An introduction to arithmetic geometry

Syllabus

Commutative algebra

• One dimensional Noetherian domains LN: 1, 2.
• Dedekind domains LN: 3, 4.
• Some local rings LN: 5, 6.
• Norm, trace,and discriminant LN: 7, 8.

Modular curves

• Recirpocity and modularity LN: 9, 10.
• Classical modular forms LN: 11, 12.
• Hecke theory
• The Jacobian

The Mordell-Weil theorem

• The weak Mordell-Weil theorem
• The Kummer pairing
• Heights on elliptic curves
• On the Birch and Swinnerton-Dyer conjecture

Abelian varieties

• Theta functions and the Riemann form
• Abelian varieties are projective
• Good and bad reduction
• Remarks on Falting's theorem

Facultad de Ciencias
Bernal Diaz del Castillo 340
28045 Colima
Mexico