We introduce and study a remarkable family of real probability measures ${{\pi }_{st}}$ that we call free Bessel laws. These are related to the free Poisson law $\pi $ via the formulae ${{\text{ }\!\!\pi\!\!\text{ }}_{s1}}={{\text{ }\!\!\pi\!\!\text{ }}^{\boxtimes s}}$ and $\text{ }\pi {{\text{ }}_{1t}}=\text{ }\pi {{\text{ }}^{\boxtimes }}^{t}$. Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

In this paper, we give several pictorial fractal representations of the ample or Kähler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth (2, 2, 2) forms in ${{\mathbb{P}}^{1}}\,\times \,{{\mathbb{P}}^{1}}\,\times \,{{\mathbb{P}}^{1}}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be 1.296 ± .010.

We prove a uniformcontrol as $z\,\to \,0$ for the resolvent ${{(P-z)}^{-1}}$ of long range perturbations $P$ of the Euclidean Laplacian in divergence form by combining positive commutator estimates and properties of Riesz transforms. These estimates hold in dimension $d\,\ge \,3$ when $P$ is defined on ${{\mathbb{R}}^{d}}$ and in dimension $d\,\ge \,2$ when $P$ is defined outside a compact obstacle with Dirichlet boundary conditions.

We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. In fact, we introduce more general vectorial zeta-functions and prove their holomorphic continuation. Moreover, we show a vectorial sum formula among those vectorial zeta-functions from which some generalizations of the classical sum formula can be deduced.

We rigorously study the resonant tunneling of fast solitons through large potential barriers for the nonlinear Schrödinger equation in one dimension. Our approach covers the case of general nonlinearities, both local and Hartree (nonlocal).

Let $X$ be a connected Riemannian manifold such that the resolvent of the free Laplacian ${{(\Delta -z)}^{-1}},\,z\,\in \,\mathbb{C}\backslash {{\mathbb{R}}^{+}}$, has a meromorphic continuation through ${{\mathbb{R}}^{+}}$. The poles of this continuation are called resonances. When $X$ has some symmetries, we construct complex-valued potentials, $V$, such that the resolvent of $\Delta \,+\,V$, which has also a meromorphic continuation, has the same resonances with multiplicities as the free Laplacian.

We investigate the number of integral solutions possessed by a pair of diagonal cubic equations in a large box. Provided that the number of variables in the system is at least fourteen, and in addition the number of variables in any non-trivial linear combination of the underlying forms is at least eight, we obtain an asymptotic formula for the number of integral solutions consistent with the product of local densities associated with the system.

The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

Parallel to the first two authors’ earlier classification of separable, unital, one-parameter, continuous fields of Kirchberg algebras with torsion free $\text{K}$-groups supported in one dimension, one-parameter, separable, unital, continuous fields of $\text{AF}$-algebras are classified by their ordered ${{\text{K}}_{0}}$-sheaves. Effros-Handelman-Shen type theorems are proved for separable unital one-parameter continuous fields of $\text{AF}$-algebras and Kirchberg algebras.

In a previous paper the authors showed that, under some technical conditions, the local Galois representations attached to the members of a non-$\text{CM}$ family of ordinary cusp forms are indecomposable for all except possibly finitely many members of the family. In this paper we use deformation theoretic methods to give examples of non-$\text{CM}$ families for which every classical member of weight at least two has a locally indecomposable Galois representation.

The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have 0, … , 4 complex-conjugate pairs of roots respectively. We show that each of these five components has a real hyperbolic structure in the sense that each is isomorphic as a real-analytic manifold to the quotient of an open dense subset of 5-dimensional real hyperbolic space $\mathbb{R}{{\mathbb{H}}^{5}}$ by the action of an arithmetic subgroup of Isom$\left( \mathbb{R}{{\mathbb{H}}^{5}} \right)$. These subgroups are commensurable to discrete hyperbolic reflection groups, and the Vinberg diagrams of the latter are computed.

In this paper we study genus 2 curves whose Jacobians admit a polarized (4, 4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion, and we derive the relation their absolute invariants satisfy.

As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered.

Our main tool is a Galois theoretic characterization of genus 2 curves admitting multiple Richelot isogenies.

We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative ${{L}^{p}}$ spaces associated with the right von Neumann algebra of $G$. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the noncommutative ${{L}^{p}}$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca–Herz algebra built out of these non-commutative ${{L}^{p}}$ spaces, say ${{A}_{p}}(\hat{G})$. It is shown that ${{A}_{2}}(\hat{G})$ is isometric to ${{L}^{1}}(G)$, generalising the abelian situation.

In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We take two different approaches, each one leading to different results that complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on $\text{CAT(0)}$ spaces where we study the particular behavior of these spaces regarding the problems we are concerned with.

Perelman established a differential Li-Yau-Hamilton $\left( \text{LHY} \right)$ type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds. As an application of the $\text{LHY}$ inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete noncompact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. The conditions are satisfied by asymptotically flatmanifolds. We also prove a long time existence result for the Kähler-Ricci flow on complete nonnegatively curved Kähler manifolds.

W. Casselman defined a basis ${{f}_{u}}$ of Iwahori fixed vectors of a spherical representation $(\pi ,\,V)$ of a split semisimple $p$-adic group $G$ over a nonarchimedean local field $F$ by the condition that it be dual to the intertwining operators, indexed by elements $u$ of the Weyl group $W$. On the other hand, there is a natural basis ${{\psi }_{u}}$, and one seeks to find the transition matrices between the two bases. Thus, let ${{f}_{u}}\,=\,{{\sum }_{v}}\overset{\tilde{\ }}{\mathop{m}}\,(u,\,v){{\psi }_{v}}$ and ${{\psi }_{u}}\,=\,{{\sum }_{v}}m(u,\,v){{f}_{v}}$. Using the Iwahori–Hecke algebra we prove that if a combinatorial condition is satisfied, then $m(u,\,v)\,=\,{{\Pi }_{\alpha }}\,\frac{1-{{q}^{-1}}\,{{z}^{\alpha }}}{1-{{z}^{\alpha }}}$ , where $\mathbf{z}$ are the Langlands parameters for the representation and $\alpha $ runs through the set $S(u,\,v)$ of positive coroots $\alpha \,\in \,\hat{\Phi }$ (the dual root systemof $G$) such that $u\,\le \,v{{r}_{\alpha }}\,<\,v$ with ${{r}_{\alpha }}$ the reflection corresponding to $\alpha $. The condition is conjecturally always satisfied if $G$ is simply-laced and the Kazhdan–Lusztig polynomial ${{P}_{{{w}_{0}}v,\,{{w}_{0}}u}}\,=\,1$ with ${{w}_{0}}$ the long Weyl group element. There is a similar formula for $\tilde{m}$ conjecturally satisfied if ${{P}_{u,\,v}}\,=\,1$. This leads to various combinatorial conjectures.

Let $f$ be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer $n$ has a factor common with the $n\text{-th}$ Fourier coefficient of $f$. It has been shown in previous papers that this happens very often. In this paper, we give an asymptotic formula for the number of integers $n$ for which $\left( n,\,a\left( n \right) \right)\,=\,1$, where $a\left( n \right)$ is the $n\text{-th}$ Fourier coefficient of a normalized Hecke eigenform $f$ of weight 2 with rational integer Fourier coefficients and having complex multiplication.

Every holomorphic function on a compact subset of a Riemann surface can be uniformly approximated by partial sums of a given series of functions. Those functions behave locally like the classical fundamental solutions of the Cauchy–Riemann operator in the plane.

The $j$-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over $\mathbf{Q}$. This paper shows that the coordinate maps given by N. Elkies for the Shimura curves associated to the quaternion algebras with discriminants 6 and 10 are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies conjectural values for the rational CM points, but also provides a way of algebraically calculating the norms of CM points with arbitrarily large negative discriminant.

In this paper we give a classification of discrete series for $SO\left( 2n,\,F \right)$, $F$$p$–adic, similar to that of Mœglin–Tadić for the other classical groups. This is obtained by taking the Mœglin–Tadić classification for $O\left( 2n,\,F \right)$ and studying how the representations restrict to $SO\left( 2n,\,F \right)$. We then extend this to an analysis of how admissible representations of $O\left( 2n,\,F \right)$ restrict.