15 research outputs found

### Geometric Bogomolov conjecture for abelian varieties and some results for those with some degeneration (with an appendix by Walter Gubler: The minimal dimension of a canonical measure)

In this paper, we formulate the geometric Bogomolov conjecture for abelian
varieties, and give some partial answers to it. In fact, we insist in a main
theorem that under some degeneracy condition, a closed subvariety of an abelian
variety does not have a dense subset of small points if it is a non-special
subvariety. The key of the proof is the study of the minimal dimension of the
components of a canonical measure on the tropicalization of the closed
subvariety. Then we can apply the tropical version of equidistribution theory
due to Gubler. This article includes an appendix by Walter Gubler. He shows
that the minimal dimension of the components of a canonical measure is equal to
the dimension of the abelian part of the subvariety. We can apply this result
to make a further contribution to the geometric Bogomolov conjecture.Comment: 30 page

### Theorie de Lubin-Tate non-abelienne et representations elliptiques

Harris and Taylor proved that the supercuspidal part of the cohomology of the
Lubin-Tate tower realizes both the local Langlands and Jacquet-Langlands
correspondences, as conjectured by Carayol. Recently, Boyer computed the
remaining part of the cohomology and exhibited two defects : first, the
representations of GL\_d which appear are of a very particular and restrictive
form ; second, the Langlands correspondence is not realized anymore. In this
paper, we study the cohomology complex in a suitable equivariant derived
category, and show how it encodes Langlands correspondance for all elliptic
representations. Then we transfer this result to the Drinfeld tower via an
enhancement of a theorem of Faltings due to Fargues. We deduce that Deligne's
weight-monodromy conjecture is true for varieties uniformized by Drinfeld's
coverings of his symmetric spaces.Comment: 54 page

### Espaces de Berkovich sur Z : \'etude locale

We investigate the local properties of Berkovich spaces over Z. Using
Weierstrass theorems, we prove that the local rings of those spaces are
noetherian, regular in the case of affine spaces and excellent. We also show
that the structure sheaf is coherent. Our methods work over other base rings
(valued fields, discrete valuation rings, rings of integers of number fields,
etc.) and provide a unified treatment of complex and p-adic spaces.Comment: v3: Corrected a few mistakes. Corrected the proof of the Weierstrass
division theorem 7.3 in the case where the base field is imperfect and
trivially value

### Degenerating families of dendrograms

Dendrograms used in data analysis are ultrametric spaces, hence objects of
nonarchimedean geometry. It is known that there exist $p$-adic representation
of dendrograms. Completed by a point at infinity, they can be viewed as
subtrees of the Bruhat-Tits tree associated to the $p$-adic projective line.
The implications are that certain moduli spaces known in algebraic geometry are
$p$-adic parameter spaces of (families of) dendrograms, and stochastic
classification can also be handled within this framework. At the end, we
calculate the topology of the hidden part of a dendrogram.Comment: 13 pages, 8 figure

### Motivic Serre invariants, ramification, and the analytic Milnor fiber

We show how formal and rigid geometry can be used in the theory of complex
singularities, and in particular in the study of the Milnor fibration and the
motivic zeta function. We introduce the so-called analytic Milnor fiber
associated to the germ of a morphism f from a smooth complex algebraic variety
X to the affine line. This analytic Milnor fiber is a smooth rigid variety over
the field of Laurent series C((t)). Its etale cohomology coincides with the
singular cohomology of the classical topological Milnor fiber of f; the
monodromy transformation is given by the Galois action. Moreover, the points on
the analytic Milnor fiber are closely related to the motivic zeta function of
f, and the arc space of X.
We show how the motivic zeta function can be recovered as some kind of Weil
zeta function of the formal completion of X along the special fiber of f, and
we establish a corresponding Grothendieck trace formula, which relates, in
particular, the rational points on the analytic Milnor fiber over finite
extensions of C((t)), to the Galois action on its etale cohomology.
The general observation is that the arithmetic properties of the analytic
Milnor fiber reflect the structure of the singularity of the germ f.Comment: Some minor errors corrected. The original publication is available at
http://www.springerlink.co

### On a Conjecture of Rapoport and Zink

In their book Rapoport and Zink constructed rigid analytic period spaces
$F^{wa}$ for Fontaine's filtered isocrystals, and period morphisms from PEL
moduli spaces of $p$-divisible groups to some of these period spaces. They
conjectured the existence of an \'etale bijective morphism $F^a \to F^{wa}$ of
rigid analytic spaces and of a universal local system of $Q_p$-vector spaces on
$F^a$. For Hodge-Tate weights $n-1$ and $n$ we construct in this article an
intrinsic Berkovich open subspace $F^0$ of $F^{wa}$ and the universal local
system on $F^0$. We conjecture that the rigid-analytic space associated with
$F^0$ is the maximal possible $F^a$, and that $F^0$ is connected. We give
evidence for these conjectures and we show that for those period spaces
possessing PEL period morphisms, $F^0$ equals the image of the period morphism.
Then our local system is the rational Tate module of the universal
$p$-divisible group and enjoys additional functoriality properties. We show
that only in exceptional cases $F^0$ equals all of $F^{wa}$ and when the
Shimura group is $GL_n$ we determine all these cases.Comment: v2: 48 pages; many new results added, v3: final version that will
appear in Inventiones Mathematica

### Big Line Bundles over Arithmetic Varieties

We prove a Hilbert-Samuel type result of arithmetic big line bundles in
Arakelov geometry, which is an analogue of a classical theorem of Siu. An
application of this result gives equidistribution of small points over
algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also
generalize Chambert-Loir's non-archimedean equidistribution