NOTE: The notes appear in separate sections, which can be accessed through the underlined titles in the table of contents.

Riemann Surfaces and Second-Order Theta Functions
Robert C. Gunning
A: Theta functions
  1. Complex tori and their period matrices
  2. Theta Functions
  3. The functional equation for theta functions
  4. Even and odd theta functions
  5. The addition theorem
  6. The theta locus
  7. Singularities of the theta locus
  8. Wirtinger varieties
B: Riemann Surfaces
  1. The topology of surfaces
  2. Abelian differentials
  3. The cross-ratio function
  4. Factors of automorphy
  5. Theta factors of automorphy
  6. The prime function
  7. Canonical meromorphic differentials
  8. Quadratic period functions
  9. Varieties of special positive divisors
  10. The canonical curve
C: First-order Riemannian theta functions
  1. Riemann's theorem
  2. Prime function expansion
  3. The Gauss mapping
  4. Semicanonical functions
  5. Prime function expansion for theta derivatives
  6. The second-order Gauss mapping
  7. Semicanonical functions of second ordeer
D: Second-order Riemannian theta functions
  1. The analogues of Riemann's theorem
  2. Translation by half-periods
  3. Fay's trisecant identity
  4. Limiting forms of the trisecant identity
  5. A special limiting form: the KP equation
  6. Multisecant identities
E: Special Riemann surfaces
  1. The hyperelliptic multisecant identities
  2. Limiting forms of the hyperelliptic multisecant identities
  3. A special limiting form: the KDV equation
  4. The trigonal multisecant identities
F: Differential expansions of Riemannian theta functions
  1. The general expansion formula
  2. Dominant terms in the expansion
  3. The primary and binary expansion: dominant terms
  4. The binary expansion: subdominant terms
  5. The ternary expansion
  6. The quaternary expansion
G: Invariants sof the differential expansion
  1. The general expansion formula
  2. Differential invariants and the addition theorem
  3. Fourth order derivatives at the origin
  4. Hyperelliptic Riemann surfaces
  5. Examples (preliminary version)