# Abstracts for MPIM/HIM-Number theory lunch seminar

Alternatively have a look at the program.

## Rational points on Picard modular surfaces

Posted in
Speaker:
Dinakar Ramakrishnan
Affiliation:
Caltech
Date:
Wed, 2018-01-24 14:30 - 15:30
Location:
MPIM Lecture Hall

Picard modular surfaces X, which are smooth compactifications of the quotients Y of the complex ball by a discrete subgroups \Gamma of SU(2,1), have been studied from various points of view. They are often defined over an imaginary quadratic field M, and we are interested in the rational points of X over finite extensions k of M. In a joint work with M.

## Euler-Kronecker constants

Posted in
Speaker:
Kumar Murty
Affiliation:
University of Toronto
Date:
Wed, 2018-01-31 14:30 - 15:30
Location:
MPIM Lecture Hall

Ihara defined and began the systematic study of the Euler-Kronecker constant
of a number field. In some cases, these constants arise in the study of periods
of Abelian varieties. For abelian number fields, they can be explicitly connected
to subtle problems about the distribution of primes. In this talk, we review some
known results and describe some joint work with Mariam Mourtada.

## Arakelov geometry, another take on L-functions

Posted in
Speaker:
Vincent Maillot
Affiliation:
Institut de Mathématiques de Jussieu, Paris
Date:
Wed, 2018-02-07 14:30 - 15:30
Location:
MPIM Lecture Hall
I will report on old and also not so old results relating Arakelov geometry and the theory of arithmetic L-functions.


## Hecke operators, buildings and Hall algebras

Posted in
Speaker:
Oliver Lorscheid
Affiliation:
IMPA, Rio de Janeiro
Date:
Wed, 2018-02-14 14:30 - 15:30
Location:
MPIM Lecture Hall

Serre's theory of trees has been applied successfully to calculations with automorphic forms for PGL(2) whenever strong approximation was sufficiently well working. This is, for instance, the case for rational function fields. In general, the class group becomes an obstruction, and a global variant of Serre's theory is needed.

## Cohomologically rigid complex local systems with finite determinant and quasi-unipotent monodromies at infinity are integral

Posted in
Speaker:
Hélène Esnault
Affiliation:
FU Berlin
Date:
Wed, 2018-02-28 14:30 - 15:30
Location:
MPIM Lecture Hall

Joint work with Michael Groechenig.

## Hecke's integral formula and Kronecker's limit formula for an arbitrary extension of number fields

Posted in
Speaker:
Hohto Bekki
Affiliation:
MPIM
Date:
Wed, 2018-03-07 14:30 - 15:30
Location:
MPIM Lecture Hall

The classical Hecke's integral formula expresses the partial zeta function of real quadratic fields as an integral of the real analytic Eisenstein series along a certain closed geodesic on the modular curve. In this talk, we present a generalization of this formula in the case of an arbitrary extension E/F of number fields. As an application, we present the residue formula and Kronecker's limit formula for an extension E/F of number fields, which gives an integral expression of the residue and the constant term at s=1 of the "relative'' partial zeta function associated to E/F.

## The Artin-Hasse isomorphism of perfectoid open unit disks and a Fourier-type theory for continuous functions on Q_p

Posted in
Speaker:
Francesco Baldassarri
Affiliation:
Date:
Wed, 2018-03-21 14:30 - 15:30
Location:
MPIM Lecture Hall

Posted in
Speaker:
David Jarossay
Affiliation:
University of Geneve
Date:
Wed, 2018-04-11 14:30 - 15:30
Location:
MPIM Lecture Hall

Multiple zeta values are periods of the pro-unipotent fundamental groupoid of the projective line minus three points. We will explain a way to compute their p-adic analogues, which keeps track of the motivic Galois action, and which has an application to the finite multiple zeta values recently studied by Kaneko and Zagier. The computation will be expressed by means of new objects which we will call p-adic pro-unipotent harmonic actions.

## First order differential equations

Posted in
Speaker:
Marius van der Put
Affiliation:
Groningen University
Date:
Wed, 2018-04-18 14:30 - 15:30
Location:
MPIM Lecture Hall

We consider a fi rst order differential equation of  the form $f(y'; y) = 0$ with $f\in K[S; T]$ and $K$ a
differential fi eld either complex or of positive characteristic.  We investigate several properties of $f$,
namely the 'Painlevé property' (PP), solvability and stratifi cation. A modern proof of the classi cation
of fi rst order equations with PP is presented for all characteristics. A version of the Grothendieck-Katz
conjecture for fi rst order equations is proposed and proven for special cases. Finally the relation with

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