In this article we show that the quotient M ∞ /B(Q p ) of the Lubin-Tate space at infinite level M ∞ by the Borel subgroup of upper triangular matrices B(Q p ) ⊂ GL 2 (Q p ) exists as a perfectoid space. As an application we show that Scholze's functoris concentrated in degree one whenever π is an irreducible principal series representation or a twist of the Steinberg representation of GL 2 (Q p ).

Abstract. We prove a p-adic Labesse-Langlands transfer from the group of units in a definite quaternion algebra to its subgroup of norm one elements. More precisely, given an eigenvariety for the first group, we show that there exists an eigenvariety for the second group and a morphism between them that extends the classical Langlands transfer. In order to find a suitable target eigenvariety for the transfer we formalise a notion of Langlands compatibility of tame levels. Proving the existence of Langlands compatible tame levels is the key to pass from the classical transfer on the level of L-packets to a map between classical points of eigenvarieties, which is then amenable for interpolation to give the p-adic transfer.

In this article, we construct examples of L-indistinguishable overconvergent eigenforms for an inner form of SL2.

We prove p-adic functoriality for inner forms of unitary groups in three variables by establishing the existence of morphisms between eigenvarieties that extend the classical Langlands functoriality.

Background: There is an urgent need to better understand the mechanisms underlying acute and long-term neurological symptoms after COVID-19. Neuropathological studies can contribute to a better understanding of some of these mechanisms. Methods: We conducted a detailed postmortem neuropathological analysis of 32 patients who died due to COVID-19 during 2020 and 2021 in Austria. Results: All cases showed diffuse white matter damage with a diffuse microglial activation of a variable severity, including one case of hemorrhagic leukoencephalopathy. Some cases revealed mild inflammatory changes, including olfactory neuritis (25%), nodular brainstem encephalitis (31%), and cranial nerve neuritis (6%), which were similar to those observed in non-COVID-19 severely ill patients. One previously immunosuppressed patient developed acute herpes simplex encephalitis. Acute vascular pathologies (acute infarcts 22%, vascular thrombosis 12%, diffuse hypoxic–ischemic brain damage 40%) and pre-existing small vessel diseases (34%) were frequent findings. Moreover, silent neurodegenerative pathologies in elderly persons were common (AD neuropathologic changes 32%, age-related neuronal and glial tau pathologies 22%, Lewy bodies 9%, argyrophilic grain disease 12.5%, TDP43 pathology 6%). Conclusions: Our results support some previous neuropathological findings of apparently multifactorial and most likely indirect brain damage in the context of SARS-CoV-2 infection rather than virus-specific damage, and they are in line with the recent experimental data on SARS-CoV-2-related diffuse white matter damage, microglial activation, and cytokine release.