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Talk Outline V1

• References:
  • Abstract and Explicit Constructions of Jacobian Varieties, David Urbanik @ U. Waterloo
  • Milne’s notes on Jacobian varieties
  • Milne’s notes on abelian varieties, chapters III and IV
  • Various Wikipedia pages
• Why care about Jacobians? A web of conjectures/theorems - The Shafarevich conjecture - Mordell’s conjecture and Faltings’ proof - Faltings theorem and the use of Torelli - The Tate conjecture - For divisors vs general - Comparison to Hodge conjecture - Equivalence to BSD, finiteness results - The standard conjectures on algebraic cycles
• What can we do with Jacobians?
• Complex-analytic construction of Jacobians
• Important things related to Jacobians
  • The Abel-Jacobi map
  • The theta divisor
  • \(\operatorname{Jac}({-}): {\mathcal{M}_g}\to {\mathcal{A}_g}\) as a map of stacks
• Applicatons of Torelli