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Abstracts for Number theory lunch seminar

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Multiplication laws for differential equations

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Speaker: 
Anton Mellit
Date: 
Wed, 2012-09-12 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

With V. Golyshev. This talk is an overview of how some cases of Langlands' correspondence for function fields can be interpreted very classically and explicitly. In particular, we see how formulas of Clausen's type and multiplication identities for Bessel functions arise this way. For example, our approach gives an explicit correspondence between solutions of the Dwork's accessory parameter problem as studied by Beukers, and automorphic forms.

Asymptotic of colored Jones polynomials, topological recursion and theta functions

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Speaker: 
Gaetan Borot
Affiliation: 
Geneva/MPI
Date: 
Wed, 2012-09-19 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

I will start with a short review of the generalized volume conjecture (cf. Pr. Murakami Oberseminar last week) about asymptotics of the colored Jones polynomial of a knot. Then, I will present (with illustration for the figure eight-knot) a conjecture aiming at computing the all-order asymptotics of the colored Jones polynomial of hyperbolic knots, solely in terms of algebraic geometry on the character variety.

Non-Abelian Zeta Functions

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Speaker: 
Lin Weng
Affiliation: 
Kyushu University/MPI
Date: 
Wed, 2012-09-26 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

On the independence of Heegner points on elliptic curves

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Speaker: 
Hatice Sahinoglu
Affiliation: 
MPI
Date: 
Wed, 2012-10-10 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar
Given an elliptic curve $E$ defined over $Q$ with a fixed modular parametrisation, we can associate Heegner points to quadratic imaginary fields on $E$. So, given a set of quadratic imaginary fields how to know whether the Heegner points associated to them are linearly independent or not? Rosen and Silverman has find a criteria on the fields for the independence of Heegner points in the case of non-CM elliptic curves. We will show how to extend their result to CM elliptic curves.

Local Maass forms and theta lifts

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Speaker: 
Ben Kane
Affiliation: 
Köln
Date: 
Wed, 2012-10-17 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

In this talk, we consider local Maass forms.  These are functions which locally
satisfy the differential equation of a Maass form away from certain geodesics
of measure zero.  In joint work with Bringmann and Kohnen, a first natural
subclass of these has been discovered and linked to the Shimura and Shintani
lifts, as will first be described in this talk.  We then determine that these
functions are theta lifts of weak Maass forms, explaining many of the
properties previously discovered.  This is based upon joint work with Bringmann

Incoherent Eisenstein series on U(n,n) and arithmetic geometry

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Speaker: 
Steve Kudla
Affiliation: 
University of Toronto
Date: 
Wed, 2012-11-07 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar 
In this lecture, I will discuss the structure of the non-singular Fourier coefficients of
the derivative at the central critical point of incoherent Eisenstein series on U(n,n). 
In certain cases, these coefficients coincide with the arithmetic degrees of
0-cycles on moduli spaces of abelian varieties.  The proof of this relation depends
on p-adic uniformization and the determination of the structure of special cycles on
Rapoport-Zink spaces. 
This is joint work with Michael Rapoport.

Darmon points: algorithms and numerical evidence

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Speaker: 
Xavier Guitart
Affiliation: 
Univ. Politécnica de Catalunya/MPI
Date: 
Thu, 2012-11-15 11:15 - 12:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar 
Darmon points are a collection of conjectural generalizations of Heegner
points on modular elliptic curves. Algorithms for their effective
calculation are useful for gathering numerical evidence supporting their
conjectured properties. In addition, in some cases they can be used in
practice as very efficient methods for computing algebraic points in such
elliptic curves. In this talk I will recall two constructions of Darmon
points in different settings, and the algorithms of Darmon-Logan and
Darmon-Green-Pollack that allow for their computation in certain elliptic
curves.

The projective translation equation: where topology, algebraic geometry, differential equations, and number theory meet

Posted in
Speaker: 
Giedrius Alkauskas
Date: 
Wed, 2012-11-21 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar 
A flow is a bivariate function (the first variable belongs to a
certain space while the second is a "time") which satisfies a natural
iterative functional equation. The aim of this talk is to describe a new
field of projective flows and its manifold aspects, including topology,
combinatorics (Polya urns), number theory (elliptic curves, elliptic and
abelian functions), linear PDEs, birational geometry (Cremona groups),
algebraic geometry, projective flows over finite fields, and group
representations (symmetries of flows).

 

The asymptotic behaviour of the torsion in the cohomology of arithmetic groups

Posted in
Speaker: 
Jonathan Pfaff
Affiliation: 
U Bonn
Date: 
Wed, 2012-11-28 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

We will study the asymptotics of the torsion in the cohomology
of certain arithmetic groups with coefficients in certain Z-modules
if the rank of the Z-modules tends to infinity. Our first main result is
that for cocompact groups the torsion grows exponentially where the
leading term is proportional to the volume of the underlying locally
symmetric space. Our method is based on studying the analytic torsion of
locally symmetric spaces with coefficients in flat vector bundles. If these space

Coverings, partitions and quasi-modular forms

Posted in
Speaker: 
Martin Möller
Affiliation: 
U Frankfurt
Date: 
Wed, 2012-12-05 13:45 - 14:45
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

 

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