Halász’s theorem gives an upper bound for the mean value of a multiplicative function $f$. The bound is sharp for general such $f$, and, in particular, it implies that a multiplicative function with $|f(n)|\leqslant 1$ has either mean value $0$, or is ‘close to’ $n^{it}$ for some fixed $t$. The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to treat short intervals and arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel’s theorem), and that there are always primes near to the start of an arithmetic progression (Linnik’s theorem).

Let $X$ be a smooth projective manifold with $\dim _{\mathbb{C}}X=n$. We show that if a line bundle $L$ is $(n-1)$-ample, then it is $(n-1)$-positive. This is a partial converse to the Andreotti–Grauert theorem. As an application, we show that a projective manifold $X$ is uniruled if and only if there exists a Hermitian metric $\unicode[STIX]{x1D714}$ on $X$ such that its Ricci curvature $\text{Ric}(\unicode[STIX]{x1D714})$ has at least one positive eigenvalue everywhere.

Let $p$ and $\ell$ be distinct primes, and let $\overline{\unicode[STIX]{x1D70C}}$ be an orthogonal or symplectic representation of the absolute Galois group of an $\ell$-adic field over a finite field of characteristic $p$. We define and study a liftable deformation condition of lifts of $\overline{\unicode[STIX]{x1D70C}}$ ‘ramified no worse than $\overline{\unicode[STIX]{x1D70C}}$’, generalizing the minimally ramified deformation condition for $\operatorname{GL}_{n}$ studied in Clozel et al. [Automorphy for some$l$-adic lifts of automorphic mod$l$Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181; MR 2470687 (2010j:11082)]. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields.

We show that the bar version of the $\text{Pin}(2)$-monopole Floer homology of a three-manifold $Y$ equipped with a self-conjugate spin$^{c}$ structure $\mathfrak{s}$ is determined by the triple cup product of $Y$ together with the Rokhlin invariants of the spin structures inducing $\mathfrak{s}$. This is a manifestation of mod $2$ index theory and can be interpreted as a three-dimensional counterpart of Atiyah’s classical results regarding spin structures on Riemann surfaces.

We globalize the derived version of the McKay correspondence of Bridgeland, King and Reid, proven by Kawamata in the case of abelian quotient singularities, to certain logarithmic algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible logarithmic blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under logarithmic blow-up, up to Morita equivalence.

We construct a dynamically convex contact form on the three-sphere whose systolic ratio is arbitrarily close to 2. This example is related to a conjecture of Viterbo, whose validity would imply that the systolic ratio of a convex contact form does not exceed 1. We also construct, for every integer $n\geqslant 2$, a tight contact form with systolic ratio arbitrarily close to $n$ and with suitable bounds on the mean rotation number of all the closed orbits of the induced Reeb flow.

Sen attached to each $p$-adic Galois representation of a $p$-adic field a multiset of numbers called generalized Hodge–Tate weights. In this paper, we discuss a rigidity of these numbers in a geometric family. More precisely, we consider a $p$-adic local system on a rigid analytic variety over a $p$-adic field and show that the multiset of generalized Hodge–Tate weights of the local system is constant. The proof uses the $p$-adic Riemann–Hilbert correspondence by Liu and Zhu, a Sen–Fontaine decompletion theory in the relative setting, and the theory of formal connections. We also discuss basic properties of Hodge–Tate sheaves on a rigid analytic variety.

We study the values of finite multiple harmonic $q$-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetric multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we prove the duality formula for these values, as an example of linear relations, which induce those among FMZVs and SMZVs simultaneously. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic $q$-series at a primitive root of unity of sufficiently large degree.

This paper introduces a new Lagrangian surgery construction that generalizes Lalonde–Sikorav and Polterovich’s well-known construction, and combines this with Biran and Cornea’s Lagrangian cobordism formalism. With these techniques, we build a framework which both recovers several known long exact sequences (Seidel’s exact sequence, including the fixed point version and Wehrheim and Woodward’s family version) in symplectic geometry in a uniform way, and yields a partial answer to a long-term open conjecture due to Huybrechts and Thomas; this also involved a new observation which relates projective twists with surgeries.

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type $A_{1}^{n}$, $C_{2}$, and $\mathbf{F}_{p}$-split groups of type $A_{2}$ (this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular $3$-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type.

Let $\unicode[STIX]{x1D70F}(\cdot )$ be the classical Ramanujan $\unicode[STIX]{x1D70F}$-function and let $k$ be a positive integer such that $\unicode[STIX]{x1D70F}(n)\neq 0$ for $1\leqslant n\leqslant k/2$. (This is known to be true for $k<10^{23}$, and, conjecturally, for all $k$.) Further, let $\unicode[STIX]{x1D70E}$ be a permutation of the set $\{1,\ldots ,k\}$. We show that there exist infinitely many positive integers $m$ such that $|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(1))|<|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(2))|<\cdots <|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(k))|$. We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.

We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties.

We combine the ideas of a Harish-Chandra–Howe local character expansion, which can be centred at an arbitrary semisimple element, and a Kim–Murnaghan asymptotic expansion, which so far has been considered only around the identity. We show that, for most smooth, irreducible representations (those containing a good, minimal K-type), Kim–Murnaghan-type asymptotic expansions are valid on explicitly defined neighbourhoods of nearly arbitrary semisimple elements. We then give an explicit, inductive recipe for computing the coefficients in an asymptotic expansion for a tame supercuspidal representation. The only additional information needed in the inductive step is a fourth root of unity, which we expect to be useful in proving stability and endoscopic-transfer identities.

Let $f$ be a quasi-homogeneous polynomial with an isolated singularity in $\mathbf{C}^{n}$. We compute the length of the ${\mathcal{D}}$-modules ${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ generated by complex powers of $f$ in terms of the Hodge filtration on the top cohomology of the Milnor fiber. When $\unicode[STIX]{x1D706}=-1$ we obtain one more than the reduced genus of the singularity ($\dim H^{n-2}(Z,{\mathcal{O}}_{Z})$ for $Z$ the exceptional fiber of a resolution of singularities). We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the quotient ${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ is nonzero when $\unicode[STIX]{x1D706}$ is a root of the $b$-function of $f$ (which Saito recently showed fails to hold in the inhomogeneous case). We obtain these results by comparing these ${\mathcal{D}}$-modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.

In this paper we establish Springer correspondence for the symmetric pair $(\text{SL}(N),\text{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at $q=-1$. Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at $q=-1$ arise in geometry.

The cardinality of the set of $D\leqslant x$ for which the fundamental solution of the Pell equation $t^{2}-Du^{2}=1$ is less than $D^{1/2+\unicode[STIX]{x1D6FC}}$ with $\unicode[STIX]{x1D6FC}\in [\frac{1}{2},1]$ is studied and certain lower bounds are obtained, improving previous results of Fouvry by introducing the $q$-analogue of van der Corput method to algebraic exponential sums with smooth moduli.

When $p$ is inert in the quadratic imaginary field $E$ and $m<n$, unitary Shimura varieties of signature $(n,m)$ and a hyperspecial level subgroup at $p$, carry a natural foliationof height 1 and rank $m^{2}$ in the tangent bundle of their special fiber $S$. We study this foliation and show that it acquires singularities at deep Ekedahl–Oort strata, but that these singularities are resolved if we pass to a natural smooth moduli problem $S^{\sharp }$, a successive blow-up of $S$. Over the ($\unicode[STIX]{x1D707}$-)ordinary locus we relate the foliation to Moonen’s generalized Serre–Tate coordinates. We study the quotient of $S^{\sharp }$ by the foliation, and identify it as the Zariski closure of the ordinary-étale locus in the special fiber $S_{0}(p)$ of a certain Shimura variety with parahoric level structure at $p$. As a result, we get that this ‘horizontal component’ of $S_{0}(p)$, as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen–Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature $(m,m)$, and a certain Ekedahl–Oort stratum that we denote $S_{\text{fol}}$. We conjecture that these are the only integral submanifolds.