Abstracts

Automorphic representations, L-functions, and periods

On the basis problem for Siegel modular forms of squarefree level

Siegfried Boecherer (Mannheim Univ.)

abstract (pdf)

Rankin-Selberg type identities and Gelfand pairs

Andre Reznikov (Bar-Ilan Univ.)

We discuss a simple method to construct Rankin-Selberg identities, including the classical identity and its anisotropic new analog. The method is based on the notion of Gelfand pairs. These identities could be used in order to obtain strong bounds on various periods of automorphic forms (including various subconvexity bounds for special values of L-functions).

p-adic multiple zeta values and double shuffle relations

Hidekazu Furusho (Nagoya Univ.)

abstract (pdf)

Hilbert modular forms with coefficients in intersection homology

Jayce Getz (Univ. of Wisconsin)

In a famous paper, Hirzebruch and Zagier considered families of homology classes $\{Z_m\}_{m \in \ZZ_{\geq 0}}$ on certain Hilbert modular surfaces and showed that the generating series $\sum_{n=0}^{\infty} Z_m \cdot Z_n q^n$ are elliptic modular forms with nebentypus. This work can be seen as giving a geometric interpretation of the Naganuma lifting.

Class one Whittaker functions on real semisimple Lie groups

Taku Ishii (Chiba Institute of Technology)

We discuss explicit formula for class one Whittaker functions on real semisimple Lie groups, especially,
(1) new expression for SL(n,R) (joint work with E.Stade),
(2) explicit formula for exceptional group G_2(R),
(3) conjecture for classical groups and its evidence.
We also report some applications to the computation of gamma factors of automorphic L-functions.

Jacobi forms of weight 1 and applications to Siegel modular forms of critical weight

Tomoyoshi Ibukiyama (Osaka Univ.)
Nils-Peter Skoruppa (University of Siegen)

The Ikeda lifting and Siegel modular forms of half integral weight

Syuichi Hayashida (University of Siegen)

By using Ikeda lifting and the Fourier-Jacobi expansion, we obtain holomorphic Jacobi forms of index 1 of general degree from elliptic modular forms. Moreover by using the isomorphism between holomorphic Jacobi forms of index 1 and Siegel modular forms of half-integral weight, we can get Siegel modular forms of weight $k-1/2$, where k is even integer. We show that if elliptic modular forms are Hecke eigen forms, then the above Siegel modular forms of weight $k-1/2$ are also Hecke eigen forms and zeta functions of these forms are written by product of L-function of elliptic modular forms.

On the zeta function for the space of binary cubic forms and distributions of discriminants of cubic ring extensions

Takashi Taniguchi (Univ. Tokyo)

Let $k$ be a number field and $O_k$ the ring of integers. In this talk we give some density theorems on the distributions of discriminants of cubic algebras of $O_k$. Our approach to derive these theorems are the use of the space of binary cubic and quadratic forms. By applying the theory of prehomogeneous vector spaces founded by M. Sato and T. Shintani, we can associate the zeta functions for these spaces and their analytic properties lead the density thorems. In the case $k$ is a quadratic field, we give a correction term as well as the main term. These are generalizations of Shintani's asymptotic formulae of the mean values of class numbers of binary cubic forms over $\mathbb Z$.

Subrepresentation theorem for p-adic symmetric spaces

Shin-ichi Kato (Kyoto Univ.)
Keiji Takano (Akashi National College of Technology)

An explicit arithmetic formula for Fourier coefficients of Siegel-Eisenstein series of degree two with a squarefree odd level

Yoshinori Mizuno (Osaka Univ.)

Sign changes of Hecke eigenvalues of Siegel cusp forms of genus 2

Winfried Kohnen (University of Heidelberg)

Commutation relation of Hecke operators for Arakawa lifting

Atsushi Murase (Kyoto Sangyo Univ.)
Hiro-aki Narita (Max-Planck-Insititute for Mathematics)

At last year's conference here, we presented the non-adelic formulation of a theta lifting from elliptic cusp forms to automorphic forms on Sp(1,q) given by Tsuneo Arakawa, and proved that such automorphic forms generate quaternionic discrete series. In this talk we reformulate Arakawa's theta lifting in the adelic setting when $q=1$ and present our result on the commutation relation of Hecke operators satisfied by the lifting.

Real Analytic Eisenstein series on the full Jacobi Group

Yoshiki Hayashi

We study a real analytic Eisenstein series on the full Jacobi group once more which Arakawa investigated and related with those associated with theta multiplier systems. This Jacobi-Eisensetin series is natural generalization of the holomorphic one of Eichler-Zagier. We calculate directly its Fourier coefficients and its constant term matrix, We want consider its applications.

Borcherds products for higher level modular forms

Yusuke Kawai (Kyoto Univ.)

Hasse invariants for some unitary Shimura varieties and applications

Tetsushi Ito (Kyoto Univ.)

On the restriction of automorphic forms on an orthogonal group to a smaller orthogonal group and the Gross-Prasad conjecture

Atsuhi Ichino (Osaka city Univ.)
Tamotsu Ikeda (Kyoto Univ.)

In this talk, we consider the restriction of automorphic forms on $SO(n+1)$ to $SO(n)$. More precisely, let $f_1$ and $f_0$ be square-integrable automorphic forms on $SO(n+1)$ and $SO(n)$, respectively. Then we formulate a conjecture which relates the inner product $< f_1|_{SO(n)}, f_0 >$ to a certain $L$-value. Our conjecture can be regarded as a refinement of the global Gross-Prasad conjecture.
abstract (pdf)

A certain Galois action on modular forms with respect to any unitary group and the arithmeticity of Petersson inner products

Atsuo Yamauchi (Nagoya Univ.)

On the zeros of Rankin-Selberg L-functions and the zeros of symmetric square L-functions.

Masatoshi Suzuki (Nagoya Univ.)

私は昨年 L.C. Lagarias氏との共同研究により有理数体に付随する階数2の非可換ゼータ関数がRiemann予想の類似を満たす事を証明した. この証明はある関数論的な補題がこの場合に適用できるという特殊事情によるものであるが, その議論の単純さから他のゼータ関数/$L$関数の零点を調べる際にも応用が可能である事が期待される. 今回の結果はSL(2,Z)上の実解析的Eisenstein級数をある関数論的命題における指数関数の類似物と見なす事により, この期待を SL(2,Z)上の正則保型形式に付随するRankin-Selberg $L$関数, 及びsymmetric square $L$関数の場合に実現したもので, これらの$L$関数を近似するある級数が広範な非零領域を持つことを述べるものである.

On the relation between the formal degree and the adjoint local factor of a discrete series representation of a reductive group over a local field

Kaoru Hiraga (Kyoto Univ.)
Atsushi Ichino (Osaka city Univ. )
Tamotsu Ikeda (Kyoto Univ.)

In this talk, we present a conjectural formula for the formal degree of a discrete series representation in terms of the adjoint local factor. Our conjecture can be regarded as a local analogue of the Gross-Prasad conjecture in the group case. We also discuss the Fourier expansion of the Dirac delta function.

CAP automorphic forms on $U_{E/F}(4)$ II. Cusp forms

Takuya Konno (Kyushu Univ.)
Kazuko Konno (Osaka pref. Univ.)

$G$を代数体上の簡約群とする。$G$の内部形式上のカスプ形式でない$L^2$保型形式とHecke指数を共有する$G$上のカスプ形式をCAP形式という。CAP形式は、明らかな Ramanujan予想の反例である以外に、志村多様体の混モチーフの計算や保型形式の一般化された周期との関連、$p$進群の超カスプ表現の構成などに重要な役割を果たす。この講演では4変数準分裂ユニタリ群$G$を考える。この場合にはCAP保型形式の (期待される)局所成分は2003年に決定しており、今回はこれらの局所成分をもつ(CAP)保型形式を可能な限り構成する。

Program