## Vice Chancellor research grant application

Goals:

- 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
Galois representations
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 $A_5$, 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.

Computer (VArStation YMP dual-Pentium II 400Mhz,
4.5-Gbyte disk, 128-Mbyte memory) ...............$3000

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.