# Misc Notes

:::{.remark}
See Iwasawa group.

:::

:::{.conjecture}
Section conjecture: $\Sec(Y/K) \cong Y(K)$, i.e. every section comes from a rational point.
:::

:::{.remark}
See the recent Lawrence-Venkatesh proof of Mordell.
See *Selmer section set* and *adelic sections*.
:::

:::{.remark}
Hyperbolic curves:

- $g=0 \leadsto \PP^1\sm Z$ where $\size Z \geq 3$
- $g=1 \leadsto$ affine
- $g\geq 2$: anything.

For $X\slice K$ for $K\in \NF$ smooth hyperbolic with good reduction away from $S$, $\size X(\OO_{K, S}) < \infty$ by Faltings.
See Bloch-Kato and Fontaine-Mazur conjectures.
:::