# Alex Smith:  Simple abelian varieties over finite fields with extreme point counts

:::{.theorem title="Howe-Kedlaya"}
Given $n > 0$, there is an $A\in \Ab\Var\slice{\FF_2}$ with $\size A(\FF_2) = n$.
:::

:::{.remark}
Recall the Weil bounds: given $A\slice{\FF_q}$,
\[
(q-2q^{1\over 2} + 1)^g \leq \size A(\FF_q) \leq (q+2q^{1\over 2} +1)^g
.\]

Let $\ts{\alpha_i}_{1\leq i\leq 2g}$ be the eigenvalues of $\Frob\actson H_{\et}^1(A \fiberprod{\FF_q} \bar{\FF_q}; \ZZladic )$.
Recall the Weil conjectures:

- All embeddings $\QQ( \alpha_i) \injects \CC$ satisfy $\abs{ \alpha_i} = q^{1\over 2}$.
- Lefschetz trace: $\size A(\FF_q) = \prod_{1\leq i\leq 2g}( \alpha_i - 1) = \prod(q+1 - \alpha_i - \bar{ \alpha_i} )^{1\over 2}$
  Note that these real numbers sit in $[-2q^{1\over 2}, 2q^{1\over 2}] \subseteq \RR$.

:::

:::{.definition title="Totally $\Sigma$"}
Given $\Sigma \subseteq \CC$ we say $\alpha$ is **totally $\Sigma$** iff all conjugates of $\alpha$ are contained in $\Sigma$.
:::

:::{.remark}
Remarkably for AVs, the $\alpha_i$ tell the entire story!
Honda-Tate theory gives a correspondence
\[
\correspond{
  \text{Abelian varieties over $\FF_q$}
}/\sim_{\scriptscriptstyle \text{isogeny}}
&\mapstofrom
\correspond{
  \text{Totally $I_q$ algebraic integers}
}/\sim_{\scriptscriptstyle\text{conjgacy}}
\]
where $I = [-2\sqrt q, 2\sqrt q]$.
Tate showed injectivity, Honda used CM theory to show surjectivity.
:::

:::{.theorem title="von Bomel-Costa-Li-Poonen-S."}
Given any $q$ and any $n \gg_q 0$, there is an $A\in\Ab\Var\slice{\FF_q}$ with $\size A(\FF_q) = n$.
:::

:::{.remark}
Idea: Howe-Kedlaya works infinitely often.
One can't attain *every* integer in the Weil bound interval, but you can get pretty close:
:::

:::{.theorem title="?"}
Given $g \gg_q 0$ and
\[
n\in [(q-2\sqrt q + 3)^g, (q+2\sqrt q - 1 - q\inv)^g]
,\]
there exists $A$ with $\size A(\FF_q) = n$.
:::

:::{.proof title="Sketch"}
Given $\alpha\in \RR$ and $\eps$, one can produce totally $I$ algebraic integers for $I = [ \alpha, 4+ \alpha+\eps]$ fitting the arcsin distribution:

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![](figures/2022-05-01_15-53-12.png)

For $\eps = \alpha = 0$, choose a large cyclotomic polynomial, whose roots are roughly equidistributed in $S^1$, then map to $[-2, 2]$.
Solving this for non-rational $\alpha$ and $\eps=0$ is a big open problem.
:::

## Schur-Segal-Smyth trace problem

:::{.problem title="?"}
Find the minimal $t$ such that for any $\eps > 0$, infinitely many totally positive algebraic integers satisfy
\[
\trace(\alpha)/\deg(\alpha) < t + \eps
.\]

:::

:::{.remark}
Idea: all conjugates greater than zero, how can you minimize the average trace?
The cyclotomic method above infinitely many whose trace is at most 2.
So using the arcsin distribution yields $t< 2$, open question: is $t=2$?
Progress has been slow and revolves around an old trick.
:::

:::{.proposition title="?"}
$t\geq 1$.
:::

:::{.proof title="?"}
Take a totally positive algebraic integer with conjugates \( \ts{\alpha_i}_{i\leq n} \) with minimal polynomial $p$.
Now apply AMGM:
\[
1\leq \abs{p(0)} = \qty{\prod a_i}^{n\over n}\leq_{\AMGM}\qty{\sum a_i \over n}^n
.\]
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:::{.remark}
We can do slightly better.
If $\abs{p(1)}\geq 1$, then $t\geq 1.05$.
If $\abs{p(a)p(\bar a)} \geq 1$ for $a\da {3+\sqrt 5\over 2}$, then $t\geq 1.1$.
More generally this is written as a resultant, i.e. $\resultant(p, x^3+3x+1)$.

Smyth shows that $t = \trace(\alpha)/\deg( \alpha)\geq 1.771$ with 14 exceptions; Wang-Wu-Wu shows $t\geq 1.793$.
Serre showed this argument can never show $t\geq 1.899$, Alex showed it can not show $t\geq 1.81$, so we're approaching the limit of Smyth's method.
:::

:::{.theorem title="Alex"}
Smyth's method limits to the right answer, and thus $t\leq 1.81$.
In particular, $t\neq 2$.
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:::{.remark}
Consequence: there are things that work better than the arcsin distribution.
:::

:::{.theorem title="?"}
Given sufficiently large square $q$, there are infinitely many $A\slice{\FF_q}$ with
\[
\size A(\FF_q) \geq (q + 2\sqrt q - 0.81)^{\dim A}
,\]
but only finitely many with
\[
\size A(\FF_q) \geq (q+2\sqrt q - 0.8)^{\dim A}
.\]
:::

:::{.definition title="?"}
Given algebraic integers $\alpha$ with conjugates $\ts{\alpha_i}_{i\leq n}$, let
\[
\mu_{\alpha}= {1\over n}\sum \delta_{ \alpha_i}
.\]
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:::{.remark}
There is a weak-$*$ topology on the space of such measures.
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:::{.theorem title="?"}
Choose $\Sigma \subseteq \RR$ with countably many components (e.g. excluding Cantor sets) of *capacity* $c > 1$.
TFAE are equivalent for a probability measure $\mu$ on $\Sigma$:

- There are totally $\Sigma$ algebraic integers $\alpha_i$ whose distributions $\mu_{\alpha_i}$ as above conver to $\mu$.
- For any integer polynomial $Q\neq 0$,
\[
\int_{\Sigma} \log\abs{Q} \dmu \geq 0
.\]

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:::{.remark}
Idea of proof: apply Minkowski's 2nd theorem as a source of promising polynomials.
Use an optimized distribution that avoids the 14 exceptions, whose average traces beat the previous averages:

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![](figures/2022-05-01_16-31-56.png)

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