Bessel Periods Research Articles

On the Gross–Prasad conjecture with its refinement for (SO(5), SO(2)) and the generalized Böcherer conjecture

We investigate the Gross–Prasad conjecture and its refinement for the Bessel periods in the case of $(\mathrm {SO}(5), \mathrm {SO}(2))$. In particular, by combining several theta correspondences, we prove the Ichino–Ikeda-type formula for any tempered irreducible cuspidal automorphic representation. As a corollary of our formula, we prove an explicit formula relating certain weighted averages of Fourier coefficients of holomorphic Siegel cusp forms of degree two, which are Hecke eigenforms, to central special values of $L$-functions. The formula is regarded as a natural generalization of the Böcherer conjecture to the non-trivial toroidal character case.

Bessel periods and anticyclotomic 𝑝-adic spinor 𝐿-functions

We construct the anticyclotomic p p -adic L L -function that interpolates a square root of central values of twisted spinor L L -functions of a quadratic base change of a Siegel cusp form of genus 2 2 with respect to a paramodular group of square-free level.

Explicit inner product formulas and Bessel period formulas for HST lifts

We explicitly give an inner product formula and a Bessel period formula for theta series on GSp(4), which was studied by Harris, Soudry and Taylor. As a consequence, we prove a non-vanishing of the theta series of small weights and we give a criterion for the non-vanishing of the theta series modulo a prime.

Refined global Gross–Prasad conjecture on special Bessel periods and Böcherer’s conjecture

In this paper we pursue the refined global Gross–Prasad conjecture for Bessel periods formulated by Yifeng Liu in the case of special Bessel periods for \mathrm{SO}(2n+1)\times\mathrm{SO}(2) . Recall that a Bessel period for \mathrm{SO}(2n+1)\times\mathrm{SO}(2) is called special when the representation of \mathrm{SO} (2) is trivial. Let \pi be an irreducible cuspidal tempered automorphic representation of a special orthogonal group of an odd-dimensional quadratic space over a totally real number field F whose local component \pi_v at any archimedean place v of F is a discrete series representation. Let E be a quadratic extension of F and suppose that the special Bessel period corresponding to E does not vanish identically on \pi . Then we prove the Ichino–Ikeda type explicit formula conjectured by Liu for the central value L (1/2, \pi) L (1/2, \pi\times\chi_E ) , where \chi_E denotes the quadratic character corresponding to E . Our result yields a proof of Böcherer’s conjecture on holomorphic Siegel cusp forms of degree two which are Hecke eigenforms.

Maass relations for Saito\u2013Kurokawa lifts of higher levels

It is known that among Siegel modular forms of degree 2 and level 1 the only functions that violate the Ramanujan conjecture are Saito–Kurokawa lifts of modular forms of level 1. These are precisely the functions whose Fourier coefficients satisfy Maass relations. More generally, the Ramanujan conjecture for mathrm {GSp}_4 is predicted to fail only in case of CAP representations. It is not known though whether the associated Siegel modular forms (of various levels) still satisfy a version of Maass relations. We show that this is indeed the case for the ones related to P-CAP representations. Our method generalises an approach of Pitale, Saha and Schmidt who employed representation–theoretic techniques to (re)prove this statement in case of level 1. In particular, we compute and express certain values of a global Bessel period in terms of Fourier coefficients of the associated Siegel modular form. Moreover, we derive a local–global relation satisfied by Bessel periods, which allows us to combine those computations with a characterization of local components of CAP representations.

Jacquet–Langlands–Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms II: An explicit formula for Bessel periods and the non-vanishing of theta lifts

This paper is a continuation of the first paper. The aim of this second paper is to discuss the non-vanishing of the theta lifts to the indefinite symplectic group $GSp(1,1)$, which have been shown to be involved in the Jacquet–Langlands–Shimizu correspondence with some theta lifts to the $\mathbb{Q}$-split symplectic group $GSp(2)$ of degree two. We study an explicit formula for the square norms of the Bessel periods of the theta lifts to $GSp(1,1)$ in terms of central $L$-values. This study involves two aspects in proving the non-vanishing of the theta lifts. One aspect is to apply the results by Hsieh and Chida–Hsieh on “non-vanishing modulo $p$” of central $L$-values for some Rankin $L$-functions. The other is to relate such non-vanishing with studies on some special values of hypergeometric functions. We also take up the theta lifts to the compact inner form $GSp^*(2)$. We provide examples of the non-vanishing theta lifts to $GSp^*(2)$, which are essentially due to Ibukiyama and Ihara.

Arthur parameters and cuspidal automorphic modules of classical groups

The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks: How to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well-expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.

Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of $L$-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan–Gross–Prasad conjecture for Bessel periods as proposed by Liu. We note several consequences of our conjecture to arithmetic and analytic properties of $L$-functions and Fourier coefficients of Siegel modular forms.

Bessel periods and the non-vanishing of Yoshida lifts modulo a prime

We give an explicit construction of vector-valued Yoshida lifts and derive a formula of the Bessel periods of Yoshida lifts, by which we prove the non-vanishing modulo a prime of Yoshida lifts attached to a pair of elliptic modular newforms. As a consequence, we obtain a new proof of the non-vanishing of Yoshida lifts.

On special Bessel periods and the Gross–Prasad conjecture for $$\mathrm {SO}(2n+1) \times \mathrm {SO}(2)$$

In this paper we investigate one instance of the global Gross–Prasad conjecture. Our main result proves that if an irreducible cuspidal automorphic representation \(\pi \) of an odd dimensional special orthogonal group, whose local component \(\pi _{w}\) at some finite place w is generic, admits the special Bessel model corresponding to a quadratic extension E over a base field F, then the central L-value \(L(1/2,\pi )L(1/2,\pi \times \chi _E)\) does not vanish. Here \(\chi _E\) denotes the quadratic character of \(\mathbb A_F^\times \) corresponding to E. As an application, we obtain the equivalence between the non-vanishing of the special Bessel period and that of the corresponding central L-value when \(\pi \) is associated to a full modular holomorphic Siegel cusp form of degree two, which is a Hecke eigenform, and E is an imaginary quadratic extension of \(\mathbb Q\).

A mean value of a triple product of L-functions

We give an explicit construction of vector-valued Yoshida lifts and derive a formula of the Bessel periods of Yoshida lifts, by which we prove the non-vanishing modulo a prime of Yoshida lifts attached to a pair of elliptic modular newforms. As a consequence, we obtain a new proof of the non-vanishing of Yoshida lifts.

A proof of the refined Gan–Gross–Prasad conjecture for non-endoscopic Yoshida lifts

Abstract We prove a precise formula relating the Bessel period of certain automorphic forms on GSp 4 ⁢ ( 𝔸 F ) ${\mathrm{GSp}_{4}(\mathbb{A}_{F})}$ to a central L-value. This is a special case of the refined Gan–Gross–Prasad conjecture for the groups ( SO 5 , SO 2 ) ${(\mathrm{SO}_{5},\mathrm{SO}_{2})}$ as set out by Ichino–Ikeda [12] and Liu [14]. This conjecture is deep and hard to prove in full generality; in this paper we succeed in proving the conjecture for forms lifted, via automorphic induction, from GL 2 ⁢ ( 𝔸 E ) ${\mathrm{GL}_{2}(\mathbb{A}_{E})}$ where E is a quadratic extension of F. The case where E = F × F ${E=F\times F}$ has been previously dealt with by Liu [14].

Fourier–Jacobi periods and the central value of Rankin–Selberg L-functions

In this paper, we propose a conjectural formula, relating the Fourier–Jacobi periods of automorphic forms on U(n)×U(n) and the central value of some Rankin–Selberg L-function. This can be viewed as a refinement of the Gan–Gross–Prasad conjecture for unitary groups. We then use the relative trace formula technique to prove this conjectural formula in some cases. We also have give applications to the conjecture of Ichino–Ikeda and N. Harris on the Bessel period of automorphic forms on unitary groups.

Refined global Gan–Gross–Prasad conjecture for Bessel periods

Abstract We formulate a refined version of the global Gan–Gross–Prasad conjecture for general Bessel models, extending the work of Ichino–Ikeda and R. N. Harris in the co-rank 1 case. It is an explicit formula relating the automorphic period of Bessel type and the central value of certain L-function. To support such conjecture, we provide two examples for pairs SO 5 × SO 2 $\operatorname{SO}_{5}\times\operatorname{SO}_{2}$ and SO 6 × SO 3 $\operatorname{SO}_{6}\times\operatorname{SO}_{3}$ (both co-rank 3) in the endoscopic case via theta lifting.

Relative trace formulae toward Bessel and Fourier–Jacobi periods on unitary groups

We propose a relative trace formula approach and state the corresponding fundamental lemma toward the global restriction problem involving Bessel or Fourier-Jacobi periods of unitary groups $\mathrm{U}_n\times\mathrm{U}_m$, extending the work of Jacquet-Rallis for $m=n-1$ (which is a Bessel period). In particular, when $m=0$, we recover a relative trace formula proposed by Flicker concerning Kloosterman/Fourier integrals on quasi-split unitary groups. As evidence for our approach, we prove the fundamental lemma for $\mathrm{U}_n\times\mathrm{U}_n$ in positive characteristics.

On central critical values of the degree four $$L$$ L -functions for $${\text {GSp}}\left( 4\right) $$ GSp 4 : a simple trace formula

We establish a simple relative trace formula for \({\text {GSp}}(4)\) and inner forms with respect to Bessel subgroups to obtain a certain Bessel identity. From such an identity, one can hope to prove a formula relating central values of degree four spinor \(L\)-functions to squares of Bessel periods as conjectured by Bocherer. Under some local assumptions, we obtain nonvanishing results, i.e., a global Gross–Prasad conjecture for \((\text {SO}(5),\text {SO}(2))\).

Geometric Waldspurger periods

AbstractLet X be a smooth projective curve. We consider the dual reductive pair $H=\mathrm {G\mathbb {O}}_{2m}$, $G=\mathrm {G\mathbb {S}p}_{2n}$ over X, where H splits on an étale two-sheeted covering $\pi :\tilde X\to X$. Let BunG (respectively, BunH) be the stack of G-torsors (respectively, H-torsors) on X. We study the functors FG and FH between the derived categories D(BunG) and D(BunH), which are analogs of the classical theta-lifting operators in the framework of the geometric Langlands program. Assume n=m=1 and H nonsplit, that is, $H=\pi _*{\mathbb {G}_m}$ with $\tilde X$ connected. We establish the geometric Langlands functoriality for this pair. Namely, we show that FG :D(BunH)→D(BunG) commutes with Hecke operators with respect to the corresponding map of Langlands L-groups LH→LG. As an application, we calculate Waldspurger periods of cuspidal automorphic sheaves on BunGL2 and Bessel periods of theta-lifts from $\mathrm {Bun}_{\mathrm {G\mathbb {O}}_4}$ to $\mathrm {Bun}_{\mathrm {G\mathbb {S}p}_4}$. Based on these calculations, we give three conjectural constructions of certain automorphic sheaves on $\mathrm {Bun}_{\mathrm {G\mathbb {S}p}_4}$ (one of them makes sense for ${\mathcal D}$-modules only).

Explicit formulas for the waldspurger and bessel models

This paper studies certain models of irreducible admissible representations of the split special orthogonal group SO(2n+1) over a nonarchimedean local field. Ifn=1, these models were considered by Waldspurger. Ifn=2, they were considered by Novodvorsky and Piatetski-Shapiro, who called them Bessel models. In the works of these authors, uniqueness of the models is established; in this paper functional equations and explicit formulas for them are obtained. As a global application, the Bessel period of the Eisenstein series on SO(2n+1) formed with a cuspidal automorphic representation π on GL(n) is computed—it is shown to be a product of L-series. This generalizes work of Böcherer and Mizumoto forn=2 and base field ℚ, and puts it in a representation-theoretic context. In an appendix by M. Furusawa, a new Rankin-Selberg integral is given for the standardL-function on SO(2n+1)×GL(n). The local analysis of the integral is carried out using the formulas of the paper.