preface to the second edition
preface to the first edition
introduction
chapter i algebraic varieties
§1. affine varieties
§2. projective varieties
§3. maps between varieties
exercises
chapter ii algebraic curves
§1. curves
§2. maps between curves
§3. divisors
§4. differentials
§5. the riemann-roch theorem
exercises
chapter iii the geometry of elliptic curves
§1. weierstrass equations
§2. the group law
§3. elliptic curves
.§4. isogenies
§5. the invariant differential
§6. the dual isogeny
§7. the tate module
§8. the weil pairing
§9. the endomorphism ring
§ 10. the automorphism group
exercises
chapter iv the formal group of an elliptic curve
§ 1. expansion around o
§2. formal groups
§3. groups associated to formal groups
§4. the invariantdifferential
§5. the formal logarithm
§6. formal groups over discrete valuation rings
§7. formal groups in characteristic p
exercises
chapter v elliptic curves over finite fields
§ 1. number of rational points
§2. the weil conjectures
§3. the endomorphism ring
§4. calculating the hasse invariant
exercises
chapter vi elliptic curves over c
§1. elliptic integrals
§2. elliptic functions
§3. construction of elliptic functions
§4. maps analytic and maps algebraic
§5. uniformization
§6. the lefschetz principle
exercises
chapter vii elliptic curves over local fields
§1. minimal weierstrass equatlons
§2. reduction modulo
§3. points of finite order
§4. the action of inertia
§5. good and bad reduction
§6. the croup ee0
§7. the criterion of n~ron-ogg-shafarevich
exercises
chapter viii elliptic curves over global fields
§1. the weak mordell-weil theorem
§2. the kummer pairing via cohomology
§3. the descent procedure
§4. the mordell-weil theorem over q
§5. heights on projective space
§6. heights on elliptic curves
§7. torsion points
§8. the minimal discriminant
§9. the canonical height
§10. the rank of an elliptic curve
§11. szpiro''s conjecture and abc
exercises
chapter ix integral points on elliptic curves
§1. diophantine approximation
§2. distance functions
§3. siegel''s theorem
§4. the s-unit equation
§5. effective methods
§6. shafarevich''s theorem
§7. the curve ye = x3 + d
§8. roth''s theorem--an overview
exercises
chapter x computing the mordell-weil group
§1. an example
§2. twisting--general theory
§3. homogeneous spaces
§4. the selmer and shafarevich-tate groups
§5. twisting--elliptic curves
§6. the curve y2 = xa + dx
exercises
chapter xi algorithmic aspects of elliptic curves
§1. double-and-add algorithms
§2. lenstra''s elliptic curve factorization algorithm
§3. counting the number of points in efq
§4. elliptic curve cryptography
§5. solving the ecdlp: the general case
§6. solving the ecdlp: special cases
§7. pairing-based cryptography
§8. computing the weil pairing
§9. the tatae-lichtenbanm pairing
exercises
appendix a elliptic curves in characteristics 2 and 3
exercises
appendix b group cohomology ho and h1
§1. cohomology of finite groups
§2. galois cohomology
§3. nonabelian cohomology
exercises
appendix c further topics: an overview
§11. complex multiplication
§12. modular functions
§13, modular curves
§14. tate curves
§15. n6ron models and tate''s algorithm
§16. l-series
§17. duality theory
§18. local height functions
§19. the image of galois
§20. function fields and specialization theorems
§21. variation of ap and the sato-tate conjecture
notes on exercises
list of notation
references
index
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