We determine the set of connected components of minuscule affine Deligne–Lusztig varieties for hyperspecial maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne–Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport–Zink spaces.

We show how some of the refined tropical counts of Block and Göttsche emerge from the wall-crossing formalism. This leads naturally to a definition of a class of putative $q$-deformed Gromov–Witten invariants. We prove that this coincides with another natural $q$-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined.

Katz’s middle convolution algorithm provides a description of rigid connections on ℙ1 with regular singularities. We extend the algorithm by adding the Fourier transform to it. The extended algorithm provides a description of rigid connections with arbitrary singularities.

We show that over polynomial extensions of normal affine domains of dimension two over perfect fields (char. ≠ 2) of cohomological dimension [les ] 1, all finitely generated projective modules are cancellative, thus answering a question of Weibel affirmatively in the case of polynomial extensions.

We prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. Applying the same technique, we also show that the Chow group of 1-cycles on a separably rationally connected Fano complete intersection of index at least 2 is generated by lines. As a consequence, we give a positive answer to a question of Professor Totaro about integral Hodge classes on rationally connected 3-folds. And by a result of Professor Voisin, the general case is a consequence of the Tate conjecture for surfaces over finite fields.

For a closed d-dimensional subvariety X of an abelian variety A and a canonically metrized line bundle L on A, Chambert-Loir has introduced measures c1(L∣X)∧d on the Berkovich analytic space associated to A with respect to the discrete valuation of the ground field. In this paper, we give an explicit description of these canonical measures in terms of convex geometry. We use a generalization of the tropicalization related to the Raynaud extension of A and Mumford’s construction. The results have applications to the equidistribution of small points.

Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of a variety $X$. Given globally generated line bundles $B_{1}, \dotsc, B_{\ell}$ on $X$ and $m_{1}, \dotsc, m_{\ell} \in \mathbb{N}$, consider the line bundle $L := B_{1}^{m_{1}} \otimes \dotsb \otimes B_{\ell}^{m_{\ell}}$. We give conditions on the $m_{i}$ which guarantee that the ideal of $X$ in $\mathbb{P}(H^{0}(X,L)^{*})$ is generated by quadrics and that the first $p$ syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.

let k be an algebraically closed field of positive characteristic p > 0 and $c \to {\mathbb p}^1_k$ a p-cyclic cover of the projective line ramified in exactly one point. we are interested in the p-sylow subgroups of the full automorphism group autkc. we prove that for curves c with genus 2 or higher, these groups are exactly the extensions of a p-cyclic group by an elementary abelian p-group. the main tool is an efficient algorithm to compute the p-sylow subgroups of autkc starting from an artin–schreier equation for the cover $c \to {\mathbb p}^1_k$. we also characterize curves c with genus $g_c\geq 2$ and a p-group action $g\subset \text{aut}_k c$ such that $2p/(p-1)<|g|/g_c$ and $4/(p-1)^2\leq |g|/g_c^2$. our methods rely on previous work by stichtenoth whose approach we have adopted.

Bondarko defines and studies the notion of weight structure and he shows that there exists a weight structure over the category of Voevodsky motives with rational coefficients (over a field of characteristic 0). In this paper we extend this weight structure to the category of Beilinson motives (for any scheme of finite type over a base scheme which is excellent of dimension at most two) introduced and studied by Cisinsky and Déglise. We also check the weight exactness of the Grothendieck operations.

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

We show that there is no square other than 122 and 7202 such that it can be written as a product of k−1 integers out of k(≥3) consecutive positive integers. We give an extension of a theorem of Sylvester that a product of k consecutive integers each greater than k is divisible by a prime exceeding k.

We show that the bi-Lipschitz equivalence of analytic function germs (${\open C}^{2}$, 0)→(${\open C}$, 0) admits continuous moduli. More precisely, we propose an invariant of the bi-Lipschitz equivalence of such germs that varies continuously in many analytic families ft: (${\open C}^{2}$, 0)→(${\open C}$, 0). For a single germ f the invariant of f is given in terms of the leading coefficients of the asymptotic expansions of f along the branches of generic polar curve of f.

Let (M,I,J,K) be a compact hyperkähler manifold, , and L a non-trivial holomorphic line bundle on (M,I). Using the quaternionic Dolbeault complex, we prove the following vanishing theorem for holomorphic cohomology of L. If c1(L) lies in the closure of the dual Kähler cone, then Hi(L)=0 for i>n. If c1(L) lies in the opposite cone , then Hi(L)=0 for i<n. Finally, if c1(L) is neither in nor in , then Hi(L)=0 for .

We show that if $A$ and $H$ are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter–Drinfeld resolution of the counit of $A$ to the same kind of resolution for the counit of $H$, exhibiting in this way strong links between the Hochschild homologies of $A$ and $H$. This enables us to obtain a finite free resolution of the counit of $\mathcal {B}(E)$, the Hopf algebra of the bilinear form associated with an invertible matrix $E$, generalizing an earlier construction of Collins, Härtel and Thom in the orthogonal case $E=I_n$. It follows that $\mathcal {B}(E)$ is smooth of dimension 3 and satisfies Poincaré duality. Combining this with results of Vergnioux, it also follows that when $E$ is an antisymmetric matrix, the $L^2$-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of $\mathcal {B}(E)$in the cosemisimple case.

Fix a prime number p [ges ] 5 and a positive integer N prime to p. We consider the projective limits of p-adic étale cohomology groups of the modular curves X1(Npr) and Y1(Npr) (r [ges ] 1), which are denoted by ESp(N) ${\biib Z}$p and GES p(N)${\biib Z}$p, respectively. Let e* ′ be the projector to the direct sum of the ωi-eigenspaces of the ordinary part, for i [nequiv ] 0, −1 mod p−1. Our main result states that e* ′ GESp (N)${\biib Z}$p has a good p-adic Hodge structure, which can be described in terms of λ-adic modular forms, extending the previously known result for e*′ ESp (N)${\biib Z}$p. We then apply the method of Harder and Pink to the Galois representation on e*′ ESp(N) ${\biib Z}$p to construct large unramified abelian p-extensions over cyclotomic ${\bib Z}$p-extensions of abelian number fields.

Under endoscopic assumptions about $L$-packets of unitary groups, we prove the local Gan–Gross–Prasad conjecture for tempered representations of unitary groups over $p$-adic fields. Roughly, this conjecture says that branching laws for $U(n-1)\subset U(n)$ can be computed using epsilon factors.

Given a smooth variety $X$ and an effective Cartier divisor $D\subset X$, we show that the cohomological Chow group of 0-cycles on the double of $X$ along $D$ has a canonical decomposition in terms of the Chow group of 0-cycles $\text{CH}_{0}(X)$ and the Chow group of 0-cycles with modulus $\text{CH}_{0}(X|D)$ on $X$. When $X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of $\text{CH}_{0}(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that $\text{CH}_{0}(X|D)$ is torsion-free and there is an injective cycle class map $\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$ if $X$ is affine. For a smooth affine surface $X$, this is strengthened to show that $K_{0}(X,D)$ is an extension of $\text{CH}_{1}(X|D)$ by $\text{CH}_{0}(X|D)$.

Soient G = exp ${\frak g}$ un groupe de Lie résoluble exponentiel et H = exp ${\frak h}$ un sous-groupe connexe de G. Soient χ un caractère unitaire de H et τ = ${\rm Ind}_{H}^{G}$χ. Soit Dτ(G/H) l'algèbre des opérateurs différentiels G-invariants sur G/H. Une question posée par Duflo et Corwin-Greenleaf consiste à voir si la finitude des multiplicités de τ est équivalente à la commutativité de Dτ(G/H). Nous répondons positivement à cette question quand H est normal dans G. Lorsque H n'est pas normal, nous préparons le terrain pour d'espaces homogènes nilpotents et nous répondons à la question dans différents cas. Nous étudions finalement l'algèbre Dπ(G)H, π∈Ĝ des opérateurs différentiels qui laissent l'espace des vecteurs C∞ de π invariant et qui commuttent avec l'action de H sur cet espace.

The aim of this article is to classify two-dimensional split trianguline representations of p-adic fields. This is a generalization of a result of Colmez who classified two-dimensional split trianguline representations of for p≠2 by using (φ,Γ)-modules over a Robba ring. In this article, for any prime p and for any p-adic field K, we classify two-dimensional split trianguline representations of using B-pairs as defined by Berger.

Let $K$ be a function field and $C$ a non-isotrivial curve of genus $g\geqslant 2$ over $K$. In this paper, we will show that if $C$ has a global stable model with only geometrically irreducible fibers, then Bogomolov conjecture over function fields holds.