In this note, we prove that the moduli stack of vector bundles on a curve with a fixed determinant is ${\mathbb A}^1$-connected. We obtain this result by classifying vector bundles on a curve up to ${\mathbb A}^1$-concordance. Consequently, we classify ${\mathbb P}^n$-bundles on a curve up to ${\mathbb A}^1$-weak equivalence, extending a result in [3] of Asok-Morel. We also give an explicit example of a variety which is ${\mathbb A}^1$-h-cobordant to a projective bundle over ${\mathbb P}^2$ but does not have the structure of a projective bundle over ${\mathbb P}^2$, thus answering a question of Asok-Kebekus-Wendt [2].

Let G denote a possibly discrete topological group admitting an open subgroup I which is pro-p. If H denotes the corresponding Hecke algebra over a field k of characteristic p, then we study the adjunction between H-modules and k-linear smooth G-representations in terms of various model structures. If H is a Gorenstein ring, we single out a full subcategory of smooth G-representations which is equivalent to the category of all Gorenstein projective H-modules via the functor of I-invariants. This applies to groups of rational points of split connected reductive groups over finite and over non-Archimedean local fields, thus generalizing a theorem of Cabanes. Moreover, we show that the Gorenstein projective model structure on the category of H-modules admits a right transfer. On the homotopy level, the right derived functor of I-invariants then admits a right inverse and becomes an equivalence when restricted to a suitable subcategory.

We introduce the quantum isomeric supercategory and the quantum affine isomeric supercategory. These diagrammatically defined supercategories, which can be viewed as isomeric analogues of the HOMFLYPT skein category and its affinisation, provide powerful categorical tools for studying the representation theory of the quantum isomeric superalgebras (commonly known as quantum queer superalgebras).

We construct an action of the affine Hecke category on the principal block $\mathrm {Rep}_0(G_1T)$ of $G_1T$-modules where G is a connected reductive group over an algebraically closed field of characteristic $p> 0$, T a maximal torus of G and $G_1$ the Frobenius kernel of G. To define it, we define a new category with a Hecke action which is equivalent to the combinatorial category defined by Andersen-Jantzen-Soergel.

Given a Galois extension $L/K$ of number fields, we describe fine distribution properties of Frobenius elements via invariants from representations of finite Galois groups and ramification theory. We exhibit explicit families of extensions in which we evaluate these invariants and deduce a detailed understanding and a precise description of the possible asymmetries. We establish a general bound on the generic fluctuations of the error term in the Chebotarev density theorem, which under GRH is sharper than the Murty–Murty–Saradha and Bellaïche refinements of the Lagarias–Odlyzko and Serre bounds, and which we believe is best possible (assuming simplicity, it is of the quality of Montgomery’s conjecture on primes in arithmetic progressions). Under GRH and a hypothesis on the multiplicities of zeros up to a certain height, we show that in certain families, these fluctuations are dominated by a constant lower order term. As an application of our ideas, we refine and generalize results of K. Murty and of Bellaïche, and we answer a question of Ng. In particular, in the case where $L/{\mathbb {Q}}$ is Galois and supersolvable, we prove a strong form of a conjecture of K. Murty on the unramified prime ideal of least norm in a given Frobenius set. The tools we use include the Rubinstein–Sarnak machinery based on limiting distributions and a blend of algebraic, analytic, representation theoretic, probabilistic and combinatorial techniques.

In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type $\mathbf {A}_1+\mathbf {A}_3$ and prove an analogue of Manin’s conjecture for integral points with respect to its singularities and its lines.

We develop a geometric method to establish the existence and uniqueness of equilibrium states associated to some Hölder potentials for center isometries (as are regular elements of Anosov actions), in particular, the entropy maximizing measure and the SRB measure. A characterization of equilibrium states in terms of their disintegrations along stable and unstable foliations is also given. Finally, we show that the resulting system is isomorphic to a Bernoulli scheme.

We introduce a category of filtered F-isocrystals and construct a symbol map from Milnor K-theory to the group of 1-extensions of filtered F-isocrystals. We show that our symbol map is compatible with the syntomic symbol map to the log syntomic cohomology by Kato and Tsuji. These are fundamental materials in our computations of syntomic regulators which we work in other papers.

Let K be an imaginary quadratic field and $p\geq 5$ a rational prime inert in K. For a $\mathbb {Q}$-curve E with complex multiplication by $\mathcal {O}_K$ and good reduction at p, K. Rubin introduced a p-adic L-function $\mathscr {L}_{E}$ which interpolates special values of L-functions of E twisted by anticyclotomic characters of K. In this paper, we prove a formula which links certain values of $\mathscr {L}_{E}$ outside its defining range of interpolation with rational points on E. Arithmetic consequences include p-converse to the Gross–Zagier and Kolyvagin theorem for E.

A key tool of the proof is the recent resolution of Rubin’s conjecture on the structure of local units in the anticyclotomic ${\mathbb {Z}}_p$-extension $\Psi _\infty $ of the unramified quadratic extension of ${\mathbb {Q}}_p$. Along the way, we present a theory of local points over $\Psi _\infty $ of the Lubin–Tate formal group of height $2$ for the uniformizing parameter $-p$.

We prove that real topological Hochschild homology $\mathrm {THR}$ for schemes with involution satisfies base change and descent for the ${\mathbb {Z}/2}$-isovariant étale topology. As an application, we provide computations for the projective line (with and without involution) and the higher-dimensional projective spaces.