---
date: 2021-11-10
tags: [ web/quick-notes ]
---

# 2021-11-10

Tags:
#web/quick-notes

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## 16:20

> Hector Pasten, UGA NT seminar.

- Mordell's conjecture: for $C$ a curve, $C(\QQ) < \infty$.
  - Chabauty: if $\rk J(\QQ) > g = \dim J(\QQ)$, then $C(\QQ)$ 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 $p\dash$adic closure of $J(\QQ)$ in $J(\QQpadic)$, which is a \(p\dash \)adic Lie subgroup of $J(\QQpadic)$.
  Interpret $\Gamma \intersect C(\QQpadic)$ as zero loci of \(p\dash \)adic analytic functions of $C(\QQpadic)$, constructed using integration.
- See [good reduction](good%20reduction.md), [hyperplane section](hyperplane%20section).
- Nice: smooth, projective, [geometrically irreducible](geometrically%20irreducible).
- Looks hyperbolic: contains no elliptic curves. 
- First Chern number: self-intersection of the canonical divisor.
- Reduction map $\red: A(\QQpadic) \to A(\FF_p)$, take residue discs $U_x \da \red\inv(x)$ for $x\in A(\FF_p)$.
  Bound the number of points in $X(\QQ_p) \intersect \Gamma \intersect U_x$.
- Fat point: $\QQpadic[z]/\gens{z^n}$ for some $n> 1$.