Contents
Preface
Chapter 1. Ellipti c curve s
1.1. Ellipti c curve s
1.2. Th e Mordell-Wei l theore m
1.3. Th e Birc h an d Swinnerton-Dye r conjectur e
1.4. L- functions
1.5. Som e know n result s
Further result s an d reference s
Exercises
Chapter 2 . Modula r form s
2.1. Modula r form s
2.2. Heck e operator s
2.3. Atkin-Lehne r theor y
2.4. L-serie s
2.5. Eichler-Shimur a theor y
2.6. Wiles ' theore m
2.7. Modula r symbol s
Further result s an d reference s
Exercises
Chapter 3 . Heegne r point s o n XQ(N)
3.1. Comple x multiplicatio n
3.2. Heegne r point s
3.3. Numerica l example s
3.4. Propertie s o f Heegner point s
3.5. Heegne r system s
3.6. Relatio n wit h th e Birc h an d Swinnerton-Dye r
3.7. Th e Cross=Zagie r formul a
3.8. Kolyvagin' s theore m
3.9. Proo f o f the Gross-Zagier-Kolyvagi n theore m
Further result s
Exercises
Chapter 4 . Heegne r point s o n Shimur a curve s
4.1. Quaternio n algebra s
4.2. Modula r form s o n quaternion algebra s
4.3. Shimur a curve s
4.4. Th e Eichler-Shimur a construction , revisite d
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