In this paper, we study the emptiness/nonemptiness and the dimension formulas of affine Deligne-Lusztig varieties for Sp 4 (L). We mainly calculate the degree of class polynomials for the Iwahori-Hecke algebra of type C 2 . Then, give an explicit description on the emptiness/nonemptiness and dimension formulas of affine Deligne-Lusztig varieties for the group Sp 4 (L).

To any connected reductive group G over a non-archimedean local field F (of characteristic p ą 0) and to any maximal torus T of G, we attach a family of extended affine Deligne-Lusztig varieties (and families of torsors over them) over the residue field of F . This construction generalizes affine Deligne-Lusztig varieties of Rapoport, which are attached only to unramified tori of G. Via this construction, we can attach to any maximal torus T of G and any character of T a representation of G. This procedure should conjecturally realize the automorphic induction from T to G.For G " GL2, we prove that our construction indeed realizes the automorphic induction from at most tamely ramified tori. Moreover, if the torus is purely tamely ramified, then the varieties realizing this correspondence are zero-dimensional and reduced, i.e., just disjoint unions of points. T of GpF q over Q . Moreover, R χ T is supercuspidal, whenever T is anisotropic modulo the center of G. Such a correspondence is a special case of the more general principle of automorphic induction for G, which is closely related to the local Langlands correspondence.Let G again be arbitrary. Roughly, one can divide all geometrical objects attached to G, in the cohomology of which one has tried to realize the automorphic induction, into two types: (i) Varieties (or rigid or adic spaces) over Spec F equipped with integral models over Spec O F and special fibers over F q . (ii) Reduced varieties over F q .Constructions of type (ii) are purely in characteristic p, i.e., over F q , and only the reduced structure is relevant. Up to now, constructions of type (ii) only existed for unramified tori of

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicate to describe the position of decomposition groups of points at the boundary of the scheme Spec O K,S

Reaction of oxygen with solutions of neodymium chloride in LiCl and 3LiCl–2KCl melts was studied at 450 °C–750 °C. The reaction resulted in the formation of neodymium oxychloride and the effect of temperature, amount of oxygen passed through the melt (oxygen-to-neodymium molar ratio), gas phase composition (O2, O2–H2O, Ar–O2, Ar–O2–H2O) on the course of the reaction were considered. Size of particles comprising solid precipitates formed in the melt was determined. High temperature electronic absorption spectroscopy was used to determine kinetic parameters of the reaction (reaction rates, rate constants, temperature coefficients). The activation energy of the reaction was evaluated.

In the present article we define coverings of affine Deligne-Lusztig varieties attached to a connected reductive group over a local field of characteristic p ą 0. In the case of GL 2 , the unramified part of the local Langlands correspondence is realized in the -adic cohomology of these varieties. We show this by giving a detailed comparison with the realization of local Langlands via cuspidal types by Bushnell-Henniart. All proofs are purely local.

We give a geometric construction of representations of parahoric subgroups P P of a reductive group G G over a local field which splits over an unramified extension. These representations correspond to characters θ \theta of unramified maximal tori and, when the torus is elliptic, are expected to give rise to supercuspidal representations of G G . We calculate the character of these P P -representations on a special class of regular semisimple elements of G G . Under a certain regularity condition on θ \theta , we prove that the associated P P -representations are irreducible. This generalizes a construction of Lusztig from the hyperspecial case to the setting of an arbitrary parahoric.

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The reaction between 15 rare earth chlorides (yttrium, lanthanum and all the lanthanides except for promethium) and sodium orthophosphate was studied in NaCl-2CsCl eutectic based melts between 550 and 750 oC under an inert atmosphere. Rare earth phosphates were precipitated from the melt and their composition determined from chemical analysis, vibrational spectroscopy and X-ray powder diffraction measurements. The effects of temperature and the initial phosphate-to-rare earth mole ratio in the melt on the phase composition of the precipitated phosphates was investigated Over 99.9% precipitation could be achieved and the mole ratio required varied between 2 and 6. The minimum excess of phosphate precipitant required for complete precipitation from the melt was determined.