Arithmetic of Modular forms and Elliptic Curves

IMPA, Rio de Janeiro, de 03/01 até 14/01

One of the principal goals of this workshop is to promote the research on arithmetic of modular forms and elliptic curves in South America. The mini courses are supposed to be accessible for students with a basic knowledge in Mathematics. It is also intended for university professors who are interested on the topic.

The goal of this workshop is to introduce students and researchers to current topics of research in Number Theory via 5 minicourses over a period of 2 weeks. The proof of Fermat’s Last Theorem (FLT) was one of the most spectacular advances in number theory, and in his proof Wiles proved a part of Serre’s conjecture which relates Galois Representations to Modular Forms. Dieulefait will give a course explaining the theory of Modular Forms and Galois representations, and a proof of Serre’s conjecture. The Birch and Swinnerton-Dyer conjecture (BSD) relating the rank of an elliptic curve to the order of its associated L-function is one of the most important open problems in mathematics (one of the 7 Clay millenium problems). One of its consequences, the so-called Selmer parity for elliptic curves over the rationals remained a conjecture for over 40 years and was solved only two years ago. Dokchitser will give a course on the BSD and his proof of the parity conjecture. Modular forms are a key ingredient for both FLT and BSD, and their study is an area of independent research. Diamantis will give a course on higher order modular forms and their associated L-functions and Hecke theory. Doran will explore the various roles played by elliptic curves in the geometric aspects of string theory. His lectures will culminate in a description of recent work on growth rates of Gromov-Witten invariants and special values of L-functions. Movasati will give a course explaining quasi-modular forms, connections with families of elliptic curves and fundamental differential equations for modular forms.

The courses will be at the level of Masters courses, and while a background of basic Algebraic Number theory and Algebraic Geometry is helpful, it is not essential. We propose to have 2- 3 lectures a day at the most, in order to leave ample time for informal discussions and a few research seminar level talks.

Due to scheduling reasons the Workshop will start Tuesday, January 04th with check-in at 09:00am

Organizing committee

Hossein Movasati (IMPA)
Ariel Martin Pacetti (Universidad de Buenos Aires)
Amilcar Pacheco (UFRJ)
Aftab Pande (UFRJ)

Speakers

Nikolaos Diamantis (University of Nottingham)
Higher-order automorphic forms

Luis Dieulefait (University of Barcelona)
Modular Forms and Serre’s conjectures

Tim Dokchitser (Cambridge)
L-functions of elliptic curves

Hossein Movasati (IMPA)
Geometric interpretation of quasi-modular forms

Lecture notes

Hossein Movasati (IMPA)
Lecture Note 1

Seminar

Ariel Pacetti (Universidad de Buenos Aires)
How to compute the local type of a modular form.

Amilcar Pacheco (Universidade Federal do Rio de Janeiro)
Rational points on abelian varieties in pro-p towers of function fields

Cecilia Salgado (Université Paris VII – Jussieu)
Comparing generic and special ranks on elliptic surfaces.

Program

Program

Conference Day

Talk 1 – 9:30 – 10:30
Roberto Miatello – Eigenvalues of Hecke operators on Hilbert modular groups

Coffee Break – 10:30 – 11:00

Talk 2 – 11:00 – 12:00
Oliver Lorscheild – Toroidal automorphic forms and the Riemann hypothesis

Lunch – 12:00 – 13:30

Talk 3 – 13:30 – 14:30
Matija Kazalicki – The Arithmetic of modular forms associated to Fermat curves

Talk 4 – 14:30 – 15:30
Martin Mereb – On the E-polynomials of a family of Character Varieties

Coffee Break – 15:30 – 16:00

Talk 5 – 16:00 – 17:00
Diamantis 3 – Higher-order automorphic forms

Cocktail – 17:30

Certificates

Lectures

Participants

Registration

Students who need financial support are requested to ask a university professor to send  a recommendation letter to modular@impa.br .

Os estudantes que necessitam de suporte financeiro, por favor , solicitem a um professor universitario uma carta de recomendação. A carta deverá ser enviar para modular@impa.br .

List of all registered partipants

List

Accommodation

IMPA has a list of suggested hotels for the event. Hopefully it can help you find accommodation in Rio during the Workshop.

If you need help, we suggest the agency of events that we use (please identify yourself as a participant in the workshop at IMPA):
Hotel reservations can be made directly with the CMOEventos agency by clicking here
The CMOEventos will reply by email (if online form was used) to confirm if the rooms are available.
Then, a credit card is required to guarantee your reservation. This information can be provided by fax, or using the CMOEventos online secure form on CMOEventos.

CMOEventos fax number: 55 (21) 2210 7282
CMOEventos phone number: 55 (21) 2210 1732 (Contact: Michele Leite)

For additional information about hotel accommodations contact Michele Leite at: micheleleite@cmoeventos.com.br

Pictures

Picture 1

Picture 2