# Tuesday, October 19

\todo[inline]{Missing some stuff! Find notes.}

:::{.remark}
Let $k$ be a number field, $X\in \Var\slice k$ a variety, and $\AA\slice k$ the adeles over $k$.
Let $F: \Sch\op \to \Set$ be a functor, we can then consider an **$F\dash$obstruction to rational points**.
A rational point is the data of a morphism $\spec k\to X$, denoted $X(k) = \Sch(\spec k, X)$, and adelic points $X(\AA\slice k) = \Sch(\AA\slice k, X)$.
Since there is a morphism $k \to \AA\slice k\in \Ring$, this yields a morphism $X(k)\to X(\AA\slice k)$ (noting contravariance). 
Note that an adelic point is a lift:

\begin{tikzcd}
	&& {\spec \AA\slice k} \\
	\\
	X && {\spec k}
	\arrow[from=3-3, to=3-1]
	\arrow[from=1-3, to=3-3]
	\arrow[dashed, from=1-3, to=3-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJYIl0sWzIsMiwiXFxzcGVjIGsiXSxbMiwwLCJcXHNwZWMgXFxBQVxcc2xpY2UgayJdLFsxLDBdLFsyLDFdLFsyLDAsIiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

Moreover, $F$ induces a diagram. 
Let $i: \spec k \to X$ denote the inclusion, and $x\in X(k)$ be a $k\dash$point.
Then writing $F(k) \da F(\spec k)$, we have the following:

\begin{tikzcd}
	{X(k)} && {X(\AA\slice k)} \\
	\\
	{F(k)} && {F(\AA\slice k)}
	\arrow[from=1-1, to=1-3]
	\arrow[from=3-1, to=3-3]
	\arrow["{x^* A \da Fi(a)}"', from=1-1, to=3-1]
	\arrow["{x_v^* A}"', from=1-3, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJYKGspIl0sWzAsMiwiRihrKSJdLFsyLDAsIlgoXFxBQVxcc2xpY2UgaykiXSxbMiwyLCJGKFxcQUFcXHNsaWNlIGspIl0sWzAsMl0sWzEsM10sWzAsMSwieF4qIEEgXFxkYSBGaShhKSIsMl0sWzIsMywieF92XiogQSIsMl1d)

:::

:::{.definition title="?"}
Let $A\in F(X)$, then
\[
X(\AA\slice k)^A &\da \ts{ (x_v) \in X(\AA\slice k) \st x_v^* A \in \im\qty{F(k) \to F(\AA\slice k)} } \\
X(\AA\slice k)^F &\da \Intersect_{A\in F(X)} X(\AA\slice k)^A
.\]
:::

:::{.example title="?"}
For $F(\wait) \da \Br(\wait)$, the resulting $X(\AA\slice k)^F$ is the **Brauer-Manin obstruction**.
Consider $A\in \Br(X)$ for $X = V(y^2 + z^2 = (x^2-2)(3-x^2))$, then any $k\dash$rational point would yield $x^* A\in \Br(k)$ and thus $x_v^* A \in \Br(\AA\slice k)$.
The claim is that $\Br(\AA\slice k)^A = \emptyset$.
So consider $\ts{ (x_v) \in \Br(\AA\slice k) \st x_v^* A \in \im\qty{\Br(k) \to \Br(\AA\slice k) }}$.
:::

:::{.exercise title="?"}
Show this set is empty!
:::

## Descent Obstruction

:::{.definition title="fppf morphisms"}
A morphism $U\to X\in \Sch$ is **fppf** if

- $f$ is flat[^flat_condition],

- $f$ is locally of finite presentation.[^finite_prese]

[^finite_prese]: 
There exist affine open covers $\mcv\covers X$ and $\mcu_i \covers f\inv(V_i)$ such that $\ro{f}{U_{ij}}: U_{ij} \to V_i$ is induced by $B_i \to A_{ij} \in \Ring$ of finite presentation.
So $B\to A\in \Ring$ yields $A = B[x_0, \cdots, x_n]/ I$ with $I$ finitely generated.

[^flat_condition]: 
Flatness is a local condition: $f_q: \OO_{X, p}\to \OO_(U, q)$ for $q\in f\inv(p)$ should be a flat morphism of algebras, where $A\to B$ is flat if $\wait\tensor_A B$ is flat as a functor on $\mods{A}$.

:::

:::{.remark}
Recall that $\Grp\Sch \leq \Sch$ is the subcategory of group objects.
:::

:::{.definition title="?"}
For $G\in \Alg\Grp$ or $\Grp\Sch\slice X$, and let 
\[
F(\wait) = H^1(\wait; G) = \ts{G\dash\text{torsors over } X \text{ locally fppf trivial} }
.\]
:::

> The fppf site is much more flexible than the etale site!
> Examples are any morphisms $X\to \Hilb(\PP^n)$.

:::{.definition title="$G\dash$torsors"}
A **$G\dash$torsor over $X$** is an object $T\in \Sch\slice X$ with a $G\dash$action $G\fp{X} T \mapsvia{\sigma} T$ such that there is an isomorphism
\[
G\fp{X} T \isovia{(\sigma, \pi_2)} T\fp{X} T
.\]
:::

:::{.remark}
How to think about torsors: a family whose fibers are abstractly isomorphic to $G$, but not canonically.

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/RationalPoints/sections/figures}{2021-10-19_10-38.pdf_tex} };
\end{tikzpicture}

So the fibers are not canonically identified with $G$, but $G$ acts naturally in a freely transitive way on them.
This is essentially the same data as a principal $G\dash$bundle.
:::

:::{.remark}
Let $L\to X$ be a line bundle, so there exists a cover $\mcu \covers X$ with $\ro{L}{U_i} \cong \AA^1\times U_i$ and transition functions
\[
\AA^1 \times (U_i \cross U_j) \mapsvia{(t_{ij} , \id)} \AA^1 \cross (U_i \cross U_j)
\]
where $t_{ij}: U_i \times U_j \to \GG_m$.
Then one can obtain a $\GG_m\dash$torsor $L\smz{0}$ by removing the zero section, and in fact all such $\GG_m\dash$torsors arise this way.
One can check here that $\GG_m(X)\fp{X} T \iso T\fiberpower{X}{2}$.
:::

:::{.example title="of $\Grp\Sch\slice X$"}
\envlist

- Take $G\da \GG_a, \GG_m$ and consider $G \times X$, then one can define a multiplication by multiplying the first factors.
This generalizes to work for any $G\in \Alg\Grp$.

- An elliptic curve over $\spec \ZZ \sm P$ for $P$ a finite set of primes.

- $\ZZ/n$ and $\mu_n$.

:::

:::{.remark}
Another formulation: $\Grp\Sch\slice X$ are representable functors 
\[
F: \Sch\slice X\op &\to \Grp \\
Y &\mapsto \Sch\slice X(Y, G)
,\]
which composes with $\Grp \mapsvia{\Forget} \Set$ in a certain way (?).
:::

:::{.remark}
Next time: descent obstructions.
:::