In this paper, we study the orthogonalities of Hecke eigenvalues of holomorphic cusp forms. An asymptotic large sieve with an unusually large main term for cusp forms is obtained. A family of special vectors formed by products of Kloosterman sums and Bessel functions is constructed for which the main term is exceptionally large. This surprising phenomenon reveals an interesting fact: that Fourier coefficients of cusp forms favor the direction of products of Kloosterman sums and Bessel functions of compatible type.

Let $V$ be a complete discrete valuation ring with residue field $k$ of characteristic $p>0$ and fraction field $K$ of characteristic zero. Let ${\cal S}$ be a formal scheme over $V$ and let $\mathfrak{X}\to {\cal S}$ be a locally projective formal abelian scheme. In this paper we prove that, under suitable natural conditions on the Hasse–Witt matrix of $\mathfrak{X}\otimes_V V/\mathit{pV}$, the kernel of the Frobenius morphism on $\mathfrak{X}_k$ can be canonically lifted to a finite and flat subgroup scheme of $\mathfrak{X}$ over an admissible blow-up of ${\cal S}$, called the ‘canonical subgroup of $\mathfrak{X}$’. This is done by a careful study of torsors under group schemes of order $p$ over $\mathfrak{X}$. We also present a filtration on ${\rm H}^1(\mathfrak{X},\mu_p)$ in the spirit of the Hodge–Tate decomposition.

Let $X$ be a smooth and projective variety such that $X$ is fibred over the projective space. We give sufficient conditions ensuring that the fibres contain adelic points satisfying Manin-like conditions.

We develop a language that makes the analogy between geometry and arithmetic more transparent. In this language there exists a base field $\mathbb{F}$, ‘the field with one element’; there is a fully faithful functor from commutative rings to $\mathbb{F}$-rings; there is the notion of the $\mathbb{F}$-ring of integers of a real or complex prime of a number field $K$ analogous to the $p$-adic integers, and there is a compactification of $\operatorname{Spec}O_K$; there is a notion of tensor product of $\mathbb{F}$-rings giving the product of $\mathbb{F}$-schemes; in particular there is the arithmetical surface $\operatorname{Spec} O_K\times\operatorname{Spec} O_K$, the product taken over $\mathbb{F}$.

Various $p$-adic versions of Littlewood's conjecture are investigated, generalizing a set-up considered recently by de Mathan and Teulié. In many cases it is shown that the sets of exceptions to these conjectures have Hausdorff dimension zero. The proof follows the measure ridigity approach of Einsiedler, Katok and Lindenstrauss.

Let $F$ be a $p$-adic field with uniformizer $\pi$. We consider $G={\rm GL}_{2}(F)/\pi$ and $I(1)$ the pro-$p$-Iwahori subgroup of $G$. The exploration of the smooth mod $p$ representations of $G$ motivates the study of the space of functions with values in $\overline{\mathbb{F}}_p$ and compact support in the set of right cosets $I(1)\backslash G$. We show that this universal module is flat over the pro-$p$-Hecke algebra if and only if the cardinal of the residue field of $F$ is equal to $p$.

This paper is a continuation of our previous work (D. Jiang and D. Soudry, On the genericity of cuspidal automorphic forms on${\rm SO}_{2n+1}$, J. reine angew. Math., to appear). We extend Moeglin's results (C. Moeglin, J. Lie Theory 7 (1997), 201–229, 231–238) from the even orthogonal groups to old orthogonal groups and complete our proof of the CAP conjecture for irreducible cuspidal automorphic representations of $\mathrm{SO}_{2n+1}(\mathbb{A})$ with special Bessel models. We also give a characterization of the vanishing of the central value of the standard $L$-function of $\mathrm{SO}_{2n+1}(\mathbb{A})$ in terms of theta correspondence. As a result, we obtain the weak Langlands functorial transfer from $\mathrm{SO}_{2n+1}(\mathbb{A})$ to $\mathrm{GL}_{2n}(\mathbb{A})$ for irreducible cuspidal automorphic representations of $\mathrm{SO}_{2n+1}(\mathbb{A})$ with special Bessel models.

In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras ${\mathbb F} = {\mathbb R}, {\mathbb C}$ or ${\mathbb H}$ which interpolate the convolution algebras of radial bounded Borel measures on a matrix space $M_{p,q}({\mathbb F})$ with $p\geq q$. Radiality in this context means invariance under the action of the unitary group $U_p({\mathbb F})$ from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize well-known structures in the rank-one case, namely the Bessel–Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper we study structures depending only on the matrix spectra. Under the mapping $r\mapsto \text{spec}(r)$, the convolutions on the underlying matrix cone induce a continuous series of hypergroup convolutions on a Weyl chamber of type $B_q$. The characters are now Dunkl-type Bessel functions. These convolution algebras on the Weyl chamber naturally extend the harmonic analysis for Cartan motion groups associated with the Grassmann manifolds $U(p,q)/(U_p\times U_q)$ over ${\mathbb F}$.

The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it has also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX relation). In addition, this relation was recently found to arise naturally in a theory of embedding obstructions for 2-spheres in 4-manifolds in terms of Whitney towers. This paper contains the first proof of the four-dimensional version of the Jacobi identity. We also expose the underlying topological unity between the three- and four-dimensional IHX relations, deriving from a beautiful picture of the Borromean rings embedded on the boundary of an unknotted genus 3 handlebody in 3-space. This picture is most naturally related to knot and 3-manifold invariants via the theory of grope cobordisms.