# Talk 1: Jordan Ellenberg, Sparsity of rational points on moduli spaces

[ADDING Conference](https://www.daniellitt.com/adding)

> [Reference for this talk](https://arxiv.org/abs/2109.01043)

:::{.question}
How many homogeneous forms in $\ZZ[x_0,\cdots, x_n]$ are there with discriminant equal to 1?
More accurately, there is a $\PGL\dash$action, so how many $\PGL$ orbits are there?
More generally: how many are there with discriminant divisible only by primes in some finite set $S$?
:::

:::{.conjecture title="Shafarevich conjecture-esque"}
There are finitely many.
:::

:::{.remark}
The general philosophy is that there should be only finitely many classes of $X$ with good reduction outside of $S$ -- this is known e.g. for abelian varieties of dimension $g$, see Faltings' proof of Mordell.
:::

:::{.remark}
In terms of rational points, there is a trichotomy:

- $g=0$,
- $g=1$,
- $g > 1$.

This in fact breaks into a dichotomy:

- $g\leq 1$,
- $g > 1$.

:::

:::{.definition title="Sparse points"}
There are bounds on rational point counts on $X$ with height at most $B$:

- $g=0 \leadsto \ll_X B^2$
- $g=1\leadsto \ll_X \log B$ (Mordell)
- $g\geq 2\leadsto \ll_X C_X$ where $C$ is a constant depending on $X$ (Faltings)

Here we'll dichotomize this in a different way: $g=0$ and $g > 0$.
We say $X$ has **sparse points** if $\size X(\QQ) \ll B^\eps$.
:::


:::{.theorem title="E, Lawrence, Venkatesh 2021"}
Let $K\in \NF, S\in \Places(K)$ finite, $U_K \injects \PP^N$ a quasiprojective variety with a geometric variation of Hodge structure with finite-to-one period map, making $U_X$ an interesting moduli space of something.
Then the periods of $U(\OO_K\invert{S})$ are sparse.
:::

:::{.remark}
The anabelian part: $U$ has large $\pi_1$, so lots of etale covers.
E.g. if $U$ is a moduli of hypersurfaces, take the universal curve $\mch\to U$, then $H^n(\mch_t; C_m)$ varies and this can be interpreted as a moduli space with level structure.
We try to show that large $\pi_1$ implies sparseness. 
This isn't quite true, since e.g. blowing up introduces lots of rational points, so a stronger condition is needed: $\pi_1(U)$ is infinite and for every finite-dimensional $V \subseteq U$, $\pi_1(V)\to \pi_1(U)$ has infinite image.
:::

:::{.remark}
Why is it useful to have lots of etale covers?
Consider $x-y=1$ for $x,y\in \ZZ\invert{S}\units$ and $S = \ts{2,3}$.
The $S\dash$units theorem of Siegel guarantees there are only finitely many solutions.
One way to approach this: if $x=2^a3^b$, we know $x$ up to squares: $x=s,2s,3s,6s$ for $s$ a square, as is $y$.
This yields a system 
\[
m^2-n^2 &=1 \\
m^2-2n^2 &= 1 \\
2m^2-n^2 &= 1 \\
\vdots & \qquad 
,\]
which e.g. some can be solved using techniques for Pell equations.
Having higher degree rigidifies the situation, so perhaps there are more techniques to solve them.
Strategy: trade one hard equation for a finite list of higher degree easier equations.
:::

:::{.fact}
A partitioning trick for integral points: $Y\to U$ is a finite etale cover, then there are a finite number of twists $\ts{Y_1,\cdots, Y_m}$ (all isomorphic over $\QQbar$) with $Y_i\to U$ such that every point in $U(\ZZ\invert{S})$is in the image of $Y(\ZZ\invert{S})$ for some $i$.
So 
\[
\disjoint_i Y_i(\ZZ\invert{S}) \covers U(\ZZ\invert{S})
.\]
:::

:::{.theorem title="Heath-Brown 2004 (determinantal methods)"}
Let $X \subset \PP^2$ be a plane curve of degree $d$, then there is a uniform bound:
\[
\size \ts{ p\in X(\QQ) \st \height(p)\leq B} \leq C_{d, \eps}B^{2d+\eps}
.\]

> Note: missed the exponent on $B$, need to fix.

:::

:::{.remark}
Useful to control numbers of points for curves you know nothing about.
Walsh removes the $\eps$ in all terms, Selberg, Brolog generalized to higher dimensions, CCDN make the constant effective.
Pitch: these theorems are useful for other theorems which are not ostensibly about uniformity!
:::

:::{.remark}
All we can control in this situation is the degree of $Y_i\to U$ and $U\subseteq \PP^N$ has a degree, so we can control the degree of the $Y_i$.
Broberg gives a bound $\ll B^{n+1\over d^{1\over n}}$ where $n=\dim U$. 
It doesn't actually matter what this is, just that it decreases in $d$, and we can take higher degree covers.
:::

:::{.remark}
"Anabelian": $\pi_1$ somehow tells the entire story.
:::

:::{.remark}
Heath-Brown's technique uses $p\dash$adic repulsion of points for $X(\QQ)\to X(\QQpadic)$ where low-height points do not end up nearby.
Recall that there is a SES
\[
1\to \pi_1^\et(X_{\QQbar}) \to \pi_1^\et(X_\QQ) \to G_\QQ\to 1
,\]
and any point $p\in X(\QQ)$ gives a section, thought of as $X(\QQ) \to H^1(G_\QQ; \pi_1^\et(X_{\QQbar}))$.
Anabelian-ness: embed into some large interesting geometric space like this.

This cohomology group has a topology where $p, q\in X(\QQ)$ are nearby iff there exists a higher degree etale cover $Y\to X$ (small subsets correspond to large index subgroups in the profinite topology) such that $p, q$ both lift to $Y(\QQ)$.
:::

:::{.remark}
How this goes for curves: $C(\QQ) \to \Jac(\QQ)$, and one can tensor up to $\Jac(\QQ) \tensor_{\ZZ} \ZZpadic$.
Modern take: points are close if they differ by a power of $p$ in the Mordell-Weil group.
Interpretation of the main theorem: Heath-Brown in more general profinite topologies.
:::