We compare the Kummer flat (resp., Kummer étale) cohomology with the flat (resp., étale) cohomology with coefficients in smooth commutative group schemes, finite flat group schemes, and Kato’s logarithmic multiplicative group. We are particularly interested in the case of algebraic tori in the Kummer flat topology. We also make some computations for certain special cases of the base log scheme.

In this article, we study a Gysin triangle in the category of motives with modulus (Theorem 1.2). We can understand this Gysin triangle as a motivic lift of the Gysin triangle of log-crystalline cohomology due to Nakkajima and Shiho. After that we compare motives with modulus and Voevodsky motives (Corollary 1.6). The corollary implies that an object in $\operatorname {\mathbf {MDM}^{\operatorname {eff}}}$ decomposes into a p-torsion part and a Voevodsky motive part. We can understand the corollary as a motivic analogue of the relationship between rigid cohomology and log-crystalline cohomology.

For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

We prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math. 192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math. 125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.

We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

In the early 1980s, Johnson defined a homomorphism $\mathcal {I}_{g}^1\to \bigwedge ^3 H_1\left (S_{g},\mathbb {Z}\right )$ , where $\mathcal {I}_{g}^1$ is the Torelli group of a closed, connected, and oriented surface of genus g with a boundary component and $S_g$ is the corresponding surface without a boundary component. This is known as the Johnson homomorphism.

We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding-pair maps, in order to compute a large quotient of $H_n\left (\mathcal {I}_{g}^1,\mathbb {Q}\right )$ in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group $\mathcal {I}_{g,1}$ of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of $H_n\left (\mathcal {I}_{g,1}\right )$ for $n\ge 2$ and g large enough.

The general position hypothesis needs strengthening.

We prove the existence and asymptotic behaviour of the transition density for a large class of subordinators whose Laplace exponents satisfy lower scaling condition at infinity. Furthermore, we present lower and upper bounds for the density. Sharp estimates are provided if an additional upper scaling condition on the Laplace exponent is imposed. In particular, we cover the case when the (minus) second derivative of the Laplace exponent is a function regularly varying at infinity with regularity index bigger than $-2$ .

Given a quiver with potential $(Q,W)$, Kontsevich–Soibelman constructed a cohomological Hall algebra (CoHA) on the critical cohomology of the stack of representations of $(Q,W)$. Special cases of this construction are related to work of Nakajima, Varagnolo, Schiffmann–Vasserot, Maulik–Okounkov, Yang–Zhao, etc. about geometric constructions of Yangians and their representations; indeed, given a quiver Q, there exists an associated pair $(\widetilde{Q}, \widetilde{W})$ whose CoHA is conjecturally the positive half of the Maulik–Okounkov Yangian $Y_{\text {MO}}(\mathfrak {g}_{Q})$.

For a quiver with potential $(Q,W)$, we follow a suggestion of Kontsevich–Soibelman and study a categorification of the above algebra constructed using categories of singularities. Its Grothendieck group is a K-theoretic Hall algebra (KHA) for quivers with potential. We construct representations using framed quivers, and we prove a wall-crossing theorem for KHAs. We expect the KHA for $(\widetilde{Q}, \widetilde{W})$ to recover the positive part of quantum affine algebra $U_{q}(\widehat {\mathfrak {g}_{Q}})$ defined by Okounkov–Smirnov.

For each $k\geq 3$, we construct a $1$-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb {H}^2\times \mathbb {R}$ with genus $1$ and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb {H}^2\times \mathbb {R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature $-4k\pi $. Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus $1$ in quotients of $\mathbb {H}^2\times \mathbb {R}$ by the action of a hyperbolic or parabolic translation.

We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck’s splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed-point functors and the motivic Adams isomorphism.

The main purpose of this article is to define a quadratic analogue of the Chern character, the so-called Borel character, that identifies rational higher Grothendieck-Witt groups with a sum of rational Milnor-Witt (MW)-motivic cohomologies and rational motivic cohomologies. We also discuss the notion of ternary laws due to Walter, a quadratic analogue of formal group laws, and compute what we call the additive ternary laws, associated with MW-motivic cohomology. Finally, we provide an application of the Borel character by showing that the Milnor-Witt K-theory of a field F embeds into suitable higher Grothendieck-Witt groups of F modulo explicit torsion.

Since Faltings proved Mordell’s conjecture in [16] in 1983, we have known that the sets of rational points on curves of genus at least $2$ are finite. Determining these sets in individual cases is still an unsolved problem. Chabauty’s method (1941) [10] is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the Jacobian, the closure of the Mordell–Weil group with the p-adic points of the curve. Under the condition that the Mordell–Weil rank is less than the genus, Chabauty’s method, in combination with other methods such as the Mordell–Weil sieve, has been applied successfully to determine all rational points in many cases.

Minhyong Kim’s nonabelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Besser, Dogra, Müller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both $3$ ).

This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only ‘simple algebraic geometry’ (line bundles over the Jacobian and models over the integers).

We present a single, connected tile which can tile the plane but only nonperiodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar–Taylor tilings. However, the new edge-to-edge orientational matching rule forces this structure in a very different way, which allows for a surprisingly simple proof of aperiodicity. We show that the hull of all tilings satisfying our rules is uniquely ergodic and that almost all tilings in the hull belong to a minimal core of tilings generated by substitution. Identifying tilings which are charge-flips of each other, these tilings are shown to have pure point dynamical spectrum and a regular model set structure.

This article deals with kinetic Fokker–Planck equations with essentially bounded coefficients. A weak Harnack inequality for nonnegative super-solutions is derived by considering their log-transform and adapting an argument due to S. N. Kružkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.

In this work we prove a Fourier–Bros–Iagolnitzer (F.B.I.) characterisation for Gevrey vectors on hypo-analytic structures and we analyse the main differences of Gevrey regularity and hypo-analyticity concerning the F.B.I. transform. We end with an application of this characterisation on a propagation of Gevrey singularities result for solutions of the nonhomogeneous system associated with the hypo-analytic structure for analytic structures of tube type.

We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

We explain an algorithm to calculate Arthur’s weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur–Kottwitz reduction and by the Harder–Narasimhan reduction. A comparison of results obtained from these two approaches gives recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur’s weighted orbital integrals. As an example, we calculate Arthur’s weighted orbital integrals for the groups ${\textrm {GL}}_{2}$ and ${\textrm {GL}}_{3}$ .

The formula of the title relates p-adic heights of Heegner points and derivatives of p-adic L-functions. It was originally proved by Perrin-Riou for p-ordinary elliptic curves over the rationals, under the assumption that p splits in the relevant quadratic extension. We remove this assumption, in the more general setting of Hilbert-modular abelian varieties.

Moduli spaces of bounded local G-shtukas are a group-theoretic generalisation of the function field analogue of Rapoport and Zink’s moduli spaces of p-divisible groups. In this article we generalise some very prominent concepts in the theory of Rapoport-Zink spaces to our setting. More precisely, we define period spaces, as well as the period map from a moduli space of bounded local G-shtukas to the corresponding period space, and we determine the image of the period map. Furthermore, we define a tower of coverings of the generic fibre of the moduli space, which is equipped with a Hecke action and an action of a suitable automorphism group. Finally, we consider the $\ell $ -adic cohomology of these towers.

Les espaces de modules de G-chtoucas locaux bornés sont une généralisation des espaces de modules de groupes p-divisibles de Rapoport-Zink, au cas d’un corps de fonctions local, pour des groupes plus généraux et des copoids pas nécessairement minuscules. Dans cet article nous définissons les espaces de périodes et l’application de périodes associés à un tel espace, et nous calculons son image. Nous étudions la tour au-dessus de la fibre générique de l’espace de modules, équipée d’une action de Hecke ainsi que d’une action d’un groupe d’automorphismes. Enfin, nous définissons la cohomologie $\ell $ -adique de ces tours.