Effective Methods for Diophantine Finiteness

We begin with a preliminary observation. To show that $\dim\overline{S(\mathcal{O}_{K,N})}^{\textrm{Zar}}\leq d$, it suffices to show, for any irreducible subscheme $T\subset S$ of dimension $>d$ and defined over $\mathcal{O}_{K,N}$, that the Zariski closure $\overline{T(\mathcal{O}_{K,N})}^{\textrm{Zar}}$ is a proper algebraic subscheme of $T$. We therefore fix such a subscheme $T\subset S$ with $\dim T>d$, and seek to show that $\dim\overline{T(\mathcal{O}_{K,N})}^{\textrm{Zar}}<\dim T$.

Fix a prime $p\in\mathbb{Z}$ not dividing $N$, and let $t\in T(\mathcal{O}_{K,N})$ be a point. It is conjectured that for any $i$ the representation of $\textrm{Gal}(\overline{K}/K)$ on $H^{i}_{\textrm{\'{e}t}}(X_{t,\overline{K}},\mathbb{Q}_{p})$ is semisimple. If this result were to be known for all such $t$, an argument of Faltings shows that for $t\in T(\mathcal{O}_{K,N})$ there are at most finitely many possibilities for the isomorphism class of the representation of $\textrm{Gal}(\overline{K}/K)$ on $H^{i}_{\textrm{\'{e}t}}(X_{t,\overline{K}},\mathbb{Q}_{p})$. To establish the non Zariski-density of $T(\mathcal{O}_{K,N})$ it would then suffice to show that the fibres of the map

are not Zariski dense. As explained by Lawrence and Venkatesh in [LV20], this is essentially the original argument for the Mordell conjecture due to Faltings.

The problem with applying this strategy in general is twofold. First, the semisimplicity of $H^{i}_{\textrm{\'{e}t}}(X_{t,\overline{K}},\mathbb{Q}_{p})$ is not known, and for most choices of $f$ appears out of reach. Secondly, without a good geometric interpretation of the étale cohomology $H^{i}_{\textrm{\'{e}t}}(X_{t,\overline{K}},\mathbb{Q}_{p})$ it is difficult to understand $\tau$. The key insight in the paper of Lawrence and Venkatesh is that one may potentially overcome both problems by passing to a $p$-adic setting where they are more managable.

Instead of considering the global Galois representation $\rho_{t}:\textrm{Gal}(\overline{K}/K)\to H^{i}_{\textrm{\'{e}t}}(X_{t,\overline{K}},\mathbb{Q}_{p})$, we consider its semisimplification $\rho^{\textrm{ss}}_{t}$, and restrict $\rho_{t}$ along the map $\textrm{Gal}(\overline{K_{v}}/K_{v})\to\textrm{Gal}(\overline{K}/K)$ induced by a fixed embedding $\overline{K}\hookrightarrow\overline{K_{v}}$ to obtain $\rho_{t,v}$, where $v$ is a fixed place above $p$. The functors of $p$-adic Hodge theory tell us that the representation $\rho_{t,v}$ determines a triple $(H^{i}_{\textrm{dR}}(X_{t,K_{v}}),\phi_{t},F^{\bullet}_{t})$, where $F^{\bullet}_{t}$ is the Hodge filtration on de Rham cohomology and $\phi_{t}$ is the crystalline Frobenius at $t$. If we somehow manage to consider the data $(H^{i}_{\textrm{dR}}(X_{t,K_{v}}),\phi_{t},F^{\bullet}_{t})$ up to “semisimplification” (in the sense that we identify such triples when the associated global Galois representations have isomorphic semisimplifications), our problem is then to study the fibres of the map

and show they lie in a Zariski closed subscheme of smaller dimension.

Next, we make the elementary observation that to bound the dimension of the Zariski closure of $T(\mathcal{O}_{K,N})$, it suffices to cover $T(\mathcal{O}_{K,v})$ by finitely many $v$-adic disks $D_{1},\ldots,D_{k}$ and bound the dimension of the Zariski closure of $D_{i}\cap T(\mathcal{O}_{K,N})$ for each $i$; here $\mathcal{O}_{K,v}$ is the ring of $v$-adic integers. It can then be shown that there exists such a cover for which the Hodge bundle $\mathcal{H}=R^{i}f_{*}\Omega^{\bullet}_{X/S}$ can be trivialized rigid-analytically over each $D_{i}$, moreover with respect to each such trivialilzation the Frobenius operator $\phi_{t}$ is independent of $t\in D_{i}$. The problem then reduces to studying a varying filtration $F^{\bullet}_{t}$ on a fixed vector space $V_{p}=H^{i}_{\textrm{dR}}(X_{t_{0}})$ for some $t_{0}\in D_{i}$. In particular, we obtain a rigid-analytic map

and those points of $\widecheck{{L}}_{p}$ arising from points $t\in T(\mathcal{O}_{K,N})\cap D_{i}$ lie inside finitely many subvarieties $O_{i1},\ldots,O_{i\ell}$ of $\widecheck{{L}}_{p}$ corresponding to the finitely many possible isomorphism classes of $\rho^{\textrm{ss}}_{t}$.

We are now faced with the problem of understanding the intersections $\psi_{p}(D_{i})\cap O_{ij}$, and showing that their inverse images under $\psi_{p}^{-1}$ lie in an algebraic subscheme of smaller dimension. One part of this problem is to understand the dimensions of the varieties $O_{ij}$, and to show that they are sufficiently small: this step is carried out successfully for the families of hypersurfaces studied both by Lawrence-Venkatesh and Lawrence-Sawin, and appears to be managable in general. The more difficult object to control is $\psi_{p}(D_{i})$, for which we need to understand the variation of the filtration $F^{\bullet}$ over $D_{i}$. This, in turn, is governed by the Gauss-Manin connection $\nabla:\mathcal{H}\otimes\Omega^{1}_{T}\to\mathcal{H}$, which exists universally over $S$ after possibly increasing $N$; we may adjust $p$ so that it does not divide $N$ if necessary. The fact that $\nabla$ exists universally over $\mathcal{O}_{K,N}$ means that the same system of differential equations satisfied by $\psi_{p}$ at $t\in T(\mathcal{O}_{K,N})\cap D_{i}$ is also satisfied by a Hodge-theoretic period map $\psi:B\to\widecheck{{L}}$ on a sufficiently small analytic neighbourhood $B\subset T(\mathbb{C})$ containing $t$, where $\widecheck{{L}}$ is a variety of Hodge flags. In particular, one can prove that the Zariski closures of $\psi_{p}(D_{i})$ and $\psi(B)$ have the same dimension.

The final step, which is to show that the Zariski closure of $T(\mathcal{O}_{K,N})\cap D_{i}$ in $T$ has smaller dimension, is completed as follows. The Ax-Schanuel Theorem [BT17] for variations of Hodge structures due to Bakker-Tsimerman shows that if $V$ is an algebraic subvariety of $\widecheck{{L}}$ of dimension at most $\dim\overline{\psi(B)}^{\textrm{Zar}}-\dim\psi(B)$,²²2The dimension $\dim\psi(B)$ can once again, at least for open neighbourhoods $B$ with a sufficiently mild geometry, be made sense of the dimension of a locally constructible analytic set. Alternatively one can replace $\dim\psi(B)$ with $\dim\varphi(Z)$, where $\varphi$ is as before and $Z\subset T_{\mathbb{C}}$ is a component containing $B$. then the inverse image under $\psi$ of the intersection $\psi(B)\cap V$ lies in an algebraic subvariety of $T_{\mathbb{C}}$ of smaller dimension. Choosing an isomorphism $\mathbb{C}\cong\overline{K_{v}}$ one can transfer this fact to the same statement for the map $\psi_{p}$ and in particular for $V=O_{ij}$. Our problem is finally reduced to giving a lower bound for the difference $\dim\overline{\psi(B)}^{\textrm{Zar}}-\dim\psi(B)$. Our main result now reads:

We note that by [Urb21b, Lem. 4.10(ii)] it is also a consequence of the Bakker-Tsimerman Theorem that when $Z$ is geometrically irreducible, we have $\overline{\psi(B)}^{\textrm{Zar}}=\mathbf{H}_{Z}\cdot\psi(t)$ for any point $t\in Z(\mathbb{C})$, which recovers the statement of 1.2.

We may observe that the computation of the bound described in 1.3 is a purely Hodge-theoretic problem, i.e., it concerns only properties of the integral variation $\mathbb{V}=R^{i}f_{*}\mathbb{Z}$ of Hodge structures on $S_{\mathbb{C}}$. Let $\mathcal{Q}:\mathbb{V}\otimes\mathbb{V}\to\mathbb{Z}$ be a polarization of $\mathbb{V}$, and let $(V,Q)$ be a fixed polarized lattice isomorphic to one (hence any) fibre of $(\mathbb{V},\mathcal{Q})$; as it causes no harm, we will assume that $V=\mathbb{Z}^{m}$ for some $m$, and therefore sometimes write $\textrm{GL}_{m}$ for $\textrm{GL}(V)$. Let $D$ be the complex manifold parametrizing polarized Hodge structures on $(V,Q)$ with the same Hodge numbers as $(\mathbb{V},\mathcal{Q})$. A point $h\in D$ we may view as a morphism $h:\mathbb{S}\to\textrm{GL}(V)_{\mathbb{R}}$, where $\mathbb{S}$ is the Deligne torus, and the Mumford-Tate group $\textrm{MT}(h)$ is the $\mathbb{Q}$-Zariski closure of $h(\mathbb{S})$.

To present our method, we introduce some terminology:

The first step in our algorithm is:

Step One: Compute a finite list of types $\mathcal{C}_{1},\ldots,\mathcal{C}_{\ell}$ such that every Hodge-theoretic type appears somewhere in the list.

The problem given in Step One is solved in [Urb21b, Prop. 5.4]; we will say little about it here. It is related to the problem of classifying Mumford-Tate groups up to conjugacy by $\textrm{GL}_{m}(\mathbb{C})$, for which one can use a constructive version of the proof in [Voi12, Thm. 4.14]. It is also similar to the problem of classifying Mumford-Tate domains as studied in [GGK12, Chap. VII]. We note that the methods of [GGK12, Chap. VII], when they can be carried out effectively, result in an approach for which $\mathcal{C}_{1},\ldots,\mathcal{C}_{\ell}$ will be exactly the set of Hodge-theoretic types.

The second step is more involved, and is the crux of our method. To describe it we need to introduce some terminology.

The construction of a local period map $\psi:B\to\widecheck{{L}}$ involves picking a basis $b^{1},\ldots,b^{m}$ of $\mathbb{V}_{\mathbb{C}}(B)$, and hence choosing an isomorphism $\mathbb{V}_{\mathbb{C}}(B)\simeq\mathbb{C}^{m}$. When working with a local period map, we will always assume that such a basis has been choosen, and hence identify subgroups of $\textrm{GL}(\mathbb{V}_{\mathbb{C}}(B))$ with subgroups of $\textrm{GL}_{m}(\mathbb{C})$. In particular, if $Z\subset S_{\mathbb{C}}$ is a geometrically irreducible subvariety which intersects $B$, we have an induced action of $\mathbf{H}_{Z}$ on $\widecheck{{L}}$.

Lastly, we need:

Step Two: For each type $\mathcal{C}_{i}$ appearing in the list, compute a differential system $\mathcal{T}(\mathcal{C}_{i})$ on $S$ characterized by the property that an algebraic subvariety $Z\subset S_{\mathbb{C}}$ is an integral subvariety for $\mathcal{T}(\mathcal{C}_{i})$ if and only if $\mathcal{C}(Z)\leq\mathcal{C}_{i}$, and determine the dimension of a maximal integral subvariety for this system.

The author thanks Brian Lawrence, Akshay Venkatesh, and Will Sawin for comments on a draft of this manuscript.

In this section we give some references for carrying out Step One as described in the introduction.

Let us comment briefly on a different approach to Step One given in [GGK12, Chap. VII]. In [GGK12, Chap. VII], the authors describe a method for classifying both Mumford-Tate groups and Mumford-Tate domains (orbits of points $h\in D$ under $\textrm{MT}(h)(\mathbb{R})$ and $\textrm{MT}(h)(\mathbb{C})$). Given an appropriate such classification, one can easily solve Step One by computing the decompositions of the groups $\textrm{MT}(h)$ that arise into $\mathbb{Q}$-simple factors. The method of [GGK12, Chap. VII] is to first classify CM Hodge structures $h_{\textrm{CM}}\in D$, and then give a criterion for deciding when a Lie subalgebra of $\mathfrak{gl}(V)$ corresponds to a Mumford-Tate group generating a Mumford-Tate domain containing $h_{\textrm{CM}}$. They carry out this classification procedure successfully when $\dim V=4$, and so for variations with Hodge numbers $(2,2)$ and $(1,1,1,1)$.

The method given for classifying CM Hodge structures given in [GGK12, Chap. VII] is to observe that CM Hodge structures up to isogeny are determined by certain data associated to embeddings of CM fields, and hence the first step of the procedure in [GGK12, Chap. VII] is to “classify all CM fields of rank up to [$\dim V$] by [their] Galois group”. We are not aware of an effective method for carrying out this step.³³3The paper [Dod84] gives a potential approach by giving a method to classify certain abstract structures associated with Galois groups of CM fields. However one still needs to determine which such structures are actually associated to a concrete CM field. It is also not clear to us precisely the sense in which the term “classify” is being used; i.e., we do not know what form the data of a “classification of CM Hodge structures” takes, and consequently what form the resulting classification of Mumford-Tate domains will have. For this reason, we were unable to apply the methods of [GGK12, Chap. VII] to prove 2.4.

Our algebro-geometric correspondences will be formulated using the language of jets. Let $A^{d}_{r}=K[t_{1},\ldots,t_{d}]/\langle t_{1},\ldots,t_{d}\rangle^{r+1}$, and define $\mathbb{D}^{d}_{r}=\textrm{Spec}\hskip 1.49994ptA^{d}_{r}$ to be the $d$-dimensional disk of order $r$; we suppress the field $K$ in the notation. A jet space associated to a space $X$ is a space which parametrizes maps $\mathbb{D}^{d}_{r}\to X$. More formally, for $X$ a finite-type $K$-scheme, we have:

The purpose of introducing jets is the following result, proven in [Urb21b], building on [Urb21a].

We note that the computability of the torsor $\mathcal{P}^{d}_{r}$ in 3.2 has in particular the following consequence: if $\mathcal{S}\subset(\textrm{GL}_{m}\backslash J^{d}_{r}\widecheck{{L}})(\mathbb{C})$ is a subset which is the image under the quotient of a constructible $L$-algebraic set $\mathcal{F}\subset J^{d}_{r}\widecheck{{L}}$, where $K\subset L$ is a computable extension, then we can compute $(\eta^{d}_{r})^{-1}(\mathcal{S})$ by computing $p^{d}_{r}((\alpha^{d}_{r})^{-1}(\mathcal{F}))$. Thus if we define

Let us now try to understand how computing the “differential constraints” induced by types $\mathcal{C}(W)$ as in 3.4 can help us carry out Step Two of subsection 1.2. Let $\Gamma=\textrm{Aut}(V,Q)(\mathbb{Z})$, and let $\varphi:S_{\mathbb{C}}\to\Gamma\backslash D$ be the canonical period map which sends a point $s\in S(\mathbb{C})$ to the isomorphism class of the polarized Hodge structure on $\mathbb{V}_{s}$. By [BBT18], the map $\varphi$ factors as $\iota\circ p$, where $p:S_{\mathbb{C}}\to T$ is a dominant map of algebraic varieties and $\iota$ is a closed embedding of analytic spaces; it follows that for each subvariety $Z\subset S_{\mathbb{C}}$ the dimension of the image $\varphi(Z)$ makes sense as the dimension of a constructible algebraic set.

Fix a sequence of compatible embeddings

of formal disks. By acting on points via pullback, we obtain natural transformations of functors $\textrm{res}^{d}_{e}:J^{d}_{r}\to J^{e}_{r}$ which produce maps $J^{d}_{r}X\to J^{e}_{r}X$ that take jets $j:\mathbb{D}^{d}_{r}\to X$ to their restrictions $j\circ\iota_{d-1}\circ\cdots\circ\iota_{e}$. We are now ready to present the key proposition for our method:

The rest of this section we devote to proving 3.6, identifying $S$ with $S_{\mathbb{C}}$ for ease of notation. To begin with, let us check that (i) implies (ii) by applying the definitions. If $g:S^{\prime}\to S$ is an étale cover and we consider the variation $\mathbb{V}^{\prime}=g^{*}\mathbb{V}$, then the maps $\eta^{d}_{r}$ and $\eta^{\prime d}_{r}$ obtained from 3.2 are related by $\eta^{\prime d}_{r}=\eta^{d}_{r}\circ(J^{d}_{r}g)$. Choosing a finite index subgroup $\Gamma^{\prime}\subset\Gamma$ and passing to such a cover, we can reduce to the case where we have a period map $\varphi:S\to\Gamma\backslash D$ with $D\to\Gamma\backslash D$ a local isomorphism. Applying [BBT18] the map $\varphi:S\to\Gamma\backslash D$ factors as $\varphi=\iota\circ p$, where $p:S\to T$ is a dominant map of algebraic varieties and $\iota$ is an analytic closed embedding. Then via $p$, the variety $Z$ is dominant over a closed subvariety $Y\subset T$ of dimension $\dim\varphi(Z)\geq e$. Shrinking $S$ (and hence $Z$) we may assume that $Z$ is smooth, and that $Z$ is surjective onto a dense open subset $Y^{\circ}\subset Y$. Shrinking $S$ even further we may assume that $Z\to Y^{\circ}$ is smooth. The smoothness of $Z\to Y^{\circ}$ implies in particular that the induced jet space maps $J^{d}_{r}Z\to J^{d}_{r}Y^{\circ}$ for all choices of $d$ and $r$ are surjective.

We may choose neighbourhoods $B\subset S(\mathbb{C})$ and $U\subset D$ such that ${\left.\kern-1.2pt\pi\vphantom{\big{|}}\right|_{U}}:U\to\pi(U)$ is an isomorphism, both $B\cap Z$ and $\pi(U)\cap Y^{\circ}$ are non-empty, and we have a local lift $\psi:B\to U$ of $\varphi$. Choose a jet $\sigma\in J^{e}_{r,nd}(Y^{\circ}\cap\pi(U))$ and lift it along $p$ to a jet $\widetilde{\sigma}\in J^{e}_{r,nd}(Z\cap B)$ landing at the point $s\in S(\mathbb{C})$. Using the fact that the germ $(Z,s)$ is smooth of dimension $\dim Z>d$ the jet $\widetilde{\sigma}$ can be extended to a jet $j\in J^{d+1}_{r,nd}(Z\cap B)$ such that $\textrm{res}^{d+1}_{e}(j)=\widetilde{\sigma}$, and hence $\textrm{res}^{d+1}_{e}(\varphi\circ j)=\sigma$. From the fact that ${\left.\kern-1.2pt\varphi\vphantom{\big{|}}\right|_{B}}=\pi\circ\psi$ and the defining property of the map $\eta^{d}_{r}$ it follows that $j$ lies inside $\mathcal{T}^{d+1}_{r}((\textrm{res}^{d+1}_{e})^{-1}(J^{e}_{r,nd}\widecheck{{L}}))\cap J^{d+1}_{r,nd}S$. We can then take $\mathcal{C}=\mathcal{C}(Z)$, and the fact that $j$ factors through $Z$ implies that $j\in\mathcal{T}^{d+1}_{r}(\mathcal{C})$ as well.

To prove the reverse implication, we review some preliminary facts relating to jets.