We prove Zhang’s dynamical Manin–Mumford conjecture and dynamical Bogomolov conjecture for dominant endomorphisms $\unicode[STIX]{x1D6F7}$ of $(\mathbb{P}^{1})^{n}$ . We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with an analysis of the symmetries of the Julia set for a rational function.

The moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich’s stable map compactification and Marian–Oprea–Pandharipande’s stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck’s Quot scheme. In this paper, we give the notion of ‘ ϵ-stable quotients’ for a positive real number ϵ, and show that stable maps and stable quotients are related by wall-crossing phenomena. We will also discuss Gromov–Witten type invariants associated to ϵ-stable quotients, and investigate them under wall crossing.

We introduce an analogue in hyperkähler geometry of the symplectic implosion, in the case of $\mathrm{SU} (n)$ actions. Our space is a stratified hyperkähler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

We give a classification of irreducible admissible modulo $p$ representations of a split$p$-adic reductive group in terms of supersingular representations. This is a generalization of a theorem of Herzig.

In the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curve A, the intersection of any finite-rank subgroup of A with the set of CM-points of A is finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to prove quantitative versions of a similar statement. The new methods apply also to certain infinite-rank subgroups, as well as to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura-curve analogues of these results.

A differential algebra of finite type over a field $\mathbb{k}$ is a filtered algebra A, such that the associated graded algebra is finite over its center, and the center is a finitely generated $\mathbb{k}$-algebra. The prototypical example is the algebra of differential operators on a smooth affine variety, when $\text{char}\mathbb{k} = 0$. We study homological and geometric properties of differential algebras of finite type. The main results concern the rigid dualizing complex over such an algebra A: its existence, structure and variance properties. We also define and study perverse A-modules, and show how they are related to the Auslander property of the rigid dualizing complex of A.

Generalizing previous results of Deligne–Serre and Taylor, Galois representations are attached to cuspidal automorphic representations of unitary groups whose Archimedean component is a holomorphic limit of discrete series. The main ingredient is a construction of congruences, using the Hasse invariant, that is independent of $q$-expansions.

We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kähler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and those of arbitrary Kähler manifolds. The consideration of certain natural ideals in the Hodge ring allows us to determine exactly which linear combinations of Hodge numbers are birationally invariant, and which are topological invariants. Combining the Hodge and unitary bordism rings, we are also able to treat linear combinations of Hodge and Chern numbers. In particular, this leads to a complete solution of a classical problem of Hirzebruch’s.

Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-$p$ extensions of imaginary quadratic number fields for $p$ an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of $\mathbb{F}_{q}(t)$, the Galois groups of the maximal unramified pro-$p$ extensions, as $q\rightarrow \infty$, have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

Let ρ be a two-dimensional modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has a large image and admits a low-weight crystalline modular deformation we show that any low-weight crystalline deformation of ρ unramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara-type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at one of the adjoint L-function associated with a Hilbert modular newform to the cardinality of the corresponding Selmer group.

Let J be the Jacobian of a smooth curve C of genus g, and let A(J) be the ring of algebraic cycles modulo algebraic equivalence on J, tensored with $\mathbb{Q}$. We study in this paper the smallest $\mathbb{Q}$-vector subspace R of A(J) which contains C and is stable under the natural operations of A(J): intersection and Pontryagin products, pull back and push down under multiplication by integers. We prove that this ‘tautological subring’ is generated (over $\mathbb{Q}$) by the classes of the subvarieties $W_1=C,\ W_2=C + C, \dots ,W_{g-1}$. If C admits a morphism of degree d onto $\mathbb{P}^1$, we prove that the last d - 1 classes suffice.

We construct a Rankin Selberg integral to represent the exterior cube L function L(π,Λ3,s) of an automorphic cuspidal module π of GL6(${\Bbb A}$F) (where F is a number field). We determine the poles of this L function and find period conditions for the special value L(π,Λ3,1/2). We use the Siegal Weil formula. We also state an analogue of the Gross–Prasad conjecture concerning a criterion for the nonvanishing of L(π,Λ3,1/2).

Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$, for $k\geqslant n$, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ($n$-dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$. By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$-dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$. We also prove similar equivalences for finite abelian covers of the $n$-dimensional pair of pants.

Gross, Hacking and Keel have constructed mirrors of log Calabi–Yau surfaces in terms of counts of rational curves. Using $q$-deformed scattering diagrams defined in terms of higher-genus log Gromov–Witten invariants, we construct deformation quantizations of these mirrors and we produce canonical bases of the corresponding non-commutative algebras of functions.

We study an analog over an imaginary quadratic field K of Serre‘s conjecture for modular forms. Given a continuous irreducible representation ρ:Gal(Q/K) →GL$_2$(F$_l$) we ask if ρ is modular. We give three examples of representations ρ obtained by restriction of even representations of Gal(Q/Q). These representations appear to be modular when viewed as representations over K, as shown by the computer calculations described at the end of the paper.

Let $(G,G^{\prime })$ be a type I irreducible reductive dual pair in Sp$(W_{\mathbb{R}})$. We assume that $(G,G^{\prime })$ is in the stable range where $G$ is the smaller member. Let $K$ and $K^{\prime }$ be maximal compact subgroups of $G$ and $G^{\prime }$ respectively. Let $\mathfrak{g}=\mathfrak{k}\bigoplus \mathfrak{p}$ and $\mathfrak{g}^{\prime }=\mathfrak{k}^{\prime }\bigoplus \mathfrak{p}^{\prime }$ be the complexified Cartan decompositions of the Lie algebras of $G$ and $G^{\prime }$ respectively. Let $\widetilde{K}$ and $\widetilde{K}^{\prime }$ be the inverse images of $K$ and $K^{\prime }$ in the metaplectic double cover $\widetilde{\text{Sp}}(W_{\mathbb{R}})$ of Sp$(W_{\mathbb{R}})$. Let ${\it\rho}$ be a genuine irreducible $(\mathfrak{g},\widetilde{K})$-module. Our first main result is that if ${\it\rho}$ is unitarizable, then except for one special case, the full local theta lift ${\it\rho}^{\prime }={\rm\Theta}({\it\rho})$ is equal to the local theta lift ${\it\theta}({\it\rho})$. Thus excluding the special case, the full theta lift ${\it\rho}^{\prime }$ is an irreducible and unitarizable $(\mathfrak{g}^{\prime },\widetilde{K}^{\prime })$-module. Our second main result is that the associated variety and the associated cycle of ${\it\rho}^{\prime }$ are the theta lifts of the associated variety and the associated cycle of the contragredient representation ${\it\rho}^{\ast }$ respectively. Finally we obtain some interesting $(\mathfrak{g},\widetilde{K})$-modules whose $\widetilde{K}$-spectrums are isomorphic to the spaces of global sections of some vector bundles on some nilpotent $K_{\mathbb{C}}$-orbits in $\mathfrak{p}^{\ast }$.

We study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic $p$ as in the tame case. Also, we link a crepant resolution with a count of Artin–Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

Building upon ideas of the second and third authors, we prove that at least $2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\unicode[STIX]{x1D700}$ is any positive real number and $s$ is large enough in terms of $\unicode[STIX]{x1D700}$. This lower bound is asymptotically larger than any power of $\log s$; it improves on the bound $(1-\unicode[STIX]{x1D700})(\log s)/(1+\log 2)$ that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.

Let X be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on X in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation and other similar identities.