2022-07-26

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2022-07-26

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Shafarevich and why it implies Mordell

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Torelli

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Punchline: nice curves are determined by their Jacobians attachments/Pasted%20image%2020220726152608.png

High-level motivations

Tate Conjecture attachments/Pasted%20image%2020220726153607.png

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

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Relation to Hodge conjecture:

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Relation to Shimura varieties: attachments/Pasted%20image%2020220726154321.png

Jacobians

For \(C\) an algebraic curve: attachments/Pasted%20image%2020220726164901.png attachments/Pasted%20image%2020220726164942.png attachments/Pasted%20image%2020220726165016.png attachments/Pasted%20image%2020220726165049.png

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Uses of Jacobians: Chabauty-Coleman

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Applications of Shaferevich: attachments/Pasted%20image%2020220726163828.png attachments/Pasted%20image%2020220726163855.png attachments/Pasted%20image%2020220726163907.png attachments/Pasted%20image%2020220726163917.png attachments/Pasted%20image%2020220726163927.png attachments/Pasted%20image%2020220726163940.png attachments/Pasted%20image%2020220726164124.png attachments/Pasted%20image%2020220726164216.png

Complex Abel-Jacobi

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Intermediate Jacobians

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Schottky problem

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Motivating divisors

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Theta functions

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