## The L-functions and Modular Forms Project

L-functions and modular forms underlie much of twentieth century number theory
and are connected to the practical applications of number theory in
cryptography. Virtually all branches of number theory have been touched by
L-functions and modular forms. Besides containing deep information concerning
the distribution of prime numbers and the structure of elliptic curves, they
feature prominently in Andrew Wiles' solution of the famous 350-year-old
Fermat's Last Theorem, and in the twentieth century classification of congruent
numbers, a problem first posed by Arab mathematicians one thousand years. In
spite of their central importance, mathematicians have only scratched the
surface of these crucial and powerful functions.
Michael Rubinstein, University of Waterloo,
and William Stein, University of Washington,
are directing a a major new project involving a dozen researchers to systematically
tabulate and study these
functions.
This project has been recommended for funding by the National Science Foundation.

The work will fall into four categories: theoretical,
algorithmic, experimental, and data gathering. The theoretical work will be
stimulated by their goal of charting the world of L-functions and modular
forms. Their experimental work will involve testing many key conjectures
concerning these functions. The project will produce a large amount of
training, with plans for three graduate student schools, an undergraduate
research experience, and support for a score of postdocs and graduate students
who will assist in research. It will result in the creation of a vast
amount of data about a wide range of modular forms and L-functions, which will
far surpass in range and depth anything computed before in this area. The data
will be organized in a freely available online data archive, along with the
actual programs that were used to generate these tables. By providing these
tables and tools online, the researchers will guarantee that the usefulness of
this project will extend far beyond the circle of researchers on this FRG. The
archive will be a rich source of examples and tools for researchers working on
L-functions and modular forms for years to come, and will allow for future
updates and expansion.