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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