We study analogues of Tate’s conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without non-trivial endomorphisms.

Given a 4-manifold, we build a non-empty ${C}^{1} $-open set of vector fields having a (chain transitive) attractor containing singularities of different indices. Then, we begin the study of the hyperbolic properties of such a robust singular attractor.

In a recent paper, Gross and Reeder study the arithmetic properties of discrete Langlands parameters for semi-simple -adic groups, and they conjecture that a special class of these – the simple wild parameters – should correspond to -packets consisting of simple supercuspidal representations. We provide a construction of this correspondence, and show that the simple wild -packets satisfy many expected properties. In particular, they admit a description in terms of the Langlands dual group, and contain a unique generic element for a fixed Whittaker datum. Moreover, we prove their stability on an open subset of the regular semi-simple elements, and show that they satisfy a natural compatibility with respect to unramified base-change.

We consider stacks of filtered $\varphi $-modules over rigid analytic spaces and adic spaces. We show that these modules parameterize $p$-adic Galois representations of the absolute Galois group of a $p$-adic field with varying coefficients over an open substack containing all classical points. Further, we study a period morphism (defined by Pappas and Rapoport) from a stack parameterizing integral data, and determine the image of this morphism.

We prove upper bounds for Hecke–Laplace eigenfunctions on certain Riemannian manifolds $X$ of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are $d$-fold products of $2$-spheres or $3$-spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a (‘Weyl-type’) saving of $\mathrm{vol} \hspace{0.167em} (X)^{- 1/ 6+ \varepsilon } $.

In a recent paper Moshe Jarden (Diamonds in torsion of Abelian varieties, J. Inst. Math. Jussieu9(3) (2010), 477–480) proposed a conjecture, later named the Kuykian conjecture, which states that if $A$ is an Abelian variety defined over a Hilbertian field $K$, then every intermediate field of $K({A}_{\mathrm{tor} } )/ K$ is Hilbertian. We prove that the conjecture holds for Galois extensions of $K$ in $K({A}_{\mathrm{tor} } )$.

We study the arithmetic of abelian varieties over where is an arbitrary field. The main result relates Mordell–Weil groups of certain Jacobians over to homomorphisms of other Jacobians over . Our methods also yield completely explicit points on elliptic curves with unbounded rank over and a new construction of elliptic curves with moderately high rank over .

In this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant $K$-theory with respect to a compact torus $G$ of various spaces associated to a linear action of $G$ in a vector space $M$ can both be described using some vector spaces of distributions, on the dual of the group $G$ or on the dual of its Lie algebra $\mathfrak{g}$. The morphism from $K$-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a $G$-transversally elliptic operator on $M$ are determined using the infinitesimal index of the symbol.

This paper concerns arithmetic families of $\varphi $-modules over reduced affinoid spaces. For such a family, we first prove that the slope polygons are lower semicontinuous around any rigid point. We further prove that if the slope polygons are locally constant around a rigid point, then around this point, the family has a global slope filtration after base change to some extended Robba ring.

Nous démontrons, sous la forme forte conjecturée par Peyre, la conjecture de Manin pour les surfaces de Châtelet dont les équations sont du type ${y}^{2} + {z}^{2} = P(x, 1)$, où $P$ est une forme binaire quartique à coefficients entiers irréductible sur $ \mathbb{Q} [i] $ ou produit de deux formes quadratiques à coefficients entiers irréductibles sur $ \mathbb{Q} [i] $. De plus, nous fournissons une estimation explicite du terme d’erreur de la formule asymptotique sous-jacente. Cela finalise essentiellement la validation de la conjecture de Manin pour l’ensemble des surfaces de Châtelet. La preuve s’appuie sur deux méthodes nouvelles, concernant, du part, les estimations en moyenne d’oscillations locales de caractères sur les diviseurs, et, d’autre part, les majorations de certaines fonctions arithmétiques de formes binaires.

Let be a field complete with respect to a discrete valuation whose residue field is perfect of characteristic . We prove that every smooth, projective, geometrically irreducible curve of genus one defined over with a non-zero divisor of degree a power of has a solvable point over .

We prove that the Brauer–Manin obstruction is the only obstruction to the existence of integral points on affine varieties over global fields of positive characteristic $p$. More precisely, we show that the only obstructions come from étale covers of exponent $p$ or, alternatively, from flat covers coming from torsors under connected group schemes of exponent $p$.

We prove a new global stability estimate for the Gel’fand–Calderón inverse problem on a two-dimensional bounded domain. Specifically, the inverse boundary value problem for the equation ${- }\Delta \psi + v\hspace{0.167em} \psi = 0$ on $D$ is analysed, where $v$ is a smooth real-valued potential of conductivity type defined on a bounded planar domain $D$. The main feature of this estimate is that it shows that the smoother a potential is, the more stable its reconstruction is. Furthermore, the stability is proven to depend exponentially on the smoothness, in a sense to be made precise. The same techniques yield a similar estimate for the Calderón problem for electrical impedance tomography.

In this note, we study an invariant associated with the zeros of the moment map generated by an action form, the infinitesimal index. This construction will be used to study the compactly supported equivariant cohomology of the zeros of the moment map and to give formulas for the multiplicity index map of a transversally elliptic operator.

Nous montrons des résultats d’existence de points fixes communs pour des homéomorphismes du plan ${ \mathbb{R} }^{2} $ ou la sphère ${ \mathbb{S} }^{2} $, qui commutent deux à deux et préservent une mesure de probabilité. Par exemple, nous montrons que des ${C}^{1} $-difféomorphismes ${f}_{1} , \ldots , {f}_{n} $ de ${ \mathbb{S} }^{2} $ suffisamment proches de l’identité, qui commutent deux à deux, et qui préservent une mesure de probabilité dont le support n’est pas réduit à un point, ont au moins deux points fixes communs.

In our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.

In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.

As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is ${H}^{s} $ with $s\gt \frac{5}{2} $. Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the lifespan ${T}_{\varepsilon } $ of the solutions is bounded below by $\log \log \log \frac{1}{\varepsilon } $, where $\varepsilon $ denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Euler system when the parameter $\varepsilon $ goes to zero. In the second part of the paper we address the same problem but for the Besov critical regularity ${ B}_{2, 1}^{\frac{5}{2} } $. This case turns out to be more subtle because of at least two features. The first one is related to the Beale–Kato–Majda criterion which is not known to be valid for rough regularities. The second one concerns the critical aspect of the Strichartz estimate ${ L}_{T}^{1} {L}^{\infty } $ for the acoustic parts $(\nabla {\Delta }^{- 1} \mathrm{div} \hspace{0.167em} {v}_{\varepsilon } , {c}_{\varepsilon } )$: it scales in the space variables like the space of the initial data.

We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general $K 3$ surface $X$ of degree $2$ over $ \mathbb{Q} $, together with a 2-torsion Brauer class $\alpha $ that is unramified at every finite prime, but ramifies at real points of $X$. With motivation from Hodge theory, the pair $(X, \alpha )$ is constructed from a double cover of ${ \mathbb{P} }^{2} \times { \mathbb{P} }^{2} $ ramified over a hypersurface of bidegree $(2, 2)$.

Linear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of . In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.