Using the ‘slice filtration’, defined by effectivity conditions on Voevodsky's triangulated motives, we define spectral sequences converging to their motivic cohomology and étale motivic cohomology. These spectral sequences are particularly interesting in the case of mixed Tate motives as their $E_2$-terms then have a simple description. In particular this yields spectral sequences converging to the motivic cohomology of a split connected reductive group. We also describe in detail the multiplicative structure of the motive of a split torus.

We investigate the large weight ($k\to\infty$) limiting statistics for the low lying zeros of a GL(4) and a GL(6) family of $L$-functions, $\{L(s,\phi \times f): f \in H_k\}$ and $\{L(s,\phi \times {\rm sym}^2 f): f \in H_k\}$; here $\phi$ is a fixed even Hecke–Maass cusp form and $H_k$ is a Hecke eigenbasis for the space $H_k$ of holomorphic cusp forms of weight $k$ for the full modular group. Katz and Sarnak conjecture that the behavior of zeros near the central point should be well modeled by the behavior of eigenvalues near 1 of a classical compact group. By studying the 1- and 2-level densities, we find evidence of underlying symplectic and SO(even) symmetry, respectively. This should be contrasted with previous results of Iwaniec–Luo–Sarnak for the families $\{L(s,f): f\in H_k\}$ and $\{L(s,{\rm sym}^2f): f\in H_k\}$, where they find evidence of orthogonal and symplectic symmetry, respectively. The present examples suggest a relation between the symmetry type of a family and that of its twistings, which will be further studied in a subsequent paper. Both the GL(4) and the GL(6) families above have all even functional equations, and neither is naturally split from an orthogonal family. A folklore conjecture states that such families must be symplectic, which is true for the first family but false for the second. Thus, the theory of low lying zeros is more than just a theory of signs of functional equations. An analysis of these families suggest that it is the second moment of the Satake parameters that determines the symmetry group.

This paper develops a general theory of canonical bases and how they arise naturally in the context of categorification. As an application, we show that Lusztig’s canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is of finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest- and highest-weight integrable representations. This generalizes past work of the author’s in the highest-weight case.

Fix a prime number [ell ]. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-[ell ] completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted completion of a profinite group with respect to a reductive representation (and other appropriate data).

This paper is a continuation of the author's previous work, where we studied one of the inequalities between the characteristic ideal of the Selmer group and the ideal of the $p$-adic $L$-function predicted by the two-variable Iwasawa main conjecture for a nearly ordinary Hida deformation $\mathcal{T}$. In this paper, we study several properties of the Selmer group and the $p$-adic $L$-function solving some of the open questions raised in the author's previous work. As applications, we have an infinite family of elliptic cuspforms where the cyclotomic Iwasawa main conjecture holds for every cuspform in the family.

After the work of Kisin, there is a good theory of canonical integral models of Shimura varieties of Hodge type at primes of good reduction. The first part of this paper develops a theory of Hodge type Rapoport–Zink formal schemes, which uniformize certain formal completions of such integral models. In the second part, the general theory is applied to the special case of Shimura varieties associated with groups of spinor similitudes, and the reduced scheme underlying the Rapoport–Zink space is determined explicitly.

We study the relationship between the equations defining a projective variety and properties of its secant varieties. In particular, we use information about the syzygies among the defining equations to derive smoothness and normality statements about SecX and also to obtain information about linear systems on the blow up of projective space along a variety X. We use these results to geometrically construct, for varieties of arbitrary dimension, a flip first described in the case of curves by M. Thaddeus via Geometric Invariant Theory.

Let $X$ be a compact Kähler manifold and let $(L,{\it\varphi})$ be a pseudo-effective line bundle on $X$. We first define a notion of numerical dimension for pseudo-effective line bundles with singular metrics, and then discuss the properties of this numerical dimension. Finally, we prove a very general Kawamata–Viehweg–Nadel-type vanishing theorem on an arbitrary compact Kähler manifold.

We present some applications of the Subspace Theorem to the investigation of the arithmetic of the values of Laurent series f(z) at S-unit points. For instance we prove that if f(qn) is an algebraic integer for infinitely many n, then h(f(qn)) must grow faster than n. By similar principles, we also prove diophantine results about power sums and transcendency results for lacunary series; these include as very special cases classical theorems of Mahler. Our arguments often appear to be independent of previous techniques in the context.

We introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne–Mumford stability. For every pair of integers 1≤m<n, we prove that the moduli problem of n-pointed m-stable curves of arithmetic genus one is representable by a proper irreducible Deligne–Mumford stack . We also consider weighted variants of these stability conditions, and construct the corresponding moduli stacks . In forthcoming work, we will prove that these stacks have projective coarse moduli and use the resulting spaces to give a complete description of the log minimal model program for .

In this work, we study the diophantine equation\renewcommand{\theequation}{1.1}\begin{equation}f(X) = Y^m,\end{equation} where $m \geqslant 2$ is an integer and $f(X)$ is a polynomial with coefficients in a number field ${\bf K}$. The first important result on this topic is due to Siegel [19], who showed that if $m = 2$ and $f$ has at least three simple roots or if $m \geqslant 3$ and $f$ has at least two simple roots, then (1.1) has only finitely many integral solutions. Three years later, he proved [20] that if the algebraic curve defined by (1.1) is of positive genus, then (1.1) has only finitely many integral solutions. The $p$-adic analogue of this theorem was established independently by Lang [9] and LeVeque [12], who showed that, under the same conditions, (1.1) has only finitely many $S$-integral solutions. After that, LeVeque [13] gave a necessary and sufficient condition for the algebraic curve defined by (1.1) to have positive genus. However, all these results are based on Thue's method, and hence are ineffective.

In Bennett et al. [BGG reciprocity for current algebras, Adv. Math. 231 (2012), 276–305] it was conjectured that a BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra. We associate a current algebra to any indecomposable affine Lie algebra and show that, in this generality, the BGG reciprocity is true for the corresponding category of representations.

In this paper, we consider the preservation of stability by using the notion of twisted stability. As applications, (1) we show that moduli spaces of stable sheaves on K3 and abelian surfaces are irreducible and (2) we compute Hodge polynomials of some moduli spaces of stable sheaves on Enriques surfaces.

We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.

Let $S$ be a Noetherian scheme of finite dimension and denote by $\unicode[STIX]{x1D70C}\in [\unicode[STIX]{x1D7D9},\mathbb{G}_{m}]_{\mathbf{SH}(S)}$ the (additive inverse of the) morphism corresponding to $-1\in {\mathcal{O}}^{\times }(S)$ . Here $\mathbf{SH}(S)$ denotes the motivic stable homotopy category. We show that the category obtained by inverting $\unicode[STIX]{x1D70C}$ in $\mathbf{SH}(S)$ is canonically equivalent to the (simplicial) local stable homotopy category of the site $S_{\text{r}\acute{\text{e}}\text{t}}$ , by which we mean the small real étale site of $S$ , comprised of étale schemes over $S$ with the real étale topology. One immediate application is that $\mathbf{SH}(\mathbb{R})[\unicode[STIX]{x1D70C}^{-1}]$ is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the $\unicode[STIX]{x1D70C}$ -local sphere (over $\mathbb{R}$ ). As further applications we show that $D_{\mathbb{A}^{1}}(k,\mathbb{Z}[1/2])^{-}\simeq \mathbf{DM}_{W}(k)[1/2]$ (improving a result of Ananyevskiy–Levine–Panin), reprove Röndigs’ result that $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9}[1/\unicode[STIX]{x1D702},1/2])=0$ for $i=1,2$ and establish some new rigidity results.

We describe an algorithm for computing parameter-test-ideals in certain local Cohen–Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Sharp’s notion of ‘special ideals’. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the celebrated result in the article Generators of D-modules in positive characteristic (J. Alvarez-Montaner, M. Blickle and G. Lyubeznik, Math. Res. Lett. 12 (2005), 459–473).

We introduce an essentially new Grothendieck topology, the Weil-étale topology, on schemes over finite fields. The cohomology groups associated with this topology should behave better than the standard étale cohomology groups. In particular there is a very natural definition of an Euler characteristic and a plausible conjecture relating the Euler characteristic of Z to the value of the zeta-function at s = 0. This conjecture is proved in certain cases.

Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a complex symmetric Kac–Moody Lie algebra. Lusztig has introduced a basis of $U(\mathfrak{n})$ called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of nilpotent modules over a preprojective algebra of the same type as $\mathfrak{n}$. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important rôle in our work on the relationship between semicanonical bases, representation theory of preprojective algebras, and Fomin and Zelevinsky's theory of cluster algebras. It was inspired by recent results of Caldero and Keller.

We define stacks of uniform cyclic covers of Brauer–Severi schemes, proving that they can be realized as quotient stacks of open subsets of representations, and compute the Picard group for the open substacks parametrizing smooth uniform cyclic covers. Moreover, we give an analogous description for stacks parametrizing triple cyclic covers of Brauer–Severi schemes of rank 1 that are not necessarily uniform, and give a presentation of the Picard group of the substacks corresponding to smooth triple cyclic covers.