algebraicnumbertheory arithmeticalgebraicgeometry(diophantinegeometry) arithmeticaspectsofmodularandshimuravarieties diophantineequations discontinuousgroupsandautomorphicforms ellipticcurvesoverglobalfields fermat'slasttheorem galoisrepresentations higherdegreeequations;fermat'sequation holomorphicmodularformsofintegralweight numbertheory