A Workshop on Modular Forms and Related Topics
February 06-10, 2012 -
AUB, CAMS
College Hall B1
Mini-Courses
Speaker 1: Peter Bruin
Affiliation: University of Zurich,
Switzerland
Title of the mini-course: Modular curves and Galois
representations
Titles of the lectures:
Lecture
1: Modular curves: complex analytic aspects
Lecture
2: Modular curves: algebraic and arithmetic aspects
Lecture 3: Galois representations in Jacobians of modular curves
Lecture
4: Computation of modular Galois representations
Description of the mini-course:
The first half of the mini-course is an introduction to various aspects
of modular curves and modular forms. We will relate the classical
description of modular curves (as quotients of the complex upper
half-plane) to moduli of elliptic curves. This leads to a ``finer''
description of modular curves as algebraic curves over the rationals. In
the second half of the mini-course, we will describe how modular curves
and their Jacobians can be used to attach two-dimensional Galois
representations to modular forms, in particular over finite fields. We
will finish with some words on how all of this can be made computable.
Speaker 2: YoungJu Choie
Affiliation: POSTECH, South Korea
Title of the mini-course: Jacobi Forms and their
Applications
Titles of the lectures:
Lecture 1: Jacobi forms -1.
Lecture 2: Jacobi forms -2.
Lecture 3: Quasi-modular forms and Jacobi forms.
Lecture 4: Mock modular forms and Jacobi forms.
Abstract
We discuss the following topics: Basic theory of Jacobi forms and
motivation, Quasi-modular forms and meromorphic Jacobi forms and mock
modular forms.
Speaker 3: Winfried Kohnen
Affiliation: University of Heidelberg, Heidelberg, Germany
Title of the mini-course: Fourier coefficients of modular
forms
Titles of the lectures:
Lecture
1: Elliptic modular forms.
Lecture 2: Siegel modular forms.
Lecture
3: Growth estimates for Fourier coefficients of Siegel cusp forms
Lecture
4: A characterization of Siegel cusp forms of degree two by the growth
of its Fourier coefficients.
Description of the
mini-course:
In the first two lectures, we will cover the basic theory of elliptic
resp. Siegel modular forms (e.g. reduction theory, examples of modular
forms, L-functions). In lecture 3, we will report on the Resnikoff-Saldana
conjecture and results in this direction, and in the last lecture, we
will report on very recent joint work with Y. Martin giving a
characterization of Siegel cusp forms of degree two in terms of the
growth of their Fourier coefficients.
Speaker 4: Fredrik Strmberg
Affiliation: TU Darmstadt
Title of the mini-course: Spectral theory and automorphic
forms for modular groups
Titles of Lectures:
Lecture
1: Introduction to hyperbolic geometry and Fuchsian groups.
Lecture
2: The spectral decomposition.
Lecture
3: Computational aspects.
Lecture
4: Selected open problems.
Description of the mini-course:
We will give an introduction to the spectral theory of a special class
of hyperbolic Riemann surfaces, corresponding to subgroups of the
modular group. The first lecture will introduce the background material
necessary to state the problem. We will then give the spectral
decomposition of the Laplace-Beltrami operator on the modular surface.
This decomposition involves both a discrete and a continuous part. In
the second half of the course we will concentrate on properties of the
discrete part, spanned by so-called Maass waveforms. In general these
cannot be expressed in closed form and the only known approach is to use
numerical computations. We will give an overview of both heuristic and
rigorous algorithms. In the last lecture we will discuss two of the
interesting (open) problems in the area: the Selberg eigenvalue
conjecture and ``arithmetic quantum chaos''.
Speaker 5: Gabor Wiese
Affiliation: University of Luxembourg
Title of the mini-course: Modular Forms mod p and Galois
Representations of Weight One.
Titles of the lectures:
Lecture
1: Integral structures in modular forms.
Lecture
2: Modular forms mod p: theory and computation.
Lecture
3: Galois representations attached to modular forms.
Lecture
4: Galois representations of weight one.
Description of the mini-course:
This mini-course will be concerned with modular forms mod p and their
attached Galois representations. We will particularly focus on those
modular forms and Galois representations that are of weight one: for a
mod p Galois representation, this means it is unramified at p. We will
first discuss integral structures in the space of modular forms, which
allow us to reduce mod p; but, we will also mention that there is an
algebraic-geometric point of view due to Katz (which - roughly speaking
- gives the same objects, except in weight one). Along the way, we will
mention algorithms to compute modular forms, which are based on the same
methods as the proof of the existence of an integral structure. Then we
will move on to Galois representations attached to modular forms, in
particular, to Galois representations taking their coefficients in Hecke
algebras. This leads us to consider Galois representations with
coefficients in the weight one Hecke algebra. We shall finally prove (in
most cases) that this Galois representation is also of weight one.
Talks
Speaker: Houria Baaziz
Affiliation: Universit USHTB d'Algrie
Title: Equations of the Modular Curve X_1(N)
Abstract: Several authors gave equations for the modular
curves X_1(N) when N <= 22, but their methods do not easily generalize
to arbitrary values of N. In this paper, we present a new algorithm for
obtaining equations for X_1(N) which at the same time enables one to
keep track explicitly of the corresponding pairs (E,P), where E is an
elliptic curve and P a point of order N on E. We use division
polynomials to characterize elliptic curves with equation
(E) y^2 + (a_1 x + a_3)y = x^3 + a_2 x^2
on which the point P = (0,0) is torsion and of given order N >= 4. We
show how to recover an explicit model for the pair (E,P) from the
corresponding point on their model of X_1(N).
We have calculated the resulting equations for all N <= 51.
Speaker: Ahmad El-Guindy
Affiliation: Cairo University
Title: Periods and Supersingularity for Elliptic Curves
and Drinfeld Modules
Abstract: In this talk, we recall some results on the
connections between the periods in families of elliptic curves and the
supersingular curves at a given prime in the same family. Guided by
this, we present some new results indicating the presence of a parallel
theory for (rank two) Drinfeld modules. We also present some general
formulas for (rank r) Drinfeld modules. This is joint work with Matt
Papanikolas.
Speaker:
Matija Kazalicki
Affiliation: University of Zagreb, Croatia
Title: Weakly Modular Forms,
de Rham Cohomology and Congruences
Abstract: In
1971 Atkin and Swinnerton-Dyer discovered remarkable congruences
involving Fourier coefficients of modular forms for noncongruence
subgroups. Substantial part of these congruences have been proved by
Scholl in 1985 (following work of Cartier, Ditter and Katz). We will
explain that similar congruences hold for weakly modular forms. Using
interaction of weakly modular forms and modular forms on certain
quotient of Fermat curve we will show that original Atkin and Swinnerton-Dyer
type of congruences for modular forms do not hold in this particular
example. This is joint work with A.J. Scholl.
Speaker: Karen
Taylor
Affiliation: Bronx University College of the City
University of New York
Title: Selberg
Trace Formula
Abstract: We
will discuss the derivation of the classical Selberg Trace Formula.