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

2011 ◽  
Vol 07 (04) ◽  
pp. 855-919
Author(s):  
YUVAL Z. FLICKER
Keyword(s):  
Quadratic Extension ◽  
Summation Formula ◽  
Unipotent Radical ◽  
Automorphic Representations ◽  
Orbital Integrals ◽  
Relative Trace Formula ◽  
Almost Everywhere ◽  
Genus 2 ◽  
Technical Advances ◽  
Ramanujan Conjecture

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.

scholarly journals On the equality case of the Ramanujan Conjecture for Hilbert modular forms

2019 ◽  
Vol 15 (10) ◽  
pp. 2107-2114
Author(s):  
Liubomir Chiriac
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Complex Multiplication ◽  
Hilbert Modular Forms ◽  
Hilbert Modular Form ◽  
Automorphic Representations ◽  
Equality Case ◽  
Hilbert Modular ◽  
Ramanujan Conjecture ◽  
Satake Parameters

The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations [Formula: see text] on [Formula: see text] posits that [Formula: see text]. We prove that this inequality is strict if [Formula: see text] is generated by a Hilbert modular form of weight two, with complex multiplication, and [Formula: see text] is a finite place of degree one. Equivalently, the Satake parameters of [Formula: see text] are necessarily distinct. We also give examples where the equality case does occur for places [Formula: see text] of degree two.

scholarly journals bel and Euler Summation Formulas for SBV(R)

2021 ◽  
Author(s):  
Sergio Venturini
Keyword(s):  
Continuous Function ◽  
Bounded Variation ◽  
Special Function ◽  
Real Variable ◽  
Singular Part ◽  
Summation Formula ◽  
Convergent Series ◽  
Function Of Bounded Variation ◽  
Summation Formulas ◽  
Almost Everywhere

The purpose of this paper is to show that the natural setting for various Abel and Euler-Maclaurin summation formulas is the class of special function of bounded variation. A function of one real variable is of bounded variation if its distributional derivative is a Radom measure. Such a function decomposes uniquely as sum of three components: the first one is a convergent series of piece-wise constant function, the second one is an absolutely continuous function and the last one is the so-called singular part, that is a continuous function whose derivative vanishes almost everywhere. A function of bounded variation is special if its singular part vanishes identically. We generalize such space of special function of bounded variation to include higher order derivatives and prove that the functions of such spaces admit a Euler-Maclaurin summation formula. Such a result is obtained by deriving in this setting various integration by part formulas which generalizes various classical Abel summation formulas.

scholarly journals TWISTED CHARACTER OF A SMALL REPRESENTATION OF GL(4)

2006 ◽  
Vol 02 (03) ◽  
pp. 329-350 ◽  
Author(s):  
YUVAL Z. FLICKER ◽  
DMITRII ZINOVIEV
Keyword(s):  
Conjugacy Class ◽  
Quadratic Extension ◽  
Maximal Parabolic Subgroup ◽  
Local Method ◽  
3 Dimensional ◽  
Almost Everywhere ◽  
Type 3 ◽  
Nontrivial Character ◽  
Central Characters

We compute by a purely local method the (elliptic) θ-twisted character χπY of the representation πY = I(3, 1)(13 × χY) of G = GL (4, F), where F is a p-adic field, p ≠ 2, and Y is an unramified quadratic extension of F; χY is the nontrivial character of F×/NY/FY×. The representation πY is normalizedly induced from [Formula: see text], mi ∈ GL (i, F), on the maximal parabolic subgroup of type (3, 1); θ is the "transpose-inverse" involution of G. We show that the twisted character χπY of πY is an unstable function: its value at a twisted regular elliptic conjugacy class with norm in CY = CY(F)="( GL (2, Y)/F×)F" is minus its value at the other class within the twisted stable conjugacy class. It is 0 at the classes without norm in CY. Moreover πY is the endoscopic lift of the trivial representation of CY. We deal only with unramified Y/F, as globally this case occurs almost everywhere. The case of ramified Y/F would require another paper. Our CY = "( R Y/F GL (2)/ GL (1))F" has Y-points CY(Y) = {(g, g′) ∈ GL (2, Y) × GL (2, Y); det (g) = det (g′)}/Y× (Y× embeds diagonally); σ(≠ 1) in Gal (Y/F) acts by σ(g, g′) = (σg′, σg). It is a θ-twisted elliptic endoscopic group of GL(4). Naturally this computation plays a role in the theory of lifting of CY and GSp(2) to GL(4) using the trace formula, to be discussed elsewhere. Our work extends — to the context of nontrivial central characters — the work of [7], where representations of PGL (4, F) are studied. In [7] we develop a 4-dimensional analogue of the model of the small representation of PGL (3, F) introduced by the first author and Kazhdan in [5] in a 3-dimensional case, and we extend the local method of computation introduced in [6]. As in [7] we use here the classification of twisted (stable) regular conjugacy classes in GL (4, F) of [4], motivated by Weissauer [13].

Stable Bi-Period Summation Formula and Transfer Factors

1999 ◽  
Vol 51 (4) ◽  
pp. 771-791
Author(s):  
Yuval Z. Flicker
Keyword(s):  
Transfer Factor ◽  
Automorphic Forms ◽  
Summation Formula ◽  
Number Fields ◽  
Cusp Forms ◽  
Weighted Sums ◽  
Standard Case ◽  
Transfer Factors ◽  
Orbital Integrals ◽  
Fundamental Lemma

AbstractThis paper starts by introducing a bi-periodic summation formula for automorphic forms on a group G(E), with periods by a subgroup G(F), where E/F is a quadratic extension of number fields. The split case, where E = F ⊕ F, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups H which occur in the case of standard conjugacy.The spectral side of the bi-period summation formula involves periods, namely integrals over the group of F-adele points of G, of cusp forms on the group of E-adele points on the group G. Our stabilization suggests that such cusp forms—with non vanishing periods—and the resulting bi-period distributions associated to “periodic” automorphic forms, are related to analogous bi-period distributions associated to “periodic” automorphic forms on the endoscopic symmetric spaces H(E)/H(F). This offers a sharpening of the theory of liftings, where periods play a key role.The stabilization depends on the “fundamental lemma”, which conjectures that the unit elements of the Hecke algebras on G and H have matching orbital integrals. Even in stating this conjecture, one needs to introduce a “transfer factor”. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case.Finally, the fundamental lemma is verified for SL(2).

scholarly journals Equidistribution in supersingular Hecke orbits

2017 ◽  
Vol 2017 (733) ◽  
Author(s):  
Arno Kret
Keyword(s):  
Rate Of Convergence ◽  
Hecke Operators ◽  
Shimura Varieties ◽  
Automorphic Representations ◽  
Ramanujan Conjecture ◽  
Cuspidal Automorphic Representations

AbstractWe prove that Hecke operators act with equidistribution on the basic stratum of certain Shimura varieties. We relate the rate of convergence to the bounds from the Ramanujan conjecture of certain cuspidal automorphic representations on

scholarly journals A local-global question in automorphic forms

2013 ◽  
Vol 149 (6) ◽  
pp. 959-995 ◽  
Author(s):  
U. K. Anandavardhanan ◽  
Dipendra Prasad
Keyword(s):  
Number Field ◽  
Automorphic Forms ◽  
Quadratic Extension ◽  
Automorphic Representation ◽  
Automorphic Representations ◽  
Formula One ◽  
Distinguished Representations ◽  
Analyze Period ◽  
Cuspidal Automorphic Representations ◽  
Periods Of Automorphic Forms

AbstractIn this paper, we consider the $\mathrm{SL} (2)$ analogue of two well-known theorems about period integrals of automorphic forms on $\mathrm{GL} (2)$: one due to Harder–Langlands–Rapoport about non-vanishing of period integrals on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$ of cuspidal automorphic representations on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{E} )$ where $E$ is a quadratic extension of a number field $F$, and the other due to Waldspurger involving toric periods of automorphic forms on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$. In both these cases, now involving $\mathrm{SL} (2)$, we analyze period integrals on global$L$-packets; we prove that under certain conditions, a global automorphic $L$-packet which at each place of a number field has a distinguished representation, contains globally distinguished representations, and further, an automorphic representation which is locally distinguished is globally distinguished.

TWISTED TRIPLE PRODUCT -ADIC L-FUNCTIONS AND HIRZEBRUCH–ZAGIER CYCLES

2019 ◽  
Vol 19 (6) ◽  
pp. 1947-1992
Author(s):  
Iván Blanco-Chacón ◽  
Michele Fornea
Keyword(s):  
Quadratic Extension ◽  
Number Fields ◽  
Triple Product ◽  
Quadratic Number ◽  
Automorphic Representations ◽  
Quadratic Number Field ◽  
Central Values ◽  
Cuspidal Automorphic Representations ◽  
Totally Real ◽  
Real Quadratic

Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a $p$-nearly ordinary family of unitary cuspidal automorphic representations of $\text{Res}_{L\times F/F}(\text{GL}_{2})$. Furthermore, when $L/\mathbb{Q}$ is a real quadratic number field and $p$ is a split prime, we prove a $p$-adic Gross–Zagier formula relating the values of the $p$-adic $L$-function outside the range of interpolation to the syntomic Abel–Jacobi image of generalized Hirzebruch–Zagier cycles.

On a Generalization of a Result of Waldspurger

1996 ◽  
Vol 48 (1) ◽  
pp. 105-142 ◽  
Author(s):  
Jiandong Guo
Keyword(s):  
Number Field ◽  
Trace Formula ◽  
Strong Evidence ◽  
Riemann Hypothesis ◽  
Symmetric Spaces ◽  
Quadratic Extension ◽  
Automorphic Representation ◽  
Cuspidal Representation ◽  
Orbital Integrals ◽  
Langlands Correspondence

AbstractWe consider a generalization of a trace formula identity of Jacquet, in the context of the symmetric spaces GL(2n)/GL(/n) × GL(n) and G′/H′. Here G′ is an inner form of GL(2n) over F with a subgroup H′ isomorphic to GL(n, E) where E/F is a quadratic extension of number field attached to a quadratic idele class character η of F. A consequence of this identity would be the following conjecture: Let π be an automorphic cuspidal representation of GL(2n). If there exists an automorphic representation π′ of G′ which is related to π by the Jacquet-Langlands correspondence, and a vector ø in the space of π′ whose integral over H′ is nonzero, then both L(1/2, π) and L(1/2,π ⊗ η) are nonvanishing. Moreover, we have L(1/2, π)L(1/2, π ⊗ η) > 0. Here the nonvanishing part of the conjecture is a generalization of a result of Waldspurger for GL(2) and the nonnegativity of the product is predicted from the generalized Riemann Hypothesis. In this article, we study the corresponding local orbital integrals for the symmetric spaces. We prove the "fundamental lemma for the unit Hecke functions" which says that unit Hecke functions have "matching" orbital integrals. This serves as the first step toward establishing the trace formula identity and in the same time it provides strong evidence for what we proposed.

scholarly journals Effective Log-Free Zero Density Estimates for Automorphic L-Functions and the Sato–Tate Conjecture

2017 ◽  
Vol 2019 (22) ◽  
pp. 6988-7036
Author(s):  
Robert J Lemke Oliver ◽  
Jesse Thorner
Keyword(s):  
Short Interval ◽  
Prime Number Theorem ◽  
Similar Estimate ◽  
Automorphic Representations ◽  
Density Estimates ◽  
Tate Conjecture ◽  
Zero Density ◽  
Dirichlet Characters ◽  
Ramanujan Conjecture ◽  
Cuspidal Automorphic Representations

Abstract Let $K/\mathbb{Q}$ be a number field. Let π and π′ be cuspidal automorphic representations of $\textrm{GL}_{d}(\mathbb{A}_{K})$ and $\textrm{GL}_{d^{\prime }}(\mathbb{A}_{K})$. We prove an unconditional and effective log-free zero density estimate for all automorphic L-functions L(s, π) and prove a similar estimate for Rankin–Selberg L-functions L(s, π × π′) when π or π′ satisfies the Ramanujan conjecture. As applications, we make effective Moreno’s analog of Hoheisel’s short interval prime number theorem and extend it to the context of the Sato–Tate conjecture; additionally, we bound the least prime in the Sato–Tate conjecture in analogy with Linnik’s theorem on the least prime in an arithmetic progression. We also prove effective log-free density estimates for automorphic L-functions averaged over twists by Dirichlet characters, which allows us to prove an “average Hoheisel” result for GLdL-functions.

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