On Some Metaplectic Eisenstein Series

We study cubic metaplectic Eisenstein series connected with the Jacobi maximal parabolic subgroup of a symplectic group. We use the so-called ``sl(2)-triples' technique in order to evaluate the Fourier coefficients of these series. In Secs. 1 and 2, we introduce the necessary notation and study the group and its subgroups in detail. In Sec. 3, we prove the main result of the present paper (Theorem 1). Section 4 is devoted to the study of the Dirichlet series appearing in Theorem 1. Bibliography: 5 titles.     

Eisenstein Series in Ramanujan's Lost Notebook

In his lost notebook, Ramanujan stated without proofs several beautifulidentities for the three classsical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.     

A Series of New Congruences for Bernoulli Numbers and Eisenstein Series

We prove congruences of shape E_k+h≡E_k·E_h (mod N) modulo powers N of small prime numbers p, thereby refining the well-known Kummer-type congruences modulo these p of the normalized Eisenstein series E_k. The method uses Serre's theory of Iwasawa functions and p-adic Eisenstein series; it presents a rather general procedure to find and verify such congruences with a modest amount of numerical calculation.     

Whittaker–Fourier Coefficients of Metaplectic Eisenstein Series

It is shown that the Fourier–Whittaker coefficients of Eisenstein series on the n-fold cover of GL(n) are L-functions, improving prior results of T. Suzuki.     

Analogues of the Hurwitz Formulas for Level 2 Eisenstein Series

In this paper, we consider certain double series of Eisenstein type involving hyperbolic functions, which can be regarded as analogues of the level 2 Eisenstein series. We prove some evaluation formulas for these series at positive integers which are analogues of both the Hurwitz formulas for the level 2 Eisenstein series and the classical results given by Cauchy, Lerch, Mellin and Ramanujan.     

The Congruences of Clausen-von Staudt and Kummer for Half-integral Weight Eisenstein Series

We extend sharp forms of the classical uncertainty principle to the context of commutative hypergroups. This hypergroup setting includes Gelfand pairs, Riemannian symmetric spaces, and locally compact abelain groups. For some Gelfand pairs our inequalities will be sharper than those in a recent paper by J. A. WOLF.     

Eisenstein Series of 3/2 Weight and Eligible Numbers of Positive Definite Ternary Forms

A general algorithm is given for the number of representations for a positive integer n by the genus of a positive definite ternary quadratic form with form ax² + by² + cz². Using this algorithm, we study several nontrivial genera of positive ternary forms with small discriminants in the paper. As a conclusion we prove that f₁ = x² + y² + 7z² represents all eligible numbers congruent to 2 mod 3 except 14 _* 7^2k which was conjectured by Kaplansky in [K]. Our method is to use Eisenstein series of weight 3/2.     

Sixteen Eisenstein series

S. Ramanujan gave fourteen families of series in his Second Notebook in Chap. 17, Entries 13–17. In each case he gave only the first few examples, giving us the motivation to find and prove a general formula for each family of series. The aim of this paper is to develop a powerful tool (four versatile functions f ₀,f ₁,f ₂, and f ₃) to collect all of Ramanujan’s examples together.      

Tame Kernels for Biquadratic Number Fields

Let and , d₁ and d₂ squarefree integers, be an imaginary field and a biquadratic field, respectively. Let S be the set consisting of all infinite primes, all dyadic primes and all finite primes which ramify in E. Suppose the 4-rank of the class group of F is zero and the S-ideal class group of F has odd order, we give the forms of all elements of order ⩽ 2 in K₂O_E and use the Hurrelbrink and Kolster’s method [Hurrelbrink, J. and Kolster, M.: J. reine angew. Math. 499 (1998), 145–188] to obtain the forms of all elements of order 4 in K₂O_E. Received: September 2004     

Eisenstein series and convolution sums

The Ramanujan Journal - We compute Fourier series expansions of weight 2 and weight 4 Eisenstein series at various cusps. Then we use results of these computations to give formulas for the...     

On shifted Eisenstein polynomials

We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials. We determine an upper bound on the maximum shift that is needed given a shifted Eisenstein polynomial and also provide a lower bound on the density of shifted Eisenstein polynomials, which is strictly greater than the density of classical Eisenstein polynomials. We also show that the number of irreducible degree \(n\) polynomials that are not shifted Eisenstein polynomials is infinite. We conclude with some numerical results on the densities of shifted Eisenstein polynomials.     

Eisenstein Quasimodes and QUE

We consider the question of quantum unique ergodicity (QUE) for quasimodes on surfaces of constant negative curvature, and conjecture the order of quasimodes that should satisfy QUE. We then show that this conjecture holds for Eisenstein series on \({SL(2,\mathbb{Z})\backslash\mathbb{H}}\) , extending results of Luo–Sarnak and Jakobson. Moreover, we observe that the equidistribution results of Luo–Sarnak and Jakobson extend to quasimodes of much weaker order—for which QUE is known to fail on compact surfaces—though in this scenario the total mass of the limit measures will decrease. We interpret this stronger equidistribution property in the context of arithmetic QUE, in light of recent joint work with Lindenstrauss (Invent Math 198(1), 219–259, 2014) on joint quasimodes.     

On integrality of Eisenstein liftings

Letf be a holomorphic Siegel modular form of integral weightk for Sp_2r (Z). Forn≥r, let[f] _r ⁿ be the lift off to Sp_2n (Z) via the Klingen type Eisenstein series, which is defined under some conditions onk. We study an integrality property of the Fourier coefficients of[f] _r ⁿ . A common denominator for them is described in terms of a critical value of the standardL-function attached tof, some Bernoulli numbers, and a certain ideal depending only onf. The result specialized to the caser=0 coincides with the Siegel-Böcherer theorem on the Siegel type Eisenstein series.