Tags: #web/quick-notes

Refs: ?


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 \(n> 1\).
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