The arithmetic of certain submotives of products of modular curves

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This project belongs to the area of Arithmetic Geometry, i.e. to the area which applies the methods of Algebraic Geometry to study Diophantine equations. A typical example of a Diophantine equation is the famous Fermat equation: x^n+y^n=z^n where n > 2. Fermat asserted in 1640 that this equation has no solution in positive integers, i.e. that the sum of two n-th powers can never be an n-th power, if n > 2. This was resolved only recently (1995) by the work of Wiles, based on ideas of Frey and work of Ribet.     An important technique used by Frey, Ribet and Wiles is the study of elliptic curves and their associated Galois representations. This project studies surfaces which classify isomorphisms between such representations. A key problem here is to understand which curves of a certain type can lie on these surfaces, and a proof of the conjectured answer would constitute a first step towards a vast generalization of Fermat's equation.     As is indicated in this proposal, this research has applications to the areas of dynamical systems (billiards) and mathematical physics (mirror symmetry) via Hurwitz spaces (counting covers of elliptic curves) and also to elliptic curve cryptography (ECC).