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Abstracts for Number theory lunch seminar

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$\sigma$, $\varphi$, $\psi$ and $\zeta$

Posted in
Speaker: 
Patrick Sole
Zugehörigkeit: 
z. Z. MPI
Datum: 
Mit, 2011-02-23 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Let $\sigma(n)$ denote the sum of divisors of $n$. Robin showed that $\sigma(n)<\exp(\gamma)*n*\log(\log n)$ for every $n>5040$ if and only if the Riemann Hypothesis holds true (with $\gamma$ Euler's constant). Nicolas derived a similar, but rather easier to prove, criterion for the Euler totient function $\varphi$. We establish the analogue of his result for the Dedekind $\psi$-function, $\psi(n)=n \prod_{p|n}(1+1/p)$. As a result we can show that the Robin inequality holds for 7-th power free integers.  (Joint work with M. Planat.)

 

Orthogonal periods of Eisenstein series

Posted in
Speaker: 
Gautam Chinta
Zugehörigkeit: 
City College of New York
Datum: 
Mit, 2011-03-02 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

We will show an identity between an orthogonal period of a minimal
parabolic Eisenstein series on GL(3) and Whittaker coefficients of an
Eisenstein series on the metaplectic double cover of GL(3), thereby
providing evidence in favor of a conjecture of Jacquet. The main tool
used in the proof is Gauss's three squares theorem. This is joint work
with O. Offen.
 

Generalized Hasse-Witt Invariants for Some Shimura Varieties

Posted in
Speaker: 
Marc-Hubert Nicole
Zugehörigkeit: 
U Paris 7/MPI
Datum: 
Mit, 2011-03-09 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

The classical Hasse invariant is defined by using the determinant of
the Hasse-Witt matrix. It allows cutting out the ordinary locus within the
special fiber of a modular curve: this is the locus where the Hasse invariant is
invertible. For more general Shimura varieties, the ordinary locus may be empty,
and the Hasse invariant thus conveys no information. This defect is already
visible in dimension one for Shimura curves at primes dividing the discriminant.
Also, except in rare circumstances, there do not exist generalized Hasse

Certain constructions of p-adic families of Siegel modular forms and their applications

Posted in
Speaker: 
Hisa-aki Kawamura
Zugehörigkeit: 
Grenoble
Datum: 
Mit, 2011-03-16 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

As an affirmative answer to the Duke-Imamoglu conjecture concerning a
generalization of the Saito-Kurokawa lifting in the case of higher genus, Ikeda
constructed a Langlands lifting from elliptic modular forms of level 1 to Siegel
modular forms of even genus and of level 1. In this talk, we would like to
introduce a similar lifting of Hida's p-adic analytic families of elliptic
modular forms to those of Siegel modular forms of arbitrary even genus. As a
consequence, we may also construct a classical lifting of ordinary elliptic

Non vanishing of Central values of modular L-functions for Hecke eigenforms of level one

Posted in
Speaker: 
YoungJu Choie
Zugehörigkeit: 
POSTECH/MPI
Datum: 
Mit, 2011-03-23 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

We discuss the behavior of the family of central values of quadratic twist modular $L-$function.
 

Experimental L-functions

Posted in
Speaker: 
Anton Mellit
Zugehörigkeit: 
Köln
Datum: 
Mit, 2011-03-30 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

An L-function is a Dirichlet series which has an Euler
product, analytic continuation and functional equation of appropriate
type. I will present numerical methods which allow, with reasonable
confidence, to find all L-functions of given type. The method is
applied to find L-functions which should correspond to Calabi-Yau
threefolds with h12=1 on the motivic side and to paramodular cusp
forms of weight 3 on the automorphic side.
 

Arithmetic Properties auf Automorphic Forms of Inner Forms of GL(n)

Posted in
Speaker: 
Harald Grobner
Zugehörigkeit: 
U Wien/MPI
Datum: 
Mit, 2011-04-06 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

In this talk we will present new results on the arithmetic of
automorphic forms of the group GL(m,D) on a central simple division
algebra D over an arbitrary number field F. Our results are
generalizations of results obtained in the split case, i.e., D=F, by
Shimura, Harder, Waldspurger and Clozel for square-integrable automorphic
forms and also of Franke and Franke-Schwermer for general automorphic
representations. The global Jacquet-Langlands Correspondence, which was
recently developed by Badulescu and Badulescu-Renard, will be an important

Zeros of the Selberg zeta-function for $\Gamma_0(4)$

Posted in
Speaker: 
Roelof Bruggeman
Zugehörigkeit: 
U Utrecht
Datum: 
Mit, 2011-04-13 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

I'll discuss an example of interplay between experimental and
theoretical Mathematics, in joint work with M.Fraczek and D.Mayer.
Markus Fraczek has done precise computations of zeros of the Selberg
zeta-function for $\Gamma_0(4)$ and a 1-parameter-group of
characters. I'll show a few observations in these computations and
will indicate how the spectral theory of automorphic forms allows us
to prove some of these observations.



 

Modular symbols for $\mathbf{F}_q(T)$ and Manin's presentation

Posted in
Speaker: 
Cécile Armana
Zugehörigkeit: 
z.Z. MPI
Datum: 
Mit, 2011-04-20 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Modular symbols are a useful theoretical and computational tool for
modular forms. In 1992, Teitelbaum introduced modular symbols for the
function field $\mathbf{F}_q(T)$. They have a presentation with a finite number of
generators and their relations, which is formally similar to Manin's
presentation of ``classical'' modular symbols for the rational field $\mathbf{Q}$.

In this talk, I will explain how the finite presentation of Teitelbaum's
modular symbols can be solved explicitly in a rather general case for

Invariant differential operators on Siegel-Jacobi space and Maass-Jacobi forms

Posted in
Speaker: 
Jae-Hyun Yang
Zugehörigkeit: 
Inha U/MPI
Datum: 
Mit, 2011-04-27 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

In this talk I will discuss differential operators on the Siegel-Jacobi space invariant under the
natural action of the Jacobi group, and using these invariant differential operators we study
Maass-Jacobi forms.
The Siegel-Jacobi space is a very important non-reductive homogeneous space
in the aspects of arithmetic and geometry. I review the works of Hans Maass and
Goro Shimura about invariant differential operators on the Siegel space roughly.
I will present my results on invariant  differential operators on the

 

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