Volume 2022, Issue 785

We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of dimension 3 or higher, thanks to the universality of the local Weyl law. The key technical step is an estimate capturing some cancellation in a triple integral of Bessel functions, which we achieve using Gegenbauer’s addition formula.

We consider the Kisin variety associated to an n -dimensional absolutely irreducible mod p Galois representation ρ¯{\bar{\rho}} of a p -adic field K together with a cocharacter μ. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin’s conjecture holds if K is totally ramified with n=3{n=3} or μ is of a very particular form. As an application, we get a connectedness result for the deformation ring associated to ρ¯{\bar{\rho}} of given Hodge–Tate weights. We also give counterexamples to show Kisin’s conjecture does not hold in general.

We establish a version “over the ring” of the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in k+n{k+n} variables, with coefficients in ℤ{\mathbb{Z}}, of positive degree in the last n variables, we show that if they are irreducible over ℤ{\mathbb{Z}} and satisfy a necessary “Schinzel condition”, then the first k variables can be specialized in a Zariski-dense subset of ℤk{\mathbb{Z}^{k}} in such a way that irreducibility over ℤ{\mathbb{Z}} is preserved for the polynomials in the remaining n variables. The Schinzel condition, which comes from the Schinzel Hypothesis, is that, when specializing the first k variables in ℤk{\mathbb{Z}^{k}}, the product of the polynomials should not always be divisible by some common prime number. Our result also improves on a “coprime” version of the Schinzel Hypothesis: under some Schinzel condition, coprime polynomials assume coprime values. We prove our results over many other rings than ℤ{\mathbb{Z}}, e.g. UFDs and Dedekind domains.

In this paper, we solve a new elliptic-parabolic system arising in geometric analysis that is motivated by the nonlinear supersymmetric sigma model of quantum field theory. The corresponding action functional involves two fields, a map from a Riemann surface into a Riemannian manifold and a spinor coupled to the map. The first field has to satisfy a second-order elliptic system, which we turn into a parabolic system so as to apply heat flow techniques. The spinor, however, satisfies a first-order Dirac-type equation. We carry that equation as a nonlinear constraint along the flow. With this novel scheme, in more technical terms, we can show the existence of Dirac-harmonic maps from a compact spin Riemann surface with smooth boundary to a general compact Riemannian manifold via a heat flow method when a Dirichlet boundary condition is imposed on the map and a chiral boundary condition on the spinor.

Let (𝔤,𝗀){\mathfrak{g},\mathsf{g})} be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with 𝗀{\mathsf{g}} being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories 𝒞𝔤{\mathscr{C}_{\mathfrak{g}}} and 𝒞𝗀{\mathscr{C}_{\mathsf{g}}} of finite-dimensional representations over the quantum loop algebras of 𝔤{\mathfrak{g}} and 𝗀{\mathsf{g}}, respectively. As a consequence, we solve long-standing problems: the positivity of the analogs of Kazhdan–Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced 𝔤{\mathfrak{g}}. In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur–Weyl dualities, we prove the analog of Kazhdan–Lusztig conjecture (formulated in [D. Hernandez, Algebraic approach to q,tq,t-characters, Adv. Math. 187 2004, 1, 1–52]) for simple modules in remarkable monoidal subcategories of 𝒞𝔤{\mathscr{C}_{\mathfrak{g}}} for any non-simply-laced 𝔤{\mathfrak{g}}, and for any simple finite-dimensional modules in 𝒞𝔤{\mathscr{C}_{\mathfrak{g}}} for 𝔤{\mathfrak{g}} of type Bn{\mathrm{B}_{n}}. In the course of the proof we obtain and combine several new ingredients. In particular, we establish a quantum analog of T -systems, and also we generalize the isomorphisms of [D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, J. reine angew. Math. 701 2015, 77–126, D. Hernandez and H. Oya, Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan–Lusztig algorithm, Adv. Math. 347 2019, 192–272] to all 𝔤{\mathfrak{g}} in a unified way, that is, isomorphisms between subalgebras of the quantum group of 𝗀{\mathsf{g}} and subalgebras of the quantum Grothendieck ring of 𝒞𝔤{\mathscr{C}_{\mathfrak{g}}}.

Let K be a number field and S a set of primes of K . We write KS/K{K_{S}/K} for the maximal extension of K unramified outside S and GK,S{G_{K,S}} for its Galois group. In this paper, we prove the following generalization of the Neukirch–Uchida theorem under some assumptions: “For i=1,2{i=1,2}, let Ki{K_{i}} be a number field and Si{S_{i}} a set of primes of Ki{K_{i}}. If GK1,S1{G_{K_{1},S_{1}}} and GK2,S2{G_{K_{2},S_{2}}} are isomorphic, then K1{K_{1}} and K2{K_{2}} are isomorphic.” Here the main assumption is that the Dirichlet density of Si{S_{i}} is not zero for at least one i . A key step of the proof is to recover group-theoretically the l -adic cyclotomic character of an open subgroup of GK,S{G_{K,S}} for some prime number l .

For a reductive group over an algebraically closed field of characteristic p>0{p>0} we construct the abelian category of perverse 𝔽p{\mathbb{F}_{p}}-sheaves on the affine Grassmannian that are equivariant with respect to the action of the positive loop group. We show this is a symmetric monoidal category, and then we apply a Tannakian formalism to show this category is equivalent to the category of representations of a certain affine monoid scheme. We also show that our work provides a geometrization of the inverse of the mod p Satake isomorphism. Along the way we prove that affine Schubert varieties are globally F -regular and we apply Frobenius splitting techniques to the theory of perverse 𝔽p{\mathbb{F}_{p}}-sheaves.

There is an error in the proof of Theorem 1.3 of the original article. Despite the problem, it is rigorously proved in joint work of the first two authors and Perales that Theorem 1.3 is true, using recent results of Allen and Perales that extend the work of Allen, Perales, and Sormani.