We study a Grothendieck topology on schemes which we call the $\mathrm{arc}$-topology. This topology is a refinement of the v-topology (the pro-version of Voevodsky’s h-topology), where covers are tested via rank $\le 1$ valuation rings. Functors which are $\mathrm{arc}$-sheaves are forced to satisfy a variety of gluing conditions such as excision in the sense of algebraic K-theory. We show that étale cohomology is an $\mathrm{arc}$-sheaf, and we deduce various pullback squares in étale cohomology. Using $\mathrm{arc}$-descent, we re-prove the Gabber–Huber affine analogue of proper base change (in a large class of examples), as well as the Fujiwara–Gabber base change theorem on the étale cohomology of the complement of a Henselian pair. As a final application, we prove a rigid analytic version of the Artin–Grothendieck vanishing theorem, extending results of Hansen.

Let p be a positive prime integer. We construct p-adic families of de Rham cohomology classes and therefore p-adic families of nearly overconvergent elliptic modular forms. As an application we define triple product p-adic L-functions attached to three finite slope families of modular forms satisfying certain assumptions.

The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations—or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices, the Gaussian unitary ensemble (GUE) being the prime example of such an ensemble. Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher–Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function.