Let $K$ be a complete discretely valued field of mixed characteristic $(0, p)$ with perfect residue field. We prove that the category of prismatic $F$-crystals on $\mathcal{O}_K$ is equivalent to the category of lattices in crystalline $G_K$-representations.
For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from classical Deligne–Lusztig theory by using the loop space functor. We study this construction in the special case that $G$ is an inner form of $\mathrm{GL}_n$ and the loop Deligne–Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its $\ell$-adic cohomology realizes many irreducible supercuspidal representations of $G$, notably almost all among those whose L‑parameter factors through an unramified elliptic maximal torus of $G$. This gives a purely local, purely geometric and—in a sense—quite explicit way to realize special cases of the local Langlands and Jacquet–Langlands correspondences.
In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $\Sigma$ is a closed minimal surface embedded in a Riemannian three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a suitable family of LD solutions on $\Sigma$, a minimal surface $\breve{M}$ resembling two copies of $\Sigma$ joined by many small catenoidal bridges can be constructed by PDE gluing methods. (An LD solution $\varphi$ on $\Sigma$ is a singular solution of the Jacobi equation with logarithmic singularities which in the construction are replaced by catenoidal bridges.) We also determine the first nontrivial term in the expansion for the area ${\lvert \breve{M} \rvert}$ of $\breve{M}$ in terms of the sizes of its catenoidal bridges and confirm that it is negative; ${\lvert \breve{M} \rvert} \lt 2 {\lvert \Sigma \rvert}$ follows.
We demonstrate the applicability of the theorem by first constructing new doublings of the Clifford torus. We then construct in Part II families of LD solutions for general $(O(2) \times \mathbb{Z}_2)$-symmetric backgrounds $(\Sigma, N, g)$. Combining with the theorem in Part I this implies the construction of new minimal doublings for such backgrounds. (Constructions for general backgrounds remain open.) This generalizes our earlier work for $\Sigma = \mathbb{S}^2 \subset N = \mathbb{S}^3$ providing new constructions even in that case.
In Part III, applying the results of Parts I and II—appropriately modified for the catenoid and the critical catenoid—we construct new self-shrinkers of the mean curvature flow via doubling the spherical self-shrinker or the Angenent torus, new complete embedded minimal surfaces of finite total curvature in the Euclidean three-space via doubling the catenoid, and new free boundary minimal surfaces in the unit ball via doubling the critical catenoid.
In this paper we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $\Gamma$ and a positive integral level $\ell$ under the assumption “$\Gamma$ preserves a Borel”. For $\Gamma = \mathbb{Z} / 2$ and double covers of $\mathbb{P}^1$, this formula was conjectured by Birke–Fuchs–Schweigert [23]. As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $\Gamma$-crossed modular fusion category as defined by Turaev.
We relate these two versions of the Verlinde formula, by formulating the notion of a $\Gamma$-crossed modular functor and show that it is very closely related to the notion of a $\Gamma$-crossed modular fusion category.We compute the Atiyah algebra and prove (with the same assumptions) that the bundles of $\Gamma$-twisted conformal blocks associated with a twisted affine Lie algebra define a $\Gamma$-crossed modular functor.
We also prove a useful criterion for rigidity of weakly fusion categories to deduce that the level $\ell$$\Gamma$-twisted conformal blocks define a $\Gamma$-crossed modular fusion category. Along the way, we prove the equivalence between a $\Gamma$-crossed modular functor and its topological analogue. We then apply these results to derive the Verlinde formula for twisted conformal blocks. We also describe the $\mathrm{S}$-matrices of the $\Gamma$-crossed modular fusion categories associated with twisted conformal blocks.
We prove that many representations $\overline{\rho} : \mathrm {Gal}(\overline{K}/K) \to \mathrm {GL}_2 (\mathbb{F}_3)$, where $K$ is a CM field, arise from modular elliptic curves. We prove similar results when the prime $p=3$ is replaced by $p=2$ or $p=5$. As a consequence, we prove that a positive proportion of elliptic curves over any CM field not containing a 5th root of unity are modular.
We study the Milnor–Witt motives which are a finite direct sum of $\mathbb{Z}(q)[p]$ and $\mathbb{Z}/\eta(q)[p]$. We show that for MW‑motives of this type, we can determine an MW‑motivic cohomology class in terms of a motivic cohomology class and a Witt cohomology class. We define the motivic Bockstein cohomology and show that it corresponds to subgroups of Witt cohomology, if the MW‑motive splits as above. As an application, we give the splitting formula of Milnor–Witt motives of Grassmannian bundles and complete flag bundles. This in particular shows that the integral cohomology of real complete flags has only 2‑torsions.
We classify a natural collection of $GL(2,\mathbb{R})$-invariant subvarieties, which includes loci of double covers, the orbits of the Eierlegende–Wollmilchsau, Ornithorynque, and Matheus–Yoccoz surfaces, and loci appearing naturally in the study of the complex geometry of Teichmüller space. This classification is the key input in subsequent work of the authors that classifies “high rank” invariant subvarieties, and in subsequent work of the first author that classifies certain invariant subvarieties with “Lyapunov spectrum as degenerate as possible”. We also derive applications to the complex geometry of Teichmüller space and construct new examples, which negatively resolve two questions of Mirzakhani and Wright and illustrate previously unobserved phenomena for the finite blocking problem.
We employ min-max techniques to show that the unit ball in $\mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
For a fixed regular cone in Euclidean space with small entropy we show that all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class.
This paper is a culmination of [CM21] on the study of multiple zeta values (MZV’s) over function fields in positive characteristic. For any finite place $v$ of the rational function field $k$ over a finite field, we prove that the $v$‑adic MZV’s satisfy the same $\overline{k}$‑algebraic relations that their corresponding $\infty$‑adic MZV’s satisfy. Equivalently, we show that the $v$‑adic MZV’s form an algebra with multiplication law given by the $q$‑shuffle product which comes from the $\infty$‑adic MZV’s, and there is a well-defined $\overline{k}$‑algebra homomorphism from the $\infty$‑adic MZV’s to the $v$‑adic MZV’s.