This will be a growing list of possible topics for final projects. It's intended to be inspirational, rather than comprehensive. Moreover, some of these topics overlap.

• Theta functions
  • Sums of squares and number theory
  • Coordinates on moduli spaces
• Curves and Jacobians
  • Uniformization of Riemann surfaces
  • Riemann relations for Jacobians
  • Torelli's theorem
  • Schottky problem
• Geometry of abelian varieties
  • Calculation of Hⁱ(X, &Omega^j_X).
  • Duality, Weil pairing
  • Endomorphisms of abelian varieties
  • Equations for abelian varieties
• Moduli spaces
  • Modular forms
  • Fundamental domains for congruence subgroups
  • Moduli spaces of PEL type
• Arithmetic
  • Complex multiplication
  • Tate modules and Galois representations
  • Group schemes
• Periods
  • Transcendence of periods
  • Deformation theory and Grassmans.
  • Hodge structures