共查询到20条相似文献,搜索用时 31 毫秒
1.
Meera Thillainatesan 《Journal of Number Theory》2008,128(4):759-780
The full multiple Dirichlet series of an automorphic cusp form is defined, in classical language, as a Dirichlet series of several complex variables over all the Fourier coefficients of the cusp form. It is different from the L-function of Godement and Jacquet, which is defined as a Dirichlet series in one complex variable over a one-dimensional array of the Fourier coefficients. In GL(2) and GL(3), the two notions are simply related. In this paper, we construct a kernel function that gives the full multiple Dirichlet series of automorphic cusp forms on GL(n,R). The kernel function is a new Poincaré series. Specifically, the inner product of a cusp form with this Poincaré series is the product of the full multiple Dirichlet series of the form times a function that is essentially the Mellin transform of Jacquet's Whittaker function. In the proof, the full multiple Dirichlet series is produced by applying the Lipschitz summation formula several times and by an integral which collapses the sum over SL(n−1,Z) in the Fourier expansion of the cusp form. 相似文献
2.
Let be a Siegel modular form of weight ?, and let be an Eichler embedding, where denotes the Siegel upper half space of degree n. We use the notion of mixed Siegel modular forms to construct the linear map of the spaces of Siegel cusp forms for the congruence subgroup and express the Fourier coefficients of the image of an element under in terms of special values of a certain Dirichlet series. We also discuss connections between mixed Siegel cusp forms and holomorphic forms on a family of abelian varieties. 相似文献
3.
Let be a Siegel modular form of weight ?, and let be an Eichler embedding, where denotes the Siegel upper half space of degree n. We use the notion of mixed Siegel modular forms to construct the linear map of the spaces of Siegel cusp forms for the congruence subgroup and express the Fourier coefficients of the image of an element under in terms of special values of a certain Dirichlet series. We also discuss connections between mixed Siegel cusp forms and
holomorphic forms on a family of abelian varieties.
(Received 28 February 2000; in revised form 11 July 2000) 相似文献
4.
Emre Alkan 《Monatshefte für Mathematik》2011,16(1):249-280
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. 相似文献
5.
Hidenori Katsurada 《Journal of Number Theory》2008,128(7):2025-2052
We consider a certain Dirichlet series of Rankin-Selberg type associated with two Siegel cusp forms of the same integral weight with respect to Spn(Z). In particular, we give an explicit formula for the Dirichlet series associated with the Ikeda lifting of cuspidal Hecke eigenforms with respect to SL2(Z). We also comment on a contribution to the Ikeda's conjecture on the period of the lifting. 相似文献
6.
M. Jutila 《Journal of Mathematical Sciences》2006,137(2):4755-4761
The inner product approach to the additive divisor problem with its cusp form analogs is surveyed, and a spectral summation
formula for convolution sums involving Fourier coefficients of Maass forms is derived. An application to subconvexity estimates
for Rankin-Selberg L-functions is announced. Bibliography: 18 titles.
Dedicated to the memory of Yu. V. Linnik
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 239–250. 相似文献
7.
Gergely Harcos 《Mathematische Annalen》2003,326(2):347-365
We establish an estimate on sums of shifted products of Fourier coefficients coming from holomorphic or Maass cusp forms
of arbitrary level and nebentypus. These sums are analogous to the binary additive divisor sum which has been studied extensively.
As an application we derive, extending work of Duke, Friedlander and Iwaniec, a subconvex estimate on the critical line for
L-functions associated to character twists of these cusp forms.
Received: 2 October 2001 / Revised version: 9 September 2002 /
Published online: 28 March 2003
Mathematics Subject Classification (2000): Primary 11F30, 11F37; Secondary 11M41. 相似文献
8.
Takumi Noda 《The Ramanujan Journal》2007,14(3):405-410
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.
相似文献
9.
Emre Alkan 《Monatshefte für Mathematik》2011,163(3):249-280
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 \({SL_2(\mathbb{Z})}\) and derivatives of such Eisenstein series. Examples of a family of Eisenstein series with a high frequency of vanishing Fourier coefficients are given. 相似文献
10.
Stephan Baier 《Monatshefte für Mathematik》2013,169(2):127-143
Under the generalized Lindelöf Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski–Shapiro primes. 相似文献
11.
WenZhi Luo 《中国科学 数学(英文版)》2010,53(9):2411-2416
In this note, we present a simple approach for bounding the shifted convolution sum involving the Fourier coefficients of half-integral weight holomorphic cusp forms and Maass cusp forms. 相似文献
12.
令λ(n)为SL_2(Z)上全纯尖形式所对应的傅里叶系数.本文研究了全纯尖形式傅里叶系数与素变量多项式的混合问题,并给出和式∑n=p_1~k+p_2~2+p_3~2≤xλ(n) and ∑ n=p_1~k+p_2~2+p_3~2≤xλ(n)Λ(n)的上界估计. 相似文献
13.
We derive for Hecke-Maass cusp forms on the full modular group a relation between the sum of the form at Heegner points (and
integrals over Heegner cycles) and the product of two Fourier coefficients of a corresponding form of half-integral weight.
Specializing to certain cycles we obtain the nonnegativity of theL-function of such a form at the center of the critical strip. These results generalize similar formulae known for holomorphic
forms.
Partially supported by NSF grant # DMS-9096262.
Partially supported by NSF grant # DMS-9102082. 相似文献
14.
N. V. Kuznetsov 《Journal of Mathematical Sciences》1984,24(2):215-238
In this work an asymptotic formula (with a powerlike residue of degree) is proved for the quadratic mean over the critical line of the Dirichlet series associated with the cusp form of weight zero for modular group. The proof is based on author's different formulas of summation for Kloosterman sums.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 109, pp. 93–130, 1981. 相似文献
15.
Y. Ichihara 《Lithuanian Mathematical Journal》2008,48(2):188-202
We consider the sum of coefficients which are in the Dirichlet series expansion of symmetric square L-functions. In this paper, we obtain two estimates of this sum in weight and level aspects. These imply two estimates of the
sum of the n
2th Fourier coefficients of cusp forms. 相似文献
16.
In this paper, we bound the square moment of the linear form in the Fourier coefficients of Hilbert modular forms by means of Poincaré series, and obtain sharp estimate on the critical line for the fourth moment of L-functions associated with Hilbert cusp forms which are primitive Hecke eigenforms. 相似文献
17.
In this article we study a Rankin‐Selberg convolution of n complex variables for pairs of degree n Siegel cusp forms. We establish its analytic continuation to ?n, determine its functional equations and find its singular curves. Also, we introduce and get similar results for a convolution of degree n Jacobi cusp forms. Furthermore, we show how the relation of a Siegel cusp form and its Fourier‐Jacobi coefficients is reflected in a particular relation connecting the two convolutions studied in this paper. As a consequence, the Dirichlet series introduced by Kalinin [7] and Yamazaki [19] are obtained as particular cases. As another application we generalize to any degree the estimate on the size of Fourier coefficients given in [14]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
18.
It is known that theL p -norms of the sums of power series can be estimated from below and above by means of their coefficients, provided these coefficients are nonnegative. In the paper we prove analogous estimates for theL p -norms of the sums of Dirichlet series. Our main result gives exact lower and upper estimates for the BMO-norm of the sums of power series and Dirichlet series, respectively, by means of their coefficients. 相似文献
19.
Zagier showed that the Galois traces of the values of j-invariant at CM points are Fourier coefficients of a weakly holomorphic modular form of weight 3/2 and Bruinier–Funke expanded
his result to the sums of the values of arbitrary modular functions at Heegner points. In this paper, we identify the Galois
traces of real-valued class invariants with modular traces of the values of certain modular functions at Heegner points so
that they are Fourier coefficients of weight 3/2 weakly holomorphic modular forms. 相似文献
20.
Takuya Miyazaki 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2014,84(1):85-122
We discuss the Fourier–Jacobi expansion of certain vector valued Eisenstein series of degree $2$ , which is also real analytic. We show that its coefficients of index $\pm 1$ can be described by using a generating series of real analytic Jacobi forms. We also describe all the coefficients of general indices in suitable manners. Our method can be applied to study another Fourier series of Saito-Kurokawa type that is associated with a cusp form of one variable and half-integral weight. Then, following the arguments in the holomorphic case, we find that the Fourier series indeed defines a real analytic Siegel modular form of degree 2. 相似文献