rational points

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To bound \(X(k)\): use descent and torsors, or the Brauer-Manin obstruction. attachments/Pasted%20image%2020220319204535.png

Open Problems

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🗓️ Timeline

2021-05-05

Want to find rational points.
2021-04-12

Unsorted/Faltings theorem : for a curves \(C\) with \(g(C) \geq 2\), the number of Unsorted/rational points is finite, i.e. \({\sharp}C({\mathbf{Q}}) < \infty\).
2021-03-17

Why is the following SES important? See Unsorted/rational points. \begin{align*} 0 \to \mathop{\mathrm{Br}}k \to \oplus_v \mathop{\mathrm{Br}}k_v \to {\mathbf{Q}}/{\mathbf{Z}}\to 0 .\end{align*}

Links to this page

unresolved links output

Brauer-Manin obstruction in Unsorted/Brauer group, -Unsorted/rational points

Manin conjecture in Unsorted/rational points

Vojta’s conjecture in Quick_Notes/2021-01-03, -Unsorted/rational points
torsor

rational point
section conjecture

Still wide open, a proof would allow checking arithmetically interesting things like existence of rational points on curves by analyzing maps between homotopy groups.
local to global

Tate-Shafarevich group for rational points
elliptic curve

Unsorted/rational points
descent

See torsor and rational point. For elliptic curves, related to the Selmer group
arithmetic geometry

Finding rational points on curves.

rational point
Strong approximation

- Tags: - #todo/untagged - Refs: - #todo/add-references - Links: - rational point
Brauer group

Used in the definition of the Brauer-Manin obstruction which obstructs lifting adelic points to Unsorted/rational points.