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

Reference for this talk

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

There are finitely many.

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.

In terms of rational points, there is a trichotomy:

• $$g=0$$,
• $$g=1$$,
• $$g > 1$$.

This in fact breaks into a dichotomy:

• $$g\leq 1$$,
• $$g > 1$$.

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 $${\sharp}X({\mathbb{Q}}) \ll B^{\varepsilon}$$.

Let $$K\in \mathsf{Field}_{/ {{\mathbb{Q}}}} , S\in {\operatorname{Places}}(K)$$ finite, $$U_K \hookrightarrow{\mathbb{P}}^N$$ 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 $$U({\mathcal{O}}_K{ \left[ { \frac{1}{S} } \right] })$$ are sparse.

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 $${\mathcal{H}}\to U$$, then $$H^n({\mathcal{H}}_t; C_m)$$ 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.

Why is it useful to have lots of etale covers? Consider $$x-y=1$$ for $$x,y\in {\mathbb{Z}}{ \left[ { \frac{1}{S} } \right] }^{\times}$$ and $$S = \left\{{2,3}\right\}$$. The $$S{\hbox{-}}$$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 \begin{align*} m^2-n^2 &=1 \\ m^2-2n^2 &= 1 \\ 2m^2-n^2 &= 1 \\ \vdots & \qquad ,\end{align*} 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.

A partitioning trick for integral points: $$Y\to U$$ is a finite etale cover, then there are a finite number of twists $$\left\{{Y_1,\cdots, Y_m}\right\}$$ (all isomorphic over $${ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }$$) with $$Y_i\to U$$ such that every point in $$U({\mathbb{Z}}{ \left[ { \frac{1}{S} } \right] })$$is in the image of $$Y({\mathbb{Z}}{ \left[ { \frac{1}{S} } \right] })$$ for some $$i$$. So \begin{align*} {\textstyle\coprod}_i Y_i({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{S} } \right] }) \rightrightarrows U({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{S} } \right] }) .\end{align*}

Let $$X \subset {\mathbb{P}}^2$$ be a plane curve of degree $$d$$, then there is a uniform bound: \begin{align*} {\sharp}\left\{{ p\in X({\mathbb{Q}}) {~\mathrel{\Big\vert}~}\operatorname{ht}(p)\leq B}\right\} \leq C_{d, {\varepsilon}}B^{2d+{\varepsilon}} .\end{align*}

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

Useful to control numbers of points for curves you know nothing about. Walsh removes the $${\varepsilon}$$ 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!

All we can control in this situation is the degree of $$Y_i\to U$$ and $$U\subseteq {\mathbb{P}}^N$$ 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.

“Anabelian”: $$\pi_1$$ somehow tells the entire story.

Heath-Brown’s technique uses $$p{\hbox{-}}$$adic repulsion of points for $$X({\mathbb{Q}})\to X({ {\mathbb{Q}}_p })$$ where low-height points do not end up nearby. Recall that there is a SES \begin{align*} 1\to \pi_1^\text{ét}(X_{{ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }}) \to \pi_1^\text{ét}(X_{\mathbb{Q}}) \to G_{\mathbb{Q}}\to 1 ,\end{align*} and any point $$p\in X({\mathbb{Q}})$$ gives a section, thought of as $$X({\mathbb{Q}}) \to H^1(G_{\mathbb{Q}}; \pi_1^\text{ét}(X_{{ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }}))$$. Anabelian-ness: embed into some large interesting geometric space like this.

This cohomology group has a topology where $$p, q\in X({\mathbb{Q}})$$ 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 $$Y({\mathbb{Q}})$$.

How this goes for curves: $$C({\mathbb{Q}}) \to \operatorname{Jac}({\mathbb{Q}})$$, and one can tensor up to $$\operatorname{Jac}({\mathbb{Q}}) \otimes_{{\mathbb{Z}}} { {\mathbb{Z}}_{\widehat{p}} }$$. 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.

# 2 Talk 2: Wanlin Li, Ceresa cycle and hyperellipticity

A hyperbolic curve is determined by its $$\pi_1^\text{ét}$$.

Recall $$y^2=f(x)$$ defines a hyperelliptic curve $$C$$, which admits an involution $$(x,y)\to(x,-y)$$ and produces a degree 2 map \begin{align*} C &\to {\mathbb{P}}^1 \\ (x,y) &\mapsto x .\end{align*}

Let $${ \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }$$ be a separable closure of $$k$$. There is a fibration induced by taking a geometric point of $$\operatorname{Spec}k$$ and pulling back: As in topology, this induces a LES in homotopy, which here splits into SESs. In particular, \begin{align*} 1\to \pi_1(C_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } })\to \pi_1(C_k) \to { \mathsf{Gal}} ( { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } /k) .\end{align*} A point induces a section and thus a map $${ \mathsf{Gal}} ( { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } /k) \to \mathop{\mathrm{Aut}}\pi_1(C_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } })$$. Take the lower central series of $$\pi \mathrel{\vcenter{:}}=\pi_1(C_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } })$$, this induces \begin{align*} 1\to L^2\pi/L^3\pi \to \pi/L^3\pi \to \pi/L^1 \pi = \pi^{\operatorname{ab}}\to 1 \end{align*} where the first term is abelian.

See Davis-Pries-Wickelgren for applications to Fermat curves.

This extension corresponds to an element in $$\mu(C) \in H^1(G_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } }; \mathop{\mathrm{Hom}}(\pi^{\operatorname{ab}}, L^2\pi/L^3\pi ) )$$, and when $$C$$ is hyperelliptic $$\mu(C) = 0$$. Does the converse hold?

There exist non-hyperelliptic curves $$C$$ over $$k$$ such that $$\mu(C)$$ is torsion – in particular, the Fricke-Macbeath curve, which is genus 7 Hurwitz. Moreover if $$C_1\to C_2$$ then $$\mu(C_1)$$ torsion implies $$\mu(C_2)$$ torsion.

$$\mu(C)$$ is the $$\ell{\hbox{-}}$$adic cycles class associated to the Ceresa cycle.

See Hain-Matsumoto and Pulte

For $$C_{/ {k}}$$ and $$p\in C(K)$$, the Abel-Jacobi map yields \begin{align*} \operatorname{AJ}: C &\hookrightarrow\operatorname{Jac}(C) \\ q &\mapsto [q-p] .\end{align*} So define the Ceresa cycle as \begin{align*} \tilde c \mathrel{\vcenter{:}}=\operatorname{AJ}(C) - \mkern 1.5mu\overline{\mkern-1.5mu \operatorname{AJ}(C) \mkern-1.5mu}\mkern 1.5mu \mathrel{\vcenter{:}}=[q-p] - [p-q] .\end{align*} Note that $$\tilde c$$ is homologically trivial in Chow, but algebraically nontrivial for a very general $$C_{/ {{\mathbb{C}}}}$$ with $$g\geq 3$$.

There is an explicit non-hyperelliptic curve $$C$$ with $$\tilde c$$ torsion: \begin{align*} x^4 + xz^3 + y^3z = 0 \subseteq {\mathbb{P}}^2, \qquad p = {\left[ {0,0,1} \right]} .\end{align*}

Consider curves over the local field $$K = {\mathbb{C}}{\left[\left[ t \right]\right] }$$. Note $${ \mathsf{Gal}} (\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{C}}{\left[\left[ t \right]\right] }\mkern-1.5mu}\mkern 1.5mu/{\mathbb{C}}{\left[\left[ t \right]\right] }) = \widehat{{\mathbb{Z}}}$$, so this resembles a circle, and one can degenerate a family over the punctured disc. Apply nonabelian Picard-Lefschetz due to Asada-Matsumoto-Oda: if $$C$$ has semistable reduction then the monodromy of $$C/{\mathbb{C}}{\left[\left[ t \right]\right] }$$ is given by a multi-twist, i.e. a product of Dehn twists about simple closed curves. One can explicitly compute the Ceresa class in this situation. The degeneration data can be encoded as a tropical curve (essentially the dual graph of the special fiber).

$$\mu(C)$$ is always torsion for $$C$$ defined over $${\mathbb{C}}{\left[\left[ t \right]\right] }$$.

There is a notion of “hyperelliptic” for tropical curves: quotienting by the involution yields a tree.

# 3 Misc Notes

See Iwasawa group.

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

See the recent Lawrence-Venkatesh proof of Mordell. See Selmer section set and adelic sections.

Hyperbolic curves:

• $$g=0 \leadsto {\mathbb{P}}^1\setminus Z$$ where $${\sharp}Z \geq 3$$
• $$g=1 \leadsto$$ affine
• $$g\geq 2$$: anything.

For $$X_{/ {K}}$$ for $$K\in \mathsf{Field}_{/ {{\mathbb{Q}}}}$$ smooth hyperbolic with good reduction away from $$S$$, $${\sharp}X({\mathcal{O}}_{K, S}) < \infty$$ by Faltings. See Bloch-Kato and Fontaine-Mazur conjectures.

# 4 Sunday, May 01

I missed the first two talks 😦

# 5 Kiran Kedlaya: Crystalline companions as an anabelian phenomenon

Setup: $$k = {\mathbb{F}}_q, q=p^n, X\in {\mathsf{Sch}}_{/ {k}}$$ smooth geometrically connected, $$\ell\neq p$$ arbitrary. Recall \begin{align*} 1\to \pi_1 X_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } } \to \pi_1 X \to G_k = { \widehat{ \mathbb{Z} } }\to 1 .\end{align*} Anabelian philosophy: everything you’d want to know about $$X$$ is contained in $$\pi_1 X$$.

If $$X$$ is an affine curve, then $$\pi_1^{{\mathrm{tame}}} X$$ determines $$X$$.

Problem: if $$X$$ is an affine genus $$g$$ curve with $$m$$ punctures, $$\pi_1^{{\scriptscriptstyle \mathrm{prime-to-}p}} X_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } }$$ is the prime-to-$$p$$ completion of $$\mathsf{Free}(2g+m-1)$$, independently of $$X$$. Since sections induce $$G_k\to \mathop{\mathrm{Out}}(\pi_1 X_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } })$$, we have lots of tame continuous $$\mkern 1.5mu\overline{\mkern-1.5mu{ {\mathbb{Q}}_\ell }\mkern-1.5mu}\mkern 1.5mu$$ reps of $$\pi_1 X_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } }$$, but very few are fixed by Frobenius.

Such representations only have “geometric origins,” i.e. if $${\mathcal{E}}$$ is a lisse $${ {\mathbb{Q}}_\ell }{\hbox{-}}$$sheaf, i.e. a lisse $$F{\hbox{-}}$$sheaf with $$[F:{ {\mathbb{Q}}_\ell }] < \infty$$, which is irreducible with determinant of finite order, then it appears on relative etale cohomology of $$\pi:Y\to X$$ for some $$Y$$.

This is known in $$\dim X = 1$$, due to Deligne, around the same time Drinfeld proved Langlands for $$\operatorname{GL}_2(k(X))$$ for $$k(X)$$ a function field (or really the adeles). So all arithmetic reps of $$\pi_1$$ come from geometry.

Note that $$Y$$ will eventually not even be a scheme. The determinant condition rules out transcendental twists. Galois side: lisse sheaves on $$X$$; automorphic side: reps values in $$\operatorname{GL}_n({\mathbb{A}}_K)$$ for $$K$$ a field. The proof above involves exhibiting the Galois objects as coming from relative etale cohomology in moduli of shtukas. A priori one only knows Frobenius traces, but this turns out to be enough to uniquely characterize things in this situation.

Later Lafforgue did this for $$\operatorname{GL}_n$$, but the corresponding statement about arithmetic reps is wide open.

If $${\mathcal{E}}$$ as above, the Frobenius traces at all closed points $$x\in {\left\lvert {X} \right\rvert}$$ are algebraic over $${\mathbb{Q}}$$.

Fix an algebraic closure $${ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }$$ and two primes $$\ell, \ell'\neq p$$ and fix embeddings $${ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }\hookrightarrow{ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }_\ell$$ and $${ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }\hookrightarrow{ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }_{\ell'}$$. Let $${\mathcal{E}},{\mathcal{E}}'$$ be lisse $${ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }_\ell, { \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }_{\ell'}$$ sheaves (resp.) on $$X$$. Say $${\mathcal{E}},{\mathcal{E}}'$$ are companions iff for every $$x\in{\left\lvert {X} \right\rvert}$$ the Frobenius traces at $$x$$ are equal in $${ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }$$.

Note that $${\mathcal{E}}$$ determines $${\mathcal{E}}'$$ up to semisimplification, using $$L{\hbox{-}}$$function techniques. Moreover properties like being irreducible or having finite determinant hold simultaneously for them.

With this setup, a lisse $${ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }_{\ell}$$ sheaf $${\mathcal{E}}_\ell$$ admits a compantion $${\mathcal{E}}'$$ which is a $${ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }_{\ell'}$$ for chosen embeddings of $${\mathbb{Q}}$$ for which all traces are in $${ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }$$, i.e. irreducible and finite determinant. This is true in arbitrary dimension.

What about when $$\ell=p$$? There are somehow too many and too few lisse $${ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }_p$$ sheaves! E.g. the Lefschetz trace formula doesn’t hold. Instead use the Riemann-Hilbert correspondence – the $$p{\hbox{-}}$$adic analogues of lisse sheaves are certain $$p{\hbox{-}}$$adic integrable connections. How to construct: start with $$X$$ affine and glue. Choose a smooth affine formal scheme over $$W(k)$$ with special fiber $$P_k \cong X$$. Let $$K = \operatorname{ff}W(k)$$ and $$P_K$$ be the Raynaud generic fiber. See Tate model, Berkovich model, etc for rigid analytic geometry. Let $${\mathbb{A}}_K^n \leadsto \widehat{{\mathbb{A}}^n}_{W(k)}$$, a closed unit disc over $$K$$.

Some definitions:

• A convergent isocrystal is a vector bundle with an integrable connection on $$P_K$$.
• A convergent $$F{\hbox{-}}$$isocrystal is this and a compatible action of (a lift of) Frobenius.
• An overconvergent $$F{\hbox{-}}$$isocrystal is this on some structural enlargment of $$P_K$$ which takes the closed unit disc to the disc of radius $$1+{\varepsilon}$$.

These are similar to $$\pi_1(X_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } }){\hbox{-}}$$representations. Issue: $$p{\hbox{-}}$$adic antidifferentiation is hard; the integral of a formal power series converging on the closed disc only converges on the open disc.

If $$X$$ is smooth this recovers Berthelot’s rigid cohomology, which is a refinement of crystalline cohomology if $$X$$ is proper. This yields a 6 functor formalism, and has the same moving parts as etale cohomology.

The Langlands correspondence extends to these when $$\dim X = 1$$.

The content here: Lafforgue’s original proof can now be run with $$p{\hbox{-}}$$adic coefficients instead of just $$\ell{\hbox{-}}$$adic coefficients.

The companion theorem extends to both $$\ell=p$$ and $$\ell'=p$$.

Deligne posited the existence of a petit camarade crystalline, a little crystalline friend. ❇🙂

## 5.1 Applications

A partial result toward a conjecture of Simpson. Let $$X_{/ {{\mathbb{C}}}}$$ be a smooth cohomologically rigid local system (so no nontrivial deformations) which is irreducible with finite determinant. Are these of geometric origin? In particular, is there a $${ {\mathbb{Z}}\mathrm{VHS} }$$? Esnault-Groecherig show that the monodromy representation factors through $$\operatorname{GL}_n({\mathcal{O}}_K) \to \operatorname{GL}_n({\mathbb{C}})$$ for some $$K\in \mathsf{Field}_{/ {{\mathbb{Q}}}}$$.

Idea: start from complex geometry, go to $$p{\hbox{-}}$$adic geometry, yields an overconvergent $$F{\hbox{-}}$$crystal. This yields integrality at $$p$$; use companions to go back to a lisse $$\ell{\hbox{-}}$$adic sheaf, then back to $${\mathbb{C}}$$ to get integrality at $$\ell$$.

Going the other way, $$\ell\to p$$: one can prove “of geometric origin” results when $$\operatorname{rank}{\mathcal{E}}= 2$$ (Krishnmorthy-Pal). Idea: go from $$\ell\to p$$, make a candidate for the crystalline Dieudonne module for some family of AVs. One will have a bound on the motivic weight, which is at most $$\operatorname{rank}{\mathcal{E}}- 1$$.

A word on the proof: define a moduli stack $$M_n$$ of mod $$p^n$$ $$F{\hbox{-}}$$crystals, which is a horrendous algebraic stack. These are roughly coherent sheaves with extra data. Study some finite-type pieces using slops, and is universally closed since one can take flat limits along curves. Take the Zariski closure of companion points, then take stable images to get some $$M_n''$$.

Show that every point in each component of it is a companion point using horizontal companions (as opposed to vertical in the fiber direction). Then show each component maps isomorphically to $$S$$, which is a pointwise condition on $$S$$. This only uses the companion on the fiber, which is easier to study.

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

Given $$n > 0$$, there is an $$A\in {\mathsf{Ab}}{\mathsf{Var}}_{/ {{\mathbb{F}}_2}}$$ with $${\sharp}A({\mathbb{F}}_2) = n$$.

Recall the Weil bounds: given $$A_{/ {{\mathbb{F}}_q}}$$, \begin{align*} (q-2q^{1\over 2} + 1)^g \leq {\sharp}A({\mathbb{F}}_q) \leq (q+2q^{1\over 2} +1)^g .\end{align*}

Let $$\left\{{\alpha_i}\right\}_{1\leq i\leq 2g}$$ be the eigenvalues of $$\operatorname{Frob}\curvearrowright H_{\text{ét}}^1(A { \underset{\scriptscriptstyle {{\mathbb{F}}_q} }{\times} } \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}_q\mkern-1.5mu}\mkern 1.5mu; { {\mathbb{Z}}_\ell })$$. Recall the Weil conjectures:

• All embeddings $${\mathbb{Q}}( \alpha_i) \hookrightarrow{\mathbb{C}}$$ satisfy $${\left\lvert { \alpha_i} \right\rvert} = q^{1\over 2}$$.
• Lefschetz trace: $${\sharp}A({\mathbb{F}}_q) = \prod_{1\leq i\leq 2g}( \alpha_i - 1) = \prod(q+1 - \alpha_i - \mkern 1.5mu\overline{\mkern-1.5mu \alpha_i\mkern-1.5mu}\mkern 1.5mu )^{1\over 2}$$ Note that these real numbers sit in $$[-2q^{1\over 2}, 2q^{1\over 2}] \subseteq {\mathbb{R}}$$.

Given $$\Sigma \subseteq {\mathbb{C}}$$ we say $$\alpha$$ is totally $$\Sigma$$ iff all conjugates of $$\alpha$$ are contained in $$\Sigma$$.

Remarkably for AVs, the $$\alpha_i$$ tell the entire story! Honda-Tate theory gives a correspondence \begin{align*} \left\{{\substack{ \text{Abelian varieties over ${\mathbb{F}}_q$} }}\right\}/\sim_{\scriptscriptstyle \text{isogeny}} &\rightleftharpoons \left\{{\substack{ \text{Totally $I_q$ algebraic integers} }}\right\}/\sim_{\scriptscriptstyle\text{conjgacy}} \end{align*} where $$I = [-2\sqrt q, 2\sqrt q]$$. Tate showed injectivity, Honda used CM theory to show surjectivity.

Given any $$q$$ and any $$n \gg_q 0$$, there is an $$A\in{\mathsf{Ab}}{\mathsf{Var}}_{/ {{\mathbb{F}}_q}}$$ with $${\sharp}A({\mathbb{F}}_q) = n$$.

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

Given $$g \gg_q 0$$ and \begin{align*} n\in [(q-2\sqrt q + 3)^g, (q+2\sqrt q - 1 - q^{-1})^g] ,\end{align*} there exists $$A$$ with $${\sharp}A({\mathbb{F}}_q) = n$$.

Given $$\alpha\in {\mathbb{R}}$$ and $${\varepsilon}$$, one can produce totally $$I$$ algebraic integers for $$I = [ \alpha, 4+ \alpha+{\varepsilon}]$$ fitting the arcsin distribution: For $${\varepsilon}= \alpha = 0$$, 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 $${\varepsilon}=0$$ is a big open problem.

## 6.1 Schur-Segal-Smyth trace problem

Find the minimal $$t$$ such that for any $${\varepsilon}> 0$$, infinitely many totally positive algebraic integers satisfy \begin{align*} \operatorname{tr}(\alpha)/\deg(\alpha) < t + {\varepsilon} .\end{align*}

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.

$$t\geq 1$$.

Take a totally positive algebraic integer with conjugates $$\left\{{\alpha_i}\right\}_{i\leq n}$$ with minimal polynomial $$p$$. Now apply AMGM: \begin{align*} 1\leq {\left\lvert {p(0)} \right\rvert} = \qty{\prod a_i}^{n\over n}\leq_{{\mathrm{AMGM}}}\qty{\sum a_i \over n}^n .\end{align*}

We can do slightly better. If $${\left\lvert {p(1)} \right\rvert}\geq 1$$, then $$t\geq 1.05$$. If $${\left\lvert {p(a)p(\mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu)} \right\rvert} \geq 1$$ for $$a\mathrel{\vcenter{:}}={3+\sqrt 5\over 2}$$, then $$t\geq 1.1$$. More generally this is written as a resultant, i.e. $${\mathrm{res}}(p, x^3+3x+1)$$.

Smyth shows that $$t = \operatorname{tr}(\alpha)/\deg( \alpha)\geq 1.771$$ 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.

Smyth’s method limits to the right answer, and thus $$t\leq 1.81$$. In particular, $$t\neq 2$$.

Consequence: there are things that work better than the arcsin distribution.

Given sufficiently large square $$q$$, there are infinitely many $$A_{/ {{\mathbb{F}}_q}}$$ with \begin{align*} {\sharp}A({\mathbb{F}}_q) \geq (q + 2\sqrt q - 0.81)^{\dim A} ,\end{align*} but only finitely many with \begin{align*} {\sharp}A({\mathbb{F}}_q) \geq (q+2\sqrt q - 0.8)^{\dim A} .\end{align*}

Given algebraic integers $$\alpha$$ with conjugates $$\left\{{\alpha_i}\right\}_{i\leq n}$$, let \begin{align*} \mu_{\alpha}= {1\over n}\sum \delta_{ \alpha_i} .\end{align*}

There is a weak-$$*$$ topology on the space of such measures.

Choose $$\Sigma \subseteq {\mathbb{R}}$$ 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 $$Q\neq 0$$, \begin{align*} \int_{\Sigma} \log{\left\lvert {Q} \right\rvert} \,d\mu\geq 0 .\end{align*}

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: 