Presentation
The purpose of this small-scale workshop is to bring together some young researchers in number theory, and give them the opportunity
to present their work. Topics will include various aspects of number theory, especially algebraic number theory, arithmetic geometry,
function field arithmetic. The atmosphere is expected to be informal and the relaxed schedule would leave time for discussions.
Organizer:
Cécile Armana (with the kind assistance of Mihran Papikian
and Ernst-Ulrich Gekeler)
New
Speakers
- Ralf Butenuth (Universität Duisburg-Essen)
An Algorithm to compute Hecke operators on Drinfeld modular curves
We will begin with a brief general introduction to Drinfeld modular
forms. We explain Teitelbaum's combinatorial description of Drinfeld cusp
forms as harmonic cochains which are invariant under certain finite index
subgroups of GL2(Fq[t]) and use this description to compute Hecke operators
on these forms. We will explain the technical details of the algorithm and
try to present some computational examples.
- Hugo Chapdelaine (Université Laval, currently in Institut de Mathématiques de Jussieu)
Archimedean and p-adic Stark conjectures
We will start by giving a quick overview of
the 12-th Hilbert problem which main objective is to provide a (complex or p-adic)
analytic description of the class fields of a given number field. We will then
recall the Gross-Stark conjecture in the Archimedean and p-adic settings. The truth
of this conjecture would provide a partial answer to the 12-th Hilbert problem. For the second
half of the talk, we will give an overview of a recent p-adic construction of Darmon and
Dasgupta which produces conjecturally, p-units in ring class fields of a real quadratic
field. Their work can be viewed as a refinement of the p-adic Gross-Stark conjecture.
-
Alina Carmen Cojocaru (University of Illinois at Chicago, currently in MPIM Bonn)
The Koblitz conjecture on average
Let E be an elliptic curve defined over Q. In 1988, Koblitz formulated a conjecture about the number of primes
p < x for which the reduction of E modulo p has prime order. This conjecture may be viewed as
a higher dimensional analogue of the classical twin prime conjecture. I will show that the Koblitz conjecture is true when
considered for a family of elliptic curves. This is joint work with Antal Balog (Renyi Institute, Budapest) and
Chantal David (Concordia University, Montreal).
-
Johannes Lengler (Universität des Saarlandes)
On the distribution of class groups of number fields
In 1801, Gauß has conjectured that there are infinitely many real quadratic number fields with class number 1. This conjecture
is still unproven, but in 1983, Cohen and Lenstra made a stunningly simple assumption about the occurences of class groups, with vast
consequences. They predicted that the sequence of class group of real quadratic fields essentially behaves like a random sequence of
finite abelian groups w.r.t. some probability measure. In particular, they predicted that the class group is trivial for about
75.446% of all real quadratic number fields. I explain the heuristic and how it can be motivated. Then I present part of my own
PhD-work about the heuristic, in particular pointing out a mysterious connection between the heuristic and partitions of natural
numbers.
-
Mihran Papikian (Penn State University, currently in Universität des Saarlandes)
Local properties of modular curves of D-elliptic sheaves
We study the existence of rational points on modular curves of D-elliptic sheaves over local fields and the structure of
fundamental domains of these curves in the Bruhat-Tits tree. We discuss some applications which include finding presentations for
arithmetic groups arising from quaternion algebras over function fields and finding the equations of modular curves of D-elliptic
sheaves.
-
Fabien Pazuki (Institut de Mathématiques de Jussieu - Université Paris 7)
Heegner points and a conjecture of Lang and Silverman
This talk will be focused on the classical Heegner point theory described in the work of Gross and Zagier.
We give some new estimates on the height of those points, and deduce that the lower-bounding constant predicted by
the Lang-Silverman conjecture is optimal.
- Enrico Varela (Universität des Saarlandes)
Fields generated by the torsion of a rank-2 Drinfeld module
Drinfeld modules offer methods to construct Galois field extensions L of
global function fields K. In this case, the associated Galois groups and
ramification behaviour can be computed. This ansatz is used to find
fields L with many rational places compared to their genus.
In this talk let K be the rational function field Fq(T). We are
studying the fields L = L(p) generated by the torsion of a particular
rank-2 Drinfeld module Phi and a polynomial p in Fq[T]. For certain
classes of polynomials p we can determine the degree of the extension, Galois group, ramification, genus etc.
We will also give an outlook on how this can hopefully be generalized for arbitrary p.
Schedule
The workshop will be held at the Department of Mathematics of the University of Saarland.
All talks will take place in Room 216. They should be at most one-hour long, and given in English.
Overhead projector and beamer are available for speakers upon request.
Tuesday, March 31
Wednesday, April 1
Lunchs will be taken at the Mensa.
Coffee will be served on Tuesday afternoon after lunch in room 310.
A dinner will be organized downtown on Tuesday night.
Participants
(if you wish to attend the workshop, please tell me
and I'll add your name to the list)
Hotel
Rooms for our guests have been reserved at Hotel Madeleine downtown (including
breakfast). The hotel is 500m from the main train station (the address of the hotel is Cecilienstrasse 5).
To go to the hotel from the train station, you can walk down Kaiserstrasse (check
this itinerary). You can also take the tramway in front of the train station, direction Römerkastell, or Brebach Bahnhof, or Güdingen, or Kleinblittersdorf, or Sarreguemines,
and stop at Johanneskirche. The hotel is just on the other side of the church (check this
map).
Getting to the Department of Mathematics (by bus)
From the Hauptbahnof:
In front of the train station, take bus either bus 102 direction Dudweiler Dudoplatz, or bus 124 direction Universität Busterminal
(bus 112 will not run during this week).
The stop near the Department is Universität Mensa. The route takes around 15 minutes (with bus 124) or 25 minutes (with bus 102).
When you get off the bus, keep walking in the same direction. The Department of Mathematics is located in a grey/black building (E2 4) on the right.
From the hotel:
You can take bus 101 (direction Dudweiler Dudoplatz), 102 (direction Dudweiler Dudoplatz) or 124 (direction Universität Busterminal).
For bus 101 and 102: the bus stop nearby the hotel is Rathaus and located on the other side of the church, near the Rathaus (check
this map). One the way back from the university, buses 101 and 102 will take a slightly different route downtown and stop at Johanneskirche, instead
of Rathaus.
For bus 124: the bus stop nearby the hotel is Haus der Zukunft, 5 minutes from the Hotel
(check
this map).
In any case, the stop near the Department is Universität Mensa. The route takes a bit more than 20 minutes.
When you get off the bus, keep walking in the same direction. The Department of Mathematics is located in a grey/black building (E2 4) on the right
(see the campus map).
General informations for bus and tramway
A one-way ticket costs 2.30 Eur (full fare). It can be bought at the vending machines (located on the tramway platforms), or in the bus (but
not in the tramway). The ticket must be validated inside the bus or tram.
Here you can find some informations on fares,
maps of the lines and a route planner.
Contact
Dr. Cécile Armana
Building E2 4 - Office 309
FR 6.1 Mathematik
Universität des Saarlandes
D-66041 Saarbrücken -- Germany
e-mail :
armana@math.jussieu.fr
phone : +49 681 302 64116
Some useful links
Last update: April 3, 2009
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