CHAPTER 1 Elliptic curve s 1.1. Ellipti c curve s DEFINITION 1.1 . A n ellipti c curv e ove r a field F i s a complete algebrai c grou p over F o f dimension 1 . Equivalently, a n ellipti c curv e i s a smoot h projectiv e curv e o f genu s on e ove r F equippe d wit h a distinguishe d F-rationa l point , th e identit y elemen t fo r th e algebraic grou p law . I t i s a consequenc e o f th e Riemann-Roc h theore m ([Si86] , chap. Ill ) tha t whe n F i s of characteristic differen t fro m 2 and 3 such a curv e ca n be describe d b y a n affin e equatio n o f the for m (1.1) E : y2 = x 3 + ax + b, wit h a , b e F , A : = -2 4 (4a 3 + 27b 2 ) ^ 0 , in whic h th e distinguishe d F-rationa l poin t i s take n t o b e th e uniqu e poin t a t infinity i n th e homogeneou s equatio n fo r th e correspondin g projectiv e curve . Ove r a field of arbitrary characteristic , a n ellipti c curv e ca n stil l be describe d a s a plan e cubic curve, given by the somewha t mor e complicated equatio n sometime s referre d to a s the generalised Weierstrass normal form (1.2) E : y2 + a\xy + a^y = x 3 + a2X 2 + a^x + a^, wit h A ^ 0 . In fact , ellipti c curve s ar e sometime s defined a s plan e cubi c curve s give n b y a n equation o f th e for m (1.1 ) o r (1.2) , th e additio n la w o n E bein g describe d geo - metrically vi a th e well-know n chord-and-tangen t law . Suc h a n approac h ha s th e virtue o f concreteness an d underscore s th e possibilit y o f doing explici t calculation s with ellipti c curve s (b y computer , o r eve n b y hand ) whic h i s on e o f th e charm s o f the subject . O n th e othe r hand , Definitio n 1. 1 i s mor e conceptua l an d explain s why ellipti c curve s shoul d b e single d ou t fo r specia l attention : the y ar e th e onl y projective curve s tha t ca n b e endowe d wit h a n algebrai c grou p law— a structur e which bot h facilitate s an d enriche s thei r diophantin e study . For tw o ellipti c curve s ove r F t o b e isomorphi c ove r F , i t i s necessar y an d sufficient tha t thei r so-calle d j-invariants, define d i n term s o f th e coefficient s o f equation (1.1 ) b y th e formul a be equal . The structure o f the group E(F) o f solutions to (1.1 ) o r (1.2 ) depends of course on th e natur e o f the field F , fo r example : 1. Whe n F i s a finite field, th e grou p E(F) i s a finite abelia n group . Th e stud y of E(F) i s a t th e origi n o f man y o f th e practica l application s o f ellipti c curve s t o cryptography an d codin g theory . l http://dx.doi.org/10.1090/cbms/101/01
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