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

2010 ◽  
Vol 10 (2) ◽  
pp. 235-324 ◽  
Author(s):  
Wee Teck Gan ◽  
Atsushi Ichino
Keyword(s):  
Explicit Formula ◽  
Automorphic Forms ◽  
Diagonal Subgroup

AbstractWe prove an explicit formula for periods of certain automorphic forms on SO5 × SO4 along the diagonal subgroup SO4 in terms of L-values. Our formula also involves a quantity from the theory of endoscopy, as predicted by the refined Gross–Prasad conjecture.

scholarly journals Periods of automorphic forms: the case of

2014 ◽  
Vol 151 (4) ◽  
pp. 665-712 ◽  
Author(s):  
Atsushi Ichino ◽  
Shunsuke Yamana
Keyword(s):  
Automorphic Form ◽  
Automorphic Forms ◽  
Selberg Integral ◽  
Selberg Integrals ◽  
Diagonal Subgroup ◽  
Periods Of Automorphic Forms

Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on $\text{GL}_{n+1}\times \text{GL}_{n}$ along the diagonal subgroup $\text{GL}_{n}$ and express it in terms of the Rankin–Selberg integral of Jacquet, Piatetski-Shapiro and Shalika. This extends the theory of Rankin–Selberg integrals to all automorphic forms on $\text{GL}_{n+1}\times \text{GL}_{n}$.

scholarly journals ON A CERTAIN LOCAL IDENTITY FOR LAPID–MAO'S CONJECTURE AND FORMAL DEGREE CONJECTURE : EVEN UNITARY GROUP CASE

2021 ◽  
pp. 1-55
Author(s):  
Kazuki Morimoto
Keyword(s):  
Explicit Formula ◽  
Unitary Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Unitary Groups ◽  
Reductive Groups ◽  
Local Identity ◽  
Metaplectic Groups ◽  
Group Case ◽  
Formal Degree

Abstract Lapid and Mao formulated a conjecture on an explicit formula of Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups and metaplectic groups as an analogue of the Ichino–Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this article, we study the even unitary-group case. Indeed, we prove this local identity over p-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over p-adic fields and get an explicit formula of Whittaker–Fourier coefficients under certain assumptions.

scholarly journals Fourier expansion of Arakawa lifting II: Relation with central L-values

2016 ◽  
Vol 27 (01) ◽  
pp. 1650001
Author(s):  
Atsushi Murase ◽  
Hiro-aki Narita
Keyword(s):  
Fourier Coefficient ◽  
Explicit Formula ◽  
Fourier Expansion ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Convolution Type ◽  
Quaternion Algebras ◽  
Central Values ◽  
Central Value ◽  
Multiplicative Groups

This is a continuation of our previous paper [Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts, Israel J. Math. 187 (2012) 317–369]. The aim of the paper here is to study the Fourier coefficients of Arakawa lifts in relation with central values of automorphic [Formula: see text]-functions. In the previous paper we provide an explicit formula for the Fourier coefficients in terms of toral integrals of automorphic forms on multiplicative groups of quaternion algebras. In this paper, after studying explicit relations between the toral integrals and the central [Formula: see text]-values, we explicitly determine the constant of proportionality relating the square norm of a Fourier coefficient of an Arakawa lift with the central [Formula: see text]-value. We can relate the square norm with the central value of some [Formula: see text]-function of convolution type attached to the lift and a Hecke character. We also discuss the existence of strictly positive central values of the [Formula: see text]-functions in our concern.

Periods of automorphic forms: The case of (U n+1×U n ,U n )

2019 ◽  
Vol 2019 (746) ◽  
pp. 1-38 ◽  
Author(s):  
Atsushi Ichino ◽  
Shunsuke Yamana
Keyword(s):  
Eisenstein Series ◽  
Automorphic Forms ◽  
Diagonal Subgroup ◽  
Periods Of Automorphic Forms

Abstract Following Jacquet, Lapid and Rogawski, we define regularized periods of automorphic forms on \mathrm{U}_{n+1} \times \mathrm{U}_{n} along the diagonal subgroup {\mathrm{U}_{n}} and compute the regularized periods of cuspidal Eisenstein series and their residues. The formula for the periods of residues has an application to the Gan–Gross–Prasad conjecture.

scholarly journals On the Bloch–Kato conjecture for Hilbert modular forms

2021 ◽  
Author(s):  
Matteo Tamiozzo
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Euler System ◽  
Central Point ◽  
Shimura Curves ◽  
Hilbert Modular Forms ◽  
Reciprocity Laws ◽  
Hilbert Modular ◽  
Special Points

AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

scholarly journals ON MULTIPLE ZETA VALUES OF EXTREMAL HEIGHT

2015 ◽  
Vol 93 (2) ◽  
pp. 186-193 ◽  
Author(s):  
MASANOBU KANEKO ◽  
MIKA SAKATA
Keyword(s):  
Explicit Formula ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Maximal Height ◽  
Zeta Values ◽  
Sum Formula

We give three identities involving multiple zeta values of height one and of maximal height: an explicit formula for the height-one multiple zeta values, a regularised sum formula and a sum formula for the multiple zeta values of maximal height.

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