Equations and Automorphic Forms

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ABSTRACTPrincipal Investigator: Wiles, Andrew J. Proposal Number: DMS - 0758379Institution: Princeton UniversityTitle: Equations and Automorphic FormsThis project aims to study the problem of solving polynomial equations. The formula for the solution of the quadratic equation, at least in special cases, was known to the Babylonians and the solutions to cubic and quartic equations were developed by Italian mathematicians in the 16th century. In the 19th century it was shown that a general equation of degree greater than or equal to five has no formula of the same kind. However it is possible that equations in more than one variable might always have simple solutions obtained by extracting roots, just as exist for the quadratic, cubic and quartic equations. Our first goal is to try to find families of equations for which there are such solutions. Our second goal is to try to describe the solutions in many cases in terms of functions with symmetries called modular forms. These symmetries are more complicated than, but are related to, the kinds of symmetries satisfied by the trigonometric functions. The particular set of curves which will be investigated are the curves of genus one. These curves have received a lot of attention in recent years. The PI hopes to extend his research into whether these curves always have solvable points. His second and principal goal is to try to prove cases of non-solvable base change. This would enable one to describe suitable representations of Galois groups in terms of automorphic forms. This is a crucial part of the Langlands program and seems to be a key stumbling block in the way of proving general results about functoriality.