One of the fundamental problems in Number Theory is that of determining the number and distribution of integral or rational points upon curves of positive genus. Falting's theorem indicates that the number of such points is finite if the genus exceeds one, but, in general, one has little specific information available. The investigator studies a variety of techniques, principally from Diophantine approximation, which may be applied to quantify Falting's theorem in special situations. In particular, Bennett considers approaches based upon rational function approximation to systems of hypergeometric functions (the generalized hypergeometric method), lower bounds for linear forms in logarithms of algebraic numbers, local methods and computational Diophantine approximation. These are applied to study families of quartic equations, Thue equations, polynomial-exponential equations and integral points on given models of elliptic curves. The investigator also discusses two effective methods for finding such ``integral points'' on higher genus curves, where the aforementioned ones fail. These involve, respectively, extensions of an old technique of Chabauty and arguments exploiting the modularity of certain Frey curves.
Number theory, one of the oldest branches of mathematics, has enjoyed a renaissance in recent years. On one hand, the remarkable proof of Fermat's Last Theorem by Andrew Wiles has reaffirmed Number theory's position as a central one in modern mathematics. On the other, numerous and striking applications to such diverse problems as data encryption and signal transmission have demonstrated the utility of the field, both for industrial and governmental purposes. In the project at hand, the investigator studies a number of questions in the subfield of Diophantine equations, including some stemming from work of Wiles. The techniques employed allow the complete resolution of a number of classic problems in this area. These equations are connected to the theory of error-correcting codes which, in turn, are of crucial importance to, for example, data processing from satellites, cd and dvd players, cell phones and smart cards.
