Fourier Coefficients Of Eisenstein Series Research Articles

Fourier expansion of light‐cone Eisenstein series

AbstractIn this work, we give an explicit formula for the Fourier coefficients of Eisenstein series corresponding to certain arithmetic lattices acting on hyperbolic ‐space. As a consequence, we obtain results on location of all poles of these Eisenstein series as well as their supremum norms. We use this information to get new results on counting rational points on spheres.

The unbounded denominators conjecture for the noncongruence subgroups of index 7

We study modular forms for the minimal index noncongruence subgroups of the modular group. Our main theorem is a proof of the unbounded denominator conjecture for these groups, and we also provide a study of the Fourier coefficients of Eisenstein series for one of these minimal groups.

Non-random behavior in sums of modular symbols

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on [Formula: see text] in the case where N is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancelation in this sum than would be predicted by the common phenomenon of “square root cancelation”, while in another case we find more cancelation.

A splitting into the double cover of SL$$(3,\mathbb {R})$$

We provide a formula for the splitting of a congruence subgroup of SL\((3,\mathbb {R})\) into the double cover of SL\((3,\mathbb {R})\) in terms of Plücker coordinates. This new formula is then used to prove that the splitting satisfies a twisted multiplicativity. This formula has applications to the computation of Fourier coefficients of Eisenstein series on the double cover of SL(3) over \(\mathbb {Q}\).

Hecke $L$-functions and Fourier coefficients of covering Eisenstein series

We consider in this paper covering groups and Fourier coefficients of Eisenstein series for induced representations from certain distinguished theta representations. It is shown that one has global factorization of such Fourier coefficients, and the local unramified Whittaker function at the identity can be computed from the local scattering matrices. For a special family of covering groups of the general linear groups, we show that the Fourier coefficients of such Eisenstein series are reciprocals of Hecke L -functions, which recovers an earlier result by Suzuki for Kazhdan-Patterson covering groups. We also consider covers of the symplectic group and carry out a detailed analysis in the rank-two case.

Poincaré and Eisenstein series for Jacobi forms of lattice index

Poincaré and Eisenstein series are building blocks for every type of automorphic forms. We define Poincaré series for Jacobi forms of lattice index and show that they reproduce Fourier coefficients of cusp forms under the Petersson scalar product. We compute the Fourier expansions of Poincaré and Eisenstein series and give an explicit formula for the Fourier coefficients of the trivial Eisenstein series in terms of values of Dirichlet L-functions at negative integers. For even weight and fixed index, finite linear combinations of Fourier coefficients of non-trivial Eisenstein series are equal to finite linear combinations of Fourier coefficients of the trivial one. This result is used to obtain formulas for the Fourier coefficients of Eisenstein series associated with isotropic elements of small order.

Kloosterman sums and Fourier coefficients of Eisenstein series

We derive explicit formulas for some Kloosterman sums on $\Gamma_0(N)$, and for the Fourier coefficients of Eisenstein series attached to arbitrary cusps, around a general Atkin-Lehner cusp.

On certain Fourier coefficients of Eisenstein series on G2

On certain Fourier coefficients of Eisenstein series on G2

Fourier coefficients of Eisenstein series formed with modular symbols and their spectral decomposition

The Fourier coefficient of a second order Eisenstein series is described as a shifted convolution sum. This description is used to obtain the spectral decomposition of and estimates for the shifted convolution sum.

The sign changes of Fourier coefficients of Eisenstein series

In this paper we prove a number of theorems that determine the extent to which the signs of the Hecke eigenvalues of an Eisenstein newform determine the newform. We address this problem broadly and provide theorems of both individual and statistical nature. Many of these results are Eisenstein series analogs of well-known theorems for cusp forms. For instance, we determine how often the pth Fourier coefficients of an Eisenstein newform begin with a fixed sequence of signs $$\varepsilon _p = \{\pm 1, 0\}$$ . Moreover, we prove the following variant of the strong multiplicity-one theorem: an Eisenstein newform is uniquely determined by the signs of its Hecke eigenvalues with respect to any set of primes with density greater than $$1/2$$ .

Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors

Motivated by string theory scattering amplitudes that are invariant under a discrete U-duality, we study Fourier coefficients of Eisenstein series on Kac– Moody groups. In particular, we analyse the Eisenstein series on E9(R), E10(R) and E11(R) corresponding to certain degenerate principal series at the values s = 3/2 and s = 5/2 that were studied in [1]. We show that these Eisenstein series have very simple Fourier coefficients as expected for their role as supersymmetric contributions to the higher derivative couplings R 4 and @ 4 R 4 coming from 1/2-BPS and 1/4-BPS instantons, respectively. This suggests that there exist minimal and next-to-minimal unipotent automorphic representations of the associated Kac–Moody groups to which these special Eisenstein series are attached. We also provide complete explicit expressions for degenerate Whittaker vectors of minimal Eisenstein series on E6(R), E7(R) and E8(R) that have not appeared in the literature before.

An explicit formula for the Fourier coefficients of Eisenstein series attached to lattices

We prove an explicit formula for local densities of inhomogeneous quadratic forms. The formula is derived by calculating explicitly the Fourier coefficients of various types of Eisenstein series.

Complex multiplication cycles and Kudla-Rapoport divisors

We study the intersections of special cycles on a unitary Shimura variety of signature (n-1,1) and show that the intersection multiplicities of these cycles agree with Fourier coefficients of Eisenstein series. The results are new cases of conjectures of Kudla and suggest a Gross-Zagier theorem for unitary Shimura varieties.

On recursive properties of certain p-adic Whittaker functions

We investigate recursive properties of certain p -adic Whittaker functions (of which representation densities of quadratic forms are special values). The proven relations can be used to compute them explicitly in arbitrary dimensions in terms of orbits under the orthogonal group acting on the representations. These relations have implications for the first and second special derivatives of the Euler product over all p of these Whittaker functions. These Euler products appear as the main part of the Fourier coefficients of Eisenstein series associated with the Weil representation. In case of signature ( l − 2 , 2 ) , we interpret these implications in terms of the theory of Borcherdsʼ products on orthogonal Shimura varieties. This gives some evidence for Kudlaʼs conjectures in higher dimensions.

Intersection theory on Shimura surfaces

AbstractKudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla, Rapoport and Yang. In the present paper we prove results in a higher dimensional setting. On the integral model of a Shimura surface we consider the intersection of a Shimura curve with a codimension two cycle of complex multiplication points, and relate the intersection to certain cycle classes constructed by Kudla, Rapoport and Yang. As a corollary we deduce that our intersection multiplicities appear as Fourier coefficients of a Hilbert modular form of half-integral weight.

On Fourier coefficients of Eisenstein series and Niebur Poincaré series of integral weight

We give a Katok–Sarnak type correspondence for Niebur type Poincaré series and Eisenstein series on the three-dimensional hyperbolic space.

Representations of integers as sums of an even number of squares

be the standard theta function. Then θ which is the generating function for rs(n) (n ∈ N), is a modular form of weight s 2 and level 4 and hence for s ≥ 4 can be expressed as the sum of a modular form in the space generated by Eisenstein series of weight s 2 and level 4 and a corresponding cusp form. Recall that the Fourier coefficients of Eisenstein series of integral weight ≥ 2 on congruence subgroups are given explicitly in terms of elementary divisor functions. Thus if the space of cusp forms happens to be zero, for even s ≥ 4 this leads to exact formulas for rs(n) in terms of those functions. In general, however, one gets only asymptotic results for rs(n) when n is large, since the coefficients of cusp forms are mysterious and in general no simple arithmetic expressions are known for them.

Fourier coefficients of Eisenstein series of the exceptional group of type G2

K(s) F (s) of the two Dedekind zeta functions is an entire function in the complex variable s. From the point of view of the trace formula, the above basic question is expected to be equivalent to a basic question in automorphic L-functions, which asks whether or not the ratio L S ( _ ;s) S F (s) is entire for all irreducible cuspidal automorphic representation of GL(n;AF ) with trivial central character, where L S ( _ ;s ) is the standard tensor product L-function of with its contragredient _ , see for example the work of Jacquet and Zagier [JaZa]. The main idea in this paper is to develop two intrinsically related methods to attack the above two questions. The work of Siegel [Sie], and of Shimura [Shi] (and of Gelbart and Jacquet [GeJa]) provided an evidence for this approach for the case of n =2 . Combined with the work of Ginzburg [Gin], the main result of this paper shows that our approach works for the case of n =3 . It is hoped that such an approach extends to at least the case of n =5 .

$p$-adic $L$-functions of Hilbert modular forms

We construct p-adic L-functions (in general case unbounded) attached to “motivic" primitive Hilbert cusp forms as a non-archimedean Mellin transform of the corresponding admissible measure. In order to prove the growth conditions of the appropriate complex-valued distributions we represent them as Rankin type representation and use Atkin–Lehner theory and explicit form of Fourier coefficients of Eisenstein series.

Local densities of quadratic forms and Fourier coefficients of Eisenstein series

Local densities of quadratic forms are important invariants in the theory of quadratic forms and they appear in Fourier coefficients of Eisenstein series. But it is not easy to evaluate them. To study their properties, it is desirable to look for relations among them, and it is known that there are many relations [3], but they are not concise. We consider a different kind of relations here and improve a result of Zharkovskaja [8, 9] in the case of Eisenstein series as an application.