The 16th Autumn Workshop on Number Theory
Harmonic analysis on spherical homogeneous spaces
HAKUBA 2013
-Abstract-
Nov. 7 (Thu)
Tamotsu Ikeda
Global motivation from arithmetic *
.
Fumihiro Sato
Spherical homogeneous spaces *
.
Nov. 8 (Fri)
Masao Tsuzuki
Harmonic analysis on G/H and Plancherel formula *
This is a survey, which consists of two parts. In the first part, I will overview the abstract Plancherel theorem for the $L^2$-space of a homogeneous manifold $H\G$ of a $p$-adic reductive group $G$. In the second part, I introduce a result by Sakellaridis about explicit formulas of spherical functions and Plancherel formulas for unramified spectrum, trying to explicate the formulas for several typical examples of spherical varieties. I also mention briefly Venkatesh-Sakellaridis's recent project.
Pierre Charollois
Integral cocycles on GLn and special values at s=0 of p-adic zeta functions
Building on earlier work by R. Sczech, we contruct an explicit integral valued cocycle on GLn. It allows for the analysis of the order of vanishing and the special value at s=0 of the p-adic partial zeta function of Cassou-Nogu¸«²s and Deligne-Ribet. In particular we recover an earlier result of Wiles (1990). Another construction, now based on the method of Shintani, is shown to lead to a cohomologous cocycle. This is joint work with S. Dasgupta and M. Greenberg. In addition, we would like to draw a connection with invariants introduced by T. Arakawa.
Keiji Takano
Relatively cuspidal/discrete/tempered representations for symmetric spaces
We are interested in representations of a group G which can be realized in the function space on a symmetric space G/H. Such representations of G are said to be H-distinguished. In this talk, I will introduce three relative notions (=symmetric-space-counterparts) in the title. Relative cuspidals (or discrete series) are expected to be the building blocks for all distinguished representations, while relatively tempered ones will constitute the support of the Plancherel measure for G/H. I will describe the criteria for these three notions in terms of (exponents of) Jacquet modules. This talk is a survey of the works of Kato-Takano, Lagier, and Delorme-Harinck.
Masataka Chida
Anticyclotomic Iwasawa main conjecture for modular forms
This is a joint work with Ming-Lun Hsieh. In the fundamental work of Bertolini-Darmon on anticyclotomic Iwasawa theory for elliptic curves, they constructed anticyclotomic p-adic L-functions for weight two modular forms and proved oneside divisibility of anticyclotomic Iwasawa main conjecture for elliptic curves over Q using an Euler system obtained by CM points on Shimura curves and the level raising of modular forms. In this talk, we will generalize their work to higher weight modular forms using an explicit construction of congruence among modular forms of different weights.
Yoshi-Hiro Ishikawa
Explicit formula for Shalika function *
As an introduction to Hironaka's lectures, we review the Casselman's argument and Hironaka's modification. We treat the case "multiplicity one property" holds, taking Shalika model as example.
Nov. 9 (Sat)
Shingo Sugiyama
A relative local trace formula for PGL(2) *
Feigon gave a local analogue of two relative trace formulas for PGL(2) and for U(2), in her PhD thesis, "A relative trace formula for PGL(2) in the local setting". I will introduce the local relative trace formula for PGL(2), the local Kuznetsov trace formula for U(2), and their comparison in terms of the unstable base change lifting.
Zhengyu Mao
A conjecture on Whittaker-Fourier coefficients of cusp forms
We state an analogue of the Ichino-Ikeda conjectures for the Whittaker-Fourier coefficients of automorphic forms on quasi-split reductive groups. We give some evidence to the conjecture.
Keiji Takano
Relative subrepresentation theorem and some related topics
In this talk, first I will describe the relative version of Jacquet's subrepresentation theorem due to Kato-Takano, which indicates how to construct all distinguished representations from relative cuspidals. Also, I will give a brief survey about distinction of induced representations due to Blanc-Delorme, as well as some recent progress on Eisenstein integrals for symmetric spaces due to Carmona-Delorme. At the end of my talk I will give some example concerning the symmetric space GL(2n)/Sp(2n).
Martin Kimball
Functoriality and distinction
Langlands' functoriality is about the notion of transferring representations of one group G to those of another group G'. Basic properties of representations of G (e.g., being cuspidal, generic or symplectic) as well as zeroes and poles of L-functions can be detected by periods on an appropriate subgroup H, which leads to the notion of H-distinction. A natural question is how does the property of being distinguished behaves under functorial transfer from G to G'. The Gan-Gross-Prasad conjectures are a special case of this. I will give an overview of this problem based on examples, highlighting different phenomena that can occur.
Yumiko Hironaka
Explicit formula for spherical functions
In the first talk, I want to introduce a typical spherical function on certain $p$-adic homogeneous space (e. g. weak spherical homogeneous satisfying some technical conditions), and give its expression formula by using data for spherical functions of groups and functional equations.
In the second talk, I will take some specific spaces and show explicit results, such as explicit formulas of spherical functions by using specialized Macdonald polynomials associated to root systems, the structure of the Schwartz space as a Hecke module, parametrizations of all the spherical functions, and Plancherel formula, etc.
There was a conference at Banff International Research Station on "Whittaker Functions: Number Theory, Geometry and Physics (13w5154)" (13--18 October 2013), and some workshop videos and files are put on their homepage. One may reach there by the keyword "BIRS Canada".
Asterisk * after the title indicate the talk is a survey talk.