1.2. TH E MORDELL-WEI L THEORE M 3 unramified extensio n o f F) t o th e curv e E q give n b y the equatio n (1.7) E q : y2 + xy = x 3 + a 4 (g)x + a 6 (g), where (1.8) a 4 (tf) = s 3 (q), a 6 (q) = , wit h s k (q) = ^ l _ n - The p-adi c analyti c isomorphis m $Tat e : ^x Eq{F) i s obtained b y settin g •» * - ( » ) - (E ( i ^ - *.«. E r S ^+ * « ) \neZ v ^ y nG Z v ^ y / (For more details, see [Si94] , ch. V. ) Thi s p-adic uniformisation theor y fo r E yield s a descriptio n o f E(F) i n th e spiri t o f th e Weierstras s theor y o f equatio n (1.6) . I t plays a n importan t rol e in th e construction s o f Chapter s 6 and 9 . 4. Th e stud y o f ellipti c curve s ove r th e finite, comple x an d p-adi c fields, whil e o f interest i n it s ow n right , i s subordinat e i n thi s monograp h t o th e cas e wher e F i s a number field —the field o f rationa l number s o r a finite extensio n o f it . Th e ke y result an d th e startin g poin t fo r th e theor y o f ellipti c curve s ove r numbe r fields i s the Mordell- Weil theorem. Thi s result wa s first established, i n certain special cases, by Fermat himsel f usin g his method o f descent, an d it s proof i s recalled i n the nex t section. 1.2. Th e Mordell-Wei l theore m Let E b e a n ellipti c curv e define d ove r a number field F. THEOREM 1. 2 (Mordell-Weil) . The Mordell-Weil group E(F) is finitely gener- ated, i.e., E{F) -17® E(F) tor , where r 0 and E(F) tOT is the finite torsion subgroup of E(F). Since (th e modern formulatio n of ) th e proo f o f the Mordell-Wei l theorem play s an important rol e in the questions studied i n this monograph, particularl y i n Chap - ter 10 , it i s worthwhile t o recal l her e th e mai n idea s which ar e behind it . PROO F O F THEORE M 1.2 . Th e proo f i s composed o f two ingredients . 1. Th e existenc e o f a height function h : E(F) R satisfyin g suitabl e properties . THEOREM 1.3 . There exists a function h : E(F) R satisfying: (1) For all points Q in E(F), there is a constant CQ depending only on Q, and an absolute constant C depending only on E, such that h(P + Q) 2h(P) + C Ql h{mP) m2h(P) + C , for allPeE(F). (2) For all B 0, {P such that h(P) B} is finite.
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