Full multiplicativity of gamma factors for SO2?+1 × GL n

In this paper we prove the full multiplicativity (in both variables) of gamma factors for generic representations of SO_2ℓ+1 × GL _n . These gamma factors are initially defined as proportionality factors of local functional equations, derived from a corresponding global theory of certain Rankin-Selberg integrals which interpolate standardL-functions for SO_2ℓ+1 × GL _n .     

On cuspidal representations of general linear groups over discrete valuation rings

We define a new notion of cuspidality for representations of GL _n over a finite quotient o _k of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups G _λ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of GL _n (F). We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of GL _n (o _k ) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups G _λ . A functional equation for zeta functions for representations of GL _n (o _k ) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL₄(o₂) are constructed. Not all these representations are strongly cuspidal.     

Period relations and p-adic measures

Let (π,σ) be a pair of cuspidal automorphic representations of GL ⁿ × GL ^{n −1} over the adele ring of ℚ having non-vanishing cohomology with constant coefficients. The p-adic distribution interpolating the critical values of the twisted corresponding Rankin–Selberg convolution is shown to be p-adically bounded thus leading to an associated p-adic L-function. Received: 18 October 2000 / Revised version: 27 July 2001     

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     

Projective bases of division algebras and groups of central type

Letk be a field. For each finite groupG and two-cocylef inZ ² (G, k ^x ) (with trivial action), one can form the twisted group algebra wherex _σ x _τ =f(σ,τ)x _στ for all σ, τ∃G. Our main result is a short list ofp-groups containing all thep-groupsG for which there is a fieldk and a cocycle such that the resulting twisted group algebra is ak-central division algebra. We also complete the proof (presented in all but one case in a previous paper by Aljadeff and Haile) that everyk-central division algebra that is a twisted group algebra is isomorphic to a tensor product of cyclic algebras.     

Self-dual representations of some dyadic groups

Let F be a non-Archimedean local field of residual characteristic two and let d be an odd positive integer. Let D be a central F-division algebra of dimension d ². Let π be one of: an irreducible smooth representation of D  ^× , an irreducible cuspidal representation of GL _d (F), an irreducible smooth representation of the Weil group of F of dimension d. We show that, in all these cases, if π is self-contragredient then it is defined over \mathbb Q{\mathbb Q} and is orthogonal. We also show that such representations exist.     

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].     

On the Selberg orthogonality for automorphic L-functions

For automorphic L-functions L(s, π) and L( s,p^￠){L( s,\pi^{\prime })} attached to automorphic irreducible cuspidal representations π and π′ of GL_m( \mathbbQ_A){GL_{m}( \mathbb{Q}_{A})} and GL_m^￠(\mathbbQ_A) {GL_{m^{\prime }}(\mathbb{Q}_{A}) }, we prove the Selberg orthogonality unconditionally for m ≤ 4 and m′ ≤ 4, and under hypothesis H of Rudnik and Sarnak if m > 4 or m′ > 4, without the additional requirement that at least one of these representations be self-contragradient.     

A spectral identity between symmetric spaces

Let π be a cuspidal automorphic representation ofGL _2n . We prove an identity between two spectral distributions onSp _2n andGL _2n respectively. The first is the spherical distribution with respect toSp _n×Sp _nof the residual Eisenstein series induced from π. The second is the weighted spherical distribution of π with respect toGL _n×GL _nand a certain degenerate Eisenstein series. A similar identity relates the pair (U _2n ,Sp _n) and (GL _n/E,GL _n/F) whereE/F is the quadratic extension defining the quasi-split unitary groupU _2n . We also have a Whittaker version of these trace identities. First-named author partially supported by NSF grant DMS 0070611. Second-named author partially supported by NSF grant DMS 9970342.     

The spinL-function on the symplectic groupGSp(6)

In this paper, we will study some essential analytic properties of the “spin”L-function on the symplectic groupGSp (6) (which is associated with the eight-dimensional spin representation of theL-group Gspin (7, ℂ), namely, uniqueness of a bilinear form on an irreducible admissible representation ofGSp (6)×GL(2), local functional equation, and meromorphic continuation, non-vanishing properties at non-archimedean places as well as at archimedean places. Consequently, we will determine the location of the possible poles of the global spinL-function of a generic automorphic cuspidal representation ofGSp(6).     

Mercer’s Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs

Given a compact metric space X and a strictly positive Borel measure ν on X, Mercer’s classical theorem states that the spectral decomposition of a positive self-adjoint integral operator T _k :L ₂(ν)→L ₂(ν) of a continuous k yields a series representation of k in terms of the eigenvalues and -functions of T _k . An immediate consequence of this representation is that k is a (reproducing) kernel and that its reproducing kernel Hilbert space can also be described by these eigenvalues and -functions. It is well known that Mercer’s theorem has found important applications in various branches of mathematics, including probability theory and statistics. In particular, for some applications in the latter areas, however, it would be highly convenient to have a form of Mercer’s theorem for more general spaces X and kernels k. Unfortunately, all extensions of Mercer’s theorem in this direction either stick too closely to the original topological structure of X and k, or replace the absolute and uniform convergence by weaker notions of convergence that are not strong enough for many statistical applications. In this work, we fill this gap by establishing several Mercer type series representations for k that, on the one hand, make only very mild assumptions on X and k, and, on the other hand, provide convergence results that are strong enough for interesting applications in, e.g., statistical learning theory. To illustrate the latter, we first use these series representations to describe ranges of fractional powers of T _k in terms of interpolation spaces and investigate under which conditions these interpolation spaces are contained in L _∞(ν). For these two results, we then discuss applications related to the analysis of so-called least squares support vector machines, which are a state-of-the-art learning algorithm. Besides these results, we further use the obtained Mercer representations to show that every self-adjoint nuclear operator L ₂(ν)→L ₂(ν) is an integral operator whose representing function k is the difference of two (reproducing) kernels.     

Mollification of the fourth moment of automorphic L-functions and arithmetic applications

We compute the asymptotics of twisted fourth power moments of modular L-functions of large prime level near the critical line. This allows us to prove some new non-vanishing results on the central values of automorphic L-functions, in particular those obtained by base change from GL ₂(Q) to GL ₂(K) for K a cyclic field of low degree. Oblatum 22-VI-1999 & 3-III-2000?Published online: 5 June 2000     

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.     

Selberg＇s Normal Density Theorem for Automorphic L-Functions for GLm

Let π be an irreducible unitary cuspidal representation of GLm（AQ） with m ≥ 2, and L（s, Tr） the L-function attached to π. Under the Generalized Riemann Hypothesis for L（s,π）, we estimate the normal density of primes in short intervals for the automorphic L-function L（s, π）. Our result generalizes the corresponding theorem of Selberg for the Riemann zeta-function.     

On the exact non-null distribution of Wilks'L VC criterion and power studies

Letx ₁, x₂, ..., x_Nbep×1 random vectors distributed independently asN(u, Σ), Σ>0;u and Σ are unknown. In this paper, we derive the exact non-null distribution of Wilks' likelihood ratio criterion,L _VC, for testingH:∑=σ ²[(1−ρ)I+ρee′], σ>0 and ρ are unknown against the alternativeA≠H,e′=(1, 1, …, 1): 1×p. The distribution has been derived in three series forms: (1) a series of Meijer'sG-functions through Mellin transform, (2) an, alternate series using contour, intergration and (3) a series of chi square distributions. Powers have been computed based on these forms of the distribution forp=2 and 3.     

On simultaneous non-vanishing of modular <Emphasis Type="Italic">L</Emphasis>-functions

For i = 1, , r, let f _i be newforms of weight 2k _i for Γ₀(N _i ) with trivial character. We consider the simultaneous non-vanishing problem for the central values of twisted L-functions of f _i . By using the Shimura correspondence, we give a certain relation between this problem and the kernel fields of 2-adic Galois representations associated to modular forms. Received: 28 January 2006     

Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on p-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or m ? [(L)\tilde]₀(t)\mu\in {\tilde L}_0(\tau) if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of [(L)\tilde]₀(t){\tilde L}_0(\tau) are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, [(L)\tilde]_m(t) é L_m(t) é L^#_m(t), 1 ￡ m ￡ ￥{\tilde L}_m(\tau)\supset L_m(\tau) \supset L^\#_m(\tau), 1\le m\le \infty , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

Représentations <Emphasis Type="Italic">p</Emphasis>-adiques de GL<Subscript>2</Subscript>(L) et Catégories Dérivées

We construct some locally ℚ _p -analytic representations of GL₂(L), L a finite extension of ℚ _p , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel’d’s upper half space has a structure of rank 2 admissible filtered (φ, N)-module. Finally, we prove that this filtered module is associated, via Fontaine’s theory, to the initial Galois representation.     

Dominated estimates of convex combinations of commuting isometries

The principal result of this paper is that the convex combination of two positive, invertible, commuting isometries ofL _p(X,F, μ) 1<p<+∞, one of which is periodic, admits a dominated estimate with constantp/p−1. In establishing this, the following analogue of Linderholm’s theorem is obtained: Let σ and ε be two commuting non-singular point transformations of a Lebesgue Space with τ periodic. Then given ε>O, there exists a periodic non-singular point transformation σ′ such that σ′ commutes with τ and μ(x:σ′x≠σx}<ε. Byan approximation argument, the principal result is applied to the convex combination of two isometries ofL _p (0, 1) induced by point transformations of the form τx=x ^k,k>0 to show that such convex combinations admit a dominated estimate with constantp/p−1. Research supported in part by NSF Grant No. GP-7475. A portion of the contents of this paper is based on the author’s doctoral dissertation written under the direction of Professor R. V. Chacon of the University of Minnesota.     

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.