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We initiate the study of affine Deligne–Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. We prove that for $${{\,\mathrm{GL}\,}}_n$$ and its inner forms, Lusztig’s semi-infinite Deligne–Lusztig construction is isomorphic to an affine Deligne–Lusztig variety at infinite level. We prove that their homology groups give geometric realizations of the local Langlands and Jacquet–Langlands correspondences in the setting that the Weil parameter is induced from a character of an unramified field extension. In particular, we resolve Lusztig’s 1979 conjecture in this setting for minimal admissible characters.

Rapoport–Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let ${{\mathscr M}}_{\infty }$ be an infinite-level Rapoport–Zink space of EL type, and let ${{\mathscr M}}_{\infty }^{\circ }$ be one connected component of its geometric fiber. We show that ${{\mathscr M}}_{\infty }^{\circ }$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^{\infty })^{\circ }$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

Molten salts have a number of potential applications including metals’ electrowinning and electrorefining, pyrochemical spent nuclear fuels’ reprocessing. Considerable efforts were devoted over decades to investigation of various electrochemical processes in molten salts, especially in fused chlorides. Silver chloride electrode is one of convenient types of reference electrodes often employed in potentiometry measurements. This electrode has several advantages (especially compactness) compare to the standard chlorine reference electrode, potential of which in chloride melts is taken as zero. Surprisingly little attention was paid so far to the systematic study (covering the effect of temperature and concentration) of silver electrode potentials in molten chlorides, with majority of the works dating back to 1950-s and 60-s. Molten LiCl–KCl eutectic was, on obvious reasons, the best studied, and several other melts (NaCl, NaCl–KCl, LiCl–KCl–CsCl, NaCl–CsCl) were also looked at. The present study was directed at the systematic study of silver electrochemistry in fused alkali chlorides and determining silver electrode potentials. Here we present the results of the experiments performed in the melts based on the ternary NaCl–KCl–CsCl (30–24.5–45.5 mol. %) eutectic mixture at 823–1073 K. Electrode potentials of silver were determined using the electromotive force measurements method and the following galvanic cell: Ag|(NaCl–KCl–CsCl)+AgCl||(NaCl–KC1–CsCl)|Cl2 (C) Concentration of silver in the electrolyte varied from 0.1 to 5 wt. %. Silver electrode quickly equilibrated with the electrolyte and the potential remained stable over long time, Figure. Experimentally obtained electrode potentials were then used to calculate silver formal standard electrode potentials. Figure. An example of silver electrode potential evolution over time in (NaCl–KCl–CsCl)–AgCl (5 wt. %) melt at 823 K. 2Cl–/Cl2 reference electrode. Figure 1

We define a stratification of Deligne–Lusztig varieties and their parahoric analogues which we call the Drinfeld stratification. In the setting of inner forms of $${{\,\mathrm{GL}\,}}_n$$ , we study the cohomology of these strata and give a complete description of the unique closed stratum. We state precise conjectures on the representation-theoretic behavior of the stratification. We expect this stratification to play a central role in the investigation of geometric constructions of representations of p-adic groups.

To any connected reductive group G over a non-archimedean local field F 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 $$G = {{\mathrm{GL}}}_2$$ in the equal characteristic case, 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 turn out to be (quite complicate) combinatorial objects: they are zero-dimensional and reduced, i.e., just disjoint unions of points.

In this article we generalize a part of Neukirch-Uchida theorem for number elds from the birational case to the case of curves SpecOK,S where S a stable set of primes of a number eld K. Such sets have positive but arbitrarily small Dirichlet density, which must be uniformly bounded from below by some > 0 in the tower KS/K. 2010 Mathematical Subject Classi cation: 11R34, 11R37, 14G32. Abstract. V to stat~e my obobwim teoremu No kirha-Uxidy dl qislovyh pole ot biracional~nogo sluqa k sluqa krivyh SpecOK,S , gde S stabil~noe mno estvo prostyh idealov v qislovom pole K. Takie mno estva ime t polo itel~nu plotnost~ Dirihle, kotora uniformno ograniqena snizu v baxne KS/K. V to stat~e my obobwim teoremu No kirha-Uxidy dl qislovyh pole ot biracional~nogo sluqa k sluqa krivyh SpecOK,S , gde S stabil~noe mno estvo prostyh idealov v qislovom pole K. Takie mno estva ime t polo itel~nu plotnost~ Dirihle, kotora uniformno ograniqena snizu v baxne KS/K.

We generalize a cohomological construction of representations due to Lusztig from the hyperspecial case to arbitrary parahoric subgroups of a reductive group over a local field, which splits over an unramified extension. We compute the character of these representations on certain very regular elements., Comment: 24 pages

Alkali chloride melts are considered as prospective working media for non-aqueous pyrochemical reprocessing of spent nuclear fuels (SNFs). Separation of fissile materials from fission products in pyrochemical reprocessing can be achieved electrochemically and amongst all the fission products zirconium has the closest electrochemical properties to uranium. Uranium and plutonium fission produces several zirconium isotopes (from Zr-91 to Zr-97) and depending on the reactor neutron spectrum, nuclear fuel type, burnup and cooling time SNF arriving for reprocessing can contain ca. 5–13 kg of zirconium (as the fission product excluding cladding) per ton. Electrochemical separation of uranium and zirconium in fused salts is a challenging task. The present work was devoted to studying the electrochemical behavior of zirconium and uranium in 3LiCl–2KCl based melts using cyclic voltammetry and cathodic polarization. The experiments were performed in LiCl–KCl–ZrCl4, LiCl–KCl–UCl4, LiCl–KCl–UCl3 and LiCl–KCl–ZrCl4–UCl4 melts. On a solid tungsten electrode zirconium(IV) ions were reduced to Zr(0) in two stages and the metal deposition potentials were between –2.17 and –2.07 V (at 532–637 oC) vs. Cl–/Cl2 couple. Changing the solid electrode to liquid zinc, gallium or gallium–zinc eutectic alloy (3.64 wt. % Zn) resulted in significant shift of zirconium deposition potential in the positive direction. Examples of the polarization curves are shown in the Fig. Polarization measurements performed in LiCl–KCl–ZrCl4–UCl4 melt on the Ga–Zn electrode showed that reduction of U(IV) to U(III) and deposition of zirconium occurred at very similar potentials and deposition potential of uranium was significantly more negative. Separation factor for uranium/zirconium couple was also determined. Fig. Polarization of Ga–Zn eutectic alloy cathode in LiCl–KCl–UCl4 (524 oC) and LiCl–KCl–ZrCl4 (550 oC) melts. Figure 1

Alkali chloride melts have numerous potential applications in nuclear fuel cycle including pyrochemical reprocessing of spent nuclear fuels, uranium electrowinning and electrorefining. Oxygen is a common technological impurity that can affect uranium speciation and behavior in fused salts. The present work was devoted to studying the reactions of oxygen with solutions of uranium tetrachloride in molten alkali chlorides. The experiments were performed in LiCl–KCl, NaCl–KCl–CsCl and NaCl–CsCl eutectic based melts at 450–750 oC. Pure oxygen and argon–oxygen mixtures (containing ca. 1 and 10 % O2) were used. Amount of oxygen passed through the melt varied from less than one to over 100 moles per mole of uranium present. Effect of moisture (0.4–2.5 % H2O) presence in oxygen or Ar–O2 mixtures was also investigated. The course of the reaction was followed by in situ electronic absorption spectroscopy measurements with the spectra recorded at the certain time intervals. Depending on temperature, cationic melt composition and oxygen-to-uranium molar ration the reaction resulted in oxidation of uranium(IV) to soluble uranyl chloride and/or precipitation of uranium dioxide. Analysis of the spectra provided the information on kinetics of U(IV) concentration change. Increasing temperature, O2 : U(IV) molar ratio or decreasing mean radius of alkali cations of the solvent melt resulted in faster decrease of U(IV) concentration in the melt. Under certain conditions U(IV) can be oxidized to UO2Cl4 2– without precipitation of UO2. Therefore sparging the melt with oxygen can be used as a way of separating uranium from certain fission products, for example rare earth elements. Rare earth chlorides react with oxygen yielding oxychlorides or oxides insoluble in alkali chloride melts. Interaction of oxygen with melts containing a mixture of uranium and rare earth chlorides was therefore also investigated and an example of the spectra recorded in LiCl–KCl–UCl4–NdCl3 melt is shown in Fig. Fig. Spectra recorded in the course of reaction of O2 with LiCl–KCl–UCl4–NdCl3 melt at 550 oC. Arrows show the direction spectra changed. Figure 1

Melts based on mixtures of alkali and/or alkaline earth halides are considered as prospective media for electrowinning rare earth metals as well as for pyrochemical reprocessing spent nuclear fuels. Fluoride or mixed fluoride-chloride baths can be operated at high temperatures yielding molten rare earth metals (REMs) thus simplifying separation of the metal and salt. One of the problems in using fluoride melts is possible formation of fluorine at the anode. This can be avoided by adding a rare earth oxide to the melt both as the source of REM and oxide ions. The latter will be oxidized to oxygen producing carbon mono- or dioxide at the anode. The limiting factor for feeding the electrolysis bath with REM oxides is their solubility in the halide melt. The aim of the present study was determining the effect of temperature and melt composition on solubility of REM oxides in fused halides. The experiments were performed in CaCl2–CaF2 mixtures containing 20 or 75 mol. % CaF2; BaCl2–BaF2 mixtures containing 15 or 73 mol. % BaF2; equimolar CaF2–BaF2 mixture and NaCl–NaF eutectic mixture (34 mol. % NaF). Solubility of REM oxides was determined by the method of isothermal saturation. Time required for reaching the equilibrium between solid REM oxides and fused salts was determined in a preliminary set of experiments. To compare the behavior of 4f- and 5f-elements, solubility of uranium dioxide was also measured. The measurements were performed at the temperatures up to 1400 oC under argon atmosphere. The lower limit of the temperature range varied from 700 to 1100 oC depending on the melting temperatures of the salt mixtures used. Oxides of yttrium, lanthanum, cerium, praseodymium, neodymium and samarium were selected for the study. To assess a possible mutual influence of rare earth elements on solubility of their oxides in fused salts the solubility of a mixture of REM oxides was determined in a separate series of experiments and concentrations of individual REMs in the melt was determined. Solubility of REM oxides increased with increasing temperature and an example of effect of temperature on solubility of neodymium oxide in various melts is shown in Fig. Fig. Concentration of neodymium in alkali and alkaline halide based melts saturated with Nd2O3. Melt: CaCl2–CaF2 20 mol. % (1); CaCl2–CaF2 75 mol. % (2); BaCl2–BaF2 73 mol. % (3); BaCl2–BaF2 15 mol. % (4); CaF2–BaF2 50 mol. % (5; and NaCl–NaF 34 mol. % (6). Figure 1

We define a new class of sets -- stable sets -- of primes in number fields. For example, Chebotarev sets $P_{M/K}(\sigma)$, with $M/K$ Galois and $\sigma \in \Gal(M/K)$, are very often stable. These sets have positive (but arbitrary small) Dirichlet density and generalize sets with density 1 in the sense that arithmetic theorems like certain Hasse principles, the Grunwald-Wang theorem, the Riemann's existence theorem, etc. hold for them. Geometrically this allows to give examples of infinite sets $S$ with arbitrary small positive density such that $\Spec \mathcal{O}_{K,S}$ is algebraic $K(\pi,1)$ (for all $p$ simultaneous)., Comment: 24 pages; minor changes and updates as suggested by the referees

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 $\Spec \caO_{K,S}$, where $K$ is a number field and $S$ a set of primes of $K$, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally with the fundamental group $\pi_1(\Spec \caO_{K,S})$: the location of decomposition groups of boundary points inside it, the $p$-part of the cyclotomic character, the number of points on the boundary of all finite etale covers, etc. Under certain finiteness hypothesis on Tate-Shafarevich groups with divisible coefficients, one can reconstruct all this quantities from the fundamental group alone., Comment: 18 pages