On p-adic lattices and Grassmannians

It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive group G over a field k, carry the geometric structure of an inductive limit of projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for G. From the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form G(W(k)[1/p])/G(W(k)), where p is a rational prime, W denotes the ring scheme of p-typical Witt vectors, k is a perfect field of characteristic p and G is a reductive group scheme over W(k). The present paper is an attempt to describe which constructions carry over from the function field case to the p-adic case, more precisely to the situation of the p-adic affine Grassmannian for the special linear group G=SL_n. We start with a description of the R-valued points of the p-adic affine Grassmannian for SL_n in terms of lattices over W(R), where R is a perfect k-algebra. In order to obtain a link with geometry we further construct projective k-subvarieties of the multigraded Hilbert scheme which map equivariantly to the p-adic affine Grassmannian. The images of these morphisms play the role of Schubert varieties in the p-adic setting. Further, for any reduced k-algebra R these morphisms induce bijective maps between the sets of R-valued points of the respective open orbits in the multigraded Hilbert scheme and the corresponding Schubert cells of the p-adic affine Grassmannian for SL_n.Comment: 36 pages. This is a thorough revision, in the form accepted by Math. Zeitschrift, of the previously published preprint "On p-adic loop groups and Grassmannians

Some genus 3 curves with many points

Using an explicit family of plane quartic curves, we prove the existence of a genus 3 curve over any finite field of characteristic 3 whose number of rational points stays within a fixed distance from the Hasse-Weil-Serre upper bound. We also provide an intrinsic characterization of so-called Legendre elliptic curves

A Light, Transmission and Scanning Electron Microscope Study of Snuff-Treated Hamster Cheek Pouch Epithelium

The effects of smokeless tobacco (snuff) on hamster cheek mucosa were studied by light microscopy, transmission (TEM) and scanning electron microscopy (SEM). Two grams of commercially available smokeless tobacco were placed into the blind end of the right cheek pouch of each experimental animal, once a day and five days a week for 24 months. The control animals did not receive smokeless tobacco. After 24 months treatment with smokeless tobacco, hamster cheek mucosal epithelium lost its translucency and had become whitish in color. By light microscopy hyperorthokeratosis, prominent granular cell layers with increased keratohyalin granules and hyperplasia were seen. At the ultrastructural level, wider intercellular spaces filled with microvilli, numerous shorter desmosomes, many thin tonofilament bundles, increased number of mitochondria, membrane coating granules and keratohyalin granules were seen in snuff-treated epithelium. The changes in the surface of the epithelium as seen by SEM were the development of an irregular arrangement of the microridges and the disappearance of the normal honeycomb pattern. The microridges were irregular, widened and surrounded the irregular elongated pits. Some smooth areas without microridges and pits were also seen. The long-term histological, TEM and SEM changes induced by smokeless tobacco treatment of the epithelium are well correlated with each other and were similar to those reported in human leukoplakia without dyskeratosis. They imply changes of pathological response resulting from topically applied snuff