We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include Whitney–Tate stratifications of Beilinson–Drinfeld Grassmannians and cellular decompositions of semi-infinite orbits. With future global applications in mind, we also achieve an equivalence relative to a power of the affine line. Finally, we use our equivalence to give Tannakian constructions of Deligne’s modification of the dual group and a modified form of Vinberg’s monoid over the integers.

We prove cohomological vanishing criteria for the Ceresa cycle of a curve C embedded in its Jacobian J: (A) if $\mathrm{H}^3(J)^{\mathrm{Aut}(C)} = 0$, then the Ceresa cycle is torsion modulo rational equivalence; (B) if $\mathrm{H}^0(J, \Omega_J^3)^{\mathrm{Aut}(C)} = 0$, then the Ceresa cycle is torsion modulo algebraic equivalence, with criterion (B) conditional on the Hodge conjecture. We then use these criteria to study the simplest family of curves where (B) holds but (A) does not, namely the family of Picard curves $C \colon y^3 = x^4 + ax^2 + bx + c$. Criterion (B) and work of Schoen combine to show that the Ceresa cycle of a Picard curve is torsion in the Griffiths group. We furthermore determine exactly when it is torsion in the Chow group. As a byproduct, we deduce that there exist one-parameter families of plane quartic curves with torsion Ceresa Chow class; that the torsion locus in $\mathcal{M}_3$ of the Ceresa Chow class contains infinitely many components; and that the order of a torsion Ceresa Chow class of a Picard curve over a number field K is bounded, with the bound depending only on $[K\colon \mathbb{Q}]$. Finally, we determine which automorphism group strata are contained in the vanishing locus of the universal Ceresa cycle over $\mathcal{M}_3$.

We study functions f on $\mathbb{Q}$ which satisfy a ‘quantum modularity’ relation of the shape $ f(x+1)=f(x)$ and $f(x) - {| {x} |}^{-k} f(-1/x) = h(x)$, where $h:\mathbb{R}_{\neq 0} \to \mathbb{C}$ is a function satisfying various regularity conditions. We study the case $\operatorname{Re}(k)\neq 0$. We prove the existence of a limiting function, denoted by $f^\triangleleft$ or $f^\triangleright$, depending on the sign of $\operatorname{Re}(k)$, which extends continuously f to $\mathbb{R}$ in some sense. This means, in particular, that in the $\operatorname{Re}(k)\neq0$ case, the quantum modular form itself has to have at least a certain level of regularity. We deduce that the values $\{f(a/q), 1\leqslant a<q, (a, q)=1\}$, appropriately normalized, tend to equidistribute along the graph of $f^\triangleleft$ or $f^\triangleright$, and we prove that, under natural hypotheses, the limiting measure is diffuse. We apply these results to obtain limiting distributions of values and continuity results for several arithmetic functions known to satisfy the above quantum modularity: higher weight modular symbols associated to holomorphic cusp forms; Eichler integral associated to Maaß forms; a function of Kontsevich and Zagier related to the Dedekind $\eta$-function; and generalized cotangent sums.

This work concerns representations of a finite flat group scheme G defined over a noetherian commutative ring R. The focus is on lattices, namely, finitely generated G-modules that are projective as R-modules, and on the full subcategory of all G-modules projective over R generated by the lattices. The stable category of such G-modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of G. Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.

We give an interpretation of the semi-infinite intersection cohomology sheaf associated with a semisimple simply connected algebraic group in terms of finite-dimensional geometry. Specifically, we describe a procedure for building factorization spaces over moduli spaces of finite subsets of a curve from factorization spaces over moduli spaces of divisors, and show that, under this procedure, the compactified Zastava space is sent to the support of the semi-infinite intersection cohomology (IC) sheaf in the factorizable Grassmannian. We define ‘semi-infinite t-structures’ for a large class of schemes with an action of the multiplicative group, and show that, for the Zastava, the limit of these t-structures recovers the infinite-dimensional version. As an application, we also construct factorizable parabolic semi-infinite IC sheaves and a generalization (of the principal case) to Kac–Moody algebras.

We improve, by a factor of 2, known homology stability ranges for the integral homology of symplectic groups over commutative local rings with infinite residue field and show that the obstruction to further stability is bounded below by Milnor–Witt K-theory. In particular, our stability range is optimal in many cases.

For a split reductive group G we realise identities in the Grothendieck group of $\widehat{G}$-representations in terms of cycle relations between certain closed subschemes inside the affine Grassmannian. These closed subschemes are obtained as a degeneration of e-fold products of flag varieties and, under a bound on the Hodge type, we relate the geometry of these degenerations to that of moduli spaces of G-valued crystalline representations of $\operatorname{Gal}(\overline{K}/K)$ for $K/\mathbb{Q}_p$ a finite extension with ramification degree e. By transferring the aforementioned cycle relations to these moduli spaces we deduce one direction of the Breuil–Mézard conjecture for G-valued crystalline representations with small Hodge type.

We show that the derived categories of perverse Nori motives investigate and mixed Hodge modules are the derived categories of their constructible hearts. This enables us to construct $\infty$-categorical lifts of the six operations. As a result, we obtain realisation functors from the category of Voevodsky étale motives to the derived categories of perverse Nori motives and mixed Hodge modules that commute with the operations. We also prove that if a motivic t-structure exists then Voevodsky étale motives and the derived category of perverse Nori motives are equivalent. Finally, we give a presentation of the indization of the derived category of perverse Nori motives as a category of modules in Voevodsky étale motives.

In this paper, we give a new geometric definition of nearly overconvergent modular forms and p-adically interpolate the Gauss–Manin connection on this space. This can be seen as an ‘overconvergent’ version of the unipotent circle action on the space of p-adic modular forms, as constructed by Gouvêa and Howe. This improves on results of Andreatta and Iovita and has applications to the construction of Rankin–Selberg and triple-product p-adic L-functions.

Counterexamples to Lagrangian Poincaré recurrence were recently found in dimensions greater than six by Broćić and Shelukhin. We construct counterexamples in dimension four using almost toric fibrations.

In this paper, we give a complete, two-way characterization, of when a noncommutative crossed product $A \rtimes_\lambda G$ is simple, in the case of G being an FC-hypercentral group. This is a large class of amenable groups that contains all virtually nilpotent groups, and in the finitely generated setting, coincides with the set of groups which have polynomial growth. We further completely characterize the ideal intersection property under the assumption that the group is FC, meaning that every element has a finite conjugacy class. Finally, for minimal actions of arbitrary discrete groups on unital C*-algebras, we are able to characterize when the crossed product $A \rtimes_\lambda G$ is prime.

We investigate the relationship between affine and Stein varieties in the context of rigid geometry. We show that the two concepts are much more closely related than in complex geometry, e.g. they are equivalent for surfaces. This rests on the density of algebraic functions in analytic functions. One key ingredient to prove such a density statement is an extension result for Cartier divisors.

We study certain categories associated to symmetric quivers with potential, called quasi-Bogomol’nyi–Prasad–Sommerfield (BPS) categories. We construct semiorthogonal decompositions of the categories of matrix factorizations for moduli stacks of representations of (framed or unframed) symmetric quivers with potential, where the summands are categorical Hall products of quasi-BPS categories. These results generalize our previous results about the three-loop quiver. We prove several properties of quasi-BPS categories: wall-crossing equivalence, strong generation, and a categorical support lemma in the case of tripled quivers with potential. We also introduce reduced quasi-BPS categories for preprojective algebras, which have trivial relative Serre functor and are indecomposable when the weight is coprime with the total dimension. In this case, we regard the reduced quasi-BPS categories as noncommutative local hyperkähler varieties and as (twisted) categorical versions of crepant resolutions of singularities of good moduli spaces of representations of preprojective algebras. The studied categories include the local models of quasi-BPS categories of K3 surfaces. In a follow-up paper, we establish analogous properties for quasi-BPS categories of K3 surfaces.

We show that smooth numbers are equidistributed in arithmetic progressions to moduli of size $x^{66/107-o(1)}$. This overcomes a longstanding barrier of $x^{3/5-o(1)}$ present in previous works of Bombieri, Friedlander and Iwaniec, Fouvry and Tenenbaum, Drappeau, and Maynard. We build on Drappeau’s variation of Linnik’s dispersion method and on exponential sum manipulations of Maynard, ultimately relying on optimized Deshouillers–Iwaniec-type estimates for sums of Kloosterman sums.

We show that each local field $\mathbb{F}_q(\!(t)\!)$ of characteristic $p > 0$ is characterised up to isomorphism within the class of all fields of imperfect exponent at most 1 by (certain small quotients of) its absolute Galois group together with natural axioms concerning the p-torsion of its Brauer group. This complements previous work by Efrat and Fesenko, who analysed fields whose absolute Galois group is isomorphic to that of a local field of characteristic p.

Let $X$ be a very general Gushel–Mukai (GM) variety of dimension $n\geq 4$, and let $Y$ be a smooth hyperplane section. There are natural pull-back and push-forward functors between the semi-orthogonal components (known as the Kuznetsov components) of the derived categories of $X$ and $Y$. In this paper, we prove that the Bridgeland stability of objects is preserved by both pull-back and push-forward functors. We then explore various applications of this result, such as constructing an eight-dimensional smooth family of Lagrangian subvarieties for each moduli space of stable objects in the Kuznetsov component of a general GM fourfold and proving the projectivity of the moduli spaces of semistable objects of any class in the Kuznetsov component of a general GM threefold, as conjectured by Perry, Pertusi, and Zhao.

A Pell–Abel equation is a functional equation of the form $P^{2}-DQ^{2} = 1$, with a given polynomial $D$ free of squares and unknown polynomials $P$ and $Q$. We show that the space of Pell–Abel equations with the degrees of $D$ and of the primitive solution $P$ fixed is a complex manifold. We describe its connected components by an efficiently computable invariant. Moreover, we give various applications of this result, including to torsion pairs on hyperelliptic curves and to Hurwitz spaces, and a description of the connected components of the space of primitive $k$-differentials with a unique zero on genus $2$ Riemann surfaces.

We study random walks on metric spaces with contracting isometries. In this first article of the series, we establish sharp deviation inequalities by adapting Gouëzel’s pivotal time construction. As an application, we establish the exponential bounds for deviation from below, central limit theorem, law of the iterated logarithms, and the geodesic tracking of random walks on mapping class groups and CAT(0) spaces.

We establish a derived geometric Satake equivalence for the quaternionic general linear group ${\textrm{GL}}_{n}({\mathbb H})$. By applying the real–symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the symmetric variety ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$. We explain how these equivalences fit into the general framework of a geometric Langlands correspondence for real groups and the relative Langlands duality conjecture. As an application, we compute the stalks of the IC-complexes for spherical orbit closures in the quaternionic affine Grassmannian and the loop space of ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$. We show that the stalks are given by the Kostka–Foulkes polynomials for ${\textrm{GL}}_n$ but with all degrees doubled.

The Hanna Neumann conjecture (HNC) for a free group G predicts that $\overline{\chi}(U\cap V)\leqslant \overline{\chi} (U)\overline{\chi}(V)$ for all finitely generated subgroups U and V, where $\overline{\chi}(H) = \max\{-\chi(H),0\}$ denotes the reduced Euler characteristic of H. A strengthened version of the HNC was proved independently by Friedman and Mineyev in 2011. Recently, Antolín and Jaikin-Zapirain introduced the $L^2$-Hall property and showed that if G is a hyperbolic limit group that satisfies this property, then G satisfies the HNC. Antolín and Jaikin-Zapirain established the $L^2$-Hall property for free and surface groups, which Brown and Kharlampovich extended to all limit groups. In this paper, we prove the $L^2$-Hall property for graphs of free groups with cyclic edge groups that are hyperbolic relative to virtually abelian subgroups. We also give another proof of the $L^2$-Hall property for limit groups. As a corollary, we show that all these groups satisfy a strengthened version of the HNC.