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After the work of Kisin, there is a good theory of canonical integral models of Shimura varieties of Hodge type at primes of good reduction. The first part of this paper develops a theory of Hodge type Rapoport–Zink formal schemes, which uniformize certain formal completions of such integral models. In the second part, the general theory is applied to the special case of Shimura varieties associated with groups of spinor similitudes, and the reduced scheme underlying the Rapoport–Zink space is determined explicitly.

In this paper, we propose a modified Kudla–Rapoport conjecture for the Krämer model of unitary Rapoport–Zink space at a ramified prime, which is a precise identity relating intersection numbers of special cycles to derivatives of Hermitian local density polynomials. We also introduce the notion of special difference cycles, which has surprisingly simple description. Combining this with induction formulas of Hermitian local density polynomials, we prove the modified Kudla–Rapoport conjecture when $n=3$. Our conjecture, combining with known results at inert and infinite primes, implies the arithmetic Siegel–Weil formula for all non-singular coefficients when the level structure of the corresponding unitary Shimura variety is defined by a self-dual lattice.

We enlarge the class of Rapoport–Zink spaces of Hodge type by modifying the centers of the associated $p$-adic reductive groups. Such obtained Rapoport–Zink spaces are said to be of abelian type. The class of Rapoport–Zink spaces of abelian type is strictly larger than the class of Rapoport–Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of Rapoport–Zink spaces of abelian type can be viewed as moduli spaces of local $G$-shtukas in mixed characteristic in the sense of Scholze. We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport–Zink spaces of abelian type. We construct and study the Ekedahl–Oort stratifications for the special fibers of Rapoport–Zink spaces of abelian type. As an application, we deduce a Rapoport–Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.

Many marine and terrestrial clades show similar latitudinal gradients in species richness, but opposite gradients in range size—on land, ranges are the smallest in the tropics, whereas in the sea, ranges are the largest in the tropics. Therefore, richness gradients in marine and terrestrial systems do not arise from a shared latitudinal arrangement of species range sizes. Comparing terrestrial birds and marine bivalves, we find that gradients in range size are concordant at the level of genera. Here, both groups show a nested pattern in which narrow-ranging genera are confined to the tropics and broad-ranging genera extend across much of the gradient. We find that (i) genus range size and its variation with latitude is closely associated with per-genus species richness and (ii) broad-ranging genera contain more species both within and outside of the tropics when compared with tropical- or temperate-only genera. Within-genus species diversification thus promotes genus expansion to novel latitudes. Despite underlying differences in the species range-size gradients, species-rich genera are more likely to produce a descendant that extends its range relative to the ancestor's range. These results unify species richness gradients with those of genera, implying that birds and bivalves share similar latitudinal dynamics in net species diversification.

Nonnative species richness typically declines along environmental gradients such as elevation. It is usually assumed that this is because few invaders possess the necessary adaptations to succeed under extreme environmental conditions. Here, we show that nonnative plants reaching high elevations around the world are not highly specialized stress tolerators but species with broad climatic tolerances capable of growing across a wide elevational range. These results contrast with patterns for native species, and they can be explained by the unidirectional expansion of nonnative species from anthropogenic sources at low elevations and the progressive dropping out of species with narrow elevational amplitudes—a process that we call directional ecological filtering. Independent data confirm that climatic generalists have succeeded in colonizing the more extreme environments at higher elevations. These results suggest that invasion resistance is not conferred by extreme conditions at a particular site but determined by pathways of introduction of nonnative species. In the future, increased direct introduction of nonnative species with specialized ecophysiological adaptations to mountain environments could increase the risk of invasion. As well as providing a general explanation for gradients of nonnative species richness and the importance of traits such as phenotypic plasticity for many invasive species, the concept of directional ecological filtering is useful for understanding the initial assembly of some native floras at high elevations and latitudes.

For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.

Aim: To determine whether the method used to build distributional maps from raw data influences the representation of two principal macroecological patterns: the latitudinal gradient in species richness and the latitudinal variation in range sizes (Rapoport's rule). Location: World-wide. Methods: All available distribution data from the Global Biodiversity Information Facility (GBIF) for those fish species that are members of orders of fishes with only marine representatives in each order were extracted and cleaned so as to compare four different procedures: point-to-grid (GBIF maps), range maps applying an α-shape [GBIF-extent of occurrence (EOO) maps], the MaxEnt method of species distribution modelling (GBIF-MaxEnt maps) and the MaxEnt method but restricted to the area delimited by the α-shape (GBIF-MaxEnt-restricted maps). Results: The location of hotspots and the latitudinal gradient in species richness or range sizes are relatively similar in the four procedures. GBIF-EOO maps and most GBIF-MaxEnt-maps provide overestimations of species richness when compared with those present in a priori well-surveyed cells. GBIF-EOO maps seem to provide more reasonable world macroecological patterns. MaxEnt can erroneously predict the presence of species in environmentally similar cells of another hemisphere or in other regions that lie outside the range of the species. Limiting this overpredictive capacity, as in the case of GBIF-MaxEnt-restricted maps, seems to mimic the frequency of observations derived from a simple point-to-grid procedure, with the utility of this procedure consequently being limited. Main conclusions: In studies of macroecological patterns at a global scale, the simple α-shape method seems to be a more parsimonious option for extrapolating species distributions from primary data than are distribution models performed indiscriminately and automatically with MaxEnt. GBIF data may be used in macroecological patterns if original data are cleaned, autocorrelation is corrected and species richness figures do not constitute obvious underestimations. Efforts therefore should focus on improving the number and quality of records that can serve as the source of primary data in macroecological studies.

Support for macroecological rules in insects is mixed, with potential confounding interrelations between patterns rarely studied. We here investigate global patterns in body and wing size, sexual size dimorphism and range size in common fruit flies (Diptera: Drosophilidae) and explore potential interrelations and the predictive power of Allen’s, Bergmann’s, Rensch’s and Rapoport’s rules. We found that thorax length (r2 = 0.05) and wing size (r2 = 0.09) increased with latitude, supporting Bergmann’s rule. Contrary to patterns often found in endothermic vertebrates, relative wing size increased towards the poles (r2 = 0.12), a pattern against Allen’s rule, which we attribute to selection for increased flight capacity in the cold. Sexual size dimorphism decreased with size, evincing Rensch’s rule across the family (r2 = 0.14). Yet, this pattern was largely driven by the virilis–repleta radiation. Finally, range size did not correlate with latitude, although a positive relationship was present in a subset of the species investigated, providing no convincing evidence for Rapoport’s rule. We further found little support for confounding interrelations between body size, wing loading and range size in this taxon. Nevertheless, we demonstrate that studying several traits simultaneously at minimum permits better interpretation in case of multiple, potentially conflicting trends or hypotheses concerning the macroecology of insects.

AIM: Three broad mechanisms have been proposed to explain geographic variation in species range size: habitat area/heterogeneity, climate seasonality and long‐term climate variability. However, it has proved difficult to disentangle their relative role, particularly as temperature seasonality often covaries with the amplitude of long‐term temperature oscillations. Here, we shed new light onto this debate by providing the first continental‐scale analysis of range size and beta diversity in groundwater habitats, where taxa are not exposed to latitudinal variation in temperature seasonality. LOCATION: Europe. METHODS: We compiled and mapped occurrence data for 1570 groundwater crustacean species. Generalized regression models were used to test for latitudinal variation in geographic range size and to assess the relative role of the three broad mechanisms in shaping present‐day patterns of range size. We partitioned beta diversity into its spatial turnover and nestedness components and analysed their latitudinal variation across Europe. RESULTS: Median range size increases with latitude above 43° N and the range size of individual species is positively correlated to latitude, even after accounting for phylogenetic effects. Long‐term temperature variability accounted for a substantially higher variation in median range size of groundwater crustaceans across Europe than precipitation seasonality and habitat heterogeneity, including aquifer area, elevation range, climatic rarity and productive energy. Spatial turnover contributes significantly more to beta diversity in southern regions characterized by stable historic climates than it does in northern Europe. MAIN CONCLUSIONS: Our findings add support to the historic climate hypothesis which suggests that patterns of increasing range size and decreasing species turnover at higher latitudes in the Palaearctic region are primarily driven by long‐term temperature oscillations rather than by climatic seasonality and the availability and heterogeneity of habitats.

Marine microbes along with microeukaryotes are key regulators of oceanic biogeochemical pathways. Herewe present a high-resolution (every 0.5° of latitude) dataset describing microbial pro- and eukaryotic richness in the surface and just below the thermocline along a 7,000-km transect from 66°S at the Antarctic ice edge to the equator in the South Pacific Ocean. The transect, conducted in austral winter, covered key oceanographic features including crossing of the polar front (PF), the subtropical front (STF), and the equatorial upwelling region. Our data indicate that temperature does not determine patterns of marine microbial richness, complementing the global model data from Ladau et al. [Ladau J, et al. (2013) ISME J 7:1669–1677]. Rather, NH₄⁺, nanophytoplankton, and primary productivity were the main drivers for archaeal and bacterial richness. Eukaryote richness was highest in the least-productive ocean region, the tropical oligotrophic province. We also observed a unique diversity pattern in the South Pacific Ocean: a regional increase in archaeal and bacterial diversity between 10°S and the equator. Rapoport’s rule describes the tendency for the latitudinal ranges of species to increase with latitude. Our data showed that the mean latitudinal ranges of archaea and bacteria decreased with latitude. We show that permanent oceanographic features, such as the STF and the equatorial upwelling, can have a significant influence on both alpha-diversity and beta-diversity of pro- and eukaryotes.