On the mean square of standard L-functions attached to Ikeda lifts

By using estimates on the frequency of large values of the Riemann zeta-function and modular L-functions attached to the full modular group SL(2, ℤ), we prove sharp upper and lower estimates of the mean square of standard L-functions attached to Siegel cusp forms which are Ikeda lifts, on boundaries and the central line of the critical strip. The mean square of spinor L-functions attached to Saito-Kurokawa lifts is also studied.     

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.     

Critical p-adic L-functions

We attach p-adic L-functions to critical modular forms and study them. We prove that those L-functions fit in a two-variables p-adic L-function defined locally everywhere on the eigencurve.     

Geometric non-vanishing

We consider L-functions attached to representations of the Galois group of the function field of a curve over a finite field. Under mild tameness hypotheses, we prove non-vanishing results for twists of these L-functions by characters of order prime to the characteristic of the ground field and more generally by certain representations with solvable image. We also allow local restrictions on the twisting representation at finitely many places. Our methods are geometric, and include the Riemann-Roch theorem, the cohomological interpretation of L-functions, and monodromy calculations of Katz. As an application, we prove a result which allows one to deduce the conjecture of Birch and Swinnerton-Dyer for non-isotrivial elliptic curves over function fields whose L-function vanishes to order at most 1 from a suitable Gross-Zagier formula.     

Non-linear Twists of L-Functions: A Survey

This is a survey of joint work with Jerzy Kaczorowski about non-linear twists of L-functions. We present the theory of non-linear twists in the general framework of the Selberg class of L-functions. Applications, mainly to the classification of the Selberg class, are also described.     

Results on L-functions and certain uniqueness questions of Gross

In this paper, we study the value distribution of L-functions in the Selberg class and concentrate on the uniqueness questions of L-functions that share one or two sets with an arbitrary meromorphic function. The results obtained in this paper improve the recent results due to Q.-Q. Yuan, X.-M. Li, and H.-X. Yi [Value distribution of L-functions and uniqueness questions of F. Gross, Lith. Math. J., 58(2):249–262, 2018] and also extend a result due to B.Q. Li [A result on value distribution of L-functions, Proc. Am. Math. Soc., 138(6):2071–2077, 2010].

     

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.     

Unique factorization results for semigroups of <Emphasis Type="Italic">L</Emphasis>-functions

In this paper we prove the unique factorization of certain subsemigroups of the extended Selberg class of L-functions. These semigroups are generated by the functions of degree 0 and 1. We also consider extensions of such results and applications to classical L-functions.     

Families of Siegel modular forms, <Emphasis Type="Italic">L</Emphasis>-functions and modularity lifting conjectures

For a prime p and a positive integer g, by making use of certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus g and associated p-adic L-functions. Describing L-functions attached to Siegel modular forms and their analytic properties from the point of view of motivic L-functions studied by Deligne and Yoshida, we discuss critical values of the L-functions and p-adic interpolation problems. In particular, we formulate a general conjecture on the existence of the modularity lifting from GSp _r × GSp_2m to GSp _r+2m for some positive integers r and m.     

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 Dirichlet L-functions with periodic coefficients and Eisenstein series

We prove a Lipschitz type summation formula with periodic coefficients. Using this formula, representations of the values at positive integers of Dirichlet L-functions with periodic coefficients are obtained in terms of Bernoulli numbers and certain sums involving essentially the discrete Fourier transform of the periodic function forming the coefficients. The non-vanishing of these L-functions at s = 1 are then investigated. There are additional applications to the Fourier expansions of Eisenstein series over congruence subgroups of SL₂(\mathbbZ){SL_2(\mathbb{Z})} and derivatives of such Eisenstein series. Examples of a family of Eisenstein series with a high frequency of vanishing Fourier coefficients are given.     

L-Functions of Jacobi forms with Shimura type

For every Jacobi form of Shimura type over H × ℂ, a system of L-functions associated to it is given. These L-functions can be analytically continued to the whole complex plane and satisfy a kind of functional equation. As a consequence, Hecke’s inverse theorem on modular forms is extended to the context of Jacobi forms with Shimura type.     

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.     

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.     

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.     

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     

Omega theorems related to the general Euler totient function

We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L-functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L-functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L-functions and shifted L-functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new.     

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