*
f(z) = ∑ _{j=0}^{∞} a_{j} (z - p)^{j}
*

about each point *p* of its domain. Equivalently, *f(z) = u(z) + i v(z)* is holomorphic
if it satisfies the Cauchy-Riemann equations

*∂ u/∂ x = ∂ v/∂ y *

*∂ u/∂ y = - ∂ v/∂ x . *

For a number theorist, a particularly useful type of holomorphic function
is a modular function. A *modular function* is a holomorphic function
defined on the upper halfplane *U = {z ∈ b: Im z > 0}* and
having particular symmetries. The

In number theory, modular functions are particularly useful in studying
Diophantine equations--polynomial equations with integer
coefficients for which we seek integer solutions. Andrew Wiles
used classical modular functions to prove Fermat's Last
Theorem (Fermat's equation is certainly a Diophantine
equation). Generalizations of modular functions, such as Hilbert modular functions,
are used to study generalizations of Fermat's equation, the *ABC* conjecture
(which is another generalization of Fermat's Last Theorem), and other
important questions in number theory.

Certainly a big subject, which goes back to the early nineteenth century and
the time of Abel (1802-1829), is the study of elliptic functions and elliptic
equations. An *elliptic*.
equation is one of the form

*
y ^{2} = p(x) ,
*

where *p* is a cubic polynomial with rational coefficients. One wishes to find and
identify the rational solutions of such an equation. Even if you *know* that
the equation has non-trivial rational solutions, it is difficult to construct them.
*Heegner points* are a technique for finding such solutions. And modular
functions can be used to construct Heegner points.

The primary goal of this workshop is to construct a Wiki-page for modular functions.
This will be an OnLine computer resource that is similar to the freeware encyclopedia
known as Wikipedia. The modular functions Wiki will have two types of
entries: **(a)** entries which are actually descriptions and properties for particular
modular functions and **(b)** data entries which consist of tables of modular functions.

Many of the workshop participants have their own well-established data- bases of modular functions which have proved to be quite useful in a variety of research projects. It is hoped that all these different software tools can be folded into the modular functions Wiki that is being produced by this workshop. In fact, even on the first day of the workshop, participants took part in entering data into the official AIM modular functions Wiki.

This is a huge project that will span years of work by many mathematicians. Much of this week was spent defining the concept of the Wiki, establishing its feasibility, and building a framework for the project. Different people need to be assigned different tasks, and a vision of the entire construction needed to be generated. The "final" product is expected to have hundreds of topics and thousands of tables.

Certainly a big vector in modern mathematical research is to enlist the computer
as a high-level tool to organize and analyze data. The *L*-Functions and Modular
Forms workshop is on the forefront of these efforts, and is changing the face
of number theory in the process.