We consider mod $p$ Hilbert modular forms associated to a totally real field of degree $d$ in which $p$ is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a $d$-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren–Oort stratification on mod $p$ Hilbert modular varieties established by Goren and Oort, and Tian and Xiao.

We compute the regulator of the Beilinson–Deninger–Scholl elements in terms of special values of $L$ -functions of modular forms. The main tool is the Rogers–Zudilin method.

Let $X$ be an irreducible complex-analytic variety, ${\mathcal{S}}$ a stratification of $X$ and ${\mathcal{F}}$ a holomorphic vector bundle on the open stratum ${X\unicode[STIX]{x0030A}}$. We give geometric conditions on ${\mathcal{S}}$ and ${\mathcal{F}}$ that produce a natural lift of the Chern class $\operatorname{c}_{k}({\mathcal{F}})\in H^{2k}({X\unicode[STIX]{x0030A}};\mathbb{C})$ to $H^{2k}(X;\mathbb{C})$, which, in the algebraic setting, is of Hodge level ${\geqslant}k$. When applied to the Baily–Borel compactification $X$ of a locally symmetric variety ${X\unicode[STIX]{x0030A}}$ and an automorphic vector bundle ${\mathcal{F}}$ on ${X\unicode[STIX]{x0030A}}$, this refines a theorem of Goresky–Pardon. In passing we define a class of simplicial resolutions of the Baily–Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake ($=$ Baily–Borel) compactification of ${\mathcal{A}}_{g}$ contains nontrivial Tate extensions.

By constructing suitable Borcherds forms on Shimura curves and using Schofer’s formula for norms of values of Borcherds forms at CM points, we determine all of the equations of hyperelliptic Shimura curves $X_{0}^{D}(N)$. As a byproduct, we also address the problem of whether a modular form on Shimura curves $X_{0}^{D}(N)/W_{D,N}$ with a divisor supported on CM divisors can be realized as a Borcherds form, where $X_{0}^{D}(N)/W_{D,N}$ denotes the quotient of $X_{0}^{D}(N)$ by all of the Atkin–Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.

We establish a connection between motivic cohomology classes over the Siegel threefold and non-critical special values of the degree-four $L$-function of some cuspidal automorphic representations of $\text{GSp}(4)$. Our computation relies on our previous work [On higher regulators of Siegel threefolds I: the vanishing on the boundary, Asian J. Math. 19 (2015), 83–120] and on an integral representation of the $L$-function due to Piatetski-Shapiro.

We prove a general formula for the $p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$. The formula is in terms of the cyclotomic derivative of a Rankin–Selberg $p$-adic $L$-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author to the context of the work of Yuan–Zhang–Zhang on the archimedean Gross–Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is $+1$ rather than $-1$, by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies one divisibility in a $p$-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

This is the first of three papers in which we give a moduli interpretation of the second flip in the log minimal model program for $\overline{M}_{g}$, replacing the locus of curves with a genus $2$ Weierstrass tail by a locus of curves with a ramphoid cusp. In this paper, for $\unicode[STIX]{x1D6FC}\in (2/3-\unicode[STIX]{x1D716},2/3+\unicode[STIX]{x1D716})$, we introduce new $\unicode[STIX]{x1D6FC}$-stability conditions for curves and prove that they are deformation open. This yields algebraic stacks $\overline{{\mathcal{M}}}_{g}(\unicode[STIX]{x1D6FC})$ related by open immersions $\overline{{\mathcal{M}}}_{g}(2/3+\unicode[STIX]{x1D716}){\hookrightarrow}\overline{{\mathcal{M}}}_{g}(2/3){\hookleftarrow}\overline{{\mathcal{M}}}_{g}(2/3-\unicode[STIX]{x1D716})$. We prove that around a curve $C$ corresponding to a closed point in $\overline{{\mathcal{M}}}_{g}(2/3)$, these open immersions are locally modeled by variation of geometric invariant theory for the action of $\text{Aut}(C)$ on the first-order deformation space of $C$.

We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker–Fourier coefficients of automorphic representations. For $\text{GL}_{n}(\mathbb{F})$ this implies that a smooth admissible representation $\unicode[STIX]{x1D70B}$ has a generalized Whittaker model ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ corresponding to a nilpotent coadjoint orbit ${\mathcal{O}}$ if and only if ${\mathcal{O}}$ lies in the (closure of) the wave-front set $\operatorname{WF}(\unicode[STIX]{x1D70B})$. Previously this was only known to hold for $\mathbb{F}$ non-archimedean and ${\mathcal{O}}$ maximal in $\operatorname{WF}(\unicode[STIX]{x1D70B})$, see Moeglin and Waldspurger [Modeles de Whittaker degeneres pour des groupes p-adiques, Math. Z. 196 (1987), 427–452]. We also express ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ as an iteration of a version of the Bernstein–Zelevinsky derivatives [Bernstein and Zelevinsky, Induced representations of reductive p-adic groups. I., Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 441–472; Aizenbud et al., Derivatives for representations of$\text{GL}(n,\mathbb{R})$and$\text{GL}(n,\mathbb{C})$, Israel J. Math. 206 (2015), 1–38]. This enables us to extend to $\text{GL}_{n}(\mathbb{R})$ and $\text{GL}_{n}(\mathbb{C})$ several further results by Moeglin and Waldspurger on the dimension of ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ and on the exactness of the generalized Whittaker functor.

Suppose that $F/F^{+}$ is a CM extension of number fields in which the prime $p$ splits completely and every other prime is unramified. Fix a place $w|p$ of $F$. Suppose that $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$ is a continuous irreducible Galois representation such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. If $\overline{r}$ is automorphic, and some suitable technical conditions hold, we show that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ can be recovered from the $\text{GL}_{3}(F_{w})$-action on a space of mod $p$ automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for $\overline{r}$, show the existence of an ordinary lifting of $\overline{r}$, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations $\overline{r}$ to which our main theorem applies.

Using a new approach starting with a residue computation, we sharpen some of the known estimates for the counting function of friable integers. The improved accuracy turns out to be crucial for various applications, some of which concern fundamental questions in probabilistic number theory.

In 2005, Knutson–Vakil conjectured a puzzle rule for equivariant $K$-theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors’ recently proved tableau rule for the same Schubert calculus problem.

In this paper we study categories ${\mathcal{O}}$ over quantizations of symplectic resolutions admitting Hamiltonian tori actions with finitely many fixed points. In this generality, these categories were introduced by Braden et al. We establish a family of standardly stratified structures (in the sense of the author and Webster) on these categories ${\mathcal{O}}$ . We use these structures to study shuffling functors of Braden et al. (called cross-walling functors in this paper). Most importantly, we prove that all cross-walling functors are derived equivalences that define an action of the Deligne groupoid of a suitable real hyperplane arrangement.

Let $A$ be a complete local ring with a coefficient field $k$ of characteristic zero, and let $Y$ be its spectrum. The de Rham homology and cohomology of $Y$ have been defined by R. Hartshorne using a choice of surjection $R\rightarrow A$ where $R$ is a complete regular local $k$-algebra: the resulting objects are independent of the chosen surjection. We prove that the Hodge–de Rham spectral sequences abutting to the de Rham homology and cohomology of $Y$, beginning with their $E_{2}$-terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of finite-dimensional $k$-spaces. These $E_{2}$-terms therefore provide invariants of $A$ analogous to the Lyubeznik numbers. As part of our proofs we develop a theory of Matlis duality in relation to ${\mathcal{D}}$-modules that is of independent interest. Some of the highlights of this theory are that if $R$ is a complete regular local ring containing $k$ and ${\mathcal{D}}={\mathcal{D}}(R,k)$ is the ring of $k$-linear differential operators on $R$, then the Matlis dual $D(M)$ of any left ${\mathcal{D}}$-module $M$ can again be given a structure of left ${\mathcal{D}}$-module, and if $M$ is a holonomic ${\mathcal{D}}$-module, then the de Rham cohomology spaces of $D(M)$ are $k$-dual to those of $M$.

We prove that a very general smooth cubic fourfold containing a plane can be embedded into an irreducible holomorphic symplectic eightfold as a Lagrangian submanifold. We construct the desired irreducible holomorphic symplectic eightfold as a moduli space of Bridgeland stable objects in the derived category of the twisted K3 surface corresponding to the cubic fourfold containing a plane.

Let $F$ be a non-discrete non-Archimedean locally compact field and ${\mathcal{O}}_{F}$ the ring of integers in $F$ . The main results of this paper are the classification of ergodic probability measures on the space $\text{Mat}(\mathbb{N},F)$ of infinite matrices with entries in $F$ with respect to the natural action of the group $\text{GL}(\infty ,{\mathcal{O}}_{F})\times \text{GL}(\infty ,{\mathcal{O}}_{F})$ and the classification, for non-dyadic $F$ , of ergodic probability measures on the space $\text{Sym}(\mathbb{N},F)$ of infinite symmetric matrices with respect to the natural action of the group $\text{GL}(\infty ,{\mathcal{O}}_{F})$ .

We study cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of $\operatorname{Ext}^{1}$ as intersection numbers of tagged curves and Auslander–Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.

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.

We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the algebra of symmetries$\mathscr{S}(\Box ^{r})$ of powers of the Laplacian $\Box$ on $\mathbb{C}^{n}$. The connection is made through the construction of a highest-weight representation of $\mathfrak{g}$ via the ring of differential operators ${\mathcal{D}}(X)$ on the singular scheme $X=(\mathtt{F}^{r}=0)\subset \mathbb{C}^{n}$, for $\mathtt{F}=\sum _{j=1}^{n}X_{i}^{2}\in \mathbb{C}[X_{1},\ldots ,X_{n}]$. In particular, we prove that $U(\mathfrak{g})/K_{r}\cong \mathscr{S}(\Box ^{r})\cong {\mathcal{D}}(X)$ for a certain primitive ideal $K_{r}$. Interestingly, if (and only if) $n$ is even with $r\geqslant n/2$, then both $\mathscr{S}(\Box ^{r})$ and its natural module ${\mathcal{A}}=\mathbb{C}[\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n},\ldots ,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n}]/(\Box ^{r})$ have a finite-dimensional factor. The same holds for the ${\mathcal{D}}(X)$-module ${\mathcal{O}}(X)$. We also study higher-dimensional analogues $M_{r}=\{x\in A:\Box ^{r}(x)=0\}$ of the module of harmonic elements in $A=\mathbb{C}[X_{1},\ldots ,X_{n}]$ and of the space of ‘harmonic densities’. In both cases we obtain a minimal $\mathfrak{g}$-representation that is closely related to the $\mathfrak{g}$-modules ${\mathcal{O}}(X)$ and ${\mathcal{A}}$. Essentially all these results have real analogues, with the Laplacian replaced by the d’Alembertian $\Box _{p}$ on the pseudo-Euclidean space $\mathbb{R}^{p,q}$ and with $\mathfrak{g}$ replaced by the real Lie algebra $\mathfrak{so}(p+1,q+1)$.

We determine the additional structure which arises on the classical limit of a DQ-algebroid.

We prove a general criterion for an algebraic stack to admit a good moduli space. This result may be considered as a generalization of the Keel–Mori theorem, which guarantees the existence of a coarse moduli space for a separated Deligne–Mumford stack. We apply this result to prove that the moduli stacks $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ parameterizing $\unicode[STIX]{x1D6FC}$-stable curves introduced in [J. Alper et al., Second flip in the Hassett–Keel program: a local description, Compositio Math. 153 (2017), 1547–1583] admit good moduli spaces.