We present a comprehensive study of the geometry of Hilbert p-adic eigenvarieties at classical parallel weight 1 intersection points of their cuspidal and Eisenstein loci. For instance, we determine all such points at which the weight map is étale. The Galois theoretic approach presents genuine difficulties due to the lack of good deformation theory for pseudo-characters irregular at p and reflects the richness of the local geometry. We believe that our geometric results lead to deeper insight into the arithmetic of Hilbert automorphic forms and we produce in support several applications in Iwasawa theory.

We investigate the realisability of the Casson–Sullivan invariant for homeomorphisms of smooth 4-manifolds, which is the obstruction to a homeomorphism being stably pseudo-isotopic to a diffeomorphism, valued in the third cohomology of the source manifold with $\mathbb{Z}/2$-coefficients. We prove that for all pairs of orientable, homeomorphic, smooth 4-manifolds this invariant can be realised fully after stabilising with a single $S^2\times S^2$. As an application, we obtain that topologically isotopic surfaces in a smooth, simply-connected 4-manifold become smoothly isotopic after sufficient external stabilisations. We further demonstrate cases where this invariant can be realised fully without stabilisation for self-homeomorphisms, which includes for manifolds with finite cyclic fundamental group. This method allows us to produce many examples of homeomorphisms which are not stably pseudo-isotopic to any diffeomorphism but are homotopic to the identity. Finally, we reinterpret these results in terms of finding examples of smooth structures on 4-manifolds which are diffeomorphic but not stably pseudo-isotopic.

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 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.

We study the classification of $\mathbb{Z}$-DGAs (differential graded algebras) with polynomial homology $\mathbb{F}_p[x]$ with $\lvert x \rvert \gt0$, motivated by computations in algebraic K-theory. This classification problem was left open in work of Dwyer, Greenlees, and Iyengar. We prove that there are infinitely many such DGAs for even $\lvert x\rvert$ and that for $\lvert x \rvert \geq 2p-2$ any such DGA is formal as a ring spectrum. Through this, we obtain examples of triangulated categories with infinitely many DG enhancements and a classification of prime DG division rings. Combining our results with earlier work of the second author and Tamme, we obtain new (relative) algebraic K-theory computations for rings such as the mixed characteristic coordinate axes $\mathbb{Z}[x]/px$ and the group ring $\mathbb{Z}[C_{p^n}]$.

Let p be a prime number and K a finite unramified extension of $\mathbb{Q}_p$. Let $\pi$ be an admissible smooth mod p representation of $\operatorname{GL}_2(K)$ occurring in some Hecke eigenspaces of the mod p cohomology and $\overline{r}$ be its underlying global two-dimensional Galois representation. When $\overline{r}$ satisfies some Taylor–Wiles hypotheses and is sufficiently generic at p, we compute explicitly certain constants appearing in the diagram associated to $\pi$, generalizing the results of Dotto and Le. As a result, we prove that the associated étale $(\varphi,\mathcal{O}_K^{\times})$-module $D_A(\pi)$ defined by Breuil, Herzig, Hu, Morra and Schraen is explicitly determined by the restriction of $\overline{r}$ to the decomposition group at p, generalizing the results of Breuil et al. and the author.

We prove the Jacquet–Rallis fundamental lemma for spherical Hecke algebras over local function fields using multiplicative Hitchin fibrations. Our work is inspired by the proof by Zhiwei Yun in the Lie algebra case and builds upon the general framework of multiplicative Hitchin fibrations by the first-named author.

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).

Let k be a perfect field of characteristic p and $\Gamma$ an infinite, first countable pro-p group. We study the behavior of the p-primary part of the ‘motivic class group’, i.e. the full p-divisible group of the Jacobian, in any $\Gamma$-tower of function fields over k that is unramified outside a finite (possibly empty) set of places $\Sigma$, and totally ramified at every place of $\Sigma$. When $\Sigma=\emptyset$ and $\Gamma$ is a torsion-free p-adic Lie group, we obtain asymptotic formulae which show that the p-torsion class group schemes grow in a remarkably regular manner. In the ramified setting $\Sigma\neq\emptyset$, we obtain a similar asymptotic formula for the p-torsion in ‘physical class groups’, i.e. the k-rational points of the Jacobian, which generalizes the work of Mazur and Wiles, who studied the case $\Gamma=\mathbf Z_p$.

The landmark quantum complexity result MIP$^*$ $=$ RE has been used to prove the existence of a non-Connes-embeddable tracial von Neumann algebra. Recently, similar ideas have been used to give a negative solution to the Aldous–Lyons conjecture: there is a non-co-sofic invariant random subgroup (IRS) on any non-abelian free group. We define a notion of hyperlinearity for an IRS and show that there is a non-co-hyperlinear IRS on any non-abelian free group. As a corollary, we prove that there is a relation whose von Neumann algebra is not Connes embeddable. We do this by significantly simplifying the reduction of Aldous–Lyons to non-local games, removing the need for subgroup tests entirely.

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].

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.

We show that the category of ind-coherent sheaves on a quasi-smooth scheme is naturally tensored over the category of sheared D-modules on its shifted cotangent bundle, commuting with its natural action of categorified Hoschschild cochains. We prove that it defines a Morita equivalence as such. We then extend these results to quasi-smooth Artin stacks. As a consequence of our formalism, we are able to articulate a precise sense in which the space of unramified automorphic functions over a function field localizes over the stack of arithmetic Arthur parameters.

We determine the A(1)-homotopy of the topological cyclic homology of the connective real K-theory spectrum ko. The answer has an associated graded that is a free $\mathbb{F}_2[v_2^4]$-module of rank 52, on explicit generators in stems $-1 \leqslant \ast \leqslant 30$. The calculation is achieved by using prismatic and syntomic cohomology of ko as introduced by Hahn, Raksit and Wilson, extending work of Bhatt, Morrow and Scholze from the case of classical commutative rings to $\mathbb{E}_\infty$-rings. A new feature in our case is that there are nonzero differentials in the motivic spectral sequence from syntomic cohomology to topological cyclic homology.

We prove that the set of limit points of the set of all spectral gaps of closed arithmetic hyperbolic surfaces equals $[0,\frac{1}{4}]$.

A classical theorem of Siegel gives the average number of lattice points in bounded subsets of Euclidean space. Motivated by this, Veech introduced the Siegel–Veech formula for translation surfaces, describing the average number of saddle connections of bounded length. However, no such formula is known for flat surfaces with cone angles that are irrational multiples of $\pi$. A convex flat cone sphere is the Riemann sphere with a flat cone metric whose cone angles lie in $(0,2\pi)$. In this paper, we extend the Siegel–Veech theory to convex flat cone spheres by introducing generalized transforms that are uniformly bounded on moduli space and give rise to Siegel–Veech measures on the positive real line. We prove that these measures are absolutely continuous and piecewise real analytic. In contrast to translation surfaces, which benefit from $GL(2,\mathbb{R})$ dynamics, our approach instead uses tools from o-minimality. Finally, we analyze the asymptotic behavior of the Siegel–Veech measure near zero, obtaining results that parallel the computation of Siegel–Veech constants in the translation surface case.

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.

We show that the $L^1$ norm of an exponential sum of length X, and with coefficients equal to the Liouville or Möbius function, is at least $\gg_{\varepsilon} X^{1/4 - \varepsilon}$ for any given $\varepsilon$. For the Liouville function, this improves on the lower bound $\gg X^{1/\log\log X}$ due to Balog and Perelli. For the Möbius function, this improves the lower bound $\gg X^{1/6}$ due to Balog and Ruzsa. The large discrepancy between these lower bounds is due to the method employed by Balog and Ruzsa, as it crucially relies on the vanishing of $\mu(n)$. Instead, our proof puts the two cases on an equal footing by exploiting the connection of these coefficients with zeros of Dirichlet L-functions. In the second paper in this series, we will obtain a lower bound $\gg X^{\delta}$ for some small $\delta$ but for general (non-pretentious) multiplicative functions.

We describe the $\mathbb{F}_p$-cohomology of the congruence subgroups $\mathrm{SL}_n(\mathbb{Z}, p^m)$ in degrees $* \lt p-1$, for all large enough n, establishing a formula proposed by F. Calegari. Along the way, we also establish a formula for the stable cohomology of $\mathrm{SL}_n(\mathbb{Z}/p)$ with certain twisted coefficients.

A Laurent polynomial in two variables is tempered if its edge polynomials are cyclotomic. Variation of coefficients leads to a family of smooth complete genus-g curves carrying a canonical algebraic $K_2$-class over a g-dimensional base S, hence to an extension of admissible variations of mixed Hodge structure (or normal function) on S. We prove that the $\mathbb{R}$-split locus of this extension is finite. Consequently, the torsion locus of the normal function and the A-factor locus for the family of curves are also finite.