Abstract of Hakuba2010
The 13th Autumn Workshop on Number Theory
Automorphic representations, automorphic forms on covering groups
HAKUBA 2010
-Abstract-
Nov. 3 (Wed)
Takuya Konno
Coverings of simple local and adelic groups *
Nov. 4 (Thu)
Seidai Yasuda
Construction of extensions by $K_2$ *
The theory of Brylinski and Deligne provides us with a systematic way to produce extensions of a reductive group by the K_2 group. In this talk, I will give a survey of the theory together with some background materials.
Keiju Sono
The matrix coefficients of the spherical and non-spherical principal series representations of $SL(3,R)$
In this talk, we see how to compute the holonomic system of rank 6 for the radial part of the matrix coefficients of class one and nonspherical principal series representations of $SL(3,R)$. We construct two kinds of differential equations; the Casimir equations and the Dirac-Schmid equation. By solving these equations, we give explicit formulas of the coefficients of six power series solutions, and express the matrix coefficients by linear combinations of these power series. Among others, the $c$-functions of non-spherical principal series are obtained.
Shuichi Hayashida
Lifting of pairs of elliptic modular forms to Siegel modular forms of half-integral weight of degree two
The purpose of this talk is to show a lifting from two elliptic modular forms to Siegel modular forms of \textit{half-integral weight} of degree $2$. More precisely, if $k$ is \textit{an even integer}, then one can obtain a lifting from two elliptic modular forms of weight $2k-2$ and $2k-4$ to Siegel modular form of weight $k-\frac12$ of degree $2$. The existence of this lifting had been conjectured by T.Ibukiyama and the speaker through numerical examination of these Euler factors.
Siegfried Boecherer
On Satake's phi-operator for squarefree levels and low weights
We show that Satake's phi operator is surjective for weight 4 and squarefree levels, mapping from Siegel modular forms of degree two to cusp forms. Methods of Poincare series or Klingen-Eisenstein series cannot be applied directly because of convergence reasons. We need sufficiently many ways to write cusp forms as linear combinations of theta series (joint work with Prof.Ibukiyama).
Shunsuke Yamana (Kyoto Univ.)
Construction of the Weil representation *
I review the basic facts about the Weil representation of the metaplectic group. The material is standard and contains the Heisenberg group, the Stone-von Neumann theorem, the Rao cocycle, Schr\"{o}dinger models, the local and global Weil representation, theta functions, etc.
Nov. 5 (Fri)
Wen-Wei Li
Towards a stable trace formula for metaplectic groups
Adams and Renard proposed a formalism of endoscopy for the real metaplectic groups and obtained the relevant character relations, which are closely related to the theta lift. In this talk, I will give another formulation and explain the corresponding results in the p-adic case. Moreover, one could even expect a stable Arthur-Selberg trace formula for metaplectic groups. I will try to explain the main technical bottleneck, namely a metaplectic version of Arthur's weighted fundamental lemma.
Takeo Okazaki
Various $\theta$-lifts from quaternionic distinguished representations
Let $L,k$ be totally real fields with $[L:k] = 2$. Let $\mathbf{B}$ be a quaternion algebra defined over $k$. We say a cuspform on $\mathbf{B}$ is weak distinguished, if its Jacquet-Langlands lift to ${\rm GL}_2(L_{\mathbb{A}})$ is a base change lift of a cuspform on ${\rm GL}_2(\mathbb{A})$. We talk about various $\theta$-lifts from a weak distinguished form to ${\rm GSp}_n(\mathbb{A}) (\subset {\rm GL}_{2n}(\mathbb{A}))$ for $n =1,2,3$. In case of $n=1$, every cuspform is obtained by a $\theta$-lift from a weak distinguished form. This result relates to the basis problem. In case of $n =3$, infinitely many CAP forms are obtained by $\theta$-lifts from one distinguished form.
Nov. 6 (Sat)
Zhengyu Mao
Ichino-Ikeda formula for Whittaker functionals
Ichino-Ikeda conjectured an identity which explicated Gross-Prasad conjecture. We discuss the analogue of Ichino-Ikeda identity in the setting of Whittaker functionals of automorphic forms on the metaplectic group.
Shunsuke Yamana
The Siegel-Weil formula for classical groups (*)
The Siegel-Weil formula is an identity between a value of a certain Eisenstein series and an integral of a theta function. Such identity was first proven by Siegel and then extended to classical dual reductive pairs by Weil under the assumption that the Eisenstein series is absolutely convergent. Later, Kudla and Rallis extended this formula for symplectic-orthogonal dual pairs in certain cases beyond the range of absolute convergence. In this talk I survey the work of Kudla and Rallis and my recent work in which the Siegel-Weil formula is extended to all quaternion dual pairs.
Tamotsu Ikeda
On the trace formula for the covering groups of SL_2
We discuss the trace formula for the $n$-fold covering group of SL_2. We explain how to stabilize the elliptic terms of the geometric side in the case $2|n$. We also discuss a possible application to the construction of the Kohnen plus subspaces for Hilbert modular forms of half integral weight.
Peter J. McNamara
The Casselman-Shalika formula in the metaplectic setting
The Casselman-Shalika formula gives a formula for the value of a Whittaker function on an unramified p-adic group. In the metaplectic situation, the corresponding vector space of Whittaker functions is no longer one-dimensional, but one may still hope to compute a basis of the Whittaker functions. Recently, Chinta and Offen showed how this task can be achieved in the case of covers of the general linear group. We will discuss their method, together with a generalisation that covers metaplectic covers of all unramified groups.
Hiroshi Sakata
On some trace relations for Hecke operators on modular forms #
This talk is a special one for the memory of late distinguished Prof. Dr. Masaru Ueda. This talk consists of the following two parts:
1.Explanation of results by Prof. Dr. Ueda (especially, description of the trace formulas for Hecke operators on modular forms of half-integral weight),
2.Description of my theory concerning the trace formula for Hecke operators on Jacobi forms and the level-index changing operator.
Nov. 7 (Sun)
Toshiaki Suzuki
Fourier coefficients of metaplectic forms
We show a method to tackle Patterson's conjecture and Shimura's problem simultaneously. We define Theta Forms and use variants of Rankin-Selberg convolutions.
Yoshi-Hiro Ishikawa
Theta correspondence for $\widetilde{\rm SO}(N)$ *
Thete correspondence by using the Weil representation is a strong tool to construct cuspidal representations of double cover of symplectic groups. It is natural to ask the orthogonal group case. But the covering splits on the orthogonal groups in any reductive dual pairs. Here we report the restriction of the "second" smallest representation of type $B$ orthogonal groups and theta-like correspondences after Bump-Friedberg-Ginzburg.
Asterisk * after the title indicate the talk is a survey talk.
The talk with hash mark # is a special one for the memory of late professor M.Ueda.