Let $\mathbb{L}\subset A\times I$ be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of $\mathfrak{sl}_{2}(\wedge )$, the exterior current algebra of $\mathfrak{sl}_{2}$. When $\mathbb{L}$ is an $m$-framed $n$-cable of a knot $K\subset S^{3}$, its sutured annular Khovanov homology carries a commuting action of the symmetric group $\mathfrak{S}_{n}$. One therefore obtains a ‘knotted’ Schur–Weyl representation that agrees with classical $\mathfrak{sl}_{2}$ Schur–Weyl duality when $K$ is the Seifert-framed unknot.

We investigate the Chow groups of zero cycles of products of curves over a p-adic field by means of the Milnor K-groups of their Jacobians as introduced by Somekawa. We prove some finiteness results for CH0(X)/m for X a product of curves over a p-adic field.

We estimate some sums of the shape S(Xβ1,…, Xβm): = [sum ]1 [les ] d1 [les ] Xβ1… [sum ]1 [les ] dm [les ] Xβm ƒ(d1,…, dm), when m ∈ $\Bbb N$ and f is a nonnegative arithmetical function. We relate them to the behaviour of the associated Dirichlet series

F(s1,…, sm) = ∞[sum ]d1 = 1 … ∞[sum ]dm = 1 f(d1,…, dm)/d1s1 … dmsm. The main aim of this work is to develop analytic tools to count the rational points of bounded height on toric varieties.

The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.

We give a full description of the Chow ring of the complex Cayley plane $\mathbb{O}\mathbb{P}^2$. For this, we describe explicitly the most interesting of its Schubert varieties and compute their intersection products. Translating our results into the Borel presentation, i.e. in terms of Weyl group invariants, we are able to compute the degree of the variety of reductions Y8 introduced by the current authors in arXiv: math.AG/0306328.

Vogan has conjectured that the cohomologically induced modules A$_ q$(λ) in the weakly fair range exhaust all unitary representations of U(p, q) with certain kinds of real integral infinitesimal character. To prove a statement like this, it is essential to identify these modules among the set of all irreducible Harish-Chandra modules. Barbasch and Vogan have parametrized this latter set in terms of their annihilators and asymptotic supports (or, equivalently, associated varieties). In this paper, we identify the weakly fair A$_ q$(λ) in this parametrization by combining known results about their asymptotic supports together with an explicit computation of their annihilators. In particular, this determines all vanishing and coincidences among the A$_ q$(λ) in the weakly fair range, and gives the Langlands parameters of these modules.

In a paper by Badulescu [Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172 (2008), 383–438], results on the global Jacquet–Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field were established, under the assumption that the local inner forms are split at archimedean places. In this paper, we extend the main local results of that article to archimedean places so that the above condition can be removed. Along the way, we collect several results about the unitary dual of general linear groups over ℝ, ℂ or ℍ which are of independent interest.

We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.

In Invent. Math.126 (2000), pp. 513–545, we gave a proof of Lang's conjecture on Abelian varieties leading to an effective bound for the number of translates involved. We show here that the method can be extended to give a similar statement for the ‘Mordell–Lang plus Bogomolov’ theorem proven by B. Poonen and independently by S. Zhang. We deal in detail with tori for which effective results have been obtained by J.-H. Evertse and H. P. Schlickewei; we improve on these mainly by providing polynomial bounds in the degree instead of doubly exponential ones. We also state a theorem for Abelian varieties. In both cases the strategy of proof is based on the approach of Mumford and Vojta–Faltings–Bombieri together with an effective Bogomolov property and therefore does not rely on either equidistribution nor subspace theorem arguments.

Let A be a homogeneous K-algebra where K is a field of characteristic 0, and h ∈ A a generic form. We bound the Hilbert function H(A/(h),-) in terms of H(A,-) which extends the bound given by M. Green for generic linear forms. We apply this to some conjectures from Higher Castelnuovo Theory and Cayley-Bacharach Theory.

According to the André–Oort conjecture, an algebraic curve in Y (1)n that is not equal to a special subvariety contains only finitely many points which correspond to ann-tuple of elliptic curves with complex multiplication. Pink’s conjecture generalizes the André–Oort conjecture to the extent that if the curve is not contained in a special subvariety of positive codimension, then it is expected to meet the union of all special subvarieties of codimension two in only finitely many points. We prove this for a large class of curves in Y (1)n. When restricting to special subvarieties of codimension two that are not strongly special we obtain finiteness for all curves defined over . Finally, we formulate and prove a variant of the Mordell–Lang conjecture for subvarieties of Y (1)n.

We construct all $\cal A$e-codimension 1 multi-germs of analytic (or smooth) maps (kn, T) → (kp, 0), with n [ges ] p − 1, (n, p) nice dimensions, k = $\mathbb C$ or $\mathbb R$, by augmentation and concatenation operations, starting from mono-germs (|T| = 1) and one 0-dimensional bi-germ. As an application, we prove general statements for multi-germs of corank [les ] 1: every one has a real form with real perturbation carrying the vanishing homology of the complexification, every one is quasihomogeneous, and when n = p − 1 every one has image Milnor number equal to 1 (this last is already known when n [ges ] p).

We determine exactly when a quadratic form is represented by a spinor genus of another quadratic form of three or four variables. We apply this to extend the embedding theorem for quaternion and also answer a question by Borovoi.

We generalize the work of Bertolini and Darmon on the anticyclotomic main conjecture for elliptic curves to modular forms of higher weight.

We prove some new evaluations for multiple polylogarithms of arbitrary depth. The simplest of our results is a multiple zeta evaluation one order of complexity beyond the well-known Broadhurst–Zagier formula. Other results we provide settle three of the remaining outstanding conjectures of Borwein, Bradley, and Broadhurst. A complete treatment of a certain arbitrary depth class of periodic alternating unit Euler sums is also given.

Let $A$ be a $d$-dimensional local ring containing a field. We will prove that the highest Lyubeznik number $\lambda_{d,d}(A)$ is equal to the number of connected components of the Hochster–Huneke graph associated to $B$, where $B=\widehat{\hat{A}^{\rm sh}}$ is the completion of the strict Henselization of the completion of $A$. This was proven by Lyubeznik in characteristic $p>0$. Our statement and proof are characteristic-free.

We prove that for any positive integers x,d and k with gcd (x,d)=1 and 3<k<35, the product x(x+d)⋯(x+(k−1)d) cannot be a perfect power. This yields a considerable extension of previous results of Győry et al. and Bennett et al., which covered the cases where k≤11. We also establish more general theorems for the case where x can also be a negative integer and where the product yields an almost perfect power. As in the proofs of the earlier theorems, for fixed k we reduce the problem to systems of ternary equations. However, our results do not follow as a mere computational sharpening of the approach utilized previously; instead, they require the introduction of fundamentally new ideas. For k>11, a large number of new ternary equations arise, which we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly with k that, in contrast with the previous proofs, it is practically impossible to handle the various cases in the usual manner. The main novelty of this paper lies in the development of an algorithm for our proofs, which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the new ternary equations that arise with several sieves based on the ternary equations already solved. In this way, we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger values of k as well, although there is, of course, a computational time constraint.

In a recent paper, Bondarko [Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), Preprint (2007), 0704.4003] defined the notion of weight structure, and proved that the category DMgm(k) of geometrical motives over a perfect field k, as defined and studied by Voevodsky, Suslin and Friedlander [Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000)], is canonically equipped with such a structure. Building on this result, and under a condition on the weights avoided by the boundary motive [J. Wildeshaus, The boundary motive: definition and basic properties, Compositio Math. 142 (2006), 631–656], we describe a method to construct intrinsically in DMgm(k) a motivic version of interior cohomology of smooth, but possibly non-projective schemes. In a sequel to this work [J. Wildeshaus, On the interior motive of certain Shimura varieties: the case of Hilbert–Blumenthal varieties, Preprint (2009), 0906.4239], this method will be applied to Shimura varieties.

The discriminant function $\Delta$ is a certain rigid analytic modular form defined on Drinfeld's upper half-plane $\Omega$. Its absolute value $\vert \Delta\vert$ may be considered as a function on the associated Bruhat–Tits tree ${\cal T}$. We compare $\log \vert \Delta\vert$ with the conditionally convergent complex-valued Eisenstein series $E$ defined on ${\cal T}$ and thereby obtain results about the growth of $\vert \Delta$ and of some related modular forms. We further determine to what extent roots may be extracted of $\Delta(z)/\Delta(nz)$, regarded as a holomorphic function on $\Omega$. In some cases, this enables us to calculate cuspidal divisor class groups of modular curves.

We study some Diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the $b$-adic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit.