Let F be a number field, $\pi$ either a unitary cuspidal automorphic representation of $\mathrm{GL}(2)/F$ or a unitary Eisenstein series, and $\chi$ a unitary Hecke character of analytic conductor $C(\chi).$ We develop a regularized relative trace formula to prove a refined hybrid subconvex bound for $L(1/2,\pi\times\chi).$ In particular, we obtain the Burgess subconvex bound

\begin{align*}L(1/2,\pi\times\chi)\ll_{\pi,F,\varepsilon}C(\chi)^{\frac{1}{2}-\frac{1}{8}+\varepsilon}.\end{align*}

We propose a method for computing generating functions of genus-zero invariants of a gauged linear sigma model (GLSM) $(V, G, \theta, w)$. We show that certain derivatives of I-functions of quasimap invariants of $[V\mathbin{/\mkern-6mu/}_\theta G]$ produce I-functions (appropriately defined) of the GLSM. When G is an algebraic torus, we obtain an explicit formula for an I-function, and check that it agrees with previously computed I-functions in known special cases. Our approach is based on a new construction of these invariants that applies whenever the evaluation maps from the moduli space are proper, and includes insertions from light marked points, which may collide with each other and with basepoints.

Mapping class groups are known to admit geometric (proper, cobounded) actions on injective spaces. Starting with such an action, and relying only on geometric arguments, we show that all finite generating sets resulting from taking large enough balls in the respective injective space yield word metrics where pseudo-Anosov maps are exponentially generic. We also show that growth tightness holds true for the Cayley graphs corresponding to these finite generating sets, providing a positive answer to a question by Arzhantseva, Cashen and Tao (Growth tight actions, Pacific J. Math. 278 (2015), 1–49).

Kreck proved that two 2q-manifolds are stably diffeomorphic if and only if they admit normally bordant normal $(q{-}1)$-smoothings over the same normal $(q{-}1)$-type $(B,\xi)$. We show that ‘stably diffeomorphic’ can be replaced by ‘diffeomorphic’ if the normal smoothings have isomorphic Q-forms (consisting of the intersection form of the manifold and the induced homomorphism on $H_q$), when the manifolds are simply-connected, $q=2k$ is even and $H_q(B)$ is free. This proves a special case of Crowley’s Q-form conjecture. The basis of the proof is the construction of an extended surgery obstruction associated to a normal bordism. As an application, we identify the inertia group of a $(2k{-}1)$-connected 4k-manifold with the kernel of a certain bordism map. By the calculations of Senger and Zhang and earlier results, these kernels are now known in all cases. For $k=2,4$, the combination of these results determines the inertia groups. We also obtain, for a simply-connected 4k-manifold M with normal $(2k{-}1)$-type $(B,\xi)$ such that $H_{2k}(B)$ is free, an algebraic description of the stable class of M, that is, the set of diffeomorphism classes of manifolds stably diffeomorphic to M. Using this description, we explicitly compute the stable class of manifolds M with rank-2 hyperbolic intersection form.

We study the basic locus of Shimura varieties associated to the group of spinor similitudes of a quadratic space over ${{\mathbb{Q}}}$ with level structure given by the stabilizer of a vertex lattice. We give a description of the underlying reduced scheme of the associated Rapoport–Zink space, generalizing results of Howard–Pappas and Oki, in the case of self-dual, and almost self-dual, level structure.

This paper is concerned with local cohomology sheaves on generalized flag varieties supported in closed Schubert varieties, which carry natural structures as (mixed Hodge) $\mathcal{D}$-modules. We employ Kazhdan–Lusztig theory and Saito’s theory of mixed Hodge modules to describe a general strategy to calculate the simple composition factors, Hodge filtration, and weight filtration on these modules. Our main tool is the Grothendieck–Cousin complex, introduced by Kempf, which allows us to relate the local cohomology modules in question to parabolic Verma modules over the corresponding Lie algebra. We show that this complex underlies a complex of mixed Hodge modules, and is thus endowed with Hodge and weight filtrations. We execute this strategy to calculate the composition factors and weight filtration for Schubert varieties in the Grassmannian; in particular, showing that the weight filtration is controlled by the admissible augmented Dyck patterns of Raicu and Weyman. As an application, upon restriction to the opposite big cell, we recover the composition factors and weight filtration on local cohomology with support in generic determinantal varieties.

Let A be an abelian variety over a number field $\mathsf{K},$ with algebraic closure $\bar{\mathsf{K}}$. Assuming the Mumford–Tate conjecture for A, we show that the isogeny class of A over $\bar{\mathsf{K}}$ contains only finitely many isomorphism classes of bounded Faltings height. As the Mumford–Tate conjecture is known for many abelian varieties, our theorem is unconditional in those cases.

We establish several surprising rigidity results for étale groupoids associated with algebraic actions. Our initial motivation comes from algebraic number theory, where our results are witnessed in a particularly striking fashion. We prove that the groupoids associated with the action of the multiplicative monoid in a ring of algebraic integers on the additive group of the ring remember the ring up to isomorphism. Already, this special case resolves an open problem about isomorphisms of Cartan pairs and leads to a dynamical analogue of the Neukirch–Uchida theorem using topological full groups.

Given an $E_1$-ring A and a class $a \in \pi_{mk}(A)$ satisfying a suitable hypothesis, we define a map of $E_1$-rings $A\to A(\!\sqrt[m]{a})$ realizing the adjunction of an mth root of a. We define a form of logarithmic topological Hochschild homology (THH) for $E_1$-rings, and show that root adjunction is log-THH-étale for tamely ramified root adjunctions, which provides a formula for $\operatorname{THH}(A(\!\sqrt[m]{a}))$ in terms of THH and log THH of A. If A is connective, we prove that the induced map $K(A) \to K(A(\!\sqrt[m]{a}))$ in algebraic K-theory is the inclusion of a wedge summand. Using this, we compute $V(1)_*K(ko_p)$ for $p\gt 3$ and also deduce that if K(A) exhibits chromatic redshift, so does $K(A(\!\sqrt[m]{a}))$. We exhibit several maps of ring spectra as root adjunctions, and use this to obtain a new proof of the fact that Lubin–Tate spectra satisfy the redshift conjecture.

We develop a unified approach for identifying spaces of stability conditions of triangulated categories arising from weighted marked surfaces with moduli spaces of quadratic differentials. This identification is based on the use of perverse schobers (perverse sheaves of triangulated categories) and a notion of positive arc system kit on a perverse schober $\mathcal F$, which provides a systematic way of assigning to a graded curve on the surface a global section of $\mathcal F$. This assignment allows us to identify mixed angulations and their flips with finite-length hearts and their tilts. As an application we obtain a generalization of the results of Bridgeland and Smith to quadratic differentials with arbitrary singularity type (zero/pole/exponential).

We prove that all p-adic period domains (and their non-minuscule analogues) are geometrically connected. This answers a question of Hartl and has consequences for the geometry of Shimura and local Shimura varieties.

Nori’s Eisenstein cohomology classes and their integral refinements due to Beilinson, Kings and Levin can be used to obtain simple proofs of the rationality and integrality properties of special values of abelian L-functions of totally real fields. Here, we introduce an adelic refinement of these constructions. This will be used to establish new divisibility properties of Stickelberger elements associated to abelian extensions of totally real fields.

In this follow-up paper we show that smooth Hodge-proper stacks over $\mathcal O_K$ are $\mathbb Q_p$-locally acyclic: namely the natural map between étale $\mathbb Q_p$-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the $\mathbb Q_p$-case of general conjectures made in D. Kubrak and A. Prikhodko [p-adic Hodge theory for Artin stacks, Mem. Amer. Math. Soc. 304 (2024), 1174]. As a corollary, we get that if a smooth Artin stack over K has a smooth Hodge-proper model over $\mathcal O_K$, its $\mathbb Q_p$-étale cohomology is a crystalline Galois representation. We then also establish a truncated version of the above results in more general setting of smooth d-de Rham-proper stacks over $\mathcal O_K$: here we only require first d-de Rham cohomology groups be finitely generated over $\mathcal O_K$. As an application, we deduce a certain purity-type statement for étale $\mathbb Q_p$-cohomology of Raynaud generic fiber, as well as crystallinity of a first several étale cohomology groups in the presence of a Cohen–Macauley model over $\mathcal O_K$ in the schematic setting.

David Vogan proved that if G is a real reductive group, and if K is a maximal compact subgroup of G, then every irreducible representation of K is included as a minimal K-type in precisely one tempered, irreducible unitary representation of G with real infinitesimal character, and that moreover it is included there with multiplicity one, and is the unique minimal K-type in that representation. We shall prove that the Connes–Kasparov isomorphism in operator K-theory is equivalent to a K-theoretic version of Vogan’s result.

We introduce the notion of a naive global 2-ring: a functor from the opposite of the $\infty$-category of global spaces to presentably symmetric monoidal stable $\infty$-categories. By passing to global sections, every naive global 2-ring decategorifies to a multiplicative cohomology theory on global spaces, i.e. a naive global ring. We suggest when a naive global 2-ring deserves to be called genuine. As evidence, we associate to such a global 2-ring a family of equivariant cohomology theories which satisfy a version of the change-of-group axioms introduced by Ginzburg, Kapranov, and Vasserot [V. Ginzburg, M. Kapranov and E. Vasserot, Elliptic algebras and equivariant elliptic cohomology, Preprint (1995), arXiv:q-alg/9505012]. We further show that the decategorified multiplicative global cohomology theory associated to a genuine global 2-ring canonically refines to an $\mathbb{E}_\infty$-ring object in global spectra. As we show, two interesting examples of genuine global 2-rings are given by quasi-coherent sheaves on the torsion points of an oriented spectral elliptic curve and Lurie s theory of tempered local systems. In particular, we obtain global spectra representing equivariant elliptic cohomology and tempered cohomology.

In this work, we establish a local-smoothing result on metrics with small curvature concentration with respect to Sobolev constants and volume growth. As an application, we prove that the Gromov–Hausdorff limit of compact manifolds with bounded curvature concentration, Ahlfors n-regularity, and bounded Sobolev constants is a topological manifold away from finitely many points. In the complete non-compact case, we show that manifolds with Euclidean-type Sobolev inequality, Euclidean volume growth, and small curvature concentration are necessarily diffeomorphic to Euclidean spaces. In contrast with all previous works, we remove the Ricci-curvature condition.

We correct the proof of Corollary 6.21 in the paper The p-cyclic McKay correspondence via motivic integration [Compositio Math. 150 (2014), 1125–1168].