Algebraic Independence of Special Points on Shimura Varieties

For $n\geq 1$, let $\boldsymbol{s}=(s_{1},\cdots,s_{n})$ be a special point in $S^{n}$ and let $\boldsymbol{g}=(g_{1},\cdots,g_{n})$ be a point in $G^{n}$ such that $(\boldsymbol{s},\boldsymbol{g})\in V^{n}$. It follows from the definition that if $(\boldsymbol{s},\boldsymbol{g})\in W$ for some dependent exemplary component then $g_{1},\cdots,g_{n}$ are algebraically dependent in $G$. Conversely, if $g_{1},\cdots,g_{n}$ are algebraically dependent in $G$, then $(\boldsymbol{s},\boldsymbol{g})$ is a distinguished point and hence contained in some dependent exemplary component. ∎

Let the Shimura datum of $S$ be $(G_{S},X)$. We may take $S$ to be a connected Shimura variety and $G$ to be a derived group. Applying [BSCFN23, Thm. 3.6] to $D\subset(X\times\mathbb{C}^{g})\times(S\times G)$ gives that the projection of $U$ in $S\times G$ is contained in a proper subvariety whose Galois group is a strict algebraic subgroup $H\subset G_{S}(\mathbb{C})\times\mathbb{G}_{a}(\mathbb{C})^{g}$. Since $G$ is a derived group, it has no abelian quotients and hence Goursat’s lemma says that the projection of $H$ to $G(\mathbb{C})$ or $\mathbb{G}_{a}(\mathbb{C})^{g}$ is not onto. By [Chi22, Thm. 3.2] for the Shimura variety side and [Ax72, Thm. 3] for the commutative group side, that means the projection of $U$ to $S$ or $G$ is contained in a proper weakly special subvariety. ∎

Let $d=\dim W$. We can find a generic point $P_{1}\times P_{2}\times\cdots\times P_{m+1}\in W^{m+1}$ such that $\operatorname{tr.deg.}_{\mathbb{Q}}\mathbb{Q}(P_{1},\dots,P_{m+1})=d(m+1)$, and we claim that these $P_{i}$ satisfy the above property. The forward direction of the if and only if is clear. Now suppose that $P_{i}\times Q\in V$ for all $i$. Viewing $P_{i}$ as a set, let $\tilde{P_{i}}\subset P_{i}$ be a minimal transcendental basis for $\operatorname{tr.deg.}_{\mathbb{Q}}\mathbb{Q}(P_{i})$. Write $Q=(q_{1},\dots,q_{n})\in\mathbb{C}^{n}$ and let $Q_{i}\subset\{1,2,\dots,n\}$ be the set of indices $j$ so that $q_{j}$ is algebraic over $\mathbb{Q}(\tilde{P_{i}})$. By construction, $\lvert\tilde{P_{i}}\rvert=d$ and the $\tilde{P_{i}}$ are algebraically independent. This implies that $Q_{i}\cap Q_{j}=\varnothing$ for $i\neq j$ and so $\sum_{i=1}^{m+1}\lvert Q_{i}\rvert\leq m$. Thus, there exists an $i$ such that $\lvert Q_{i}\rvert=0$ and for this $i$ we have

Let $X=\overline{\{P_{i}\}}^{\mathrm{Zar}}$ and $Y=\overline{\{Q\}}^{\mathrm{Zar}}$ be the $\overline{\mathbb{Q}}$-Zariski closure of $P_{i}$ and $Q$. Then $\dim X=\operatorname{tr.deg.}_{\mathbb{Q}}\mathbb{Q}(P_{i})=d$ and $\dim Y=\operatorname{tr.deg.}_{\mathbb{Q}}\mathbb{Q}(Q)$. Note that the Zariski closure of $P_{i}\times Q$ has dimension $\operatorname{tr.deg.}_{\mathbb{Q}}\mathbb{Q}(P_{i},Q)=\dim X+\dim Y$, showing that the $\overline{\mathbb{Q}}$-Zariski closure of $P_{i}\times Q$ is $X\times Y$. But $X\times Y\subset V$ by definition and $X=W$ showing inclusion in the other direction. ∎

The result for a fixed $V\subset S\times G$ is proven by [PT22, Prop. 3.4] and [DR18, Prop. 6.10]. The version for family is given by the uniform Ax–Schanuel theorem, which is implied by the Ax–Schanuel theorem. This is proven in [ES23, Prop. 2.20]. ∎

Let $\pi_{S}\colon V\to S$ and $\pi_{G}\colon V\to G$ be the projections onto the two factors. Let $W\subset V$ be an exemplary component and let $S^{\prime}\coloneqq\pi_{S}(W)$ be the special subvariety of $S$ which $W$ maps onto, and let $G^{\prime}\supset\pi_{G}(W)$ be the smallest special subvariety of $G$ containing $\pi_{G}(W)$. $S^{\prime}$ is a special subvariety of $S$ and hence there is a sub-Shimura datum $(H,X_{H})$ of $(G_{S},X)$, a decomposition $(H^{\operatorname{ad}},X_{H}^{\operatorname{ad}})=(H_{1},X_{1})\times(H_{2},X_{2})$, and a point $y_{2}\in X_{2}$ so that $S^{\prime}$ is the image of $X_{1}\times\{y_{2}\}$. By Lemma 4.2, there are only finitely many choices of $X_{H}$ and $X_{1}$. Thus, to show finiteness of exemplary components, it suffices to show that for a fixed $X_{1}$ and fixed dimension of $G^{\prime}$, there are only finitely many points $y_{2}\in X_{2}$ arising from exemplary subvarieties.

Let $\tilde{q}\colon X_{1}\times X_{2}\to X\to S$ denote the uniformization map of $S$ restricted to $X_{1}\times X_{2}$ and let $q\colon F_{1}\times F_{2}\to S$ denote the restriction of $\tilde{q}$ to a fundamental domain $F_{1}\times F_{2}$ of $X_{1}\times X_{2}$. Let $\exp\colon\mathbb{C}^{g}\to G$ denote the uniformization map of the algebraic group and let $e\colon F_{G}\to G$ denote the restriction of that map to a fundamental domain. Using Lemma 4.1, let $x_{1},\dots,x_{r}\in F_{1}$ denote a set of points to determine if $q(X_{1}\times\{x_{2}\})\times\exp(z)\subset V$. Let $\xi_{1},\dots,\xi_{g}$ denote the coordinates in $G$ and let $\tau_{1},\dots,\tau_{k}\in\operatorname{End}(G)$ be a $\mathbb{Z}$-basis for $\operatorname{End}(G)$. Take a generating set of all equations of the form

that all the points of $G^{\prime}$ satisfy, and let $G_{0}$ be the identity component of the algebraic subgroup of $G$ defined by these equations. We can extend each map $\tau_{i}\colon G\to G$ to an endomorphism of its covering space $\tilde{\tau_{i}}\colon\mathbb{C}^{g}\to\mathbb{C}^{g}$ satisfying $\tilde{\tau_{i}}(0)=0$. Suppose that $G_{0}$ is cut out by $\ell$ such equations. Let

and set $Z$ to be the projection of $Y$ to $F_{2}\times\mathbb{R}^{gk\ell}\times\mathbb{R}^{\ell}$. Both $Y$ and $Z$ are definable sets. The set $Y$ parametrizes points $y\in F_{2}$ so that a $V$-image of $X_{1}\times\{y\}$ lies within a proper special subvariety cut out by the $\boldsymbol{m}$ of $G$ (but not all choices of $\boldsymbol{m}$ and $\boldsymbol{b}$ correspond to algebraic subvarieties).

Suppose for the sake of contradiction that there were infinitely many $\overline{\mathbb{Q}}$-algebraic exemplary subvarieties $W^{\prime}$ with fibers over $X_{1}\times X_{2}$, and with dimension of the smallest special subvariety of $G$ containing $\pi_{G}(W^{\prime})$ equal to a fixed dimension $d=\dim G^{\prime}$. Each one gives a $\overline{\mathbb{Q}}$-point $(y^{\prime},\boldsymbol{m},\boldsymbol{b})\in Z$, and the $\operatorname{Gal}(\overline{\mathbb{Q}}/K)$-orbits of $W^{\prime}$ also lie in $Z$ for $K$ the defining field of $S,G,\operatorname{End}(G)$. Over $\overline{\mathbb{Q}}$, any commutative connected group $G$ can be written as a product of a semi-abelian variety with affine space $G^{\prime}\times\mathbb{G}_{a}^{n}$ (see [NW14, Prop. 5.1.12]). The tuple $\boldsymbol{m}$ consists integers and for semi-abelian varieties, we may take the fundamental domain so that the real part of each $z_{i}$ is in the interval $[0,1]$. Thus, each $b_{l}$ corresponding to an equation on the semi-abelian variety is an integer bounded by $\sum\lvert m_{ijl}\rvert$, which by Proposition 3.5, is bounded by $C\lvert\Delta(y^{\prime})\rvert^{c}$. Special subvarieties of $\mathbb{G}_{a}^{n}$ are given by the equations $\sum m_{ij}\tilde{\tau_{j}}(z_{i})=0$, so $b_{l}=0$ for those equations. Thus, taking the Galois orbit of $W^{\prime}$ gives a point with different $y^{\prime}$ as well as different $b_{l}$, but the same $\boldsymbol{m}$. View $Z$ is fibered over the $\boldsymbol{m}$ variable and let $\Sigma\subset Z_{\boldsymbol{m}}$ be the set of points arising from exemplary subvarieties $W^{\prime}$ as well as their Galois conjugates. By Proposition 3.2, there are at least $C^{\prime}\lvert\Delta(y^{\prime})\rvert^{c^{\prime}}$ points of height less than $C\lvert\Delta(y^{\prime})\rvert^{c}$. Then Theorem 4.3 gives the existence of a set $R\subset Z_{\boldsymbol{m}}$ of positive dimension whose projection to $F_{2}$ is connected semialgebraic and whose projection to $\mathbb{R}^{\ell}$ is non-constant.

Let $\overline{\exp}\colon\mathbb{C}^{g}/\tilde{G_{0}}\to G/G_{0}$ be the exponentiation map of $G/G_{0}$. Let $F_{G_{0}}$ denote the image of the fundamental domain $F_{G}$ under the quotient map, which will serve as a fundamental domain for $\overline{\exp}$. First, take the preimage of $R\subset Z_{\boldsymbol{m}}$ under the projection map $Y\to Z$ and then project the preimage under the map

Let the image of $R$ under these transformations be $R^{\prime}\subset F_{2}\times F_{G_{0}}$. By applying the point counting theorem if necessary, we may take $R^{\prime}$ to be connected and semialgebraic. Since the projection of $R$ to $\mathbb{R}^{\ell}$ was non-constant, the projection of $R^{\prime}$ to $F_{G_{0}}$ is also non-constant. Moreover, we may assume that $G/G_{0}$ is a semi-abelian variety by expanding $G_{0}$ if necessary because the projection to the affine space $\mathbb{G}_{a}$ is constant ($b_{l}=0$). By applying the Ax–Schanuel theorem (Theorem 2.5) to the Zariski closure of $(F_{1}\times R^{\prime})\times(q\times\exp)(F_{1}\times R^{\prime})$, we get that $(q\times\exp)(F_{1}\times R^{\prime})\subset S\times G/G_{0}$ is contained in a proper weakly special subvariety. Since $R^{\prime}$ contains preimages of special points, its image must lie in a proper special subvariety $S^{\prime}\times G^{\prime}\subset S\times G/G_{0}$.

Let $V^{\prime}\subset S\times G/G_{0}$ be the image of $V$ under the quotient map $G\to G/G_{0}$. By construction, we have that $(q\times\exp)(F_{1}\times R^{\prime})\subset V^{\prime}$, and so $V^{\prime}$ contains a proper special subvariety $S^{\prime}\times G^{\prime}$. However, the map $V\to S$ is finite, and hence $V^{\prime}\to S$ is also finite. Hence, $G^{\prime}$ is a single point, meaning that $R$ was originally of the form $R^{\prime\prime}\times\{g\}$, with $g$ corresponding to a torsion translation of $G_{0}$, which by abuse of notation we will also denote $G_{0}$. Let $\tilde{G_{0}}\subset F_{G}$ be the preimage of $G_{0}$. Applying Ax–Schanuel to the Zariski closure of

we get that $q(F_{1}\times R^{\prime\prime})\times G_{0}$ is contained in a proper special subvariety of $V$, which must be of the form $S^{\prime}\times G_{0}\subset V$. This shows that our original exemplary component $W$ was not exemplary because there is a larger Shimura variety $S^{\prime}$ properly containing $\pi_{S}(W)$ whose $V$ image lies within $G_{0}=G^{\prime}$. ∎

Let $F$ be the finitely generated subfield over $\mathbb{Q}$ which $S,G,\operatorname{End}(G)$, and $V$ are all defined. Then, there exists a quasi-projective geometrically irreducible variety $Z$ over a number field $K$ such that $F$ is the function field of $Z$, and $S$ is defined over $K$. By taking a Zariski open subset of $Z$, we may assume that $G$ extends to an abelian scheme $\mathcal{G}$ over $Z$ and $V$ extends to a variety $\mathcal{V}$ that is flat over $Z$. Choose a generic closed point $z_{0}\in Z(\mathbb{C})$ such that the field gotten by localizing at $z_{0}$ is isomorphic to $F$. Choose a simply connected Euclidean open neighborhood $U\ni z_{0}$ of $z_{0}$ so that the homology of $\mathcal{G}$ can be trivialized over $U$, and so we have $\mathcal{G}_{U}\cong G\times U$ as analytic varieties.

Suppose for the sake of contradiction that there are infinitely many exemplary subvarieties of $V\cong\mathcal{V}_{z_{0}}$. By Lemma 4.2, special subvarieties that arise as projections of the exemplary subvarieties of $V$ to $S$ come from finitely many splittings of special subvarieties of $S$. So, there is a splitting of sub-Shimura datum $(H_{1},X_{1})\times(H_{2},X_{2})$ with image $S_{1}\times S_{2}\subset S$ and infinitely many special points $p_{i}\in S_{2}$ such that there are exemplary subvarieties $W_{i}$ of $V$ mapping surjectively to $S_{1}\times\{p_{i}\}$. By the André–Oort conjecture, proven by [PST⁺21], the Zariski closure of these $S_{1}\times\{p_{i}\}$ is a finite union $R_{1},\dots,R_{n}\subset S$ of special subvarieties of $S$. For each $i$, let $G_{i}\subset G$ be the smallest special subvariety containing $\pi_{G}(W_{i})\subset\mathcal{G}_{z_{0}}$, and we extend $G_{i}$ to a family $\mathcal{G}_{i}$ over $U$. For every $z\in U$, we have that the $\mathcal{V}_{z}$ image of $S_{1}\times\{p_{i}\}$ lies within $G_{i,z}$. However, for each $z\in U(\overline{\mathbb{Q}})$, Theorem 4.4 says that there are only finitely many exemplary components and hence the image of each $R_{i}$ must also lie within a proper special subvariety of $G$.

The $\mathcal{V}$-images of each $R_{i}$ gives a family of subvarieties $\mathcal{R}$ of $\mathcal{G}$ such that for each $z\in U(\overline{\mathbb{Q}})$, the fiber $\mathcal{R}_{z}$ lies within a proper special subvariety of $\mathcal{G}_{z}$. We claim that this holds at $z_{0}$ as well. By replacing $\mathcal{R}$ with a $g$ self-sum, where $g=\dim G$, we may assume the $\mathcal{R}$ is a coset of an abelian subscheme of $\mathcal{G}$ and we still have that $\mathcal{R}_{z}$ lies within a proper special subvariety of $G$. By quotienting by the identity component of $\mathcal{R}$, we may assume that $\mathcal{R}$ is finite over $U$ and after finite base change that $\mathcal{R}$ is a section of $U$. Applying the Main Theorem of [Mas89] and the extension to semi-abelian varieties given in Section $5$ of loc. cit. to $\mathcal{R}\times\tau_{1}\mathcal{R},\dots,\tau_{k}\mathcal{R}\subset G^{k+1}$, we get that $\mathcal{R}_{z}$ must lie within a proper special subvariety of $\mathcal{G}_{z}$. ∎