list of figures
list of tables
preface
standard notation
i. overview
ii. curves in projective space
1. projective space
2. curves and tangents
3. flexes
4. application to cubics
5. bezout''s theorem and resultants
iii. cubic curves in weierstrass form
1. examples
2. weierstrass form, discriminant, j-invariant
3. group law
4. computations with the group law
5. singular points
iv. mordell''s theorem
1. descent
2. condition for divisibility by 2
3. eq2eq, special case
4. eq2eq, general case
5. height and mordell''s theorem
6. geometric formula for rank
7. upper bound on the rank
8. construction of points in eq
9. appendix on algebraic number theory
v. torsion subgroup of eq
1. overview
2. reduction modulo p
3. p-adic filtration
4. lutz-nagell theorem
5. construction of curves with prescribed torsion
6. torsion groups for special curves
vi. complex points
1. overview
2. elliptic functions
3. weierstrass p function
4. effect on addition
5. overview of inversion problem
6. analytic continuation
7. riemann surface of the integrand
8. ''an elliptic integral
9. computability of the correspondence
vii. dirichlet''s theorem
1. motivation
2. dirichlet series and euler products
3. fourier analysis on finite abelian groups
4. proof of dirichlet''s theorem
5. analytic properties of dirichlet l functions
viii. modular forms for sl2, z
1. overview
2. definitions and examples
3. geometry of the q expansion
4. dimensions of spaces of modular forms
5. l function of a cusp form
6. petersson inner product
7. hecke operators
8. interaction with petersson inner product
ix. modular forms for hecke subgroups
1. hecke subgroups
2. modular and cusp forms
3. examples of modular forms
4. l function of a cusp form
5. dimensions of spaces of cusp forms
6. hecke operators
7. oldforms and newforms
x. l function of an elliptic curve
1. global minimal weierstrass equations
2. zeta functions and l functions
3. hasse''s theorem
xl. eichler-shimura theory
1. overview
2. riemann surface xon
3. meromorphic differentials
4. properties of compact riemann surfaces
5. hecke operators on integral homology
6. modular function jr
7. varieties and curves
8. canonical model of xon
9. abstract elliptic curves and isogenies
10. abelian varieties and jacobian variety
11. elliptic curves constructed from s2γ0n
12. match of l functions
xii. taniyama-weil conjecture
1. relationships among conjectures
2. strong wei] curves and twists
3. computations of equations of well curves
4. connection with fermat''s last theorem
notes
references
index of notation
index
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