Tata Lectures on Theta II

IIIa:. An Elementary Construction of Hyperelliptic Jacobians

1. Review of background in algebraic geometry
David Mumford

2. Divisors on hyperelliptic curves
David Mumford

3. Algebraic construction of the Jacobian of a hyperelliptic curve
David Mumford

4. The translation-invariant vector fields
David Mumford

5. Neumann’s dynamical system
David Mumford

6. Tying together the analytic Jacobian and algebraic Jacobian
David Mumford

7. Theta characteristics and the fundamental Vanishing Property
David Mumford

8. Frobenius’ theta formula
David Mumford

9. Thomae’s formula and moduli of hyperelliptic curves
David Mumford

10. Characterization of hyperelliptic period matrices
David Mumford

11. The hyperelliptic p-function
David Mumford

12. The Korteweg-deVries dynamical system
David Mumford

IIIb:. Fay’s Trisecant Identity for Jacobian theta functions

13. The Prime Form E(x,y).
David Mumford

14. Fay’s Trisecant Identity
David Mumford

15. Corollaries of the identity
David Mumford

16. Applications to solutions of differential equations
David Mumford

17. The Generalized Jacobian of a Singular Curve and its Theta Function
David Mumford

18. Resolution of algebraic equations by theta constants
Hiroshi Umemura

Keywords: Mathematics, Special Functions, Algebraic Geometry, Mathematical Methods in Physics, Functions of a Complex Variable, Algebraic Topology, Partial Differential Equations