We fix a prime p and consider a connected reductive algebraic group G over a perfect field k which is defined over 𝔽p. Let M be a finite-dimensional rational G-module M, a comodule for k[G]. We seek to somewhat unravel the relationship between the restriction of M to the finite Chevalley subgroup G(𝔽p)⊂G and the family of restrictions of M to Frobenius kernels G(r) ⊂G. In particular, we confront the conundrum that if M is the Frobenius twist of a rational G-module N,M=N(1), then the restrictions of M and N to G(𝔽p) are equal whereas the restriction of M to G(1) is trivial. Our analysis enables us to compare support varieties (and the finer non-maximal support varieties) for G(𝔽p) and G(r) of a rational G-module M where the choice of r depends explicitly on M.

We prove a conjecture of Kontsevich, which asserts that the iterations of the non-commutative rational map Fr:(x,y)→(xyx−1,(1+yr)x−1) are given by non-commutative Laurent polynomials with non-negative integer coefficients.

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus $g$ must have slope $2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

Let K be a finite unramified extension of Qp. We parametrize the (φ,Γ)-modules corresponding to reducible two-dimensional -representations of GK and characterize those which have reducible crystalline lifts with certain Hodge–Tate weights.

A Pfaffian equation on $X$ is an invertible subsheaf $L \hookrightarrow \Omega^1_X$ of the sheaf of differential 1-forms on $X$. It can thus be thought of as a global section of $\Omega^1_X \otimes L^{-1}$ or, by choosing an isomorphism $L^{-1} \simeq {\cal O}_X(E) \subset K$, as a meromorphic differential form $\omega$ on $X$. We say that the Pfaffian equation has a first integral if there is a non-empty open subset $U\subset X$ and a smooth map $f: U\rightarrow {\rm P}^1$ such that $L_U \simeq f^* \Omega^1_{{\rm P}^1}$ as subsheaves of $\Omega^1_X$; that is, if $L$ ‘comes from’ a rational map to a curve. In this case, $f$, viewed as a rational function, is called a first integral. Note that as rational differential forms, ${\rm d} f \wedge \omega = 0$.

We present an explicit formula for computing toric residues of ample divisors as a quotient of two determinants, à la Macaulay, where the numerator is a minor of the denominator. We present a combinatorial construction of a specific element of residue 1. We also give an irreducible representation of toric residues by extending the theory of subresultants to monomials of critical degree in the homogeneous coordinate ring of the corresponding toric variety.

We investigate the relationship between the usual and general Hodge conjectures for abelian varieties. For certain abelian varieties $A$, we show that the usual Hodge conjecture for all powers of $A$ implies the general Hodge conjecture for $A$.

In this note, we will apply the results of Gross–Zagier, Gross–Kohnen–Zagier and their generalizations to give a short proof that the differences of singular moduli are not units. As a consequence, we obtain a result on isogenies between reductions of CM elliptic curves.

We define a multiple Dirichlet series whose group of functional equations is the Weyl group of the affine Kac–Moody root system $\widetilde{A}_{n}$, generalizing the theory of multiple Dirichlet series for finite Weyl groups. The construction is over the rational function field $\mathbb{F}_{q}(t)$, and is based upon four natural axioms from algebraic geometry. We prove that the four axioms yield a unique series with meromorphic continuation to the largest possible domain and the desired infinite group of symmetries.

Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$, there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level-$m$ structure. To this end, we develop a version of Vojta’s conjecture for Deligne–Mumford stacks, which we deduce from Vojta’s conjecture for schemes.

Let 𝒱 be a complete discrete valuation ring of mixed characteristic. We classify arithmetic 𝒟-modules on Spf(𝒱[[t]]) up to certain kind of ‘analytic isomorphism’. This result is used to construct canonical extensions (in the sense of Katz and Gabber) for objects of this category.

We introduce Poisson boundaries of II$_1$ factors with respect to density operators that give the traces. The Poisson boundary is a von Neumann algebra that contains the II$_1$ factor and is a particular example of the boundary of a unital completely positive map as introduced by Izumi. Studying the inclusion of the II$_1$ factor into its boundary, we develop a number of notions, such as double ergodicity and entropy, that can be seen as natural analogues of results regarding the Poisson boundaries introduced by Furstenberg. We use the techniques developed to answer a problem of Popa by showing that all finite factors satisfy his MV property. We also extend a result of Nevo by showing that property (T) factors give rise to an entropy gap.

We show a rigidity theorem for the Seiberg–Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of $4$-manifolds $M$ for which the inclusion maps $\operatorname {Diff}(M) \hookrightarrow \operatorname {Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin $4$-manifolds.

In this paper we prove that most ropes of arbitrary multiplicity supported on smooth curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely that finite covers onto smooth irreducible curves embedded in projective space can be deformed to a family of 1:1 maps. We apply our general theory to prove the smoothing of ropes of multiplicity 3 on P1. Even though this paper focuses on ropes of dimension 1, our method yields a general approach to deal with the smoothing of ropes of higher dimension.

In the present paper we introduce and study the twisted γ-filtration on K0(Gs), where Gs is a split simple linear algebraic group over a field k of characteristic prime to the order of the center of Gs. We apply this filtration to construct nontrivial torsion elements in γ-rings of twisted flag varieties.

For cohomological (respectively homological) coefficient systems ${\mathcal F}$ (respectively ${\mathcal V}$) on affine buildings X with Coxeter data of type $\widetilde{A}_d$, we give for any $k\ge1$ a sufficient local criterion which implies $H^k(X,{\mathcal F})=0$ (respectively $H_k(X,{\mathcal V})=0$). Using this criterion we prove a conjecture of de Shalit on the acyclicity of coefficient systems attached to hyperplane arrangements on the Bruhat–Tits building of the general linear group over a local field. We also generalize an acyclicity theorem of Schneider and Stuhler on coefficient systems attached to representations.

In this paper we establish a version of homological mirror symmetry for punctured Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured surface $\unicode[STIX]{x1D6F4}$ via the topological Fukaya category. We prove that the topological Fukaya category of $\unicode[STIX]{x1D6F4}$ is equivalent to the category of matrix factorizations of a certain mirror LG model $(X,W)$. Along the way we establish new gluing results for the topological Fukaya category of punctured surfaces which are of independent interest.

We give a Rankin–Selberg integral representation for the Spin (degree eight) $L$-function on $\operatorname{PGSp}_{6}$ that applies to the cuspidal automorphic representations associated to Siegel modular forms. If $\unicode[STIX]{x1D70B}$ corresponds to a level-one Siegel modular form $f$ of even weight, and if $f$ has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D70B},\text{Spin},s)$ of $\unicode[STIX]{x1D70B}$.