Building on Beilinson’s work, ‘constructible sheaves are holonomic’, we introduce the notion of holonomicity for étale sheaves, without assuming a priori constructibility. We establish the converse of Beilinson’s result, showing that holonomic sheaves are indeed constructible. This can be seen as an étale analogue of Kashiwara’s theorem on holonomic ${\mathcal D}_X$-modules.

We compute the $K$-theory of the ${C}^{*}$-category generated by order zero, equivariant, properly supported, classical pseudodifferential operators acting on sections of homogeneous bundles over the symmetric space of a real reductive Lie group $G$. Our result uses the Connes-Kasparov isomorphism for $G$, and in fact it is equivalent to the Connes-Kasparov isomorphism. We relate our computation to David Vogan’s well-known parametrization of the tempered irreducible representations of $G$ with real infinitesimal character. When the reductive group $G$ has real rank one, we formulate and prove a Fourier isomorphism theorem for equivariant order zero pseudodifferential operators on the symmetric space, and use it to prove a $K$-theoretic version of Vogan’s theorem.

We compute a Hirzebruch-Riemann-Roch type formula for the invariant Riemann-Roch number of a quantizable Hamiltonian $S^1$-manifold $(M,\omega ,{ \mathcal J})$, allowing $0$ to be a singular value of the moment map ${ \mathcal J}:M\to {\mathbb R}$. Our formula represents an instance of the Guillemin-Sternberg principle, which states that quantization should commute with reduction. The conceptual novelty of our result is that the involved reduced system only depends on the symplectic data of M. To establish this, we derive a complete singular stationary phase expansion of the Witten integral without appealing to any kind of desingularization. As a consequence, our formula expresses the invariant Riemann-Roch number purely in terms of symplectic invariants of the singular symplectic quotient. In particular, it involves a new explicit symplectic invariant of the singularities.

This work is the second in a series, following Part I [Kwa24] (Algebra Number Theory 18.10 (2024)) and preceding Part III [Kwa25] (Math. Ann. 391.1 (2025)). We continue our investigation of spectral moments of $\text {GL}(3)\times \text {GL}(2)\ L$-functions from the perspective of period integrals. Using an identity between two distinct periods for the $\text {GL}(3)$ Eisenstein series, we establish an exact Motohashi-type identity linking the shifted cubic moment of $\text {GL}(2)\ L$-functions to the shifted fourth moment of $\text {GL}(1) L$-functions. In addition, we offer a novel, intrinsic and automorphic account for the sources and symmetries of the full set of main terms for both moments, in agreement with the CFKRS Moment Conjectures (Proc. Lond. Math. Soc. (3) 91 (2005)).

Given a smooth variety X over $\mathbb {C}$, a smooth divisor $i:Y\hookrightarrow X$ and a global function f on X which vanishes on Y and on its critical locus, we compute the map induced on Hochschild homology by the pushforward functor $i_{\ast }:D^b(Y)\to D^{abs}(MF(X,f))$ in terms of the Hochschild-Kostant-Rosenberg isomorphisms.

The rational homology of the IA-automorphism group $\operatorname {IA}_n$ of the free group $F_n$ is still mysterious. We study the quotient of the rational homology of $\operatorname {IA}_n$ that is obtained as the image of the map induced by the abelianization map, which we call the Albanese homology of $\operatorname {IA}_n$. We obtain a representation-stable $\operatorname {GL}(n,\mathbb {Q})$-subquotient of the Albanese homology of $\operatorname {IA}_n$, which conjecturally coincides with the entire Albanese homology of $\operatorname {IA}_n$. In particular, we obtain a lower bound of the dimension of the Albanese homology of $\operatorname {IA}_n$ for each homological degree in a stable range. Moreover, we determine the entire third Albanese homology of $\operatorname {IA}_n$ for $n\ge 9$. We also study the Albanese homology of an analogue of $\operatorname {IA}_n$ for the outer automorphism group of $F_n$ and the Albanese homology of the Torelli groups of surfaces. Moreover, we study the relation between the Albanese homology of $\operatorname {IA}_n$ and the cohomology of $\operatorname {Aut}(F_n)$ with twisted coefficients.

In the Morel-Voevodsky motivic stable homotopy category of a quasi-compact quasi-separated scheme S, several candidates exist for a motivic spectrum representing hermitian K-theory. This note shows that the cellular absolute motivic spectrum constructed in [9] via the geometry of orthogonal and hyperbolic Grassmannians over S coincides with the motivic ring spectrum constructed in [4].

We further develop the abstract representation theory of affine Hecke algebras with arbitrary positive parameters. We establish analogues of several results that are known for reductive p-adic groups. These include: the relation between parabolic induction/restriction and Hermitian duals, Bernstein’s second adjointness and generalizations of the Langlands classification. We check that, in the known cases of equivalences between module categories of affine Hecke algebras and Bernstein blocks for reductive p-adic groups, such equivalences preserve Hermitian duality.

We also initiate the study of generic representations of affine Hecke algebras. Based on an analysis of the Hecke algebras associated to generic Bernstein blocks for quasi-split reductive p-adic groups, we propose a fitting definition of genericity for modules over affine Hecke algebras. With that notion we prove special cases of the generalized injectivity conjecture, about generic subquotients of standard modules for affine Hecke algebras.

We prove that the positive-dimensional part of the torsion locus of the Ceresa normal function in $\mathcal {M}_g$ is not Zariski dense when $g\geq 3$. Moreover, it has only finitely many components with generic Mumford-Tate group equal to $\mathrm {GSp}_{2g}$; these components are defined over $\overline {\mathbb Q}$, and their union is closed under the action of $\mathrm {Gal}(\overline {\mathbb Q}/\mathbb Q)$. More generally, we study the distribution of the torsion locus of arbitrary admissible normal functions.

We give a new proof of a result by Fathi, which states that, to any homeomorphism of a closed surface which is isotopic to a pseudo-Anosov homeomorphism, we can associate a stable and an unstable invariant partition of the surface with properties which are similar to the unstable and the stable foliation of a pseudo-Anosov homeomorphism.

Let M be a pinched negatively curved Riemannian orbifold, whose fundamental group has torsion of order $2$. Generalising results of Sarnak and Erlandsson-Souto for constant curvature oriented surfaces, and with very different techniques, we give an asymptotic counting result on the number of strongly reversible periodic orbits of the geodesic flow in $T^1M$, and prove their equidistribution towards the Bowen-Margulis measure. The result holds in the more general setting with weights coming from thermodynamic formalism, and also in the analogous setting of graphs of groups with $2$-torsion. We give new examples in real hyperbolic Coxeter groups, complex hyperbolic orbifolds and graphs of groups.

The integral identity conjecture of Kontsevich and Soibelman plays an important role in proving the existence of motivic Donaldson-Thomas invariants for three-dimensional noncommutative Calabi-Yau manifolds. There are a number of different formulations of this conjecture in different contexts, and accordingly, there are corresponding solutions to them. The methods devoted to solving this conjecture are diverse, ranging from $\ell $-adic cohomology of rigid analytic varieties to Hrushovski-Kazhdan motivic integration and motivic Fubini theorem for tropicalization maps,... In [Ivo24], Ivorra deduces a functorial version of the integral identity in the motivic stable homotopy categories of schemes, from the Braden hyperbolic localization theorem. This functorial version concerns Ayoub’s nearby cycles functor associated with a $\mathbb {G}_m$-equivariant function $f \colon \mathbb {V}(\mathcal {E}) \longrightarrow \mathbb {A}^1$ on a vector bundle $\mathbb {V}(\mathcal {E})$ over a field of characteristic zero. In the present work, we follow the functorial approach from [Ivo24] and extend the scope of the original conjecture by Kontsevich and Soibelman by studying more generally the case of $\mathbb {G}_m$-equivariant functions on algebraic S-spaces with a $\tau $-locally linearizable action of $\mathbb {G}_m$ over a noetherian base scheme S.

An exoflop takes a gauged Landau-Ginzburg (LG) model, partially compactifies it, and then performs certain birational transformations on it. When certain criteria hold, this can provide a crepant categorical resolution or equivalence of derived categories associated to the gauged LG models. We provide sufficient criteria for when this provides categorical resolutions for (or equivalences between) certain complete intersections in toric stacks.

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.

We continue our study of Ulam’s measure problem. In contrast to our previous works, we shift our focus from measures stratified by their additivity, to measures stratified by their indecomposability. The breakthrough here is obtained by replacing the classical ‘least’ function associated with ideals by a two-dimensional ‘last’ function associated with walks on ordinals. Consequently, we obtain conditions under which a measure admits not just infinite pairwise disjoint families of positive sets, but in fact families of maximum possible size. As an application we solve a problem left open in Shelah’s Cardinal Arithmetic book, proving that for every weakly inaccessible cardinal $\kappa $, if there exists a stationary subset of $\kappa $ that does not reflect at regulars, then the strong Ramsey relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ holds.

For a finite extension F of ${\mathbb Q}_p$ and $n \geq 1$, we show that the category of Lubin-Tate bundles on the $(n-1)$-dimensional Drinfeld symmetric space is equivalent to the category of finite-dimensional smooth representations of the group of units of the division algebra of invariant $1/n$ over F.

We establish several new properties of the p-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze’s basic finiteness theorems, prove a duality theorem, and show a kind of partial Künneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results.

Our key new tool is the six functor formalism with solid almost $\mathcal {O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the domain of validity of the $!$-functor formalism developed in [Man22] to allow certain ‘stacky’ maps. In the language of this extended formalism, we show that if G is a p-adic Lie group, the structure map of the classifying small v-stack $B\underline {G}$ is p-cohomologically smooth.

A closed Riemannian three-manifold $(Y,g)$ equipped with a torsion spin$^c$ structure determines a family of Dirac operators $\{D_B\}$ parametrized by a $b_1(Y)$-dimensional torus $\mathbb {T}_Y$. In this paper, we develop techniques to study how the topology of the locus $\mathsf {K}\subset \mathbb {T}_Y$ corresponding to operators with non-trivial kernel (the three-dimensional analogue of the theta divisor of a Riemann surface) depends on the geometry of the metric. As a concrete example of our methods, we show that for any metric on the three-torus $Y=T^3$ for which the spectral gap $\lambda _1^*$ on coexact $1$-forms is large, after a small perturbation of the family, the locus $\mathsf {K}$ is a two-sphere.

While the result only involves linear operators, its proof relies on the non-linear analysis of the Seiberg-Witten equations. It follows from a more general understanding of transversality in the context of the monopole Floer homology of a torsion spin$^c$ three-manifold $(Y,\mathfrak {s})$ with a large $\lambda _1^*$. When $b_1>0$, this gives rise to a very rich setup and we discuss a framework to describe explicitly in certain situations the Floer homology groups of $(Y,\mathfrak {s})$ in terms of the topology of the family of Dirac operators $\{D_B\}$.

In recent years, b-symplectic manifolds have emerged as important objects in symplectic geometry. These manifolds are Poisson manifolds that exhibit symplectic behaviour away from a distinguished hypersurface, where the symplectic form degenerates in a controlled manner. Inspired by this rich landscape, E-structures were introduced by Nest and Tsygan in [NT01] as a comprehensive framework for exploring generalizations of b-structures. This paper initiates a deeper investigation into their Poisson facets, building on foundational work by [MS21]. We also examine the closely related concept of almost regular Poisson manifolds, as studied in [AZ17], which reveals a natural Poisson groupoid associated with these structures.

In this article, we investigate the intricate relationship between E-structures and almost regular Poisson structures. Our comparative analysis not only scrutinizes their Poisson properties but also offers explicit formulae for the Poisson structure on the Poisson groupoid associated to the E-structures as both Poisson manifolds and singular foliations. In doing so, we reveal an interesting link between the existence of commutative frames and Darboux-Carathéodory-type expressions for the relevant structures.

The compactly supported $\mathbb {A}^1$-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm {K}_0(\mathrm {Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm {GW}(k)$ of the base field k. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on $\mathrm {GW}(k)$ and show that the compactly supported $\mathbb {A}^1$-Euler characteristic respects these two power structures for $0$-dimensional varieties, or equivalently étale k-algebras. In this paper, we define the class $\mathrm {Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb {A}^1$-Euler characteristic respects the power structures and study the algebraic properties of the subring $\mathrm {K}_0(\mathrm {Sym}_k)$ of symmetrisable varieties. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb {A}^1$-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.