Showing papers in "Annales de l'Institut Fourier in 2019"
TL;DR: In this paper, the authors analyzed the long-time asymptotics for the Degasperis-procesi equation on the half-line by applying nonlinear steepest descent techniques to an associated 3 x 3-matrix valued Riemann-Hilbert pr
Abstract: We analyze the long-time asymptotics for the Degasperis- Procesi equation on the half-line. By applying nonlinear steepest descent techniques to an associated 3 x 3-matrix valued Riemann-Hilbert pr ...
36 citations
TL;DR: In this article, the authors studied geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms and showed that almost all these shifts have pure discrete spectrum, which proves a conjecture of Arnoux and Rauzy going back to the early nineties in a metric sense.
Abstract: This paper studies geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iterating infinitely many substitutions in an adic way. Pure discrete spectrum for $S$-adic shifts and tiling properties of associated Rauzy fractals are established under a generalized Pisot assumption together with a geometric coincidence condition. These general results extend the scope of the Pisot substitution conjecture to the $S$-adic framework. They are applied to families of $S$-adic shifts generated by Arnoux-Rauzy as well as Brun substitutions (related to the respective continued fraction algorithms). It is shown that almost all of these shifts have pure discrete spectrum, which proves a conjecture of Arnoux and Rauzy going back to the early nineties in a metric sense. We also prove that each linearly recurrent Arnoux-Rauzy shift with recurrent directive sequence has pure discrete spectrum. Using $S$-adic words related to Brun's continued fraction algorithm, we exhibit bounded remainder sets and natural codings for almost all translations on the two-dimensional torus.
Due to the lack of a dominant eigenvector and the fact that we lose the self-similarity properties present for substitutive systems we cannot follow the known arguments from the substitutive case and have to develop new proofs to obtain our results in the $S$-adic setting.
24 citations
TL;DR: This work estimates Gowers uniformity norms for some classical automatic sequences, such as the Thue-Morse and Rudin-Shapiro sequences, and asymptotically counts arithmetic progressions in the set of integers.
Abstract: We estimate Gowers uniformity norms for some classical automatic sequences, such as the Thue-Morse and Rudin-Shapiro sequences. The methods can also be extended to other automatic sequences. As an application, we asymptotically count arithmetic progressions in the set of integers $\leq N$ where the Thue-Morse (resp. Rudin-Shapiro) sequence takes the value $+1$.
23 citations
TL;DR: In this paper, the authors extend many results on Selmer groups for elliptic curves and modular forms to the non-ordinary setting and show that if two modular forms are congruent to each other, then the Iwasawa invariants of their signed Selmer group are related in an explicit manner.
Abstract: We extend many results on Selmer groups for elliptic curves and modular forms to the non-ordinary setting. More precisely, we study the signed Selmer groups defined using the machinery of Wach modules over $\mathbf{Z}_p$-cyclotomic extensions. First, we provide a definition of residual and non-primitive Selmer groups at non-ordinary primes. This allows us to extend techniques developed by Greenberg (for $p$-ordinary elliptic curves) and Kim ($p$-supersingular elliptic curves) to show that if two $p$-non-ordinary modular forms are congruent to each other, then the Iwasawa invariants of their signed Selmer groups are related in an explicit manner. Our results have several applications. First of all, this allows us to relate the parity of the analytic ranks of such modular forms generalizing a recent result of the first-named author for $p$-supersingular elliptic curves. Second, we can prove a Kida-type formula for the signed Selmer groups generalizing results of Pollack and Weston.
23 citations
TL;DR: In this paper, the de Rham cohomology of Oeljeklaus-Toma (OT) manifolds is computed in terms of invariants associated to the background number field.
Abstract: Oeljeklaus-Toma (OT) manifolds are complex non-Kahler manifolds whose construction arises from specific number fields. In this note, we compute their de Rham cohomology in terms of invariants associated to the background number field. This is done by two distinct approaches, one using invariant cohomology and the other one using the Leray-Serre spectral sequence. In addition, we compute also their Morse-Novikov cohomology. As an application, we show that the low degree Chern classes of any complex vector bundle on an OT manifold vanish in the real cohomology. Other applications concern the OT manifolds admitting locally conformally Kahler (LCK) metrics: we show that there is only one possible Lee class of an LCK metric, and we determine all the possible Morse-Novikov classes of an LCK metric, which implies the nondegeneracy of certain Lefschetz maps in cohomology.
19 citations
TL;DR: In this article, a sheaf-theoretic formulation of Shimura's theory of nearly holomorphic Siegel modular forms and differential operators is introduced, which is used to define and study nearly overconvergent Siegel forms and their p-adic families.
Abstract: We introduce a sheaf-theoretic formulation of Shimura’s theory of nearly holomorphic Siegel modular forms and differential operators. We use it to define and study nearly overconvergent Siegel modular forms and their p-adic families.
18 citations
TL;DR: In this article, counterexamples to uniqueness for the Calderon problem on Riemannian manifolds with boundary when the Dirichlet and Neumann data are measured on disjoint sets of the boundary are given.
Abstract: In this paper, we give some simple counterexamples to uniqueness for the Calderon problem on Riemannian manifolds with boundary when the Dirichlet and Neumann data are measured on disjoint sets of the boundary. We provide counterexamples in the case of two and three dimensional Riemannian manifolds with boundary having the topology of circular cylinders in dimension two and toric cylinders in dimension three. The construction could be easily extended to higher dimensional Riemannian manifolds.
17 citations
TL;DR: The theory of balanced co-polarised deformations of compact complex n-fold X satisfying the ∂∂ ¯-lemma was introduced in this paper, which is a natural extension of the classical theory of Kahler polarized deformations in the context of Calabi-Yau or holomorphic compact complex manifolds.
Abstract: Given a compact complex n-fold X satisfying the ∂∂ ¯-lemma and supposed to have a trivial canonical bundle K X and to admit a balanced (=semi-Kahler) Hermitian metric ω, we introduce the concept of deformations of X that are co-polarised by the balanced class [ω n-1 ]∈H n-1,n-1 (X,ℂ)⊂H 2n-2 (X,ℂ) and show that the resulting theory of balanced co-polarised deformations is a natural extension of the classical theory of Kahler polarised deformations in the context of Calabi–Yau or holomorphic symplectic compact complex manifolds. The concept of Weil–Petersson metric still makes sense in this strictly more general, possibly non-Kahler context, while the Local Torelli Theorem still holds.
17 citations
TL;DR: In this article, it was shown that Courant's nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in the case of an open, bounded and connected set in R n with a C 1,1 boundary.
Abstract: In this paper, we show that equality in Courant's nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in the case of an open, bounded and connected set in R n with a C 1,1 boundary. This result is analogous to Pleijel's nodal domain theorem for the Dirichlet Laplacian (1956). It confirms, in all dimensions, a conjecture formulated by Pleijel, which had already been solved by I. Polterovich for a two-dimensional domain with a piecewise-analytic boundary (2009). We also show that the argument and the result extend to a class of Robin boundary conditions.
17 citations
TL;DR: In this article, it was shown that the semipositive envelope is a continuous semipoSitive metric on L-an and that the non-archimedean Monge-Ampere equation has a solution.
Abstract: Let L be an ample line bundle on a smooth projective variety X over a non-archimedean field K. For a continuous metric on L-an, we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L-an and that the non-archimedean Monge-Ampere equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that X is a surface defined geometrically over the function field of a curve over a perfect field k of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over k. The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps.
16 citations
TL;DR: In this article, the authors introduce horizontal holonomy groups, which are groups defined using parallel transport along curves tangent to a given subbundle of the tangent bundle of a foliation.
Abstract: We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle $D$ of the tangent bundle. We provide explicit means of computing these holonomy groups by deriving analogues of Ambrose-Singer's and Ozeki's theorems. We then give necessary and sufficient conditions in terms of the horizontal holonomy groups for existence of solutions of two problems on foliated manifolds: determining when a foliation can be either (a) totally geodesic or (b) endowed with a principal bundle structure.
The subbundle $D$ plays the role of an orthogonal complement to the leaves of the foliation in case (a) and of a principal connection in case (b).
TL;DR: In this article, it was shown that for every finite extension of a Galois group, there exists a rational prime ideal that satisfies the Lagarias, Montgomery, and Odlyzko result.
Abstract: Lagarias, Montgomery, and Odlyzko proved that there exists an effectively computable absolute constant $A_1$ such that for every finite extension $K$ of ${\mathbb{Q}}$, every finite Galois extension $L$ of $K$ with Galois group $G$ and every conjugacy class $C$ of $G$, there exists a prime ideal $\mathfrak{p}$ of $K$ which is unramified in $L$, for which $\left[\frac{L/K}{\mathfrak{p}}\right]=C$, for which $N_{K/{\mathbb Q}}\,\mathfrak{p}$ is a rational prime, and which satisfies $N_{K/{\mathbb Q}}\,{\mathfrak{p}} \leq 2 {d_L}^{A_1}$. In this paper we show without any restriction that $N_{K/{\mathbb Q}}\,{\mathfrak{p}} \leq {d_L}^{12577}$ if $L
eq {\mathbb Q}$, using the approach developed by Lagarias, Montgomery, and Odlyzko.
TL;DR: The Long-Moody functor as mentioned in this paper gives an endofunctor between the category of functors from the homogeneous category associated with the braid groupoid to a module category.
Abstract: In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of Bn with a representation of Bn+1. In this paper, we prove that this construction is functorial: it gives an endofunctor, called the Long-Moody functor, between the category of functors from the homogeneous category associated with the braid groupoid to a module category. Then we study the effect of the Long-Moody functor on strong polynomial functors: we prove that it increases by one the degree of strong polynomiality.
TL;DR: In this paper, positive loops of Legendrian embeddings are examined from the point of view of the Floer homology of Lagrangian cobordisms, which leads to new obstructions to the existence of a positive loop containing...
Abstract: Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing ...
TL;DR: In this paper, the authors extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular L-functions at the central point s = 1/2 and for prime moduli q.
Abstract: We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular L-functions at the central point s = 1/2 and for prime moduli q. As an application, we exploit our recent result on the mollification of the fourth moment of Dirichlet L-functions to derive that for any pair $(\omega_1,\omega_2)$ of multiplicative characters modulo q, there is a positive proportion of $\chi$ (mod q) such that $L(\chi, 1/2 ), L(\chi\omega_1, 1/2 )$ and $L(\chi\omega_2, 1/2)$ are simultaneously not too small.
TL;DR: The seminal work of Eskin-Masur-Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections as mentioned in this paper.
Abstract: The seminal work of Eskin-Masur-Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections. In this paper we describe the principal boundary for each configuration in terms of twisted differentials over Deligne-Mumford pointed stable curves. We also describe similarly the principal boundary of moduli spaces of quadratic differentials originally studied by Masur-Zorich. Our main technique is the flat geometric degeneration and smoothing developed by Bainbridge-Chen-Gendron-Grushevsky-Moller.
TL;DR: In this paper, a conformal scattering theory for Maxwell fields in the static exterior region of a Reissner-Nordstrom-de Sitter black hole spacetime is presented.
Abstract: We construct a complete conformal scattering theory for Maxwell fields in the static exterior region of a Reissner–Nordstrom–de Sitter black hole spacetime. We use uniform energy decay results, which we obtain in a separate paper, to show that the trace operators are injective and have closed ranges. We then solve the Goursat problem (characteristic Cauchy problem) for Maxwell fields on the null boundaries showing that the trace operators are also surjective.
TL;DR: In this article, the systole function along Weil-Petersson geodesics has been studied in the context of the growth of moduli space of Riemann surfaces.
Abstract: In this paper we study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichmuller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of moduli space of Riemann surfaces of genus $g$ with $n$ punctures as a function of $g$ and $n$. We show that the Weil-Petersson inradius is comparable to $\sqrt{\ln{g}}$ with respect to $g$, and is comparable to $1$ with respect to $n$.
Moreover, we also study the asymptotic behavior, as $g$ goes to infinity, of the Weil-Petersson volumes of geodesic balls of finite radii in Teichmuller space. We show that they behave like $o((\frac{1}{g})^{(3-\epsilon)g})$ as $g\to \infty$, where $\epsilon>0$ is arbitrary.
TL;DR: In this paper, it was shown that bordered Heegaard Floer homology detects incompressible surfaces and bordered-sutured Floer invariance detects partly boundary parallel tangles and bridges, in natural ways.
Abstract: We show that bordered Heegaard Floer homology detects incompressible surfaces and bordered-sutured Floer homology detects partly boundary parallel tangles and bridges, in natural ways. For example, there is a bimodule Lambda so that the tensor product of CFD(Y) and Lambda is Hom-orthogonal to CFD(Y) if and only if the boundary of Y admits an essential compressing disk. In the process, we sharpen a nonvanishing result of Ni's. We also extend Lipshitz-Ozsvath-Thurston's "factoring" algorithm for computing HF-hat to compute bordered-sutured Floer homology, to make both results on detecting incompressibility practical. In particular, this makes Zarev's tangle invariant manifestly combinatorial.
TL;DR: In this paper, a generalization of the pentagram map to a map on twisted polygons in the Grassmannian space Gr(n,mn) was proposed. But the invariants of Grassmannians under the natural action of SL(nm) were not defined.
Abstract: In this paper we define a generalization of the pentagram map to a map on twisted polygons in the Grassmannian space Gr(n,mn). We define invariants of Grassmannian twisted polygons under the natural action of SL(nm), invariants that define coordinates in the moduli space of twisted polygons. We then prove that when written in terms of the moduli space coordinates, the pentagram map is preserved by a certain scaling. The scaling is then used to construct a Lax representation for the map that can be used for integration.
TL;DR: In this article, the essential characteristic set and equivariant Euclidean embeddings were used to construct new quasimorphisms on many groups acting on CAT(0) cube complexes.
Abstract: We construct new families of quasimorphisms on many groups acting on CAT(0) cube complexes. These quasimorphisms have a uniformly bounded defect of 12, and they “see” all elements that act hyperbolically on the cube complex. We deduce that all such elements have stable commutator length at least 1/24. The group actions for which these results apply include the standard actions of right-angled Artin groups on their associated CAT(0) cube complexes. In particular, every non-trivial element of a right-angled Artin group has stable commutator length at least 1/24. These results make use of some new tools that we develop for the study of group actions on CAT(0) cube complexes: the essential characteristic set and equivariant Euclidean embeddings.
TL;DR: In this article, the authors classify elements of a cluster modular group into three types, and characterize them in terms of fixed point property of the action on the tropical compactifications associated with the corresponding cluster ensemble.
Abstract: We classify elements of a cluster modular group into three types. We characterize them in terms of fixed point property of the action on the tropical compactifications associated with the corresponding cluster ensemble. The characterization gives an analogue of the Nielsen-Thurston classification theory on the mapping class group of a surface.
TL;DR: For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties as discussed by the authors, and the David-Semmes uniformization theorem states that if a compact metric space satisfies all the three properties, then it is quasi symmetric equivalent to the middle-third Cantor set.
Abstract: For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties. The David-Semmes uniformization theorem states that if a compact metric space satisfies all the three properties, then it is quasi-symmetrically equivalent to the middle-third Cantor set. We say that a Cantor metric space is standard if it satisfies all the three properties; otherwise, it is exotic. In this paper, we conclude that for each of exotic types the class of all the conformal gauges of Cantor metric spaces has continuum cardinality. As a byproduct of our study, we state that there exists a Cantor metric space with prescribed Hausdorff dimension and Assouad dimension.
TL;DR: In this article, the free product construction of infinitely many identical half-sided modular inclusions with ergodic canonical endomorphism and trivial relative commutant was applied to various local algebras in algebraic quantum field theory.
Abstract: We apply the free product construction to various local algebras in algebraic quantum field theory.
If we take the free product of infinitely many identical half-sided modular inclusions with ergodic canonical endomorphism, we obtain a half-sided modular inclusion with ergodic canonical endomorphism and trivial relative commutant. On the other hand, if we take Mobius covariant nets with trace class property, we are able to construct an inclusion of free product von Neumann algebras with large relative commutant, by considering either a finite family of identical inclusions or an infinite family of inequivalent inclusions. In two dimensional spacetime, we construct Borchers triples with trivial relative commutant by taking free products of infinitely many, identical Borchers triples. Free products of finitely many Borchers triples are possibly associated with Haag-Kastler net having S-matrix which is nontrivial and non asymptotically complete, yet the nontriviality of double cone algebras remains open.
TL;DR: In this article, the authors proved Gaussian estimates for the heat kernel of Schrodinger operators whose potential V is "small at infinity" in an integral sense, and showed sharp boundedness result for the associated Riesz trans-form with potential d(�+V) −1/2.
Abstract: The goal of this article is two-fold: in a first part, we prove Gaussian estimates for the heat kernel of Schrodinger operators � + V whose potential V is "small at infinity" in an integral sense. In a second part, we prove sharp boundedness result for the associated Riesz trans- form with potential d(�+V) −1/2 . A characterization of p-hyperbolicity, which is of independent interest, is also proved. 2010 MSC. Primary 35K; Secondary 31E, 58J. Keywords. Heat kernel, Schrodinger operators, Riesz transform, p- hyperbolicity.
TL;DR: In this article, the authors studied the relationship between the growth of eigenfunctions and their concentration as measured by defect measures, and showed that the defect measures must be neither more concentrated nor more diffuse than the zonal harmonics.
Abstract: We study the relationship between $L^\infty$ growth of eigenfunctions and their $L^2$ concentration as measured by defect measures. In particular, we characterize the defect measures of any sequence of eigenfunctions with maximal $L^\infty$ growth, showing that they must be neither more concentrated nor more diffuse than the zonal harmonics. As a consequence, we obtain new proofs of results on the geometry manifolds with maximal eigenfunction growth obtained by Sogge--Zelditch, and Sogge--Toth--Zelditch.
TL;DR: In this article, the authors revisited Fourier analysis on the Heisenberg group H^d and established an explicit extension of the Fourier transform for smooth functions on H * that are independent of the vertical variable.
Abstract: We here revisit Fourier analysis on the Heisenberg group H^d. Whereas, according to the standard definition, the Fourier transform of an integrable function f on H^d is a one parameter family of bounded operators on L 2 (R^d), we define (by taking advantage of basic properties of Hermite functions) the Fourier transform f_H of f to be a uniformly continuous mapping on the set N^d × N^d ×R \ {0} endowed with a suitable distance. This enables us to extend f_H to the completion of that space, and to get an explicit asymptotic description of the Fourier transform when the 'vertical' frequency tends to 0. We expect our approach to be relevant for adapting to the Heisenberg framework a number of classical results for the Euclidean case that are based on Fourier analysis. As an example, we here establish an explicit extension of the Fourier transform for smooth functions on H^d that are independent of the vertical variable.
TL;DR: In this article, the authors study the variation of a smooth volume form along extremals of a variational problem with nonholonomic constraints and an action-like Lagrangian and introduce a new invariant describing the interaction of the volume with the dynamics.
Abstract: We study the variation of a smooth volume form along extremals of a variational problem with nonholonomic constraints and an action-like Lagrangian. We introduce a new invariant describing the interaction of the volume with the dynamics and we study its basic properties. We then show how this invariant, together with curvature-like invariants of the dynamics, appear in the expansion of the volume at regular points of the exponential map. This generalizes the well-known expansion of the Riemannian volume in terms of Ricci curvature to a wide class of geometric structures, including all sub-Riemannian manifolds.