arXiv subjects
Publications and source records attributed to Jean-Benoît Bost.
Let ${\bf C}$ be a smooth geometrically connected projective curve over a finite field, and let $A$ be the affine algebra of its regular functions outside a fixed place of ${\bf C}$. We give precise relationships between the Mahler successive minima of normed $A$-lattices and the Harder-Narasimhan slopes of vector bundles over ${\bf C}$ using their category equivalence.
This memoir is devoted to the study of formal-analytic arithmetic surfaces. These are arithmetic counterparts, in the context of Arakelov geometry, of germs of smooth complex-analytic surfaces along a projective complex curve. Formal-analytic surfaces provide a natural framework for arithmetic algebraization theorems, old and new. Formal-analytic arithmetic surfaces admit a rich geometry which parallels the geometry of complex analytic surfaces. Notably the dichotomy between pseudoconvexity and pseudoconcavity plays a central role in their geometry. Our study of formal-analytic arithmetic surfaces relies crucially on the use of real-valued invariants. Some of these are intersection-theoretic, in the spirit of Arakelov intersection theory. Some other invariants involve infinite-dimensional geometry of numbers. Relating our new intersection-theoretic invariants to more classical invariants of Arakelov geometry leads us to investigate a new invariant, the Archimedean overflow, attached to an analytic map from a pointed compact Riemann surface with boundary to a Riemann surface. It is related to the characteristic functions of Nevanlinna theory. Our results on the geometry of formal-analytic arithmetic surfaces admit applications to concrete problems of arithmetic geometry. Notably we generalize the arithmetic holonomicity theorem of Calegari-Dimitrov-Tang regarding the dimension of spaces of power series with integral coefficients satisfying some convergence conditions. We also establish an arithmetic counterpart of theorems of Lefschetz and Nori by providing a bound on the index, in the étale fundamental group of an arithmetic surface, of the closed subgroup generated by the étale fundamental groups of some arithmetic curve and of some compact Riemann surfaces mapping to the arithmetic surface.
These are the notes of lectures delivered at Grenoble's summer school on \emph{Arakelov Geo\-me\-try and Diophantine Applications}, in June 2017. They constitute an introduction to the study of Euclidean lattices and of their invariants defined in terms of theta series. Recall that Euclidean lattice is defined as a pair $\bar{E}:= (E, \Vert .\Vert)$ where $E$ is some free $\mathbb{Z}$-module of finite rank $E$ and $\Vert. \Vert$ is some Euclidean norm on the real vector space $E_\mathbb{R} := E \otimes \mathbb{R}$. The most basic of these invariants is the non-negative real number: $$h^0_θ(\bar{E}) := \log \sum_{v \in E} e^{- π\Vert v \Vert^2}.$$ In these notes, we explain how such invariants naturally arise when one investigates basic questions concerning classical invariants of Euclidean lattices, such as their successive minima, their covering radius, or the number of lattice points in balls of a given radius. We notably discuss their significance from the perspective of Arakelov geometry and of the analogy between number fields and function fields, their role (discovered by Banaszczyk) in the derivation of optimal transference estimates, and their interpretation in terms of the formalism of statistical thermodynamics. These notes have been primarily written for an audience of arithmetic geometers, but should also be suited to a wider circle of mathematicians and theoretical physicists with some interest in Euclidean lattices or in the mathematical foundations of statistical physics.
In this monograph, we lay some foundations of a theory of infinite dimensional Euclidean lattices - and more generally, of infinite dimensional Hermitian vector bundles over some "arithmetic curve" ${\rm Spec}\,\mathcal{O}_K$ attached to the ring of integers $\mathcal{O}_K$ of some number field $K$ - with a view towards applications to transcendence theory and Diophantine geometry. In the first chapters of this monograph, we study the properties of the invariant $h^0_θ(\overline{E})$ attached to some Euclidean lattice $\overline{E}:= (E, \Vert.\Vert)$, defined by the expression $$h^0_θ(\overline{E}) := \log \sum_{v \in E} e^{- π\Vert v \Vert^2},$$ and, more generally, attached to some finite rank Hermitian vector bundle $\overline{E}$ over an arithmetic curve. Then we construct categories of infinite dimensional Hermitian vector bundles and we show that it is possible to associate generalized $θ$-invariants to these objects, so that they satisfy suitable subadditivity and summability properties. In the last chapter, we present a first application of this formalism to Diophantine geometry: we show how it allows one to establish some algebraicity criterion à la Chudnovsky concerning formal curves over number fields embedded in some projective space, by arguments that are direct counterparts of classical algebraization proofs in complex analytic and formal geometry.
We discuss various results and questions around the Grothendieck period conjecture, which is a counterpart, concerning the de Rham-Betti realization of algebraic varieties over number fields, of the classical conjectures of Hodge and Tate. These results give new evidence towards the conjectures of Grothendieck and Kontsevich-Zagier concerning transcendence properties of the torsors of periods of varieties over number fields. We notably establish that the Grothendieck period conjecture holds in degree 1 for products of curves, of abelian varieties, and of K3 surfaces, and that it holds in degree 2 for smooth cubic fourfolds.
We study the semistability of the tensor product of hermitian vector bundles by using the $\varepsilon$-tensor product and the geometric (semi)stability of vector subspaces in the tensor product of two vector spaces.
We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of Borel-Dwork and Pólya-Bertrandias valid over the projective line to arbitrary algebraic curves over a number field. The formulation and the proof of these criteria involve some basic notions in Arakelov geometry, combined with complex and rigid analytic geometry (notably, potential theory over complex and $p$-adic curves). We also discuss geometric analogues, pertaining to the algebraic geometry of projective surfaces, of these arithmetic criteria.