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2022-07-26

Shafarevich and why it implies Mordell

Torelli

Punchline: nice curves are determined by their Jacobians

High-level motivations

Tate Conjecture

Etale cohomology carries interesting extra structure as a \(G_k\) module:

Relation to Hodge conjecture:

Relation to Shimura varieties:

Jacobians

For \(C\) an algebraic curve:

Uses of Jacobians: Chabauty-Coleman

Applications of Shaferevich:

Complex Abel-Jacobi

Intermediate Jacobians

Schottky problem

Motivating divisors

Theta functions