Diophantine problems : modern and classical perspectives

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Diophantine problems have a classical appeal that reaches beyond research mathematics, in some cases such as Fermat's Last Theorem, even as far as popular culture. For a practising mathematician, they provide test-cases for the latest theorems and conjectures from Arithmetic and, indeed, Algebraic Geometry, for understanding the arithmetic of curves and surfaces, and for observing the interplay between analytic and algebraic techniques, between the theoretical and the computational. This proposal is focussed upon the application of three rather diverse techniques to a variety of Diophantine problems, sometimes solely, sometimes in combination; in most cases, these are classical problems that have proved particularly resilient to prior attempts via other means. Our first two techniques (lower bounds for linear forms in logarithms and the hypergeometric method of Thue and Siegel) derive from Diophantine approximation, while the third (that of Frey-Hellegouarch curves associated with modular abelian varieties) arises from the deep interconnection between Galois representations and modular forms. The problems where we intend to apply these methods range from classical Diophantine equations, through more analytic questions about gaps between k-free numbers, to problems with the flavour of additive combinatorics.