Prasad Conjecture Research Articles

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.

The local Gan–Gross–Prasad conjecture for U(3)×U(2): The non-generic case

In this paper, we investigate the local Gan–Gross–Prasad conjecture for some pair of representations of U(3)×U(2) involving a non-generic representation. For a pair of generic L-parameters of (U(n),U(n−1)), it is known that there is a unique pair of representations in their associated Vogan L-packets which produces the unique Bessel model of these L-parameters. We showed that this is not true for some pair of L-parameters involving a non-generic one.On the other hand, we give the precise local theta correspondence for (U(1),U(3)) not at the level of L-parameters but of individual representations in the framework of the local Langlands correspondence for unitary group. As an application of these results, we prove an analog of Ichino–Ikeda conjecture for some non-tempered case. The main tools in this work are the see-saw identity, local theta correspondence for (almost) equal rank cases and recent results on the local Gan–Gross–Prasad conjecture both on the Fourier–Jacobi and the Bessel case.

PERIODS OF AUTOMORPHIC FORMS: THE TRILINEAR CASE

Following Jacquet, Lapid and Rogawski, we regularize trilinear periods. We use the regularized trilinear periods to compute Fourier–Jacobi periods of residues of Eisenstein series on metaplectic groups, which has an application to the Gan–Gross–Prasad conjecture.

Endoscopie et conjecture locale raffinée de Gan–Gross–Prasad pour les groupes unitaires

Under endoscopic assumptions about $L$-packets of unitary groups, we prove the local Gan–Gross–Prasad conjecture for tempered representations of unitary groups over $p$-adic fields. Roughly, this conjecture says that branching laws for $U(n-1)\subset U(n)$ can be computed using epsilon factors.

The non-tempered [formula omitted] Arthur parameter and Gross–Prasad conjectures

We provide a construction of local and automorphic non-tempered Arthur packets AΨ of the group SO(3,2) and its inner form SO(4,1) associated with Arthur's parameterΨ:LF×SL2(C)→O2(C)×SL2(C)→Sp4(C) and prove a multiplicity formula. We further study the restriction of the representations in AΨ to the subgroup SO(3,1). In particular, we discover that the local Gross–Prasad conjecture, formulated for generic L-packets, does not generalize naively to a non-generic A-packet. We also study the non-vanishing of the automorphic SO(3,1)-period on the group SO(4,1)×SO(3,1) and SO(3,2)×SO(3,1) for the representations above. The main tool is the local and global theta correspondence for unitary quaternionic similitude dual pairs.

Fourier transform and the global Gan--Gross--Prasad conjecture for unitary groups

By the relative trace formula approach of Jacquet�Rallis, we prove the global Gan�Gross�Prasad conjecture for unitary groups under some local restrictions for the automorphic representations

Expression d'un facteur epsilon de paire par une formule intégrale

AbstractLet E/F be a quadratic extension of p–adic fields and let d, m be nonnegative integers of distinct parities. Fix admissible irreducible tempered representations π and σ of GLd(E) and GLm(E) respectively. We assume that π and σ are conjugate–dual. That is to say and where c is the nontrivial F–automorphism of E. This implies that we can extend π to an unitary representation π of a nonconnected group GLd(E) . Define the same way. We state and prove an integral formula for involving the characters of and . ˜˜This formula is related to the local Gan–Gross–Prasad conjecture for unitary groups.

The Gan–Gross–Prasad conjecture for [formula omitted

We prove the Gan–Gross–Prasad conjecture for U(n)×U(n) under some local conditions using a relative trace formula. We deduce some new cases of the Gan–Gross–Prasad conjecture for U(n+1)×U(n) from the case of U(n)×U(n).

Distinction of the Steinberg representation II: An equality of characters

Abstract In a previous paper, Broussous and the author prove that for symmetric spaces of the form G(E)/G(F), where G is a reductive group defined and split over a p-adic field F and E is an unramified quadratic extension of F, when the residual field of F is large enough, the Steinberg representation is ε-distinguished for a unique quadratic character ε of G(F). This proves a conjecture stated by Dipendra Prasad in 2001, under the assumption that ε is the same character as the character χ used in the statement of that conjecture. This article completes the proof of Prasad's conjecture in the unramified case assuming that G is split over F, by proving that we actually have ε = χ ${\varepsilon =\chi }$ , and also extends the aforementioned distinguishedness result by removing the condition on the residual field.

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.

Recent progress on the Gross–Prasad conjecture

We report on recent progress on both the local and global Gross–Prasad conjectures for unitary groups.

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 \((\text {SO}(5),\text {SO}(2))\).

On the arithmetic fundamental lemma in the minuscule case

AbstractThe arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of$p$-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture. We prove this conjecture in the minuscule case.

L-functions and periods of adjoint motives

The article studies the compatibility of the refined Gross–Prasad (or Ichino–Ikeda) conjecture for unitary groups, due to Neal Harris, with Deligne’s conjecture on critical values of L-functions. When the automorphic representations are of motivic type, it is shown that the L-values that arise in the formula are critical in Deligne’s sense, and their Deligne periods can be written explicitly as products of Petersson norms of arithmetically normalized coherent cohomology classes. In some cases this can be used to verify Deligne’s conjecture for critical values of adjoint type (Asai) L-functions.

The Refined Gross–Prasad Conjecture for Unitary Groups

Let F be a number field, AF its ring of adeles, and let [pi]n and [pi]n+1 be irreducible, cuspidal, automorphic representations of SOn(AF) and SOn+1AF), respectively. In 1991, Benedict Gross and Dipendra Prasad conjectured the non-vanishing of a certain period integral attached to [pi]n and [pi]n+1 is equivalent to the non-vanishing of L(1/2, [pi]n [X] [pi]n+1). More recently, Atsushi Ichino and Tamotsu Ikeda gave a refinement of this conjecture, as well as a proof of the first few cases (n = 2,3). Their conjecture gives an explicit relationship between the aforementioned L-value and period integral. We make a similar conjecture for unitary groups, and prove the first few cases. The first case of the conjecture will be proved using a theorem of Waldspurger, while the second case will use the machinery of the [theta] correspondence

On endoscopy and the refined Gross–Prasad conjecture for (SO5, SO4)

AbstractWe prove an explicit formula for periods of certain automorphic forms on SO5× SO4along the diagonal subgroup SO4in terms ofL-values. Our formula also involves a quantity from the theory of endoscopy, as predicted by the refined Gross–Prasad conjecture.

Une formule intégrale reliée à la conjecture locale de Gross–Prasad

AbstractLet V be a vector space over a p-adic field F, of finite dimension, let q be a non-degenerate quadratic form over V and let D be a non-isotropic line in V. Denote by W the hyperplane orthogonal to D, and by G and H the special orthogonal groups of V and W. Let π, respectively σ, be an irreducible admissible representation of G(F) , respectively H(F) . The representation σ appears as a quotient of the restriction of π to H(F) with a certain multiplicity m(π,σ) . We know that m(π,σ)≤1 . We assume that π is supercuspidal. Then we prove a formula that computes m(π,σ) as an integral of functions deduced from the characters of π and σ. Let Π, respectively Σ, be an L-packet of tempered irreducible representations of G(F) , respectively H(F) . Here we use the sophisticated notion of L-packet due to Vogan and we assume some usual conjectural properties of those packets. A weak form of the local Gross–Prasad conjecture says that there exists a unique pair (π,σ)∈Π×Σ such that m(π,σ)=1 . Assuming that the elements of Π are supercuspidal, we prove this assertion.

On the Periods of Automorphic Forms on Special Orthogonal Groups and the Gross–Prasad Conjecture

In this paper, we would like to formulate a conjecture on a relation between a certain period of automorphic forms on special orthogonal groups and some L-value. Our conjecture can be considered as a refinement of the global Gross–Prasad conjecture.

Periods of residual representations of SO (2 l )

In this paper, we extend the results on G2-period of residual representations of SO8 in [Jng98a], [Jng98b] to a generalized G2-period of residual representations of SO2l (l≥4). The period is expected to detect the nonvanishing of the relevant tensor product L-function at the center of the symmetry, which is a special case of the Gross- Prasad conjecture. More general cases will be considered in our forth-coming work [GJRa], [GJRb].