Elementary Modular Iwasawa Theory

1. Curves over a field 3 1.1. Plane curves 3 1.2. Tangent space and local rings 5 1.3. Projective space 8 1.4. Projective plane curve 9 1.5. Divisors 11 1.6. The theorem of Riemann–Roch 12 1.7. Regular maps from a curve into projective space 13 2. Elliptic curves 14 2.1. Abel’s theorem 14 2.2. Weierstrass Equations of Elliptic Curves 15 2.3. Moduli of Weierstrass Type 17 3. Modular forms and functions 20 3.1. Geometric modular forms 20 3.2. Topological Fundamental Groups 21 3.3. Classical Weierstrass ℘-function 23 3.4. Complex Modular Forms 24 3.5. Weierstarss ζ and σ functions 26 3.6. Product q-expansion 28 3.7. Klein forms 29 4. Modular units 32 4.1. Siegel units 33 4.2. Distribution on p-divisible groups 34 4.3. Stickelberger distribution 35 4.4. Rank of distribution 36 4.5. Cusps of X(N) 37 4.6. Finiteness of ClX(N) 38 4.7. Siegel units generate Apm 38 4.8. Fricke–Wohlfahrt theorem 42 4.9. Siegel units and Stickelberger’s ideal 44 4.10. Cuspidal class number formula 46 4.11. Cuspidal class number formula for X1(N). 49