Number theory, particularly analytic number theory, uses many tools to attack its problems. Chief among these is complex variable theory. A function f(z) of a complex variable is said to be holomorphic if it has a local power series expansion

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 elliptic modular function is used classically to prove Picard's Little Theorem: An entire function that omits two values is constant.

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

y² = 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.