Abstract. We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using non-archimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the local-global principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system. 1. Amoebas 1.1. Generalities. Let k be a field. Recall [6, VI.6.1] that a norm (or absolute value) on k is a function a ↦ → |a | from k to R�0 such that

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For a Noetherian scheme X, let Xreg denote the regular locus of X. The scheme X is said to admit a resolution of singularities if there exists a blowup X ′ → X with center disjoint from Xreg and regular X ′. More generally, for a closed subscheme Z ⊂ X, let (X,Z)reg denote the set of points x ∈ Xreg

Let us recall the classical definition of a p-adic zeta function, as it was given by Igusa [29]. A survey of the theory of p-adic zeta functions can be found in Denef’s Bourbaki report [19].

Abstract. We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on its étale cohomology. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal R-scheme X of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. Our trace formula yields a cohomological interpretation of this Weil generating series. When Xis the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f. When X is the formal completion of f at a closed point x of the special fiber f −1 (0), we obtain the local motivic zeta function of f at x. In the latter case, the generic fiber of X is the so-called analytic Milnor fiber of f at x; we show that it completely determines the formal germ of f at x. 1.

1.1. Motivation. This paper is largely concerned with constructing quotients by étale equivalence relations. We are inspired by questions in classical rigid geometry, but to give satisfactory answers in that category we have to first solve quotient problems within the framework of Berkovich’s k-analytic spaces. One source of motivation is the relationship between algebraic spaces and analytic spaces over C, as follows. If X

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Abstract. The purpose of these notes is to introduce the basic notions of rigid analytic geometry, with the aim of discussing the non-archimedean uniformizations of certain abelian varieties. 1.

Abstract. We continue the development of the theory of higher-genus Möbius periodicity that was studied in Part I for odd characteristic, now treating asymptotic questions and the case of characteristic 2. The extra difficulties in characteristic 2 are overcome via rigid geometry in characteristic 0. The results on Möbius periodicity in any positive characteristic are used to incorporate a correction factor into the false naive conjecture of Bateman–Horn type concerning how often a polynomial with a higher-genus coefficient ring takes prime values; numerical evidence is provided to support the suitability of this correction factor. We also prove some asymptotic and non-triviality properties of the correction factor. 1.