Cohomology of quaternionic Shimura varieties at parahoric levels

 Some semi-simple L-functions which are associated with the cohomology of a quaternionic Shimura variety are compared with semi-simple automorphic L-functions. Assuming a certain purity condition this yields a similar result for the usual L-functions. The main theorem of the present paper extends previous results of the author to a more general case. Received: 19 July 2000     

A criterion for the Generalized Riemann hypothesis

In this paper, we study the automorphic L-functions attached to the classical automorphic forms on GL（2）, i.e. holomorphic cusp form. And we also give a criterion for the Generalized Riemann Hypothesis （GRH） for the above L-functions.     

Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms

The theory of zeta functions associated with prehomogeneous vector spaces (p.v. for short) provides us a unified approach to functional equations of a large class of zeta functions. However the general theory does not include zeta functions related to automorphic forms such as the HeckeL-functions and the standardL-functions of automorphic forms on GL(n), even though they can naturally be considered to be associated with p.v.’s. Our aim is to generalize the theory to zeta functions whose coefficients involve periods of automorphic forms, which include the zeta functions mentioned above. In this paper, we generalize the theory to p.v.’s with symmetric structure ofK _ε-type and prove the functional equation of zeta functions attached to automorphic forms with generic infinitesimal character. In another paper, we have studied the case where automorphic forms are given by matrix coefficients of irreducible unitary representations of compact groups. Dedicated to the memory of Professor K G Ramanathan     

On zeros of Hecke <Emphasis Type="Italic">L</Emphasis>-functions and their linear combinations on the critical line

In this paper two theorems were obtained. In the first theorem it is proved that a positive proportion of non-trivial zeros lie on the critical line for L-functions attached to automorphic cusp forms for congruence-subgroups. Therefore, the class of functions satisfying a variant of Selberg’s theorem was extended. In the second theorem a new lower bound was obtained for the number of zeros of linear combinations of Hecke L-functions on the intervals of the critical line. This theorem essentially improves the previously known S.A. Gritsenko’s result of 1997.     

Siegel modular forms and representations

This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n,?) with a Siegel modular form of degree n for the full modular group Γ _n =Sp(2n,ℤ), generalizing the well-known procedure for n=1. This will show that the so-called “standard” and ldquo;spinor”L-functions associated with such forms are obtained as Langlands L-functions. The theory of Euler products, developed by Langlands, applied to a Levi subgroup of the exceptional group of type F _<4, is then used to establish meromorphic continuation for the spinor L-function when n=3. Received: 28 March 2000 / Revised version: 25 October 2000     

Uniform subconvexity and symmetry breaking reciprocity

A non-symmetric reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q and primitive central character twisted by the ?-th Hecke eigenvalue as a twisted mixed moment of automorphic L-functions of level ? and trivial central character. As an application, uniform subconvexity bounds for L-functions in the level and the eigenvalue aspect are derived.     

Value-distribution of twisted automorphic <Emphasis Type="Italic">L</Emphasis>-functions

In this paper, we prove a limit theorem for twisted with character automorphic L-functions with an increasing modulus of the character.     

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].     

Value-distribution of twisted automorphic <Emphasis Type="Italic">L</Emphasis>-functions. II

In this paper, we prove a limit theorem for the argument of twisted with character automorphic L-functions with an increasing modulus of the character.     

L-Functions for Jacobi Forms of Arbitrary Degree

A finite number ofL-functions are associated to every Jacobi cusp form of degreen. TheseL-functions are infinite series constructed with the Fourier coefficients of the form and a variables in ℂⁿ. It is proved that eachL-function has an integral representation, admits a holomorphic continuation to the whole space ℂⁿ, and the row vector formed with them satisfies a particular matrix functional equation.     

An explicit formula of the Koecher-Maass series associated with Hermitian modular forms belonging to the Maass space

We give an explicit form of the Koecher-Maass series for Hermitian modular forms belonging to the Maass space. We express the Koecher-Maass series as a finite sum of products of two L-functions associated with automorphic forms of one variable. In particular the Koecher-Maass series associated with the Hermitian-Eisenstein series of degree two can be described by a finite sum of products of four shifted Dirichlet L-functions associated with some quadratic characters under the assumption that the class number of imaginary quadratic fields is one.     

Selberg’s integral for Ramanujan automorphic representation

Let π _Δ be the automorphic representation of GL(2,ℚ_A) associated with Ramanujan modular form Δ and L(s, π _Δ) the global L-function attached to π _Δ. We study Selberg’s integral for the automorphic L-function L(s, π _Δ) under GRH. Our results give the information for the number of primes in short intervals attached to Ramanujan automorphic representation.     

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     

Langlands-shahidi method and poles of automorphicL-functions II

We use Langlands-Shahidi method and the observation that the local components of residual automorphic representations are unitary representations, to study the Rankin-SelbergL-functions of GL _k × classical groups. Especially we prove thatL(s, σ ×τ) is holomorphic, except possibly ats=0, 1/2, 1, whereσ is a cuspidal representation of GL _k which satisfies weak Ramanujan property in the sense of Cogdell and Piatetski-Shapiro andτ is any generic cuspidal representation of SO_2l+1. Also we study the twisted symmetric cubeL-functions, twisted by cuspidal representations of GL₂. Partially supported by NSF grant DMS9610387.     

The subconvexity problem for GL2

Generalizing and unifying prior results, we solve the subconvexity problem for the L-functions of GL ₁ and GL ₂ automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino–Ikeda.     

A note on the non-holomorphic Eisenstein series

We give a sufficient condition of bounded growth for the non-holomorphic Eisenstein series on SL ₂(ℤ). The C ^∞-automorphic forms of bounded growth are introduced by Sturm (Duke Math. J. 48(2), 327–350, 1981) in the study of automorphic L-functions. We also give a Laplace-Mellin transform of the Fourier coefficients of the Eisenstein series. The transformation constructs a projection of the Eisenstein series to the space of holomorphic cusp forms.      

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     

On automorphicL-functions for the Jacobi group of degree one and a relation withL-functions for Jacobi forms

We try to attach anL-function to an automorphic representations of the Jacobi group by defining local factors via certain zeta-integrals. We come up with two kinds of factors which are compared to factors appearing in theL-functions associated to Jacobi forms (of index 1).     

Deux théorèmes de non-annulation de valeurs spéciales de fonctions L

In answer to questions recently raised by Merel [Mer], we prove two non-vanishing theorems for the central value of automorphic L-functions: let p be prime and let χ be a primitive character modulo p. Then for all p large enough 1. If χ is not quadratic and even, there exists a primitive weight 2 form f of level p with . 2. If χ is quadratic and even, then there exists a primitive weight 2 form f of level p with . Received: 12 March 2000 / Revised version: 26 September 2000     

The <Emphasis Type="Italic">L</Emphasis>-functions of Witt coverings

Results on L-functions of Artin–Schreier coverings by Dwork, Bombieri and Adolphson–Sperber are generalized to L-functions of Witt coverings.