The main result of this paper concerns the positivity of the Hodge bundles of abelian varieties over global function fields. As applications, we obtain some partial results on the Tate–Shafarevich group and the Tate conjecture of surfaces over finite fields.

For a Coxeter system and a representation $V$ of this Coxeter system, Soergel defined a category which is now called the category of Soergel bimodules and proved that this gives a categorification of the Hecke algebra when $V$ is reflection faithful. Elias and Williamson defined another category when $V$ is not reflection faithful and proved that this category is equivalent to the category of Soergel bimodules when $V$ is reflection faithful. Moreover, they proved the categorification theorem for their category with fewer assumptions on $V$. In this paper, we give a bimodule description of the Elias–Williamson category and re-prove the categorification theorem.

We prove the longstanding physics conjecture that there exists a unique two-parameter ${\mathcal {W}}_{\infty }$-algebra which is freely generated of type ${\mathcal {W}}(2,3,\ldots )$, and generated by the weights $2$ and $3$ fields. Subject to some mild constraints, all vertex algebras of type ${\mathcal {W}}(2,3,\ldots , N)$ for some $N$ can be obtained as quotients of this universal algebra. As an application, we show that for $n\geq 3$, the structure constants for the principal ${\mathcal {W}}$-algebras ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ are rational functions of $k$ and $n$, and we classify all coincidences among the simple quotients ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ for $n\geq 2$. We also obtain many new coincidences between ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ and other vertex algebras of type ${\mathcal {W}}(2,3,\ldots , N)$ which arise as cosets of affine vertex algebras or nonprincipal ${\mathcal {W}}$-algebras.

We show that, up to strong cocycle conjugacy, every countable exact group admits a unique equivariantly $\mathcal {O}_{2}$-absorbing, pointwise outer action on the Cuntz algebra $\mathcal {O}_{2}$ with the quasi-central approximation property (QAP). In particular, we establish the equivariant analogue of the Kirchberg $\mathcal {O}_{2}$-absorption theorem for these groups.

Consider the algebraic function $\Phi _{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi _{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $M$ is proper. As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq ~2$ we associate to an annular link $L$ a naive $\mathbb {Z}/r\mathbb {Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$-canonical bases for the highest weight integrable $\textbf U$-modules and their tensor products regarded as $\textbf {U}^\imath$-modules, as well as an $\imath$-canonical basis for the modified form of the $\imath$-quantum group $\textbf {U}^\imath$. A key new ingredient is a family of explicit elements called $\imath$-divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$. We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi-$K$-matrix and the constructions of $\imath$-canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.

Using the theory of ${\mathbf {FS}} {^\mathrm {op}}$ modules, we study the asymptotic behavior of the homology of ${\overline {\mathcal {M}}_{g,n}}$, the Deligne–Mumford compactification of the moduli space of curves, for $n\gg 0$. An ${\mathbf {FS}} {^\mathrm {op}}$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of ${\overline {\mathcal {M}}_{g,n}}$ the structure of an ${\mathbf {FS}} {^\mathrm {op}}$ module and bound its degree of generation. As a consequence, we prove that the generating function $\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$ is rational, and its denominator has roots in the set $\{1, 1/2, \ldots, 1/p(g,i)\},$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$. We also obtain restrictions on the decomposition of the homology of ${\overline {\mathcal {M}}_{g,n}}$ into irreducible $\mathbf {S}_n$ representations.

We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left-orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left-orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition.

We exploit the Galois symmetries of the little disks operads to show that many differentials in the Goodwillie–Weiss spectral sequences approximating the homology and homotopy of knot spaces vanish at a prime $p$. Combined with recent results on the relationship between embedding calculus and finite-type theory, we deduce that the $(n+1)$th Goodwillie–Weiss approximation is a $p$-local universal Vassiliev invariant of degree $\leq n$ for every $n \leq p + 1$.

We study the depth filtration on multiple zeta values, on the motivic Galois group of mixed Tate motives over $\mathbb {Z}$ and on the Grothendieck–Teichmüller group, and its relation to modular forms. Using period polynomials for cusp forms for $\mathrm {SL} _2(\mathbb {Z})$, we construct an explicit Lie algebra of solutions to the linearized double shuffle equations, which gives a conjectural description of all identities between multiple zeta values modulo $\zeta (2)$ and modulo lower depth. We formulate a single conjecture about the homology of this Lie algebra which implies conjectures due to Broadhurst and Kreimer, Racinet, Zagier, and Drinfeld on the structure of multiple zeta values and on the Grothendieck–Teichmüller Lie algebra.

La formule des traces relative de Jacquet–Rallis (pour les groupes unitaires ou les groupes linéaires généraux) est une identité entre des périodes des représentations automorphes et des distributions géométriques. Selon Jacquet et Rallis, une comparaison de ces deux formules des traces relatives devrait aboutir à une démonstration des conjectures de Gan–Gross–Prasad et Ichino–Ikeda pour les groupes unitaires. Les termes géométriques des groupes unitaires ou des groupes linéaires sont indexés par les points rationnels d'un espace quotient commun. Nous établissons que ces termes géométriques peuvent être vus comme des fonctionnelles sur des espaces d'intégrales orbitales semi-simples régulières locales. En outre, nous montrons que point par point ces distributions sont en fait égales, via l'identification des espaces d'intégrales orbitales locales donnée par le transfert et le lemme fondamental (essentiellement connus dans cette situation). Cela donne leur comparaison et cela clôt la partie géométrique du programme de Jacquet–Rallis. Notre résultat principal est donc un analogue de la stabilisation de la partie géométrique de la formule des traces due à Langlands, Kottwitz et Arthur.

In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support it by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $\mathrm {A}_n$ and $\mathrm {D}_n$, that is, flag varieties $\operatorname {Fl}(1,n;n+1)$ and isotropic orthogonal Grassmannians $\operatorname {OG}(2,2n)$; in particular, we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $\operatorname {OG}(2,2n)$ this is the first exceptional collection proved to be full.

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi–Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim (Y)\leq 5$ and $Y$ is not of product type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of Kawamata log terminal pairs $(X, \Delta )$ with $K_X+\Delta$ numerically trivial and not of product type, in dimension at most four.

Let $L$ be a very ample line bundle on a projective scheme $X$ defined over an algebraically closed field $\Bbbk$ with ${\rm char}\,\Bbbk \neq 2$. We say that $(X,L)$ satisfies property $\mathsf {QR}(k)$ if the homogeneous ideal of the linearly normal embedding $X \subset {\mathbb {P}} H^{0} (X,L)$ can be generated by quadrics of rank less than or equal to $k$. Many classical varieties, such as Segre–Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property $\mathsf {QR}(4)$. In this paper, we first prove that if ${\rm char}\,\Bbbk \neq 3$ then $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (d))$ satisfies property $\mathsf {QR}(3)$ for all $n \geqslant 1$ and $d \geqslant 2$. We also investigate the asymptotic behavior of property $\mathsf {QR}(3)$ for any projective scheme. Specifically, we prove that (i) if $X \subset {\mathbb {P}} H^{0} (X,L)$ is $m$-regular then $(X,L^{d} )$ satisfies property $\mathsf {QR}(3)$ for all $d \geqslant m$, and (ii) if $A$ is an ample line bundle on $X$ then $(X,A^{d} )$ satisfies property $\mathsf {QR}(3)$ for all sufficiently large even numbers $d$. These results provide affirmative evidence for the expectation that property $\mathsf {QR}(3)$ holds for all sufficiently ample line bundles on $X$, as in the cases of Green and Lazarsfeld's condition $\mathrm {N}_p$ and the Eisenbud–Koh–Stillman determininantal presentation in Eisenbud et al. [Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), 513–539]. Finally, when ${\rm char}\,\Bbbk = 3$ we prove that $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (2))$ fails to satisfy property $\mathsf {QR}(3)$ for all $n \geqslant 3$.

We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6, and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter-example of Malle's conjecture given by Klüners. We also prove new local uniformity results for ramified $S_5$ quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.

We investigate families of minimal rational curves on Schubert varieties, their Bott–Samelson desingularizations, and their generalizations constructed by Nicolas Perrin in the minuscule case. In particular, we describe the minimal families on small resolutions of minuscule Schubert varieties.

In the present article, we study the following problem. Let $\boldsymbol {G}$ be a linear algebraic group over $\mathbb {Q}$, let $\Gamma$ be an arithmetic lattice, and let $\boldsymbol {H}$ be an observable $\mathbb {Q}$-subgroup. There is a $H$-invariant measure $\mu _H$ supported on the closed submanifold $H\Gamma /\Gamma$. Given a sequence $(g_n)$ in $G$, we study the limiting behavior of $(g_n)_*\mu _H$ under the weak-$*$ topology. In the non-divergent case, we give a rather complete classification. We further supplement this by giving a criterion of non-divergence and prove non-divergence for arbitrary sequence $(g_n)$ for certain large $\boldsymbol {H}$. We also discuss some examples and applications of our result. This work can be viewed as a natural extension of the work of Eskin–Mozes–Shah and Shapira–Zheng.

The aim of this corrigendum is to correct an error in Corollary 10.7 to Theorem 10.6, one of the main results in the paper ‘On the cuspidal cohomology of $S$-arithmetic subgroups of reductive groups over number fields’. This makes necessary a thorough investigation of the conditions under which a Cartan-type automorphism exists on $G_1=\mathrm {Res}_{\mathbb {C}/\mathbb {R}}G_0$, where $G_0$ is a connected semisimple algebraic group defined over $\mathbb {R}$.