This monograp h i s based o n a n NSF-CBM S lectur e serie s give n b y th e autho r
at th e Universit y o f Centra l Florid a i n Orland o fro m Augus t 8 to 12, 2001.
The goa l o f thi s lectur e serie s wa s t o surve y som e recen t development s i n th e
arithmetic o f modular ellipti c curves , with specia l emphasi s o n
(1) th e Birc h an d Swinnerton-Dye r conjecture ;
(2) th e constructio n o f rational point s o n modula r ellipti c curves ;
(3) th e crucial role played by modularity in shedding light on these two closely
related issues .
The tex t i s divided int o thre e part s o f roughly equa l length .
The first consist s o f Chapter s 1-3 an d Chapte r 10. Th e first thre e chapter s in -
troduce the background an d prerequisites fo r what follows : ellipti c curves, modula r
forms an d th e Shimura-Taniyama-Wei l conjecture , comple x multiplication, an d th e
fundamental Heegner point construction whos e study an d generalisation i s the mai n
theme o f the monograph . Th e notio n o f "Heegne r system" , whic h i s spelled ou t i n
Chapter 3 , i s use d i n Chapte r 10 to prov e Kolyvagin' s theore m relatin g Heegne r
points t o th e arithmeti c o f elliptic curves , givin g strong evidenc e fo r th e Birc h an d
Swinnerton-Dyer conjectur e fo r ellipti c curve s o f analytic ran k a t mos t one . Whil e
more advance d tha n Chapter s 1-3, Chapte r 10 i s independen t o f th e materia l i n
Chapters 4- 9 an d coul d b e rea d immediatel y afte r Chapte r 3 .
Chapters 4-6 introduc e variants of modular parametrisation s i n which modula r
curves ar e replace d b y Shimur a curve s attache d t o certai n indefinit e quaternio n
algebras. A stud y o f thes e parametrisation s reveal s a n importan t ne w structure :
the rigi d analyti c uniformisatio n o f Shimur a curve s discovere d b y Ceredni k an d
Drinfeld, givin g rise to p-adic uniformisations o f modular ellipti c curves by discret e
arithmetic subgroup s o f SL
(QP) arisin g fro m definit e quaternio n algebras .
The mai n ne w contribution s o f this monograph ar e contained i n Chapter s 7-9 .
These Chapter s giv e a n overvie w o f th e author' s attempt s t o exten d th e theor y
of Heegner point s an d comple x multiplicatio n t o certai n situation s wher e th e bas e
field is not a CM field. Th e notion s o f rigi d analysi s develope d i n Chapter s 5 an d
6 pla y a ke y rol e i n suggestin g a p-adic varian t o f th e theorie s o f Chapter s 7 an d
8. Thi s leads , i n Chapte r 9 , t o a conjectura l constructio n o f point s o n a modula r
elliptic curv e ove r Q define d ove r rin g clas s fields o f a real quadratic field, whic h
are expected to behave much like classical Heegner points attache d t o an imaginar y
quadratic field.
The reade r i s cautione d tha t man y proof s giv e onl y th e mai n ideas ; detail s
have often bee n lef t ou t o r relegated t o exercises, retaining (fo r bette r o r for worse )
the flavou r o f th e origina l lectur e series . O f necessity , a numbe r o f importan t
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