Central values of L-functions over CM fields

We prove an explicit formula for the central values of certain Rankin L-functions. These L-functions are the L-functions attached to Hilbert newforms over a totally real field F, twisted by unitary Hecke characters of a totally imaginary quadratic extension of F. This formula generalizes our former result on L-functions twisted by finite CM characters.     

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.     

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.     

Moments of the products of quadratic twists of automorphic L-functions

In this paper, assuming the Generalized Riemann Hypothesis and some other hypotheses, we give sharp upper bounds for the moments of the products of central values of automorphic L-functions twisted by quadratic characters and averaged over fundamental discriminants.     

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 effective determination of modular forms by twists of critical L-values

We address the problem of identifying a newform f from the central values of the twisted L-functions ${L(1/2,f\otimes \chi)}We address the problem of identifying a newform f from the central values of the twisted L-functions L(1/2,f?c){L(1/2,f\otimes \chi)} where χ runs through the set of real characters. We prove a quantitative result in this direction.     

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 eighth moment of Dirichlet L-functions

We prove an asymptotic for the eighth moment of Dirichlet L-functions averaged over primitive characters χ modulo q  , over all moduli q?Q$q?Q$ and with a short average on the critical line, conditionally on GRH. We derive the analogous result for the fourth moment of Dirichlet twists of GL(2)$\mathit{GL}(2)$L-functions. Our results match the moment conjectures in the literature; in particular, the constant 24 024 appears as a factor in the leading order term of the eighth moment.     

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.     

A generalized Shimura correspondence for newforms

We associate a set of half integral weight forms to an integral weight newform of odd level. We prove an explicit identity relating the central values of the twist L-functions of the newform to the Fourier coefficients of the half integral weight forms.     

Arakawa-Kaneko L-functions and generalized poly-Bernoulli polynomials

We introduce and investigate generalized poly-Bernoulli numbers and polynomials. We state and prove several properties satisfied by these polynomials. The generalized poly-Bernoulli numbers are algebraic numbers. We introduce and study the Arakawa-Kaneko L-functions. The non-positive integer values of the complex variable s of these L-functions are expressed rationally in terms of generalized poly-Bernoulli numbers and polynomials. Furthermore, we prove difference and Raabe?s type formulae for these L-functions.     

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.     

On the low-lying zeros of Hasse-Weil L-functions for elliptic curves

In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. This upper bound for the average rank enables us to deduce that, under the same assumption, a positive proportion of elliptic curves have algebraic ranks equaling their analytic ranks and finite Tate-Shafarevich group. Statements of this flavor were known previously [M.P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (1) (2005) 205-250] under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper.     

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 the oscillatory behavior of certain arithmetic functions associated with automorphic forms

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin-Selberg convolutions (both “naive” and “modified”), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.     

Congruences for Hecke eigenvalues of Siegel modular forms

We prove some congruences for Hecke eigenvalues of Klingen-Eisenstein series and those of cusp forms for Siegel modular groups modulo special values of automorphic L-functions.     

Pairings of automorphic distributions

We present a pairing of automorphic distributions that applies in situations where a Lie group acts with an open orbit on a product of generalized flag varieties. The pairing gives meaning to an integral of products of automorphic distributions on these varieties. This generalizes classical integral representations or “Rankin–Selberg integrals” of L-functions, and gives new constructions and analytic continuations of automorphic L-functions.     

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.     

Estimates of generalized Chebyshev function on GL m

In this paper, we study the generalized Chebyshev function related to automorphic L-functions of $GL_m \left( {\mathbb{A}_\mathbb{Q} } \right)$ , and estimate its asymptotic behavior with respect to the parameters of the original automorphic objects.     

Values of automorphic L-functions with prime power level at the point $$$$

We discuss in this work the distributions of values of L(1, f), where f is a primitive cusp form whose level is a prime power. We prove the upper bound part of the Montgomery–Vaughan’s first conjecture and give a weaker version of the lower bound part for automorphic L-functions in this case. We establish an unweighted trace formula in aspect of prime power level in our proof.