# Abstracts (Kobe Number Theory Workshop)

Location : Dept. of Mathematics, Graduate School of Science, Kobe Univ.
Date : January 13(Wed) - 15(Fri), 2010
Room : Room 602, Building 3, Graduate School of Science and Technology

## Takashi Hara (Univ. of Tokyo)

Non-commutative Iwasawa main conjecture for totally real number fields and induction method

After brief review of the formulation of the non-commutative Iwasawa main conjecture for totally real number fields, we discuss how to construct the p-adic zeta functions and verify the conjecture via congruences among abelian p-adic zeta functions. If time permits, we will explain that non-commutative Iwasawa main conjecture should imply parts of the equivariant Tamagawa number conjecture (or "non-commutative Tamagawa number conjecture" in the sense of Fukaya and Kato) by virtue of Burns-Venjakob's descent theory.

## Kenji Hashimoto (Univ. of Tokyo)

K3 surfaces and positive definite ternary quadratic forms

Let G be a group in the Mukai's list of maximal finite symplectic actions on K3 surfaces. The number of primitively polarized K3 surfaces of degree d with a symplectic action of G is equal to the number of equivalent classes of integral primitive vectors of norm d in certain quadratic spaces. We compute this number as a function of d by using the theory of quadratic forms.

## Shin Hattori (Kyushu Univ.)

On a ramification bound of torsion semi-stable representations over a local field

Let k be a perfect field of characteristic p>0, K_0=Frac(W(k)) and K be a finite totally ramified extension of K_0 of degree e. Let r be a non- negative integer with r < p-1. In this paper, we give a bound of the upper ramification of the torsion semi-stable representations with Hodge-Tate weights in {0,...r}.

## Tetsushi Ito (Kyoto Univ.)

Supersingular abelian surfaces and the etale cohomology of Siegel threefolds in characteristic p

..............

## Hisa-aki Kawamura (Hokkaido)

Some constructions of $p$-adic families of Siegel modular forms of even genus

Some $p$-adic families of automorphic forms on reductive algebraic groups and their applications have been studied from various points of view by several researchers. In this context, we shall explain some constructions of $p$-adic families of Siegel modular forms of arbitrary even genus $2n$, that is, automorphic forms on the symplectic group ${\rm Sp}(4n)$, which are connected with the Duke-Imamoglu-Ikeda lifting of the $p$-adic analytic family of ordinary elliptic modular forms due to Hida.

## Hidetaka Kitayama (Osaka Univ.)

An explicit dimension formula for Siegel cusp forms of degree two with respect to the non-split symplectic groups

Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$, $W$ be a free left $B$-module of rank 2, and $f$ be a non-degenerate quaternion hermitian form on $W$. We define $U(2;B)$ as the unitary group with respect to the hermitian space $(W,f)$. Siegel modular forms with respect to discrete subgroups of $Sp(2;\mathbb{R})$ defined from $U(2;B)$ were studied by T.Arakawa, K.Hashimoto, T.Sugano, Y.Hirai and others. The purpose of this talk is to give an explicit dimension formula for vector valued Siegel cusp forms with respect to the above groups. This is a generalization of K.Hashimoto's formula.

## Yoichi Mieda (Kyushu Univ.)

Lefschetz trace formula for open adic spaces and its applications

..............

## Michitaka Miyauchi (Kyoto Univ.)

Whittaker models of supercuspidal representations of unramified U(3)

Genericity of supercuspidal representations of p-adic U(3)@relates to the local Langlands conjecture. In fact, it is known that an discrete L-packet on p-adic U(3) contains exactly one generic representation. In this talk, we review the construction of the supercuspidal representations of unramified U(3), and describe the generic supercuspidal representations in terms of supercuspidal types.

## Takeo Okazaki (Kyoto Univ.)

Theta correspondence between GSp(2) and GO(4) or GO(6)

We disscuss the theta correspondence from the veiw point of the recent work by B. Roberts and R.Schmidt `Local newforms for GSp(4)' in L.N.M. 1918 (2007), Springer.

## Frank Thorne (Stanford Univ. / Univ. of South Carolina)

Analytic properties of Shintani zeta functions

The Shintani zeta function is a Dirichlet series which counts cubic rings, or equivalently, orbits of SL_2(Z) on the lattice of integral binary cubic forms. Shintani proved that this zeta function has an analytic continuation with a functional equation, without any Euler product or simple representation in terms of Euler products.
In this talk, we will examine the Shintani zeta function from an analytic point of view. In particular we will ask questions about the distribution of its zeroes. The talk will concentrate on open questions and possible approaches, as well as a couple of preliminary results. Where applicable, we will also consider these questions in a more general setting.

## Takuya Yamauchi (Osaka Pref. Univ.)

A Calabi-Yau family associated to some hypergeometric sheaf and its applications

In this talk, we introduce a Calabi-Yau family associated to some hypergeometric sheaf by using convolution starting from some local system of rank 2 on $\mathbb{P}^1\setminus\{0,1,\infty\}$ and discuss its applications to potential modularity and unit root formula.

## Satoshi Wakatsuki (Kanazawa Univ.)

On congruences for dimensions of spaces of cusp forms

First, we review some known results for congruences for dimensions of spaces of cusp forms and class numbers of imaginary quadratic fields. Next, we consider dimensions of spaces of Siegel cusp forms of degree two. We conjecture that certain relations of such dimensions become non- negative even numbers. This conjecture is supported by numerical experiments. It is concerned with Arthur's conjecture for PGSp(2). From the conjecture, we obtain some congruences for dimensions of Siegel cusp forms and class numbers of imaginary quadratic fields.