Journal für die reine und angewandte Mathematik

Huyghe, Christine; Schmidt, Tobias

doi: 10.1515/crelle-2017-0021pmid: N/A

AbstractSoient p un nombre premier,V un anneau de valuation discrète complet d’inégales caractéristiques(0,p)(0,p), et G un groupe réductif et deployé sur Spec⁡V\operatorname{Spec}V.Nous obtenons un théorème de localisation, en utilisant les distributions arithmétiques, pour le faisceau des opérateurs différentiels arithmétiques sur la variété de drapeaux formelle de G.Nous donnons une application à la cohomologie rigide pour des ouverts dans la variété de drapeaux en caractéristique p.Let p be a prime number, V a complete discrete valuation ring of unequal characteristics (0,p)(0,p), and G a connected split reductive algebraic group over Spec⁡V\operatorname{Spec}V. We obtain a localizationtheorem, involving arithmetic distributions, for the sheaf of arithmetic differential operators on the formal flag variety of G.We give an application to the rigid cohomology of open subsets in the characteristic p flag variety.

doi: 10.1515/crelle-2019-frontmatter754pmid: N/A

Polterovich, Iosif; Sher, David A.; Toth, John A.

doi: 10.1515/crelle-2017-0018pmid: N/A

AbstractWe prove sharp upper and lower bounds for the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary. The argument involves frequency function methods for harmonic functions in the interior of the surface as well as the construction of exponentially accurate approximations for the Steklov eigenfunctions near the boundary.

Druţu, Cornelia; Nowak, Piotr W.

doi: 10.1515/crelle-2017-0002pmid: N/A

AbstractIn this paper we investigate generalizations of Kazhdan’s property (T) to the setting of uniformly convex Banach spaces. We explain the interplay between the existence of spectral gaps and that of Kazhdan projections.Our methods employ Markov operators associated to a random walk on the group, for which we provide new norm estimates and convergence results.This construction exhibits useful properties and flexibility, and allowsto view Kazhdan projections in Banach spaces as natural objects associated to random walks on groups.We give a number of applications of these results. In particular, we address several open questions.We give a direct comparison of properties (TE) and FE with Lafforgue’s reinforced Banach property (T);we obtain shrinking target theorems for orbits of Kazhdan groups; finally, answering a question of Willett and Yu we constructnon-compact ghost projections for warped cones. In this last case we conjecture that such warped cones providecounterexamples to the coarse Baum–Connes conjecture.

Gekeler, Ernst-Ulrich

doi: 10.1515/crelle-2017-0012pmid: N/A

AbstractWe construct Galois covers Xr,k⁢(N){X^{r,k}(N)}over ℙ1/𝔽q⁢(T){{\mathbb{P}}^{1}/{\mathbb{F}}_{q}(T)}with Galois groups close toGL⁢(r,𝔽q⁢[T]/(N)){{\rm GL}(r,{\mathbb{F}}_{q}[T]/(N))}(r≥3{r\geq 3}) and rationality and ramification properties similar to those ofclassical modular curves X⁢(N){X(N)}over ℙ1/ℚ{{\mathbb{P}}^{1}/{\mathbb{Q}}}. As application we find plenty of goodtowers (with lim sup⁡number of rational pointsgenus>0\limsup{\frac{\text{number~{}of~{}rational~{}points}}{{\rm genus}}>0}) of curves overthe field 𝔽qr{{\mathbb{F}}_{q^{r}}}with qr{q^{r}}elements.

Meinhardt, Sven; Reineke, Markus

doi: 10.1515/crelle-2017-0010pmid: N/A

AbstractThe main result of this paper is the statement that the Hodge theoretic Donaldson–Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson–Thomas “function” to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson–Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.

Brevig, Ole Fredrik; Perfekt, Karl-Mikael; Seip, Kristian

doi: 10.1515/crelle-2016-0069pmid: N/A

AbstractFor a Dirichlet series symbol g⁢(s)=∑n≥1bn⁢n-s{g(s)=\sum_{n\geq 1}b_{n}n^{-s}}, the associated Volterra operator 𝐓g{\mathbf{T}_{g}}acting on a Dirichlet series f⁢(s)=∑n≥1an⁢n-s{f(s)=\sum_{n\geq 1}a_{n}n^{-s}}is defined by the integralf↦-∫s+∞f⁢(w)⁢g′⁢(w)⁢𝑑w.{f\mapsto-\int_{s}^{+\infty}f(w)g^{\prime}(w)\,dw}.We show that 𝐓g{\mathbf{T}_{g}}is a bounded operator on the Hardy space ℋp{\mathcal{H}^{p}}of Dirichlet series with 0<p<∞{0<p<\infty}if and only if the symbol g satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of BMOA⁡(𝔻){{\operatorname{BMOA}}(\mathbb{D})}. A further analogy with classical BMO{{\operatorname{BMO}}}is that exp⁡(c⁢|g|){\exp(c|g|)}is integrable (on the infinite polytorus) for some c>0{c>0}whenever 𝐓g{\mathbf{T}_{g}}is bounded. In particular, such g belong to ℋp{\mathcal{H}^{p}}for every p<∞{p<\infty}. We relate the boundedness of 𝐓g{\mathbf{T}_{g}}to several other BMO{{\operatorname{BMO}}}-type spaces: BMOA{{\operatorname{BMOA}}}in half-planes, the dual of ℋ1{\mathcal{H}^{1}}, and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for m-homogeneous symbols as well as for general symbols. Finally, we consider the action of 𝐓g{\mathbf{T}_{g}}on reproducing kernels for appropriate sequences of subspaces of ℋ2{\mathcal{H}^{2}}. Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbols g.

Isenberg, James; Wu, Haotian

doi: 10.1515/crelle-2017-0019pmid: N/A

AbstractWe study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the “vanishing” time T: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter γ>12{\gamma>\frac{1}{2}}, there is a solution with the highest curvature blowing up at the rate (T-t)-(γ+12){(T-t)^{{-(\gamma+\frac{1}{2})}}}. (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the “Grim Reaper” solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ.

Deraux, Martin

doi: 10.1515/crelle-2017-0005pmid: N/A

AbstractWe give an algebro-geometric construction of some of thenon-arithmetic ball quotients constructed by the author, Parker andPaupert. The new construction reveals a relationshipbetween the corresponding orbifold fundamental groups and theautomorphism group of the Klein quartic, and also with groupsconstructed by Barthel–Hirzebruch–Höfer and Couwenberg–Heckman–Looijenga.

Bursztyn, Henrique; Lima, Hudson; Meinrenken, Eckhard

doi: 10.1515/crelle-2017-0014pmid: N/A

AbstractAccording to the Weinstein splitting theorem, any Poisson manifold is locally, near any given point, a product of a symplectic manifold with another Poisson manifold whose Poisson structure vanishes at the point. Similar splitting results are known, e.g., for Lie algebroids, Dirac structures and generalized complex structures. In this paper, we develop a novel approach towards these results that leads to various generalizations, including their equivariant versions as well as their formulations in new contexts.