Ramanujan Conjecture Research Articles

On Riesz means of the coefficients of spinor zeta functions in genus two

Let F be a Siegel cusp form of integral weight k on the Siegel modular group Sp2(ℤ) of genus 2. The Fourier coefficients of the spinor zeta function ZF(s) are denoted by cn. Let Dρ(x;ZF) be the Riesz mean of cn. In this paper, we obtain the truncated Voronoi-type formula of Dρ(x;ZF) under the Ramanujan–Petersson conjecture.

Erratum to “Mass equidistribution for automorphic forms of cohomological type on $GL_{2}$”

Corollary 3 of [4] is not known unconditionally, as cohomological automorphic forms on GL2 over an imaginary quadratic field are not known to satisfy the Ramanujan conjecture. We shall briefly describe the reason for this and discuss what information Theorem 1 of [4] does give in the case of imaginary quadratic fields. Let K be an imaginary quadratic field with nontrivial automorphism c, and let π be a cuspidal automorphic representation of GL2(AK) with unitary central character ω. Suppose that ω = ω and that π∞ has Langlands parameter WC = C× → GL2(C) given by z → diag(z1−k, z1−k) for some integer k ≥ 2 (i.e. so that π is any cohomological representation up to twist). It is then known (see Theorem 1.1 of [1]) that for any one may associate a continuous irreducible representation ρ : Gal(K/K) → GL2(Q ) to π such that the characteristic polynomial of ρ(Frobv) agrees with the Hecke polynomial of πv at all places v which do not divide and at which K/Q, π, and π are unramified. However, because ρ is constructed via an -adic limiting process, it is not known to arise from a motive and so is not known to be pure. To construct ρ, one first makes a theta lift from π to a holomorphic limit of discrete series representation Π on Sp4/Q as in [3]. Weissauer [6] has proven that if Π′ is a holomorphic discrete series representation of Sp4/Q which is not a CAP representation, then one may associate a Galois representation to it which is pure and locally compatible with Π′ at all unramified places, so that Π′ satisfies Ramanujan wherever it is unramified. These results are not known for the limit of discrete series representation Π, and to associate a Galois representation to it one must apply techniques of Taylor [5] which are similar to those used by Deligne and Serre to associate Galois representations to classical weight 1 modular forms. These involve multiplying a holomorphic form in Π by a well understood regular holomorphic form of large weight, applying the results of Weissauer and recovering ρ from these products by an -adic limiting process. Any Archimedean information about the Frobenius eigenvalues of ρ is lost during this, and neither does one know that ρ arises from a motive. As a result, the algebraic information we obtain about π is insufficient to deduce Ramanujan for it. We may still draw interesting conclusions from Theorem 1 of [4] in the imaginary quadratic case. Cohomological forms on GL2/K which are base changes from Q will satisfy Ramanujan, and so Theorem 1 establishes their equidistribution as their weight becomes large. Moreover, the experimental results of [2] suggest that all

Fractional moments of automorphic L-functions on GL(m)

Let π be an irreducible unitary cuspidal representation of \(GL_m \left( {\mathbb{A}_\mathbb{Q} } \right)\), m ≥ 2. Assume that π is self-contragredient. The author gets upper and lower bounds of the same order for fractional moments of automorphic L-function L(s, π) on the critical line under Generalized Ramanujan Conjecture; the upper bound being conditionally subject to the truth of Generalized Riemann Hypothesis.

CUSP FORMS ON GSp(4) WITH SO(4)-PERIODS

The Saito–Kurokawa lifting of automorphic representations from PGL(2) to the projective symplectic group of similitudes PGSp(4) of genus 2 is studied using the Fourier summation formula (an instance of the "relative trace formula"), thus characterizing the image as the representations with a nonzero period for the special orthogonal group SO(4, E/F) associated to a quadratic extension E of the global base field F, and a nonzero Fourier coefficient for a generic character of the unipotent radical of the Siegel parabolic subgroup. The image is nongeneric and almost everywhere nontempered, violating a naive generalization of the Ramanujan conjecture. Technical advances here concern the development of the summation formula and matching of relative orbital integrals.

Arthur packets and the Ramanujan conjecture

The purpose of this article is to show that under a part of the generalized Arthur’s A-packet conjecture, locally generic cuspidal automorphic representations of a quasi-split group over a number field are of Ramanujan type, that is, are tempered at almost all places. The A-packet conjecture allows one to reduce the problem to a special case of a general local question about the components of the corresponding Langlands L-packet which is then answered here in its generality.

Universality for L-functions in the Selberg class

We prove universality for L-functions \( \mathcal{L} \) from the Selberg class satisfying some mild condition on the Dirichlet coefficients (which might be considered as a prime number theorem for \( \mathcal{L} \)). This generalizes a previous universality theorem by the second author, where the L-function was assumed to have a polynomial Euler product satisfying the Ramanujan hypothesis.

Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem

Let K be a totally real number field, π an irreducible cuspidal represen- tation of GL2(K) GL2(AK) with unitary central character, and χ a Hecke character of conductor q .T henL(1/2 ,π ⊗ χ) � (N q) 1 2 − 1 8 (1−2θ)+e ,w here 0 θ 1/ 2i s any exponent towards the Ramanujan-Petersson conjecture (θ =1 /9 is admissible). The proof is based on a spectral decomposition of shifted convolution sums and a gener- alized Kuznetsov formula.

REPRESENTATIONS OF INTEGERS BY TERNARY QUADRATIC FORMS

We investigate the representation of integers by quadratic forms whose theta series lie in Kohnen's plus space [Formula: see text], where p is a prime. Conditional upon certain GRH hypotheses, we show effectively that every sufficiently large discriminant with bounded divisibility by p is represented by the form, up to local conditions. We give an algorithm for explicitly calculating the bounds. For small p, we then use a computer to find the full list of all discriminants not represented by the form. Finally, conditional upon GRH for L-functions of weight 2 newforms, we give an algorithm for computing the implied constant of the Ramanujan–Petersson conjecture for weight 3/2 cusp forms of level 4N in Kohnen's plus space with N odd and squarefree.

Sieving for mass equidistribution

We approach the holomorphic analogue to the Quantum Unique Ergodicity conjecture through an application of the Large Sieve. We deal with shifted convolution sums as in [Ho109], with various simplifications in our analysis due to the knowledge of the Ramanujan-Petersson conjecture in this holomorphic case.

Prime and almost prime integral points on principal homogeneous spaces

We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular, we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers. When the orbit is the set of integral matrices of a fixed determinant, we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups, and sharp and uniform counting of points on such orbits when ordered by various norms.

On Li’s coefficients for the Rankin–Selberg L-functions

We define a generalized Li coefficient for the L-functions attached to the Rankin–Selberg convolution of two cuspidal unitary automorphic representations π and π′ of \(GL_{m}(\mathbb{A}_{F})\) and \(GL_{m^{\prime }}(\mathbb{A}_{F})\) . Using the explicit formula, we obtain an arithmetic representation of the n th Li coefficient \(\lambda _{\pi ,\pi ^{\prime }}(n)\) attached to \(L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime })\) . Then, we deduce a full asymptotic expansion of the archimedean contribution to \(\lambda _{\pi ,\pi ^{\prime }}(n)\) and investigate the contribution of the finite (non-archimedean) term. Under the generalized Riemann hypothesis (GRH) on non-trivial zeros of \(L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime })\) , the nth Li coefficient \(\lambda _{\pi ,\pi ^{\prime }}(n)\) is evaluated in a different way and it is shown that GRH implies the bound towards a generalized Ramanujan conjecture for the archimedean Langlands parameters μπ(v,j) of π. Namely, we prove that under GRH for \(L(s,\pi _{f}\times \widetilde{\pi}_{f})\) one has \(|\mathop {\mathrm {Re}}\mu_{\pi}(v,j)|\leq \frac{1}{4}\) for all archimedean places v at which π is unramified and all j=1,…,m.

Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

Functoriality for the Classical Groups over Function Fields

Langlands' functoriality for generic representations from the split classical groups to an appropriate GL N is established. The functorial lift or transfer to GL N is obtained with the help of a converse theorem once the analytic properties of L-functions are studied using the Langlands-Shahidi approach. This paper is mostly devoted to understanding L-functions for the classical groups over a global function field, since the Langlands-Shahidi method has only been developed over number fields. To overcome many difficulties, stability of γ-factors under twists by highly ramified characters is used together with multiplicativity. Finally, by analyzing the image of functoriality, a proof of the Ramanujan conjecture for generic representations is obtained.

Classical interpretation of the Ramanujan conjecture for Siegel cusp forms of genus n

We obtain a classical interpretation of the representation theoretic statement of the Generalized Ramanujan Conjecture for Siegel cusp forms of genus n in terms of estimates on Hecke eigenvalues.

The generalized prime number theorem for automorphic L-functions

Let π and π′ be automorphic irreducible cuspidal representations of GLm(ℚ\( \mathbb{Q}_\mathbb{A} \)) and GLm′ (ℚ\( \mathbb{Q}_\mathbb{A} \)), respectively, and L(s, π × \( \tilde \pi ' \)) be the Rankin-Selberg L-function attached to π and π′. Without assuming the Generalized Ramanujan Conjecture (GRC), the author gives the generalized prime number theorem for L(s, π × \( \tilde \pi ' \)) when π ≅ π′. The result generalizes the corresponding result of Liu and Ye in 2007.

Central Value Of Automorphic L-Functions

We prove a generalization to the totally real field case of the Waldspurger’s formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger’s formula as a combination of two ingredients – an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen–Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelof hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting.

A Burgess-like subconvex bound for twisted L-functions

Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, χ a primitive character of conductor q , and s a point on the critical line ℜ s = ½. It is proved that , where ε > 0 is arbitrary and θ = is the current known approximation towards the Ramanujan–Petersson conjecture (which would allow θ = 0); moreover, the dependence on s and all the parameters of g is polynomial. This result is an analog of Burgess' classical subconvex bound for Dirichlet L -functions. In Appendix 2 the above result is combined with a theorem of Waldspurger and the adelic calculations of Baruch–Mao to yield an improved uniform upper bound for the Fourier coefficients of holomorphic half-integral weight cusp forms.

An explicit formula and estimations for Hecke L-functions: Applying the Li criterion

We advance the use of a criterion for the validity of the generalized Riemann hypothesis. We rewrite and then estimate various terms of an explicit formula for Hecke L-functions very recently obtained by Li. We demonstrate how additional information on a coefficient function appearing in the logarithmic derivative of the L function would lead directly to verification of a generalized Riemann hypothesis. More generally, we mention a direct link between the generalized Ramanujan conjectures and the validity of generalized Riemann hypotheses. Mathematics Subject Classification: 11M26, 11M36

Generic transfer from GSp(4) to GL(4)

We establish the Langlands functoriality conjecture for the transfer from the generic spectrum of GSp(4) to GL(4) and give a criterion for the cuspidality of its image. We apply this to prove results toward the generalized Ramanujan conjecture for generic representations of GSp(4).

Subconvexity bounds for Rankin–Selberg L-functions for congruence subgroups

Estimation of shifted sums of Fourier coefficients of cusp forms plays crucial roles in analytic number theory. Its known region of holomorphy and bounds, however, depend on bounds toward the general Ramanujan conjecture. In this article, we extended such a shifted sum meromorphically to a larger half plane Re s > 1 / 2 and proved a better bound. As an application, we then proved a subconvexity bound for Rankin–Selberg L-functions which does not rely on bounds toward the Ramanujan conjecture: Let f be either a holomorphic cusp form of weight k, or a Maass cusp form with Laplace eigenvalue 1 / 4 + k 2 , for Γ 0 ( N ) . Let g be a fixed holomorphic or Maass cusp form. What we obtained is the following bound for the L-function L ( s , f ⊗ g ) in the k aspect: L ( 1 / 2 + i t , f ⊗ g ) ≪ k 1 − 1 / ( 8 + 4 θ ) + ε , where θ is from bounds toward the generalized Ramanujan conjecture. Note that a trivial θ = 1 / 2 still yields a subconvexity bound.