This paper is a continuation of our previous work (D. Jiang and D. Soudry, On the genericity of cuspidal automorphic forms on${\rm SO}_{2n+1}$, J. reine angew. Math., to appear). We extend Moeglin's results (C. Moeglin, J. Lie Theory 7 (1997), 201–229, 231–238) from the even orthogonal groups to old orthogonal groups and complete our proof of the CAP conjecture for irreducible cuspidal automorphic representations of $\mathrm{SO}_{2n+1}(\mathbb{A})$ with special Bessel models. We also give a characterization of the vanishing of the central value of the standard $L$-function of $\mathrm{SO}_{2n+1}(\mathbb{A})$ in terms of theta correspondence. As a result, we obtain the weak Langlands functorial transfer from $\mathrm{SO}_{2n+1}(\mathbb{A})$ to $\mathrm{GL}_{2n}(\mathbb{A})$ for irreducible cuspidal automorphic representations of $\mathrm{SO}_{2n+1}(\mathbb{A})$ with special Bessel models.

This paper addresses several questions related to the Hodge conjecture. First of all we consider the question, asked by Maillot and Soulé, whether the Hodge conjecture can be reduced to the case of varieties defined over number fields. We show that this is the case for the Hodge classes whose corresponding Hodge locus is defined over a number field. We also give simple criteria for this last condition to be satisfied. Finally we discuss the relation between this condition and the notion of absolute Hodge class.

We introduce a homotopy 2-category structure on the collection of dg-categories, dg-functors, and their derived transformations. This construction provides for a conceptual proof of Deligne's conjecture on Hochschild cochains.

Let k be a complete, non-Archimedean valued field (the trivial absolute value is allowed) and let φ:X→Y be a morphism between two Berkovich k-analytic spaces; we show that, for any integer n, the set of points of X at which the local dimension of φ is at least equal to n is a Zariski-closed subset of X. In order to establish it, we first prove an analytic analogue of Zariski’s Main Theorem, and we also introduce, and study, the notion of an analytic system of parameters at a point.

Applying the main results of the author's previous work (J. Wildeshaus, The boundary motive: definition and basic properties, Compositio Math. 142 (2006), 631–656), we identify the motives occurring as cones in the co-localization and localization filtrations of the boundary motive of a Shimura variety.

Nichols algebras naturally appear in the theory of quantized enveloping algebras of Kac–Moody algebras and in the classification of finite-dimensional pointed Hopf algebras. Assuming that the base field has characteristic zero a list of finite-dimensional rank 2 Nichols algebras of diagonal type is given. Each of them is described in terms of generators and relations. Kharchenko's restricted Poincaré–Birkhoff–Witt basis is visualized using full binary trees.

We describe an equivalence between the notion of balanced twisted curve introduced by Abramovich and Vistoli, and a new notion of log twisted curve, which is a nodal curve equipped with some logarithmic data in the sense of Fontaine and Illusie. As applications of this equivalence, we construct a universal balanced twisted curve, prove that a balanced twisted curve over a general base scheme admits étale locally on the base a finite flat cover by a scheme, and also give a new construction of the moduli space of stable maps into a Deligne–Mumford stack and a new proof that it is bounded.

In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras ${\mathbb F} = {\mathbb R}, {\mathbb C}$ or ${\mathbb H}$ which interpolate the convolution algebras of radial bounded Borel measures on a matrix space $M_{p,q}({\mathbb F})$ with $p\geq q$. Radiality in this context means invariance under the action of the unitary group $U_p({\mathbb F})$ from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize well-known structures in the rank-one case, namely the Bessel–Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper we study structures depending only on the matrix spectra. Under the mapping $r\mapsto \text{spec}(r)$, the convolutions on the underlying matrix cone induce a continuous series of hypergroup convolutions on a Weyl chamber of type $B_q$. The characters are now Dunkl-type Bessel functions. These convolution algebras on the Weyl chamber naturally extend the harmonic analysis for Cartan motion groups associated with the Grassmann manifolds $U(p,q)/(U_p\times U_q)$ over ${\mathbb F}$.

We prove the Kneser-Tits conjecture for groups of index $^2E^{29}_{6,1}$ using an argument inspired by a 1968 paper by Veldkamp. We also prove that these groups are stably rational varieties.

The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it has also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX relation). In addition, this relation was recently found to arise naturally in a theory of embedding obstructions for 2-spheres in 4-manifolds in terms of Whitney towers. This paper contains the first proof of the four-dimensional version of the Jacobi identity. We also expose the underlying topological unity between the three- and four-dimensional IHX relations, deriving from a beautiful picture of the Borromean rings embedded on the boundary of an unknotted genus 3 handlebody in 3-space. This picture is most naturally related to knot and 3-manifold invariants via the theory of grope cobordisms.

We compute the rational cohomology of the moduli space of non-singular complex plane quartic curves with two marked points. This allows us to calculate the rational cohomology of the moduli space of non-singular complex projective curves of genus three with two marked points.

Let F be a non-Archimedean local field and let p be the residual characteristic of F. Let G=GL2(F) and let P be a Borel subgroup of G. In this paper we study the restriction of irreducible smooth representations of G on -vector spaces to P. We show that in a certain sense P controls the representation theory of G. We then extend our results to smooth -modules of finite length and unitary K-Banach space representations of G, where is the ring of integers of a complete discretely valued field K with residue field .

The Gromov–Witten theory of Deligne–Mumford stacks is a recent development, and hardly any computations have been done beyond three-point genus 0 invariants. This paper provides explicit recursions which, together with some invariants computed by hand, determine all genus 0 invariants of the stack $\mathbb{P}^2_{D,2}$. Here $D$ is a smooth plane curve and $\mathbb{P}^2_{D,2}$ is locally isomorphic to the stack quotient $[U/(\mathbb{Z}/(2))]$, where $U\to V\subseteq \mathbb{P}^2$ is a double cover branched along $D\cap V$. The introduction discusses an enumerative application of these invariants.

We recover the Newton diagram (modulo a natural ambiguity) from the link for any surface hypersurface singularity with non-degenerate Newton principal part whose link is a rational homology sphere. As a corollary, we show that the link determines the embedded topological type, the Milnor fibration, and the multiplicity of such a germ. This proves (an even stronger version of) Zariski's conjecture about the multiplicity for such a singularity.

We give counterexamples to Okounkov’s log-concavity conjecture for Littlewood–Richardson coefficients.

We define integral formulas which produce certain matrix coefficients of cohomologically induced representations of real reductive groups. They are analogous to Harish-Chandra's Eisenstein integrals for matrix coefficients of ordinary induced representations, and generalize Flensted-Jensen's fundamental functions for discrete series.

We use the geometry of characteristic cycles of Harish-Chandra modules for a real semi-simple Lie group $G_{\mathbb{R}}$ to prove an upper triangularity relationship between two bases of each special representation of a classical Weyl group. One basis consists of Goldie rank polynomials attached to primitive ideals in the enveloping algebra of the complexified Lie algebra ${\mathfrak{g}}$; the other consists of polynomials that measure the Euler characteristic of the restriction of an equivariant line bundle on the flag variety for ${\mathfrak{g}}$ to an irreducible component of the Springer fiber. While these two bases are defined only using the structure of the complex Lie algebra ${\mathfrak{g}}$, the relationship between them is closely tied to the real group $G_{\mathbb{R}}$. More precisely, the order leading to the upper triangularity result is a suborder of closure order for the orbits of the complexification of a maximal compact subgroup of $G_{\mathbb{R}}$ on the flag variety for ${\mathfrak{g}}$.

In this paper we describe the intersections between the balls of maximal symplectic packings of . This analysis shows the existence of singular points for maximal packings of by more than three equal balls. It also yields a construction of a class of very regular examples of maximal packings by five balls.

For the groups $G={\mathrm{Sp}}(p,q),\ \mathrm{SO}^\ast(2n)$, and $\mathrm{U}(m,n)$, we consider degenerate principal series whose infinitesimal character coincides with a finite-dimensional representation of $G$. We prove that each irreducible constituent of maximal Gelfand–Kirillov dimension is a derived functor module. We also show that at an appropriate ‘most singular’ parameter, each irreducible constituent is weakly unipotent and unitarizable. Conversely we show that any weakly unipotent representation associated to a real form of the corresponding Richardson orbit is unique up to isomorphism and can be embedded into a degenerate principal series at the most singular integral parameter (apart from a handful of very even cases in type D). We also discuss edge-of-wedge-type embeddings of derived functor modules into degenerate principal series.

Allcock, Carlson and Toledo defined a period map for cubic threefolds which takes values in a ball quotient of dimension 10. A theorem of Voisin implies that this is an open embedding. We determine its image and show that on the algebraic level this amounts to identification of the algebra of $\operatorname{SL}(5,\mathbb{C})$-invariant polynomials on the representation space $\operatorname{Sym}^3(\mathbb{C}^5)^*$ with an explicitly described algebra of meromorphic automorphic forms on the complex 10-ball.