On Li’s coefficients for the Rankin–Selberg <Emphasis Type="Italic">L</Emphasis>-functions

We define a generalized Li coefficient for the L-functions attached to the Rankin–Selberg convolution of two cuspidal unitary automorphic representations π and π ^′ of GL_m(\mathbbA_F)GL_{m}(\mathbb{A}_{F}) and GL_m^￠(\mathbbA_F)GL_{m^{\prime }}(\mathbb{A}_{F}) . Using the explicit formula, we obtain an arithmetic representation of the n th Li coefficient l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) attached to L(s,p_f×[(p)\tilde]_f^￠)L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime }) . Then, we deduce a full asymptotic expansion of the archimedean contribution to l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) and investigate the contribution of the finite (non-archimedean) term. Under the generalized Riemann hypothesis (GRH) on non-trivial zeros of L(s,p_f×[(p)\tilde]_f^￠)L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime }) , the nth Li coefficient l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) is evaluated in a different way and it is shown that GRH implies the bound towards a generalized Ramanujan conjecture for the archimedean Langlands parameters μ _π (v,j) of π. Namely, we prove that under GRH for L(s,p_f×[(p)\tilde]_f)L(s,\pi _{f}\times \widetilde{\pi}_{f}) one has |Rem_p(v,j)| ￡ \frac14|\mathop {\mathrm {Re}}\mu_{\pi}(v,j)|\leq \frac{1}{4} for all archimedean places v at which π is unramified and all j=1,…,m.     

Fractional Moments of Automorphic L-Functions on GL（m）

Let π be an irreducible unitary cuspidal representation of GLm(AQ), m ≥ 2. Assume that π is self-contragredient. The author gets upper and lower bounds of the same order for fractional moments of automorphic L-function L(s, π) on the critical line under Generalized Ramanujan Conjecture; the upper bound being conditionally subject to the truth of Generalized Riemann Hypothesis.     

Schlicht envelopes of holomorphy and topology

Let Ω be a domain in ${\mathbb{C}^{2}}Let Ω be a domain in \mathbbC²{\mathbb{C}^{2}}, and let p: [(W)\tilde]? \mathbbC²{\pi: \tilde{\Omega}\rightarrow \mathbb{C}^{2}} be its envelope of holomorphy. Also let W￠=p([(W)\tilde]){\Omega'=\pi(\tilde{\Omega})} with i: W\hookrightarrow W￠{i: \Omega \hookrightarrow \Omega'} the inclusion. We prove the following: if the induced map on fundamental groups i_*:p₁(W) ? p₁(W￠){i_{*}:\pi_{1}(\Omega) \rightarrow \pi_{1}(\Omega')} is a surjection, and if π is a covering map, then Ω has a schlicht envelope of holomorphy. We then relate this to earlier work of Fornaess and Zame.     

Spectral theorem for multipliers on {L_{\omega}^{2}(\mathbb{R})}

We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces L_w²(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with f ? C_c(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L²_w(\mathbb R^k){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form.     

On the means of Fourier integrals and Bernstein inequality in the two-weighted setting

The norm estimation problem for Cesaro and Abel–Poisson operators acting from L_w^p(\mathbbR){L_{w}^{p}(\mathbb{R})} to L_v^q (\mathbbR){L_{v}^{q} (\mathbb{R})} where 1 < p ≤ q < ∞ was investigated. These results were generalized to the multidimensional case and applied to obtain generalizations of the Bernstein inequality for integral functions of finite degree of one and several variables.     

A proof of Selberg's orthogonality for automorphic L-functions

Let π and π′ be automorphic irreducible cuspidal representations of GL_m(Q_A) and GL_m′(Q_A), respectively. Assume that π and π′ are unitary and at least one of them is self-contragredient. In this article we will give an unconditional proof of an orthogonality for π and π′, weighted by the von Mangoldt function Λ(n) and 1−n/x. We then remove the weighting factor 1−n/x and prove the Selberg orthogonality conjecture for automorphic L-functions L(s,π) and L(s,π′), unconditionally for m≤4 and m′≤4, and under the Hypothesis H of Rudnick and Sarnak [20] in other cases. This proof of Selberg's orthogonality removes such an assumption in the computation of superposition distribution of normalized nontrivial zeros of distinct automorphic L-functions by Liu and Ye [12].     

An inverse spectral theorem for Kre?n strings with a negative eigenvalue

A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L²(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D_\mathfrakmD_x{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as

On Some Rational Triangles

We prove that every set of n ≥ 3 points in \mathbbR²{\mathbb{R}^2} can be slightly perturbed to a set of n points in \mathbbQ²{\mathbb{Q}^2} so that at least 3(n − 2) of mutual distances between those new points are rational numbers. Some special rational triangles that are arbitrarily close to a given triangle are also considered. Given a triangle ABC, we show that for each ε > 0 there is a triangle A′B′C′ with rational sides and at least one rational median such that |AA′|, |BB′|, |CC′| < ε and a Heronian triangle A′′B′′C′′ with three rational internal angle bisectors such that A^￠￠, B^￠￠, C^￠￠ ? \mathbbQ²{A^{\prime\prime}, B^{\prime\prime}, C^{\prime\prime} \in \mathbb{Q}^2} and |AA′′|, |BB′′|, |CC′′| < ε.     

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

Composition Operators and Endomorphisms

If b is an inner function, then composition with b induces an endomorphism, β, of L^￥(\mathbbT){L^\infty({\mathbb{T}})} that leaves H^￥(\mathbbT){H^\infty({\mathbb{T}})} invariant. We investigate the structure of the endomorphisms of B(L²(\mathbbT)){B(L^2({\mathbb{T}}))} and B(H²(\mathbbT)){B(H^2({\mathbb{T}}))} that implement β through the representations of L^￥(\mathbbT){L^\infty({\mathbb{T}})} and H^￥(\mathbbT){H^\infty({\mathbb{T}})} in terms of multiplication operators on L²(\mathbbT){L^2({\mathbb{T}})} and H²(\mathbbT){H^2({\mathbb{T}})} . Our analysis, which is based on work of Rochberg and McDonald, will wind its way through the theory of composition operators on spaces of analytic functions to recent work on Cuntz families of isometries and Hilbert C*-modules.     

A New Class of Inner Functions Uniformly Approximable by Interpolating Blaschke Products

We study the class of inner functions Q{\Theta} whose zero set Z(Q){Z(\Theta)} stays hyperbolically close to [`(Z_\mathbbD(Q))]{\overline{Z_\mathbb{D}(\Theta)}} on the corona of H ^∞ and show that these functions are uniformly approximable by interpolating Blaschke products.     

Additive polylogarithms and their functional equations

Let ${k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})}Let k[e]₂:=k[e]/(e²){k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})} . The single valued real analytic n-polylogarithm L_n: \mathbbC ? \mathbbR{\mathcal{L}_{n}: \mathbb{C} \to \mathbb{R}} is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in ünver (Algebra Number Theory 3:1–34, 2009) to define additive n-polylogarithms li_n:k[e]₂? k{li_{n}:k[\varepsilon]_{2}\to k} and prove that they satisfy functional equations analogous to those of L_n{\mathcal{L}_{n}}. Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group B_n￠(k[e]₂){B_{n}' (k[\varepsilon]_{2})} defined by Goncharov (Adv Math 114:197–318, 1995). We hope that these functions will be useful in the study of weight n motivic cohomology over k[ε]₂.     

Local–global divisibility by 4 in elliptic curves defined over {\mathbb{Q}}

Let ${\mathcal{E}}Let E{\mathcal{E}} be an elliptic curve defined over \mathbbQ{\mathbb{Q}} . Let P ? E(\mathbb Q){P\in {\mathcal{E}}(\mathbb {Q})} and let q be a positive integer. Assume that for almost all valuations v ? \mathbbQ{v\in \mathbb{Q}} , there exist points D_v ? E(\mathbb Q_v){D_v\in {\mathcal{E}}(\mathbb {Q}_v)} such that P = qD _v . Is it possible to conclude that there exists a point D ? E(\mathbb Q){D\in {\mathcal{E}}(\mathbb {Q})} such that P = qD? A full answer to this question is known when q is a power of almost all primes p ? \mathbbN{p\in \mathbb{N}} , but some cases remain open when p ? S={2,3,5,7,11,13,17,19,37,43,67,163}{p\in S=\{2,3,5,7,11,13,17,19,37,43,67,163\}} . We now give a complete answer in the case when q = 4.     

A Property of Isometric Mappings Between Dual Polar Spaces of Type {DQ (2n, {\mathbb K})}

Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into D￠ = DQ(2n, \mathbb K￠){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let e￠: D￠? S￠ @ PG(2ⁿ - 1, \mathbb K￠){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H￠{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry S @ PG(2ⁿ - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e￠°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e￠°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.     

Composition Followed by Differentiation Between H ∞ and Zygmund Spaces

Let j{\varphi} be an analytic self-map of the unit disk \mathbbD{\mathbb{D}}, H(\mathbbD){H(\mathbb{D})} the space of analytic functions on \mathbbD{\mathbb{D}} and g ? H(\mathbbD){g \in H(\mathbb{D})}. The boundedness and compactness of the operator DC_j : H^￥ ? Z{DC_\varphi : H^\infty \rightarrow { \mathcal Z}} are investigated in this paper.     

Functoriality of automorphic <Emphasis Type="Italic">L</Emphasis>-functions through their zeros

Let E be a Galois extension of ℚ of degree l, not necessarily solvable. In this paper we first prove that the L-function L(s, π) attached to an automorphic cuspidal representation π of cannot be factored nontrivially into a product of L-functions over E. Next, we compare the n-level correlation of normalized nontrivial zeros of L(s, π₁)…L(s, π _k ), where π _j , j = 1,…, k, are automorphic cuspidal representations of , with that of L(s,π). We prove a necessary condition for L(s, π) having a factorization into a product of L-functions attached to automorphic cuspidal representations of specific , j = 1,…,k. In particular, if π is not invariant under the action of any nontrivial σ ∈ Gal _E/ℚ, then L(s, π) must equal a single L-function attached to a cuspidal representation of and π has an automorphic induction, provided L(s, π) can factored into a product of L-functions over ℚ. As E is not assumed to be solvable over ℚ, our results are beyond the scope of the current theory of base change and automorphic induction. Our results are unconditional when m,m ₁,…,m _k are small, but are under Hypothesis H and a bound toward the Ramanujan conjecture in other cases. The first author was supported by the National Basic Research Program of China, the National Natural Science Foundation of China (Grant No. 10531060), and Ministry of Education of China (Grant No. 305009). The second author was supported by the National Security Agency (Grant No. H98230-06-1-0075). The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein     

Complex cycles on algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to the set X(\mathbbR){X(\mathbb{R})} of real points of a nonsingular projective real algebraic variety X, which is called an algebraic model of M. Each algebraic cycle of codimension k on the complex variety X_\mathbbC=X×_\mathbbR\mathbbC{X_{\mathbb{C}}=X\times_{\mathbb{R}}\mathbb{C}} determines a cohomology class in H^2k(X(\mathbbR);\mathbbD){H^{2k}(X(\mathbb{R});\mathbb{D})} , where \mathbbD{\mathbb{D}} denotes \mathbbZ{\mathbb{Z}} or \mathbbQ{\mathbb{Q}} . We investigate the behavior of such cohomology classes as X runs through the class of algebraic models of M.     

Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let Co(α) denote the class of concave univalent functions in the unit disk \mathbbD{\mathbb{D}}. Each function f ? Co(a){f\in Co(\alpha)} maps the unit disk \mathbbD{\mathbb{D}} onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional (1-|z|²)( f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional (1-|z|²)(f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)} whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H ^p space for p < 1/α.     

The Generalized Prime Number Theorem for Automorphic L-Functions

Let π and π＇ be automorphic irreducible cuspidal representations of GLm（QA） and GLm′ （QA）, respectively, and L（s, π×π′） be the Rankin-Selberg L-function attached to π and π＇. Without assuming the Generalized Ramanujan Conjecture （GRC）, the author gives the generalized prime number theorem for L（s, π × π′） when π =π＇. The result generalizes the corresponding result of Liu and Ye in 2007.