We investigate exponential sums over those numbers ${\leqslant}x$ all of whose prime factors are ${\leqslant}y$. We prove fairly good minor arc estimates, valid whenever $\log ^{3}x\leqslant y\leqslant x^{1/3}$. Then we prove sharp upper bounds for the $p$th moment of (possibly weighted) sums, for any real $p>2$ and $\log ^{C(p)}x\leqslant y\leqslant x$. Our proof develops an argument of Bourgain, showing that this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our bounds with major arc estimates of Drappeau, we obtain an asymptotic for the number of solutions of $a+b=c$ in $y$-smooth integers less than $x$ whenever $\log ^{C}x\leqslant y\leqslant x$. Previously this was only known assuming the generalised Riemann hypothesis. Combining them with transference machinery of Green, we prove Roth’s theorem for subsets of the $y$-smooth numbers whenever $\log ^{C}x\leqslant y\leqslant x$. This provides a deterministic set, of size ${\approx}x^{1-c}$, inside which Roth’s theorem holds.

Let $G$ be a real reductive group and $Z=G/H$ a unimodular homogeneous $G$ space. The space $Z$ is said to satisfy VAI (vanishing at infinity) if all smooth vectors in the Banach representations $L^{p}(Z)$ vanish at infinity, $1\leqslant p<\infty$. For $H$ connected we show that $Z$ satisfies VAI if and only if it is of reductive type.

We show that there are no non-trivial stratified bundles over a smooth simply connected quasi-projective variety over an algebraic closure of a finite field if the variety admits a normal projective compactification with boundary locus of codimension greater than or equal to $2$.

We classify the horizontal $\text{SL}(2)$s and $\mathbb{R}$-split polarized mixed Hodge structures on a Mumford–Tate domain.

Using twisted nearby cycles, we define a new notion of slopes for complex holonomic ${\mathcal{D}}$ -modules. We prove a boundedness result for these slopes, study their functoriality and use them to characterize regularity. For a family of (possibly irregular) algebraic connections ${\mathcal{E}}_{t}$ parametrized by a smooth curve, we deduce under natural conditions an explicit bound for the usual slopes of the differential equation satisfied by the family of irregular periods of the ${\mathcal{E}}_{t}$ . This generalizes the regularity of the Gauss–Manin connection proved by Griffiths, Katz and Deligne.

Let ${\rm\Gamma}$ be a finite subgroup of $\text{Sp}(V)$. In this article we count the number of symplectic resolutions admitted by the quotient singularity $V/{\rm\Gamma}$. Our approach is to compare the universal Poisson deformation of the symplectic quotient singularity with the deformation given by the Calogero–Moser space. In this way, we give a simple formula for the number of $\mathbb{Q}$-factorial terminalizations admitted by the symplectic quotient singularity in terms of the dimension of a certain Orlik–Solomon algebra naturally associated to the Calogero–Moser deformation. This dimension is explicitly calculated for all groups ${\rm\Gamma}$ for which it is known that $V/{\rm\Gamma}$ admits a symplectic resolution. As a consequence of our results, we confirm a conjecture of Ginzburg and Kaledin.

The purpose of this work is to give a definition of a topological K-theory for dg-categories over $\mathbb{C}$ and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cyclic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps: taking the topological realization of algebraic K-theory and inverting the Bott element. The topological realization is the left Kan extension of the functor ‘space of complex points’ to all simplicial presheaves over complex algebraic varieties. Our first main result states that the topological K-theory of the unit dg-category is the spectrum $\mathbf{BU}$. For this we are led to prove a homotopical generalization of Deligne’s cohomological proper descent, using Lurie’s proper descent. The fact that the Chern character descends to topological K-theory is established by using Kassel’s Künneth formula for periodic cyclic homology and the proper descent. In the case of a dg-category of perfect complexes on a separated scheme of finite type, we show that we recover the usual topological K-theory of complex points. We show as well that the Chern map tensorized with $\mathbb{C}$ is an equivalence in the case of a finite-dimensional associative algebra – providing a formula for the periodic homology groups in terms of the stack of finite-dimensional modules.

Let $X\subset \mathbb{P}^{4}$ be a terminal factorial quartic $3$-fold. If $X$ is non-singular, $X$ is birationally rigid, i.e. the classical minimal model program on any terminal $\mathbb{Q}$-factorial projective variety $Z$ birational to $X$ always terminates with $X$. This no longer holds when $X$ is singular, but very few examples of non-rigid factorial quartics are known. In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface $X\subset \mathbb{P}^{4}$. A singular point on such a hypersurface is of type $cA_{n}$ ($n\geqslant 1$), or of type $cD_{m}$ ($m\geqslant 4$) or of type $cE_{6},cE_{7}$ or $cE_{8}$. We first show that if $(P\in X)$ is of type $cA_{n}$, $n$ is at most $7$ and, if $(P\in X)$ is of type $cD_{m}$, $m$ is at most $8$. We then construct examples of non-rigid factorial quartic hypersurfaces whose singular loci consist (a) of a single point of type $cA_{n}$ for $2\leqslant n\leqslant 7$, (b) of a single point of type $cD_{m}$ for $m=4$ or $5$ and (c) of a single point of type $cE_{k}$ for $k=6,7$ or $8$.

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$ , motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some where $K$ is of characteristic $2$ , and we complete its proof by reducing the positive characteristic case to characteristic $0$ . For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_{p}$ : we find an elliptic curve $E^{\prime }$ defined over a $p$ -adic field such that all the terms in the Kramer–Tunnell formula for $E^{\prime }$ are equal to those for $E$ .

We generalize Siegel’s theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree $d$ or less over some number field. Generalizing Picard’s theorem, we prove an analogous result characterizing complex affine curves admitting a nonconstant holomorphic map from a degree $d$ (or less) analytic cover of $\mathbb{C}$.

The notion of Berman–Gibbs stability was originally introduced by Berman for $\mathbb{Q}$-Fano varieties $X$. We show that the pair $(X,-K_{X})$ is K-stable (respectively K-semistable) provided that $X$ is Berman–Gibbs stable (respectively semistable).

We start developing a notion of reciprocity sheaves, generalizing Voevodsky’s homotopy invariant presheaves with transfers which were used in the construction of his triangulated categories of motives. We hope that reciprocity sheaves will eventually lead to the definition of larger triangulated categories of motivic nature, encompassing non-homotopy invariant phenomena.

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\bar{\unicode[STIX]{x1D70C}}$ be a continuous, absolutely irreducible representation of its absolute Galois group with values in a finite field of characteristic $p$. We prove that the Galois representations that become crystalline of a fixed regular weight after an abelian extension are Zariski-dense in the generic fiber of the universal deformation ring of $\bar{\unicode[STIX]{x1D70C}}$. In fact we deduce this from a similar density result for the space of trianguline representations. This uses an embedding of eigenvarieties for unitary groups into the spaces of trianguline representations as well as the corresponding density claim for eigenvarieties as a global input.

Recently Houdayer and Isono have proved, among other things, that every biexact group $\unicode[STIX]{x1D6E4}$ has the property that for any non-singular strongly ergodic essentially free action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ on a standard measure space, the group measure space von Neumann algebra $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(X)$ is full. In this paper, we prove the same property for a wider class of groups, notably including $\text{SL}(3,\mathbb{Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $\unicode[STIX]{x1D6E4}\leqslant G$, and any closed non-amenable subgroup $H\leqslant G$, the non-singular action $\unicode[STIX]{x1D6E4}\curvearrowright G/H$ is strongly ergodic and the von Neumann factor $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(G/H)$ is full.

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$, the moduli space of $n$-pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$. This result is obtained via a more general study of the cohomology groups of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^{k}({\mathcal{M}}_{2,n}^{\mathsf{ct}})$ for any $k$ and $n$ considered both as $\mathbb{S}_{n}$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline{{\mathcal{M}}}_{2,n}$ is tautological for $n<20$, and that the tautological ring of $\overline{{\mathcal{M}}}_{2,n}$ fails to have Poincaré duality for all $n\geqslant 20$. This improves and simplifies results of the author and Orsola Tommasi.

We define a theory of étale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of these categories coincides with the triangulated categories of Beilinson motives (and is thus strongly related to algebraic $K$-theory). We extend the rigidity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion étale motives essentially coincide with the usual complexes of torsion étale sheaves (at least if we restrict ourselves to torsion prime to the residue characteristics). As a consequence, we obtain the expected results of absolute purity, of finiteness, and of Grothendieck duality for étale motives with integral coefficients, by putting together their counterparts for Beilinson motives and for torsion étale sheaves. Following Thomason’s insights, this also provides a conceptual and convenient construction of the $\ell$-adic realization of motives, as the homotopy $\ell$-completion functor.

We give a method for constructing a Legendrian representative of a knot in $S^{3}$ which realizes its maximal Thurston–Bennequin number under a certain condition. The method utilizes Stein handle decompositions of $D^{4}$ , and the resulting Legendrian representative is often very complicated (relative to the complexity of the topological knot type). As an application, we construct infinitely many knots in $S^{3}$ each of which yields a reducible 3-manifold by a Legendrian surgery in the standard tight contact structure. This disproves a conjecture of Lidman and Sivek.

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmüller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apéry-like integrality statement for solutions of Picard–Fuchs equations. We illustrate the theory on a worked example, giving explicit Fourier expansions of twisted modular forms and the equation of a Teichmüller curve in a Hilbert modular surface. In Part III we show that genus two Teichmüller curves are cut out in Hilbert modular surfaces by a product of theta derivatives. We rederive most of the known properties of those Teichmüller curves from this viewpoint, without using the theory of flat surfaces. As a consequence we give the modular embeddings for all genus two Teichmüller curves and prove that the Fourier developments of their twisted modular forms are algebraic up to one transcendental scaling constant. Moreover, we prove that Bainbridge’s compactification of Hilbert modular surfaces is toroidal. The strategy to compactify can be expressed using continued fractions and resembles Hirzebruch’s in form, but every detail is different.

In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$ . To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity.