We show that the $L^1$ norm of an exponential sum of length X, and with coefficients equal to the Liouville or Möbius function, is at least $\gg_{\varepsilon} X^{1/4 - \varepsilon}$ for any given $\varepsilon$. For the Liouville function, this improves on the lower bound $\gg X^{1/\log\log X}$ due to Balog and Perelli. For the Möbius function, this improves the lower bound $\gg X^{1/6}$ due to Balog and Ruzsa. The large discrepancy between these lower bounds is due to the method employed by Balog and Ruzsa, as it crucially relies on the vanishing of $\mu(n)$. Instead, our proof puts the two cases on an equal footing by exploiting the connection of these coefficients with zeros of Dirichlet L-functions. In the second paper in this series, we will obtain a lower bound $\gg X^{\delta}$ for some small $\delta$ but for general (non-pretentious) multiplicative functions.

Let p be a prime number and K a finite unramified extension of $\mathbb{Q}_p$. Let $\pi$ be an admissible smooth mod p representation of $\operatorname{GL}_2(K)$ occurring in some Hecke eigenspaces of the mod p cohomology and $\overline{r}$ be its underlying global two-dimensional Galois representation. When $\overline{r}$ satisfies some Taylor–Wiles hypotheses and is sufficiently generic at p, we compute explicitly certain constants appearing in the diagram associated to $\pi$, generalizing the results of Dotto and Le. As a result, we prove that the associated étale $(\varphi,\mathcal{O}_K^{\times})$-module $D_A(\pi)$ defined by Breuil, Herzig, Hu, Morra and Schraen is explicitly determined by the restriction of $\overline{r}$ to the decomposition group at p, generalizing the results of Breuil et al. and the author.

We prove the vanishing of bounded cohomology with separable dual coefficients for many groups of interest in geometry, dynamics, and algebra. These include compactly supported structure-preserving diffeomorphism groups of certain manifolds; the group of interval exchange transformations of the half line; piecewise linear and piecewise projective groups of the line, giving strong answers to questions of Calegari and Navas; direct limit linear groups of relevance in algebraic K-theory, thereby answering a question by Kastenholz and Sroka and a question of two of the authors and Löh; and certain subgroups of big mapping class groups, such as the stable braid group and the stable mapping class group, proving a conjecture of Bowden. Moreover, we prove that in the recently introduced framework of enumerated groups, the generic group has vanishing bounded cohomology with separable dual coefficients. At the heart of our approach is an elementary algebraic criterion called the commuting cyclic conjugates condition that is readily verifiable for the aforementioned large classes of groups.

We prove a monoidal equivalence, called universal Koszul duality, between genuine equivariant K-motives on a Kac–Moody flag variety and constructible monodromic sheaves on its Langlands dual. The equivalence is obtained by a Soergel-theoretic description of both sides which extends results for finite-dimensional flag varieties by Taylor and the first author. Universal Koszul duality bundles together a whole family of equivalences for each point of a maximal torus. At the identity, it recovers an ungraded version of Beilinson, Ginzburg and Soergel’s and Bezrukavnikov and Yun’s Koszul duality for equivariant and unipotently monodromic sheaves. It also generalizes Soergel-theoretic descriptions for monodromic categories on finite-dimensional flag varieties by Lusztig and Yun, Gouttard and the second author. For affine Kac–Moody groups, our work sheds new light on the conjectured quantum Satake equivalences by Cautis and Kamnitzer as well as Gaitsgory. On our way, we establish foundations on six functors for reduced K-motives and introduce a formalism of constructible monodromic sheaves.

We study the injectivity of the Kudla–Millson lift of genus-2 Siegel cusp forms, vector-valued with respect to the Weil representation associated to an even lattice L. We prove that if L splits off two hyperbolic planes and is of sufficiently large rank, then the lift is injective. As an application, we deduce that the image of the lift in the degree-4 cohomology of the associated orthogonal Shimura variety has the same dimension as the lifted space of cusp forms. Our results also cover the case of moduli spaces of quasi-polarized K3 surfaces. To prove the injectivity, we introduce vector-valued indefinite Siegel theta functions of genus 2 and of Jacobi type attached to L. We describe their behavior with respect to the split of a hyperbolic plane in L. This generalizes the results of Borcherds to genus higher than 1.

We define relative Donaldson–Thomas invariants of local log Calabi–Yau 4-folds, under the assumption that the derived Hilbert scheme on the anti-canonical divisor is a derived critical locus. We prove a degeneration formula and apply it to compute zero-dimensional invariants on $\mathbb{C}^4$ and on any local curve. This verifies a conjecture proposed by the first author and Kool.

The symplectic spectral metric on the set of Lagrangian submanifolds or Hamiltonian maps can be used to define a completion of these spaces. For an element of such a completion, we define its $\gamma$-support. We also define the notion of a $\gamma$-coisotropic set and prove that every $\gamma$-support is $\gamma$-coisotropic. We then establish numerous properties of $\gamma$-supports and $\gamma$-coisotropic sets. We construct examples of Lagrangians in the completion having large $\gamma$-support and we study those (called ‘regular Lagrangians’) having small $\gamma$-support. Finally, we try to understand which singular Hamiltonians (i.e. Hamiltonians continuous on the complement of a set) have a well-defined flow in the Humilière completion.

We show that the category of ind-coherent sheaves on a quasi-smooth scheme is naturally tensored over the category of sheared D-modules on its shifted cotangent bundle, commuting with its natural action of categorified Hoschschild cochains. We prove that it defines a Morita equivalence as such. We then extend these results to quasi-smooth Artin stacks. As a consequence of our formalism, we are able to articulate a precise sense in which the space of unramified automorphic functions over a function field localizes over the stack of arithmetic Arthur parameters.

We prove a Poincaré duality for arithmetic p-adic pro-étale cohomology of smooth dagger curves over finite extensions of $\mathbf{Q}_p$. We deduce it, via the Hochschild–Serre spectral sequence, from geometric comparison theorems combined with Tate and Serre dualities. The compatibility of all the products involved is checked via reduction to the ghost circle, for which we also prove a Poincaré duality (showing that it behaves like a proper smooth analytic variety of dimension $1/2$). Along the way we study functional analytic properties of arithmetic p-adic pro-étale cohomology and prove that the usual cohomology is nuclear Fréchet and the compactly supported one: of compact type.

For all Zarski dense Anosov subgroups of a semisimple real algebraic group, we prove that their limit sets are Ahlfors regular for intrinsic conformal premetrics. As a consequence, we obtain that a Patterson–Sullivan measure is Ahlfors regular (and hence equal to the Hausdorff measure) if and only if the associated linear form is symmetric. We also discuss several applications, including analyticity of (p,q)-Hausdorff dimensions on the Teichmüller spaces, new upper bounds on the growth indicator, and $L^2$-spectral properties of associated locally symmetric manifolds.

Let k be a perfect field of characteristic p and let W(k) be its ring of Witt vectors. We construct an equivalence of categories between the full subcategory of the derived category of quasi-coherent sheaves on the syntomification of W(k) spanned by objects whose Hodge–Tate weights are between $[0,p-2]$ and an appropriate derived category of Fontaine–Laffaille modules.

We study the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology 3-sphere from the point of view of gauge theory. Namely, we introduce a class of perturbations of the $SL(2,\mathbb{C})$ Chern–Simons functional and prove a localization result: the perturbed critical points either approach a compact subset of the $SL(2, \mathbb{C})$ character variety or else ‘escape to infinity’. Furthermore, the Euler characteristic and Poincaré polynomial of the stable locus of the character variety are obtained by suitably counting the localizing critical points. As an application, we obtain formulas for the Euler characteristic and Poincaré polynomial of the stable locus of the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology 3-sphere. In particular, we prove that the Euler characteristic equals the Milnor number (divided by 4) of any weighted-homogeneous isolated complete intersection singularity whose link is the given 3-manifold.

We give a proof of Mukai’s theorem on the existence of certain exceptional vector bundles on prime Fano threefolds. To our knowledge this is the first complete proof in the literature. The result is essential for Mukai’s biregular classification of prime Fano threefolds, and for the existence of semiorthogonal decompositions in their derived categories. Our approach is based on Lazarsfeld’s construction that produces vector bundles on a variety from globally generated line bundles on a divisor, on Mukai’s theory of stable vector bundles on K3 surfaces, and on Brill–Noether properties of curves and (sensu Mukai) of K3 surfaces.

We investigate the realisability of the Casson–Sullivan invariant for homeomorphisms of smooth 4-manifolds, which is the obstruction to a homeomorphism being stably pseudo-isotopic to a diffeomorphism, valued in the third cohomology of the source manifold with $\mathbb{Z}/2$-coefficients. We prove that for all pairs of orientable, homeomorphic, smooth 4-manifolds this invariant can be realised fully after stabilising with a single $S^2\times S^2$. As an application, we obtain that topologically isotopic surfaces in a smooth, simply-connected 4-manifold become smoothly isotopic after sufficient external stabilisations. We further demonstrate cases where this invariant can be realised fully without stabilisation for self-homeomorphisms, which includes for manifolds with finite cyclic fundamental group. This method allows us to produce many examples of homeomorphisms which are not stably pseudo-isotopic to any diffeomorphism but are homotopic to the identity. Finally, we reinterpret these results in terms of finding examples of smooth structures on 4-manifolds which are diffeomorphic but not stably pseudo-isotopic.

We introduce and explore the Uniform Izumi–Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if R is a normal domain essentially of finite type over a field, there exists a constant C such that for all prime ideals $\mathfrak{p}\subseteq \mathfrak{q}\in\mathrm{Spec}(R)$, if $\mathfrak{p}\subseteq \mathfrak{q}^{(t)}$, then for all $n\in\mathbb{N}$, there is a containment of symbolic powers $\mathfrak{p}^{(Cn)}\subseteq \mathfrak{q}^{(tn)}$.

We prove that the set of limit points of the set of all spectral gaps of closed arithmetic hyperbolic surfaces equals $[0,\frac{1}{4}]$.

We study the geometry of Calabi–Yau conifold transitions. This deformation process is known to possibly connect a Kähler threefold to a non-Kähler threefold. We use balanced and Hermitian Yang–Mills metrics to geometrize the conifold transition and show that the whole operation is continuous in the Gromov–Hausdorff topology.

A Laurent polynomial in two variables is tempered if its edge polynomials are cyclotomic. Variation of coefficients leads to a family of smooth complete genus-g curves carrying a canonical algebraic $K_2$-class over a g-dimensional base S, hence to an extension of admissible variations of mixed Hodge structure (or normal function) on S. We prove that the $\mathbb{R}$-split locus of this extension is finite. Consequently, the torsion locus of the normal function and the A-factor locus for the family of curves are also finite.

We prove an arithmetic refinement of the Yau–Zaslow formula by replacing the classical Euler characteristic in Beauville’s argument by a motivic Euler characteristic, related to the work of Levine. Our result implies similar formulas for other related invariants, including a generalisation of a formula of Kharlamov and Răsdeaconu on counting real rational curves on real K3 surfaces, and Saito’s determinant of cohomology.

Let k be a perfect field of characteristic p and $\Gamma$ an infinite, first countable pro-p group. We study the behavior of the p-primary part of the ‘motivic class group’, i.e. the full p-divisible group of the Jacobian, in any $\Gamma$-tower of function fields over k that is unramified outside a finite (possibly empty) set of places $\Sigma$, and totally ramified at every place of $\Sigma$. When $\Sigma=\emptyset$ and $\Gamma$ is a torsion-free p-adic Lie group, we obtain asymptotic formulae which show that the p-torsion class group schemes grow in a remarkably regular manner. In the ramified setting $\Sigma\neq\emptyset$, we obtain a similar asymptotic formula for the p-torsion in ‘physical class groups’, i.e. the k-rational points of the Jacobian, which generalizes the work of Mazur and Wiles, who studied the case $\Gamma=\mathbf Z_p$.