lead up to properties related to commutative algebra but also to the inverse Galois problem. Knowing that the coherent cohomology of polydisks vanishes also opens the road towards computing global cohomology groups for projective analytic spaces over ring of integers of number fields.

In the setting of Berkovich geometry (one among the several notions of p-adic geometry), recent developments have enabled to get a fine understanding of the topology of the spaces (work of Berkovich and Hrushovski-Loeser) and to define the basic tools of potential theory (work of Baker-Rumely, Thuillier, Boucksom-Favre-Jonsson and Chambert-Loir-Ducros). The conditions for a comprehensive study of p-adic Stein spaces are now met; this will be our first goal. The theory will then be used to investigate envelopes of holomorphy and meromorphy. As an application, we plan to derive rationality criteria for power series over function fields.

The second part of the project is devoted to the theory of Stein spaces for Berkovich spaces over rings of integers of number fields (where all the places appear on an equal footing). Those spaces have hardly been studied and only a very small part of the usual analytic machinery is available in this setting. Here, our main goal will consist in proving the basic and fundamental fact that relative polydisks are Stein spaces (in the cohomological sense). This will allow a deeper investigation of rings of convergent arithmetic power series (i.e. with integral coefficients) and will lead up to properties related to commutative algebra but also to the inverse Galois problem. Knowing that the coherent cohomology of polydisks vanishes also opens the road towards computing global cohomology groups for projective analytic spaces over ring of integers of number fields.

Marco Maculan and Jérôme Poineau investigated the notion of Stein space in the p-adic (and more generally non-archimedean) setting. In this context, the basic definition is due to R. Kiehl and requires exhausting the space by affinoid spaces. In complex geometry, on the other hand, the definition is usually given in more topological terms through the notions of holomorphic separability and convexity. Finally, in the applications, one is especially interested in the vanishing of the cohomology of coherent sheaves, and this can also be taken as a definition. Marco Maculan and Jérôme Poineau first managed to prove that those three definitions actually give rise to the same notion in the case of spaces with no boundary. In a next step, they extended the theory (and Kiehl's definition) to handle the case of spaces with boundary.

Velibor Bojkovic and Jérôme Poineau focused on the case of p-adic analytic curves. In this setting, Michael Temkin recently proved deep results on the structure of morphisms. Jérôme Poineau and Velibor Bojkovic used them in order to relate in a precise way, by means of an explicit formula, the ramification invariants of a morphism of p-adic curves to the radii of convergence of the associated differential equation. This was previously known only in the case of a tame morphism (where the result is trivial) and of the Frobenius morphism. They also derive a general formula computing the Laplacian of the height of the Newton polygon of a p-adic differential equation, a conjectured formula whose proof in the general case was still missing.

Pablo Cubides Kovacsics and Jérôme Poineau also studied non-Archimedean analytic curves, but from a model-theoretic point of view, following the lead of Ehud Hrushovski and François Loeser. They focused on the case of curves and provided an alternative approach to endow the analytification of an algebraic curve with a definable structure. Their method allowed them to deal with analytic morphisms, which were not covered by the previous instances of the theory. Along the way, they gave definable versions of several usual notions of Berkovich analytic geometry: branch emanating from a point, residue curve at a point of type 2, etc. In addition, they derived a complete description of the definable subsets of analytic curves.

In the setting of global analytic geometry, i.e. for Berkovich spaces over Z, Thibaud Lemanissier and Jérôme Poineau managed to get a result of cohomological vanishing for higher-dimensional spaces. More precisely, they proved that open and closed Berkovich disks over Z have no higher coherent cohomology. This is a first step towards developing a full-fledged theory of Stein spaces in this context. Thibaud Lemanissier and Jérôme Poineau then derived results for rings of convergent arithmetic power series. The latter, roughly speaking series with coefficients in Z with positive radii of convergence, were introduced by David Harbater in a series of work related to the inverse Galois problem and he had proven several results about them in the case of one variable. Using cohomological techniques, Thibaud Lemanissier and Jérôme Poineau proved the noetherianity of rings of convergent arithmetic power series with an arbitrary number of variables.

In complex geometry, any Riemann surface of genus g admits a so-called Schottky uniformization, i.e. a uniformization by the complex projective line with 2g disks removed and whose group is the free group on g generators. David Mumford showed that this theory may partially be extended to a non-archimedean setting. Jérôme Poineau and Daniele Turchetti managed to give a uniform construction generalizing the previous ones. More precisely, they defined a universal Schottky space S as a Berkovich spaces over Z (an open subset of the affine space of dimension 3g-3 over Z). To each point of this space, one may naturally associate a Riemann surface if the point is archimedean or a Mumford curve if it is not. This gives rise to a projective curve over S, called the universal Mumford curve, which admits a global uniformization.

Vlerë Mehmeti introduced Berkovich spaces in the study of local-global principles over function fields of p-adic curves. General techniques to deal with this kind of problems were developed a few years ago by David Harbater, Julia Hartmann and Daniel Krashen and relied heavily on models of curves. By working directly on the generic fiber, Vlerë Mehmeti managed to strengthen their results and extend them to compact non-necessarily projective analytic curves and to non-necessarily discretely valued fields. As a corollary, she reproves that quadratic forms over p-adic fields in at least 9 variables have non-trivial zeroes.

Velibor Bojkovic and Jérôme Poineau focused on the case of p-adic analytic curves. In this setting, Michael Temkin recently proved deep results on the structure of morphisms. Jérôme Poineau and Velibor Bojkovic used them in order to relate in a precise way, by means of an explicit formula, the ramification invariants of a morphism of p-adic curves to the radii of convergence of the associated differential equation. This was previously known only in the case of a tame morphism (where the result is trivial) and of the Frobenius morphism. They also derive a general formula computing the Laplacian of the height of the Newton polygon of a p-adic differential equation, a conjectured formula whose proof in the general case was still missing.

Pablo Cubides Kovacsics and Jérôme Poineau also studied non-Archimedean analytic curves, but from a model-theoretic point of view, following the lead of Ehud Hrushovski and François Loeser. They focused on the case of curves and provided an alternative approach to endow the analytification of an algebraic curve with a definable structure. Their method allowed them to deal with analytic morphisms, which were not covered by the previous instances of the theory. Along the way, they gave definable versions of several usual notions of Berkovich analytic geometry: branch emanating from a point, residue curve at a point of type 2, etc. In addition, they derived a complete description of the definable subsets of analytic curves.

In the setting of global analytic geometry, i.e. for Berkovich spaces over Z, Thibaud Lemanissier and Jérôme Poineau managed to get a result of cohomological vanishing for higher-dimensional spaces. More precisely, they proved that open and closed Berkovich disks over Z have no higher coherent cohomology. This is a first step towards developing a full-fledged theory of Stein spaces in this context. Thibaud Lemanissier and Jérôme Poineau then derived results for rings of convergent arithmetic power series. The latter, roughly speaking series with coefficients in Z with positive radii of convergence, were introduced by David Harbater in a series of work related to the inverse Galois problem and he had proven several results about them in the case of one variable. Using cohomological techniques, Thibaud Lemanissier and Jérôme Poineau proved the noetherianity of rings of convergent arithmetic power series with an arbitrary number of variables.

In complex geometry, any Riemann surface of genus g admits a so-called Schottky uniformization, i.e. a uniformization by the complex projective line with 2g disks removed and whose group is the free group on g generators. David Mumford showed that this theory may partially be extended to a non-archimedean setting. Jérôme Poineau and Daniele Turchetti managed to give a uniform construction generalizing the previous ones. More precisely, they defined a universal Schottky space S as a Berkovich spaces over Z (an open subset of the affine space of dimension 3g-3 over Z). To each point of this space, one may naturally associate a Riemann surface if the point is archimedean or a Mumford curve if it is not. This gives rise to a projective curve over S, called the universal Mumford curve, which admits a global uniformization.

Vlerë Mehmeti introduced Berkovich spaces in the study of local-global principles over function fields of p-adic curves. General techniques to deal with this kind of problems were developed a few years ago by David Harbater, Julia Hartmann and Daniel Krashen and relied heavily on models of curves. By working directly on the generic fiber, Vlerë Mehmeti managed to strengthen their results and extend them to compact non-necessarily projective analytic curves and to non-necessarily discretely valued fields. As a corollary, she reproves that quadratic forms over p-adic fields in at least 9 variables have non-trivial zeroes.

The work by Maculan and Poineau qualifies as an important first step in the theory of non-Archimedean Stein spaces. It gives a clear and thorough account of the first definitions and properties, but, at the same time, leaves several interesting questions open and paves the way for further study.

The work by Lemanissier and Poineau laid solid foundations for the theory of Berkovich spaces over Z. They developed several aspects: categorical, topological, cohomological,... so as to make possible to use in several situations. Concrete applications should soon follow in various areas: dynamical systems, unlikely intersections, differential equations, etc.

The work by Lemanissier and Poineau laid solid foundations for the theory of Berkovich spaces over Z. They developed several aspects: categorical, topological, cohomological,... so as to make possible to use in several situations. Concrete applications should soon follow in various areas: dynamical systems, unlikely intersections, differential equations, etc.