Arithmetic degree and its application to Zariski dense orbit conjecture

Since $\lambda_{3}(f)=1$, the assumption of Theorem 1.6 is satisfied. ∎

By replacing $K$ with a finite extension and replacing $A$ with an appropriate subset, we may assume $A$ is a basic adelic subset over $K$ with respect to $X_{K}$. Moreover, we may assume

where $\tau_{i}\colon K\longrightarrow{\mathbb{C}}_{p_{\tau_{i}}}$ are field embeddings such that $|\ |_{i}:=|\tau_{i}(\ )|_{{\mathbb{C}}_{p_{\tau_{i}}}}$ are distinct absolute values on $K$, and as usual $U_{i}\subset X_{K}({\mathbb{C}}_{p_{\tau_{i}}})$ are non-empty $p_{\tau_{i}}$-adic open subsets. (cf. the beginning of the proof of [29, Proposition 3.9]. ) Let $K_{p_{i}}$ be the closure of $\tau_{i}(K)$ in ${\mathbb{C}}_{p_{\tau_{i}}}$. By further replacing $K$ with a finite extension, we may assume $U_{i}\cap X_{K}(K_{p_{i}})\neq\emptyset$. Note that this in particular implies $U_{i}\cap X_{K}(K_{p_{i}})$ is Zariski dense in $(X_{K})_{K_{p_{i}}}$.

By Noether normalization, there is a non-empty open subscheme $X_{K}^{\circ}\subset X_{K}$ with finite étale morphism

to an open subscheme $V\subset{\mathbb{A}}^{d}_{K}$ of an affine space. By taking a connected Galois étale covering of $V$ dominating $X_{K}^{\circ}$ (cf. [10, Proposition 3.2.10]) and applying it to [25, Proposition 3.3.1], there is a thin subset $Z\subset V(K)$ such that for all $x\in V(K)\setminus Z$, the scheme theoretic inverse image $\pi^{-1}(x)$ is integral, i.e. it is of the form $\operatorname{Spec}\text{(field)}$.

Let $W_{i}=\pi(U_{i}\cap X_{K}^{\circ}(K_{p_{i}}))$, which is a non-empty open subset of $V(K_{p_{i}})$.

Let $x\in(V(K)\setminus Z)\cap\bigcap_{i=1}^{m}W_{i}$. Then $\pi^{-1}(x)=\operatorname{Spec}L$ for some finite field extension $L$ of $K$. Note that $[L:K]\leq\deg\pi$. Fixing a field embedding $L\to\overline{\mathbb{Q}}$ over $K$ and we get a point $z\in X_{K}^{\circ}(\overline{\mathbb{Q}})\subset X_{K}(\overline{\mathbb{Q}})$:

Since $x\in W_{i}=\pi(U_{i}\cap X_{K}^{\circ}(K_{p_{i}}))$, there is $y_{i}\in U_{i}\cap X_{K}^{\circ}(K_{p_{i}})$ such that $\pi(y_{i})=x$. Then we get the following diagram

where the dashed arrow is induced by extending $L\longrightarrow K_{p_{i}}\hookrightarrow{\mathbb{C}}_{p_{\tau_{i}}}$ to $\overline{\mathbb{Q}}\longrightarrow{\mathbb{C}}_{p_{\tau_{i}}}$. This embedding restricted on $K$ agrees with $\tau_{i}$. This means $z\in X_{K}^{\circ}(\tau_{i},U_{i}\cap X_{K}^{\circ}({\mathbb{C}}_{p_{\tau_{i}}}))\subset X_{K}(\tau_{i},U_{i})$. Therefore we proved $z\in A\cap X_{K}(\deg\pi)$. Since $x$ is arbitrary element of $(V(K)\setminus Z)\cap\bigcap_{i=1}^{m}W_{i}$, which is Zariski dense in $V$, these $z$’s are Zariski dense in $X$ and we are done.

∎

We may assume $A$ is a general adelic subset, i.e. there is a flat morphism $\pi\colon Y\longrightarrow X$ from a reduced algebraic scheme over $\overline{\mathbb{Q}}$ and a basic adelic subset $B\subset Y(\overline{\mathbb{Q}})$ such that $A=\pi(B)$. By replacing $Y$ with a small open affine subscheme of an irreducible component intersecting with $B$, we may assume $Y$ is a quasi-projective variety. Let $K\subset\overline{\mathbb{Q}}$ be a number field such that $X,Y$, and $\pi$ are defined over $K$. Let $\pi_{K}\colon Y_{K}\longrightarrow X_{K}$ be their model. Now suppose $A$ is height bounded. Then for all $d\in{\mathbb{Z}}_{\geq 1}$, $A\cap X_{K}(d)$ are finite sets because of Northcott’s theorem. Since $B\cap Y_{K}(d)\subset\pi^{-1}(A\cap X_{K}(d))$, $\pi$ is flat, and $\dim X\geq 1$, $B\cap Y_{K}(d)$ is not Zariski dense in $Y$ for all $d\in{\mathbb{Z}}_{\geq 1}$. This contradicts to Lemma 2.5. ∎

By replacing $X$ with its smooth locus, we may assume $X$ is smooth. Let us take a projective closure $\iota\colon X\hookrightarrow X^{\prime}$, i.e. $X^{\prime}$ is a projective variety over $\overline{\mathbb{Q}}$ and $\iota$ is an open immersion. By replacing $X^{\prime}$ with its normalization, we may assume $X^{\prime}$ is normal. Let $L$ be a very ample divisor on $X^{\prime}$. We take $L$ so that the embedding $X^{\prime}\hookrightarrow{\mathbb{P}}^{N}_{\overline{\mathbb{Q}}}$ by the complete linear system $|L|$ is not an isomorphism. We regard $f$ as a dominant rational self-map on $X^{\prime}$. Let us write $\lambda_{i}=\lambda_{i}(f)$ and $\mu_{i}=\mu_{i}(f)$. To prove the theorem, we may assume $\lambda_{1}>1$. Take $p\in\{1,\dots,\dim X\}$ such that

Let $\varepsilon>0$ be arbitrary positive number. Let $A\subset X(\overline{\mathbb{Q}})$ be an arbitrary non-empty adelic open subset. We will construct a point $x\in X_{f}(\overline{\mathbb{Q}})\cap A$ such that $\alpha_{f}(x)\geq\lambda_{1}-\varepsilon$.

Take $\zeta\in(0,1)$, which is close to $1$, such that

By [30, Remark 3.7], there is $m_{\zeta}\geq 1$ such that for all $m\geq m_{\zeta}$,

is big as elements of $\widetilde{\operatorname{Pic}}(X^{\prime})_{{\mathbb{R}}}$. Here $\widetilde{\operatorname{Pic}}(X^{\prime})_{{\mathbb{R}}}$ is the colimit of $\operatorname{Pic}(X^{\prime\prime})_{{\mathbb{R}}}$ where $X^{\prime\prime}$ runs over birational models of $X^{\prime}$. See [30] for the detail. We fix an $m\geq m_{\zeta}$ so that

holds. Such $m$ exists because of eq. 3.3. Let us fix a Weil height function $h_{L}\colon X^{\prime}(\overline{\mathbb{Q}})\longrightarrow{\mathbb{R}}$ associated with $L$. We choose $h_{L}$ so that $h_{L}\geq 1$. By eq. 3.4, there are $c\in{\mathbb{R}}$ and a Zariski open dense subset $V\subset X$ such that

Then by eq. 3.5, we have

or, equivalently

on $V(\overline{\mathbb{Q}})$. If we take $c_{1}\in{\mathbb{R}}$ so that $c_{1}-\zeta^{2m}\mu_{p}^{m}c_{1}=c$, then we have

on $V(\overline{\mathbb{Q}})$. This recursive inequality almost shows that the arithmetic degree is at least $\zeta^{2m}\mu_{p}^{m}$. What we need to show is that there is at least one initial point at which $h_{L}\circ f^{m}-\zeta^{-2m}\mu_{p+1}^{m}h_{L}-c_{1}$ is strictly positive. We will find such point on a curve whose forward iterates by $f^{m}$ have maximal degree growth. But we first need to guarantee that there are plenty of points whose orbits are well-defined and have nice properties.

By [29, Proposition 3.24, Proposition 3.27] (cf. proof of [21, Proposition 3.2]), there is a non-empty adelic open subset $A^{\prime}\subset V(\overline{\mathbb{Q}})$ such that for all $x\in A^{\prime}$, we have

Here the last condition means that for any closed set $W\subset X^{\prime}$, the return set $\{n\geq 0\mid f^{n}(x)\in W\}$ is a finite union of arithmetic progressions.

Now set $g=f^{m}$. Let $d=\dim X$. Since $\lambda_{1}(g)=\lambda_{1}(f)^{m}=\mu_{1}^{m}=\mu_{p}^{m}$, we have

(Here $(g^{n})^{*}L$ is the one defined as an element of $\widetilde{\operatorname{Pic}}(X^{\prime})_{{\mathbb{R}}}$. So is the intersection number. ) We choose $\eta\in(0,1)$ close to $1$ and $l\in{\mathbb{Z}}_{\geq 1}$ large enough so that

We can choose such $\eta$ and $l$ because of eq. 3.3.

Let us pick a point $a\in A\cap A^{\prime}$ such that

• •

  $X^{\prime}$ is smooth at $a$;

• •

  the projection from $a$, $p_{a}\colon{\mathbb{P}}^{N}_{\overline{\mathbb{Q}}}\dashrightarrow{\mathbb{P}}^{N-1}_{\overline{\mathbb{Q}}}$, is generically finite on $X^{\prime}\setminus\{a\}$.

(Recall we chose $L$ so that the embedding $X^{\prime}\hookrightarrow{\mathbb{P}}^{N}_{\overline{\mathbb{Q}}}$ defined by $|L|$ is not an isomorphism. In particular, $X^{\prime}$ is not contained in a hyperplane of ${\mathbb{P}}^{N}_{\overline{\mathbb{Q}}}$ and thus projection from a general point of $X^{\prime}$ is generically finite on $X^{\prime}$. Such $a$ exists because a non-empty adelic open set is Zariski dense.) By the choice of $A^{\prime}$, we have $a\notin I_{g^{l}}$. Note also that $\operatorname{codim}I_{g^{l}}\geq 2$ since $X^{\prime}$ is normal projective. Let $\Gamma\subset|L|$ be the sub-linear system consisting of all hypersurfaces passing through $a$. If $\dim X^{\prime}\geq 2$, by Lemma 3.3, there is $H_{1}\in\Gamma$ such that $H_{1}$ is irreducible and reduced, smooth at $a$, $\dim H_{1}\cap I_{g^{l}}<\dim I_{g^{l}}$, and $p_{a}$ is generically finite on $H_{1}\setminus\{a\}$. If $\dim H_{1}\geq 2$, apply the same argument to the restriction of $\Gamma$ to $H_{1}$ and get $H_{2}\in\Gamma$ such that $H_{1}\cap H_{2}$ is irreducible and reduced, smooth at $a$, $\dim H_{1}\cap H_{2}\cap I_{g^{l}}<\dim I_{g^{l}}\cap H_{1}$, and $p_{a}$ is generically finite on $H_{1}\cap H_{2}\setminus\{a\}$. Repeat this and we get $H_{1},\dots,H_{d-1}\in|L|$ passing through $a$ such that

Moreover, the local equations of $H_{1},\dots,H_{d-1}$ form a regular sequence at each point of $C$.

Let us consider

where $\Gamma_{g^{l}}$ is the graph of the rational map $g^{l}$. Then we have

(Here for the second equality, we use the equality of schemes $\pi^{*}H_{1}\cap\cdots\cap\pi^{*}H_{d-1}=\pi^{-1}(C)$ to see that the cycle class $c_{1}(\pi^{*}L)^{d-1}\cap[\Gamma_{g^{l}}]$ is represented by the cycle $[\pi^{-1}(C)]$. )

Thus we get (use [11, Theorem B.5.9] on the normalization of $C$)

on $C^{\circ}(\overline{\mathbb{Q}})$ where $C^{\circ}$ is the normal locus of $C$ and $c^{\prime}$ is a constant depends on $C,(g^{l}|_{C})^{*}L,L|_{C},h_{L}$.

By the construction of $C$ (namely $a\in C$), $C^{\circ}(\overline{\mathbb{Q}})\cap A\cap A^{\prime}$ is a non-empty adelic open set of $C(\overline{\mathbb{Q}})$. By Proposition 2.3, there is a point $x\in C^{\circ}(\overline{\mathbb{Q}})\cap A\cap A^{\prime}$ such that

where $M\in{\mathbb{R}}_{\geq 1}$ is a large constant that we choose below. Then we have