Compute, and make available via the Internet, a database of
eigenforms for the action of the Hecke operators on spaces of modular
forms. Both the database, and the software developed in creating it, will
significantely fill a gap in the resources currently
available to researchers in the area of modular forms. The software
and data I have already developed has been of use
to many of my fellow Ph.D. students,
as well as to mathematicians at other institutes, such as Barry Mazur.
Computationally find new examples of two dimensional
satisfying the Artin conjecture on holomorphicity of the corresponding
L-function. Certain cases of this conjecture,
proved by Langlands and Tunnel, played a key role in Andrew Wiles's
recent proof of Fermat's Last Theorem. Our knowledge about the
remaining open case, in which the projective image is the alternating
group A5, is still limited. Essentially only seven examples
are currently known, and any technique which can produce more
is of interest.
Use formulas established by Coleman (and myself) to
numerically compute p-adic characteristic series in order
to begin to understand the ``Eigencurve'' recentely discovered
by Coleman and Mazur.
Each of these projects involves significant use of the computer.
In making the database it is essential to have a fast
machine, otherwise ``cutting edge'' tables can not be constructed.
Computing new examples satisfying Artin's conjecture
requires a machine with a large memory. Though I've been
working with some success for the past several months using
the department's general use computers and my home PC, I think my project
would be considerably more successful and comprehensive if I had
access to more powerful and dedicated computing equipment.
The computer is the only lab equipment relevant to mathematics. After
my graduation it will remain in the mathematics department to support
other computation-intensive research.
Computationally intensive research to date has taken place on the
public use computers available to graduate students in the Mathematics
department, my home computer, and a computer purchased
by Roland Dreier last year using his research grant.