Vice Chancellor research grant application

William A. Stein


  1. 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.
  2. 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.
  3. 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.