▁to ▁algebraic ▁geometry , ▁he ▁moved ▁into ▁areas ▁of ▁di oph antine ▁geometry . ▁Maz ur ' s ▁t ors ion ▁theorem , ▁which ▁gives ▁a ▁complete ▁list ▁of ▁the ▁possible ▁t ors ion ▁sub groups ▁of ▁elli ptic ▁curves ▁over ▁the ▁rational ▁numbers , ▁is ▁a ▁deep ▁and ▁important ▁result ▁in ▁the ▁arithmetic ▁of ▁elli ptic ▁curves . ▁ ▁Maz ur ' s ▁first ▁proof ▁of ▁this ▁theorem ▁dep ended ▁upon ▁a ▁complete ▁analysis ▁of ▁the ▁rational ▁points ▁on ▁certain ▁mod ular ▁curves . ▁ ▁This ▁proof ▁was ▁carried ▁in ▁his ▁sem inal ▁paper ▁" Mod ular ▁curves ▁and ▁the ▁Eisen stein ▁ideal ". ▁The ▁ideas ▁of ▁this ▁paper ▁and ▁Maz ur ' s ▁notion ▁of ▁Gal ois ▁de form ations , ▁were ▁among ▁the ▁key ▁ing red ients ▁in ▁Wil es ' s ▁proof ▁of ▁F erm at ' s ▁Last ▁Theorem . ▁Maz ur ▁and ▁Wil es ▁had ▁earlier ▁worked ▁together ▁on ▁the ▁main ▁conject ure ▁of ▁I was awa ▁theory . ▁ ▁In ▁an ▁ex pository ▁paper , ▁Number ▁Theory ▁as ▁G ad fly , ▁Maz ur ▁describes ▁number ▁theory ▁as ▁a ▁field ▁which ▁ ▁He ▁expanded ▁his ▁thoughts ▁in ▁the ▁ 2 0 0 3 ▁book ▁Im ag ining ▁Num bers ▁and ▁Cir cles ▁Dist urbed , ▁a ▁collection ▁of ▁ess ays ▁on ▁mathematics ▁and ▁narr ative ▁that ▁he ▁edited ▁with ▁writer ▁Apost ol os ▁Do xi ad is . ▁ ▁Awards ▁and ▁hon ors ▁In ▁ 1 9 8 2 ▁he ▁was ▁elected ▁a ▁member ▁of ▁the