# 2021-11-10

Tags: #web/quick-notes

Refs: ?

## 16:20

Hector Pasten, UGA NT seminar.

• Mordell’s conjecture: for $$C$$ a curve, $$C({\mathbf{Q}}) < \infty$$.
• Chabauty: if $${\operatorname{rank}}J({\mathbf{Q}}) > g = \dim J({\mathbf{Q}})$$, then $$C({\mathbf{Q}})$$ is finite.
• Faltings: Proof using heights on moduli spaces
• Vojta: Proof by Diophantine approximation.
• Abel-Jacobi map: $$C\to J_C$$ by $$x \mapsto [x-x_0]$$.
• Chabauty’s proof: let $$\Gamma$$ be the $$p{\hbox{-}}$$adic closure of $$J({\mathbf{Q}})$$ in $$J({ {\mathbf{Q}}_p })$$, which is a $$p{\hbox{-}}$$adic Lie subgroup of $$J({ {\mathbf{Q}}_p })$$. Interpret $$\Gamma \cap C({ {\mathbf{Q}}_p })$$ as zero loci of $$p{\hbox{-}}$$adic analytic functions of $$C({ {\mathbf{Q}}_p })$$, constructed using integration.
• See good reduction, hyperplane section.
• Nice: smooth, projective, geometrically irreducible.
• Looks hyperbolic: contains no elliptic curves.
• First Chern number: self-intersection of the canonical divisor.
• Reduction map $${ \text{red} }: A({ {\mathbf{Q}}_p }) \to A({ \mathbf{F} }_p)$$, take residue discs $$U_x \coloneqq{ \text{red} }^{-1}(x)$$ for $$x\in A({ \mathbf{F} }_p)$$. Bound the number of points in $$X({\mathbf{Q}}_p) \cap\Gamma \cap U_x$$.
• Fat point: $${ {\mathbf{Q}}_p }[z]/\left\langle{z^n}\right\rangle$$ for some $$n> 1$$.