# Abstracts for 3rd Japanese-German Number Theory Workshop, November 20 - 24, 2017

Alternatively have a look at the program.

## Number theory with Magma: for performant and flexible computation

Posted in
Speaker:
Shun'ichi Yokoyama
Affiliation:
Kyushu University
Date:
Mon, 2017-11-20 09:50 - 10:40
Location:
MPIM Lecture Hall

Magma (distributed at the U. Sydney) is a software designed for computations in algebra, number theory,
and arithmetic geometry. In this talk, we will introduce Magma and extensive projects with packages we
created (or collaborated) with short demos.

## Selberg's orthonormality conjecture and joint universality of L-functions

Posted in
Speaker:
Takashi Nakamura
Affiliation:
Tokyo University of Science
Date:
Mon, 2017-11-20 11:10 - 12:00
Location:
MPIM Lecture Hall

In this talk, we introduce the new approach how to use an orthonormality relation
of coecients of Dirichlet series de ning given L-functions from the Selberg class to
prove joint universality (joint work with Yoonbok Lee, Lukasz Pankowski).

## Explicit trace formulas for symmetric square L-functions for GL(2) and some applications

Posted in
Speaker:
Shingo Sugiyama
Affiliation:
Kyushu University
Date:
Mon, 2017-11-20 14:00 - 14:50
Location:
MPIM Lecture Hall

Zagier exhibited an indentity between meromorphic functions as a generalization of the Eichler-Selberg
trace formula for elliptic modular forms. Later, Jacquet and Zagier described such a kind of trace formula
with complex parameter for adelic GL(2), containing abstractness of its spectral and geometric terms. In
this talk, we give a generalization of Zagier's formula to the case of holomorphic Hilbert modular forms
in a new way. As applications, we discuss an equidistribution result for Hecke eigenvalues of Hilbert

## A certain level-index changing operator on Jacobi cusp new forms

Posted in
Speaker:
Hiroshi Sakata
Affiliation:
Waseda University Senior High School
Date:
Mon, 2017-11-20 15:00 - 15:50
Location:
MPIM Lecture Hall

Let $N$ be an odd square-free integer. We give a Hecke isomorphism map from the space of Jacobi cusp new forms of level $1$ and index $N$ satisfying a certain condition into the space of Jacobi cusp new forms of level $N$, index $1$ and the character after reviewing about some known properties of level-index changing operators.

## Arithmetic formulas for the coefficients of the McKay-Thompson series

Posted in
Speaker:
Toshiki Matsusaka
Affiliation:
Kyushu University
Date:
Mon, 2017-11-20 16:30 - 17:10
Location:
MPIM Lecture Hall

By the work of R. Borcherds we know the Fourier coefficients of the elliptic modular $j$-function are closely related
to the Monster group. As a new perspective, M. Kaneko gave an arithmetic formula for the Fourier coefficients expressed
in terms of the traces of the CM values of the $j$-function in 1996. In this talk, we consider analogues of this formula for
the McKay-Thompson series. Our main result is an explicit formula for the coefficients expressed in terms of the CM
values of certain hauptmoduln. Furthermore, we give some applications.

## On Charlton's conjecture about the multiple zeta values

Posted in
Speaker:
Nabuo Sato
Affiliation:
NCTS of National Taiwan University
Date:
Tue, 2017-11-21 09:50 - 10:40
Location:
MPIM Lecture Hall

In this talk, we give a proof of a special case of the generalized cyclic insertion conjecture on the MZVs, which was formulated by Steven Charlton in his thesis. The conjecture is stated in terms of the block notation for MZVs introduced by himself. Charlton's conjecture is a broad generalization of several long unproven families of identities such as Borwein-Bradley-Broadhurst-Lisoněk's cyclic insertion conjecture and certain conjectural identities posed by Hoffman.

## Average values of higher Green's functions and their factorizations.

Posted in
Speaker:
Yingkun Li
Affiliation:
Technical University of Darmstadt
Date:
Tue, 2017-11-21 11:10 - 12:00
Location:
MPIM Lecture Hall

By the classical theory of complex multiplication, the Klein $j$-invariant takes algebraic
values at CM points. In their seminal work on singular moduli, Gross and Zagier gave a
factorization of the difference between two such values, which can be viewed as the exponential
of a special value of a Green's function on the upper half plane. From numerical computations,
they conjectured that the special values of certain higher Green's functions also enjoy similar
algebraicity property. We will revisit this conjecture and discuss some recent progress.

## Explicit constructions of non-tempered cusp forms on orthogonal groups of low split ranks

Posted in
Speaker:
Hiro-aki Narita
Affiliation:
Kumamoto University
Date:
Tue, 2017-11-21 14:00 - 14:50
Location:
MPIM Lecture Hall

The aim of this talk is to report a recent research on explicit constructions of cusp forms on orthogonal groups of split rank one or two by some lifts from cusp forms on the complex upper half plane. We also discuss cuspidal representations generated by them in terms of the explicit determination of their local components. As for the representation theoretic treatment, the point is to use Sugano’s non-archimedean local theory of Jacobi form formulation” of Oda-Rallis-Schiffman lifting to orthogonal groups of rank two.

## On the Miyawaki lifts of hermitian modular forms

Posted in
Speaker:
Hiraku Atobe
Affiliation:
Tokyo University
Date:
Tue, 2017-11-21 15:00 - 15:50
Location:
MPIM Lecture Hall

We construct a lifting from a elliptic modular cusp form f and a hermitian modular cusp
form g of degree r to a hermitian modular cusp form of degree n + r. This lift is the
so-called Miyawaki lift. When f and g are "Hecke eigenforms", its lift is also a "Hecke eigenform"
(if it is not identically zero). Moreover, we determine explicitly the standard L-function and
A-parameter associated to the Miyawaki lift.

## The Hermitian modular group and the orthogonal group

Posted in
Speaker:
Annalena Wernz
Affiliation:
RWTH Aachen
Date:
Tue, 2017-11-21 16:30 - 17:10
Location:
MPIM Lecture Hall

The Hermitian modular group of degree $n$ over an imaginary quadratic field
$K=\mathbb{Q}(\sqrt{-m})$ was introduced by Hel Braun in the 1940s
as an analogue for the well known Siegel modular group. It acts on the
Hermitian half space and the accociated Hermitian modular forms have
been studied thoroughly in the past. However, this talk does not concentrate
on the modular forms but on the modular group itself. For $n=2$ and $m \neq 1,3$,
$m$ squarefree, we will prove that the Hermitian modular group $U(2,2;\mathcal{O}_K)$,

 © MPI f. Mathematik, Bonn Impressum & Datenschutz