Resources:

- <https://math.mit.edu/classes/18.783/2017/Lecture1.pdf>

Topics to discuss:

- [ ] For $X = S^1$, finding $X(\QQ)$
	- [ ] Finding one point in $X(\QQ)$ and using it to "generate" all of $X(\QQ)$.
	- [ ] $\AA^1\slice \QQ$ as a moduli space for $X(\QQ)$.
- [ ] Defining projective space
	- [ ] In coordinates: e.g. $\PP^2\slice \CC$ is $\tv{a:b:c} = \ts{\tv{a,b,c} \in \CC^3}/ \sim$ where $\tv{a,b,c}\sim \tv{\lambda a, \lambda b, \lambda c}$ for $\lambda \in U(\CC)$, so the space of lines through $\vector 0$.
	- [ ] Topological description: $\RP^n = \RR^{n+1}/C_2$ and $\CP^n = \CC^{n+1}/C_2$ for $C_2 \actson k^n$ the antipodal action.
		- [ ] E.g. $\RP^1 \PP^1(\RR) = \PP(\RR^1) = \RR^2/\GG_m \cong S^1$ is the space of lines in $\RR^2$ and $\CP^1 = \PP^1(\CC) = \PP(\CC^1) = \CC^2/\GG_m \cong S^2$.
			Here $\GG_m(\RR) = \RR\units = U(\RR) = \RR\smz$ and similarly $\GG_m(\CC) = \CC\units = \CC\smz$.
		- [ ] $\AA^1\embeds \PP^1$ by $\tv{x} \mapsto \tv{x: 1}$ or $\tv{x}\mapsto \tv{1: x}$, so $\PP^1 = \AA^1\glue{x\mapsto 1/x} \AA^1$.
			Here we think of $\tv{1: 0}$ as the "point (hyperplane) at infinity" and $\tv{0: 1}$ as zero.
	- [ ] For $\PP^2$: to parameterize all lines, fix $\vector 0\in \AA^3$, cast a ray and intersect with the plane $\ts{z=1} \cong \AA^2$; this is the embedding $\tv{a, b}\mapsto \tv{a,b,1}$.
		You get every line this way except for an $\PP^1$ worth in the $z=0$ plane, so $\PP^2 \cong \AA^2\disjoint \PP^1$.
		Similarly $\PP^1 \cong \AA^1 \disjoint \PP^0 = \AA^1\disjoint \pt$ by projecting onto the $y=1$ line. Leads to a **CW decomposition**.
- [ ] Examples:
	- [ ] Count points in coordinates in $\AA^1(K)$ for $K = \FF_q$ to get $\AA^1(\FF_q) = q$.
	- [ ] Count points in coordinates to get $\PP^1(K) = q+1$. Need decomposision $\PP^1 = \AA^1 \disjoint \tv{1: 0}$.
- [ ] Polynomial functions on projective space: the homogenization procedure
	- [ ] Homogeneous of degree $d$ iff $F(\lambda x, \lambda y, \lambda z) = \lambda^2 F(x, y, z)$.

# Rational points on circles

![](attachments/Pasted%20image%2020220214114725.png)

![](attachments/Pasted%20image%2020220214114742.png)

![](attachments/Pasted%20image%2020220214114800.png)

# Projective Space

![](attachments/Pasted%20image%2020220214120039.png)
![](attachments/Pasted%20image%2020220214120104.png)

![](attachments/Pasted%20image%2020220214120126.png)
![](attachments/Pasted%20image%2020220214120206.png)


# Elliptic Curves

![](attachments/Pasted%20image%2020220214120245.png)
![](attachments/Pasted%20image%2020220214120305.png)

![](attachments/Pasted%20image%2020220214120322.png)
![](attachments/Pasted%20image%2020220214120452.png)

# Mordell-Weil

![](attachments/Pasted%20image%2020220214120549.png)

![](attachments/Pasted%20image%2020220214120625.png)


# Mazur's theorem

![](attachments/Pasted%20image%2020220214120722.png)
![](attachments/Pasted%20image%2020220214120729.png)


# Examples

![](attachments/Pasted%20image%2020220214120833.png)
![](attachments/Pasted%20image%2020220214120853.png)
![](attachments/Pasted%20image%2020220214120858.png)