*Note:
These are some sketchy personal notes from the ADDING 2022 conference at UGA.
*

dzackgarza@gmail.com

Last updated: 2022-05-01

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 \(U({\mathcal{O}}_K{ \left[ { \scriptstyle \frac{1}{S} } \right] })\) 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 \(x,y\in {\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{S} } \right] }^{\times}\) 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 \(U({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{S} } \right] })\)is in the image of \(Y({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{S} } \right] })\) 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.

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.

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.

I missed the first two talks 😦

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. ❇🙂

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.

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.

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: