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We use canonically twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for the vector-valued mock modular forms attached to automorphisms of this lattice by umbral moonshine. We also characterize the vector-valued mock modular forms arising, in which four of Ramanujan’s fifth-order mock theta functions appear as components.
We prove, for quasicompact separated schemes over ground fields, that Čech cohomology coincides with sheaf cohomology with respect to the Nisnevich topology. This is a partial generalization of Artin’s result that for noetherian schemes such an equality holds with respect to the étale topology, which holds under the assumption that every finite subset admits an affine open neighborhood (AF-property). Our key result is that on the absolute integral closure of separated algebraic schemes, the intersection of any two irreducible closed subsets remains irreducible. We prove this by establishing general modification and contraction results adapted to inverse limits of schemes. Along the way, we characterize schemes that are acyclic with respect to various Grothendieck topologies, study schemes all local rings of which are strictly henselian, and analyze fiber products of strict localizations.
Let be a triangulated homology ball whose boundary complex is . A result of Hochster asserts that the canonical module of the Stanley–Reisner ring of is isomorphic to the Stanley–Reisner module of the pair . This result implies that an Artinian reduction of is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of . We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the -numbers of Buchsbaum complexes and use it to prove the monotonicity of -numbers for pairs of Buchsbaum complexes as well as the unimodality of -vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold -conjecture.
Let be a discretely valued nonarchimedean field. We give an explicit description of analytic functions whose norm is bounded by a given real number on tubes of reduced -analytic spaces associated to special formal schemes (including -affinoid spaces as well as open polydiscs). As an application we study the connectedness of these tubes. In the discretely valued case, this generalizes a result of Siegfried Bosch. We use as a main tool a result of Aise Johan de Jong relating formal and analytic functions on special formal schemes and a generalization of de Jong’s result which is proved in the joint appendix with Christian Kappen.
We prove that if a is an algebraic power series of degree , height , and genus , then the sequence a is generated by an automaton with at most states, up to a vanishingly small error term. This is a significant improvement on previously known bounds. Our approach follows an idea of David Speyer to connect automata theory with algebraic geometry by representing the transitions in an automaton as twisted Cartier operators on the differentials of a curve.
In previous papers we formulated an analogue of the Ichino–Ikeda conjectures for Whittaker–Fourier coefficients of automorphic forms on quasisplit classical groups and the metaplectic group of arbitrary rank. In the latter case we reduced the conjecture to a local identity. In this paper we prove the local identity in the -adic case, and hence the global conjecture under simplifying conditions at the archimedean places.