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We explain which Weierstrass
${\wp}$
-functions are locally definable from other
${\wp}$
-functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit functional transcendence theorems in a systematic way.
Standard results from non-abelian cohomology theory specialize to a theory of torsors and stacks for cosimplicial groupoids. The space of global sections of the stack completion of a cosimplicial groupoid
$G$
is weakly equivalent to the Bousfield–Kan total complex of
$BG$
for all cosimplicial groupoids
$G$
. The
$k$
-invariants for the Postnikov tower of a cosimplicial space
$X$
are naturally elements of stack cohomology for the stack associated to the fundamental groupoid
${\it\pi}(X)$
of
$X$
. Cocycle-theoretic ideas and techniques are used throughout the paper.
Soit
$G$
un groupe réductif connexe déployé sur une extension finie
$F$
de
$\mathbb{Q}_{p}$
. Nous déterminons les extensions entre séries principales continues unitaires
$p$
-adiques et lisses modulo
$p$
de
$G(F)$
dans le cas générique. Pour cela, nous calculons le delta-foncteur
$\text{H}^{\bullet }\text{Ord}_{B(F)}$
des parties ordinaires dérivées d’Emerton relatif à un sous-groupe de Borel sur certaines représentations induites de
$G(F)$
en utilisant une filtration de Bruhat. Ces extensions interviennent dans le programme de Langlands
$p$
-adique et modulo
$p$
.
We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand–Naimark duality we conclude that the assertion that the Stone–Čech remainder of the half-line has only trivial automorphisms is independent from ZFC (Zermelo-Fraenkel axiomatization of set theory with the Axiom of Choice). Consistency of this statement follows from the Proper Forcing Axiom, and this is the first known example of a connected space with this property.
We consider the distribution of
$p$
-power group schemes among the torsion of abelian varieties over finite fields of characteristic
$p$
, as follows. Fix natural numbers
$g$
and
$n$
, and let
${\it\xi}$
be a non-supersingular principally quasipolarized Barsotti–Tate group of level
$n$
. We classify the
$\mathbb{F}_{q}$
-rational forms
${\it\xi}^{{\it\alpha}}$
of
${\it\xi}$
. Among all principally polarized abelian varieties
$X/\mathbb{F}_{q}$
of dimension
$g$
with
$X[p^{n}]_{\bar{\mathbb{F}}_{q}}\cong {\it\xi}_{\bar{\mathbb{F}}_{q}}$
, we compute the frequency with which
$X[p^{n}]\cong {\it\xi}^{{\it\alpha}}$
. The error in our estimate is bounded by
$D/\sqrt{q}$
, where
$D$
depends on
$g$
,
$n$
, and
$p$
, but not on
${\it\xi}$
.
Given a separably closed field
$K$
of characteristic
$p>0$
and finite degree of imperfection, we study the
$\sharp$
functor which takes a semiabelian variety
$G$
over
$K$
to the maximal divisible subgroup of
$G(K)$
. Our main result is an example where
$G^{\sharp }$
, as a ‘type-definable group’ in
$K$
, does not have ‘relative Morley rank’, yielding a counterexample to a claim in Hrushovski [J. Amer. Math. Soc.9 (1996), 667–690]. Our methods involve studying the question of the preservation of exact sequences by the
$\sharp$
functor, and relating this to issues of descent as well as model-theoretic properties of
$G^{\sharp }$
. We mention some characteristic 0 analogues of these ‘exactness-descent’ results, where differential algebraic methods are more prominent. We also develop the notion of an iterative D-structure on a group scheme over an iterative Hasse field, which is interesting in its own right, as well as providing a uniform treatment of the characteristic 0 and characteristic
$p$
cases of ‘exactness descent’.
We provide asymptotics for the range
$R_{n}$
of a random walk on the
$d$
-dimensional lattice indexed by a random tree with
$n$
vertices. Using Kingman’s subadditive ergodic theorem, we prove under general assumptions that
$n^{-1}R_{n}$
converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension
$4$
, and in the case of a symmetric random walk with exponential moments, we prove that
$R_{n}$
grows like
$n/\!\log n$
. We apply our results to asymptotics for the range of a branching random walk when the initial size of the population tends to infinity.
We construct
$p$
-adic families of Klingen–Eisenstein series and
$L$
-functions for cusp forms (not necessarily ordinary) unramified at an odd prime
$p$
on definite unitary groups of signature
$(r,0)$
(for any positive integer
$r$
) for a quadratic imaginary field
${\mathcal{K}}$
split at
$p$
. When
$r=2$
, we show that the constant term of the Klingen–Eisenstein family is divisible by a certain
$p$
-adic
$L$
-function.
We consider étale motivic or Lichtenbaum cohomology and its relation to algebraic cycles. We give an geometric interpretation of Lichtenbaum cohomology and use it to show that the usual integral cycle maps extend to maps on integral Lichtenbaum cohomology. We also show that Lichtenbaum cohomology, in contrast to the usual motivic cohomology, compares well with integral cohomology theories. For example, we formulate integral étale versions of the Hodge and the Tate conjecture, and show that these are equivalent to the usual rational conjectures.
We study the propagation of wave packets for a one-dimensional system of two coupled Schrödinger equations with a cubic nonlinearity, in the semiclassical limit. Couplings are induced by the nonlinearity and by the potential, whose eigenvalues present an avoided crossing: at one given point, the gap between them reduces as the semiclassical parameter becomes smaller. For data which are coherent states polarized along an eigenvector of the potential, we prove that when the wave function propagates through the avoided crossing point there are transitions between the eigenspaces at leading order. We analyze the nonlinear effects, which are noticeable away from the crossing point, but see that in a small time interval around this point the nonlinearity’s role is negligible at leading order, and the transition probabilities can be computed with the linear Landau–Zener formula.
We continue the research programme of comparing the complex exponential with Zilberś exponential. For the latter, we prove, using diophantine geometry, various properties about zero sets of exponential functions, proved for
$\mathbb{C}$
using analytic function theory, for example, the Identity Theorem.
Let
${\mathcal{K}}$
be an imaginary quadratic field. Let
${\rm\Pi}$
and
${\rm\Pi}^{\prime }$
be irreducible generic cohomological automorphic representation of
$\text{GL}(n)/{\mathcal{K}}$
and
$\text{GL}(n-1)/{\mathcal{K}}$
, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, and the other is given in terms of the Whittaker model. The ratio between these rational structures is called a Whittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if
${\rm\Pi}$
is cuspidal and the weights of
${\rm\Pi}$
and
${\rm\Pi}^{\prime }$
are in a standard relative position, the critical values of the Rankin–Selberg product
$L(s,{\rm\Pi}\times {\rm\Pi}^{\prime })$
are essentially algebraic multiples of the product of the Whittaker periods of
${\rm\Pi}$
and
${\rm\Pi}^{\prime }$
. We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal
${\rm\Pi}$
can be given a motivic interpretation, and can also be related to a critical value of the adjoint
$L$
-function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin–Selberg and adjoint
$L$
-functions are compatible with Deligne’s conjecture.
It is well known that a finitely generated group
${\rm\Gamma}$
has Kazhdan’s property (T) if and only if the Laplacian element
${\rm\Delta}$
in
$\mathbb{R}[{\rm\Gamma}]$
has a spectral gap. In this paper, we prove that this phenomenon is witnessed in
$\mathbb{R}[{\rm\Gamma}]$
. Namely,
${\rm\Gamma}$
has property (T) if and only if there exist a constant
${\it\kappa}>0$
and a finite sequence
${\it\xi}_{1},\ldots ,{\it\xi}_{n}$
in
$\mathbb{R}[{\rm\Gamma}]$
such that
${\rm\Delta}^{2}-{\it\kappa}{\rm\Delta}=\sum _{i}{\it\xi}_{i}^{\ast }{\it\xi}_{i}$
. This result suggests the possibility of finding new examples of property (T) groups by solving equations in
$\mathbb{R}[{\rm\Gamma}]$
, possibly with the assistance of computers.
In order to develop the foundations of derived logarithmic geometry, we introduce a model category of logarithmic simplicial rings and a notion of derived log-étale maps, and use them to define derived log stacks.
For an infinite cardinal
${\it\kappa}$
, let
$\text{ded}\,{\it\kappa}$
denote the supremum of the number of Dedekind cuts in linear orders of size
${\it\kappa}$
. It is known that
${\it\kappa}<\text{ded}\,{\it\kappa}\leqslant 2^{{\it\kappa}}$
for all
${\it\kappa}$
and that
$\text{ded}\,{\it\kappa}<2^{{\it\kappa}}$
is consistent for any
${\it\kappa}$
of uncountable cofinality. We prove however that
$2^{{\it\kappa}}\leqslant \text{ded}(\text{ded}(\text{ded}(\text{ded}\,{\it\kappa})))$
always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as large as in the class of all countable theories.
In this paper, we prove that cyclic homology, topological cyclic homology, and algebraic
$K$
-theory satisfy a pro Mayer–Vietoris property with respect to abstract blow-up squares of varieties, in both zero and finite characteristic. This may be interpreted as the well-definedness of
$K$
-theory with compact support.
Suppose that
$G$
is a connected reductive algebraic group defined over
$\mathbf{R}$
,
$G(\mathbf{R})$
is its group of real points,
${\it\theta}$
is an automorphism of
$G$
, and
${\it\omega}$
is a quasicharacter of
$G(\mathbf{R})$
. Kottwitz and Shelstad defined endoscopic data associated to
$(G,{\it\theta},{\it\omega})$
, and conjectured a matching of orbital integrals between functions on
$G(\mathbf{R})$
and its endoscopic groups. This matching has been proved by Shelstad, and it yields a dual map on stable distributions. We express the values of this dual map on stable tempered characters as a linear combination of twisted characters, under some additional hypotheses on
$G$
and
${\it\theta}$
.
Cet article est une contribution à la fois au calcul du nombre de fibrés de Hitchin sur une courbe projective et à l’explicitation de la partie nilpotente de la formule des traces d’Arthur-Selberg pour une fonction test très simple. Le lien entre les deux questions a été établi dans [Chaudouard, Sur le comptage des fibrés de Hitchin. À paraître aux actes de la conférence en l’honneur de Gérard Laumon]. On décompose cette partie nilpotente en une somme d’intégrales adéliques indexées par les orbites nilpotentes. Pour les orbites de type «régulières par blocs», on explicite complètement ces intégrales en termes de la fonction zêta de la courbe.
We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set
$K$
is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects
$K$
at exactly one point.
We construct a cohomology theory using quasi-smooth derived schemes as generators and an analog of the bordism relation using derived fiber products as relations. This theory has pull-backs along all morphisms between smooth schemes independent of any characteristic assumptions. We prove that, in characteristic zero, the resulting theory agrees with algebraic cobordism as defined by Levine and Morel. We thus obtain a new set of generators and relations for algebraic cobordism.