An Inner Product on Adelic Measures
With Applications to the Arakelov–Zhang Pairing

Let $K$ be a number field and let $M_{K}$ be the set of places of $K$. By an adelic measure, we mean a sequence $\mu=(\mu_{v})_{v\in M_{K}}$ of suitably regular complex-valued measures indexed by the places of $K$. At the finite places of $K$, each measure $\mu_{v}$ is supported on the Berkovich projective line $\mathbf{P}_{v}^{1}$ and $\mu_{v}$ is supported on the Riemann sphere at the infinite places. We also impose two global conditions on adelic measures, namely that the collection of measures $(\mu_{v})_{v\in M_{K}}$ has constant total mass, and that $\mu_{v}$ is trivial for all but finitely many places; precise definitions are deferred to Section 3. The collection of such adelic measures forms complex vector space, which we denote by $\mathcal{A}_{K}$.

The principal example of an adelic measure is the canonical adelic measure $\mu_{f}=(\mu_{f,v})_{v\in M_{K}}$ of a rational map $f$ defined over $K$. This adelic measure encodes a good amount of information on the $v$-adic Julia sets of $f$, and so defines an important dynamical invariant. For example, the preperiodic points of $f$ distribute according to $\mu_{f,v}$ at each place. When $v$ is an Archimedean place, $\mu_{f,v}$ is the measure investigated for polynomials by Brolin [Bro65], and for rational maps by Ljubich [Lju83] and by Freire, Lopes, and Mañé, [FLM83]. For non-Archimedean places, this measure was introduced independently by several authors, including Baker–Rumely [BR06], Chambert-Loir [Cha06], and Favre–Rivera-Letelier [FR06].

We equip $\mathcal{A}_{K}$ with the following pairing. Fix an adelic probability measure $\lambda=(\lambda_{v})_{v\in M_{K}}$. At each place $v$ we let $g_{\lambda,v}(x,y)$ be an Arakelov–Green’s function of $\lambda_{v}$. (See Section 2 for the definition). We choose a normalization of $g_{\lambda_{v}}$ so that

and require $t_{v}=0$ for all but finitely many $v$, and that $\sum_{v\in M_{K}}r_{v}t_{v}=:t>0$, where $r_{v}=[K_{v}:{\mathbb{Q}}_{v}]/[K:{\mathbb{Q}}]$ is the ratio of local to global degrees. Inspired by the work of Favre–Rivera-Letelier and of Baker-Rumely (discussed below), given two adelic measures $\mu,\nu\in\mathcal{A}_{K}$ we define

For example, take the Arakelov adelic measure, $\lambda_{\mathrm{Ar}}=(\lambda_{\mathrm{Ar},v})_{v\in M_{K}}$. At Archimedean places $v$, $\lambda_{\mathrm{Ar},v}$ is the spherical measure, and is trivial at finite places (see (2.2)). An Arakelov–Green’s function for $\lambda_{\mathrm{Ar},v}$ which satisfies the above conditions is $g_{\lambda_{\mathrm{Ar}},v}(x,y)=\log||x,y||_{v}^{-1}$, where $||\cdot,\cdot||_{v}$ is a continuous extension of the projective metric on ${\mathbb{P}}^{1}({\mathbb{C}}_{v})$ to $\mathbf{P}_{v}^{1}$. (This is called the spherical kernel in the book by Baker and Rumely [BR10]). It is not hard to check that $\int_{\mathbf{P}_{v}^{1}}\int_{\mathbf{P}_{v}^{1}}\log||x,y||_{v}^{-1}\,d\lambda_{\mathrm{Ar},v}(x)\,d\lambda_{\mathrm{Ar}}(y)=1/2$ for Archimedean $v$ and is $0$ for non-Archimedean $v$; thus $\sum_{v\in M_{K}}r_{v}t_{v}=1/2$ and so

Our first theorem is that this pairing is an inner product, and that, in most situations of interest, convergence in the norm $\|\cdot\|_{\lambda,t}$ induced by this inner product gives weak convergence at each place.

Similar potential theoretic pairings are used by both Favre–Rivera-Letelier and Baker–Rumely in connection with their respective proofs of the adelic equidistribution theorem [FR06],[BR06]. The novelty of Theorem 1.1 is that $\langle\cdot,\cdot\rangle_{\lambda,t}$ is positive definite for adelic measures of arbitrary mass. In particular, the pairing $\langle\cdot,\cdot\rangle_{\lambda,t}$ should perhaps be viewed as a modest extension of the potential theoretic pairings investigated by Favre–Rivera-Letelier and Baker–Rumely. The following comparisons between the height of a rational map $f$ and the norm $\|\mu_{f}\|^{2}_{\lambda,t}$ of the canonical adelic measure of $f$ are the primary new results of this article.

The proof of Theorem 1.2 relies on studying the $v$-adic Julia sets of $f$ at each place, and uses escape rate arguments to bound $\frac{d}{2}\|\mu_{f}\|^{2}_{\lambda,t}$ in terms of the height of the coefficients of $f$. Such arguments do not apply to rational maps. However, our main theorem shows that $f\mapsto\frac{d}{2}\|\mu_{f}\|_{\lambda,t}^{2}$ is at least commensurate with a height on $\mathrm{Rat}_{d}(K)$, the space of rational maps with degree $d$ defined over $K$.

These constants are given explicitly in the case of the Arakelov height of the coefficients of $f$ and $\|\mu_{f}\|^{2}_{\lambda_{\mathrm{Ar}},1/2}$ (see (5.9)). The main idea of the proof of Theorem 1.3 is that while $\mu_{f}$ can be quite complicated, $\|\mu_{f}\|_{\lambda_{\mathrm{Ar}},1/2}^{2}$ cannot be too far from the norm of the pull-back of the Arakelov measure $f^{*}\lambda_{\mathrm{Ar}}=(f^{*}\lambda_{\mathrm{Ar},v})_{v\in M_{K}}$. It turns out that $\frac{d}{2}\|d^{-1}f^{*}\lambda_{\mathrm{Ar}}\|_{\lambda_{\mathrm{Ar}},1/2}^{2}$ is a Weil height on $\mathrm{Rat}_{d}(K)$, a result of potential interest in its own right (see Section 5 for details). Combining these two results with the observation that $\|\cdot\|_{\lambda,t}^{2}$ is never too far from $\|\cdot\|^{2}_{\lambda_{\mathrm{Ar}},1/2}$ (Proposition 3.2) gives the theorem.

Suppose we have two rational maps $f,g:{\mathbb{P}}^{1}\to{\mathbb{P}}^{1}$ of degree at least two. The Arakelov–Zhang pairing (also called the dynamical height pairing), denoted here by $(f,g)_{\mathrm{AZ}}$, is a symmetric, non-negative pairing which in some sense is a dynamical distance between $f$ and $g$. It is closely related to the Call–Silverman canonical heights of $f$ and $g$: the pairing $(f,g)_{\mathrm{AZ}}$ vanishes precisely when the canonical heights of $f$ and $g$ are equal, or, equivalently, when $f$ and $g$ have the same canonical adelic measure (see [PST12], Theorem 3). To some extent, the Arakelov–Zhang pairing measures how similar the $v$-adic Julia sets of two rational maps $f$ and $g$ are at all places of a number field.

The Arakelov–Zhang pairing is not a metric on rational maps; however, Fili (Theorem 9, [Fil17]) has discovered that it is expressible as the square of a metric on adelic measures. Specifically, Fili showed that the Arakelov-Zhang pairing is equal to the mutual energy pairing:

where

Here, $\delta_{v}(x,y)$ is a continuous extension of $|x-y|_{v}$ on ${\mathbb{C}}_{v}\times{\mathbb{C}}_{v}$ to $\mathbf{A}_{v}^{1}\times\mathbf{A}_{v}^{1}$ and $\mathrm{Diag}_{v}$ is the diagonal of ${\mathbb{C}}_{v}\times{\mathbb{C}}_{v}$ in $\mathbf{A}_{v}^{1}\times\mathbf{A}_{v}^{1}$. (For Archimedean $v$, $\delta_{v}(x,y)$ is the usual affine distance $|x-y|_{v}$.) This formulation of the dynamical height pairing has been useful in studying unlikely intersection questions; see, for example, [DKY20]. The mutual energy pairing itself was introduced by Favre and Rivera-Letelier in connection with a quantitative form of the adelic equidistribution theorem [FR06]. Our inner product could be understood as a mutual energy pairing relative to a fixed adelic probability measure $\lambda$.

While quite closely related, our inner product and the mutual energy pairing are distinct on $\mathcal{A}_{K}$. For instance, the mutual energy pairing $(\!(\cdot,\cdot)\!)$ is not an inner product on adelic measures of arbitrary mass. Indeed, it follows from a result of Baker–Hsia that if $f$ is a polynomial defined over $K$, then $(\!(\mu_{f},\mu_{f})\!)=0$ (Theorem 4.1 of [BH05]). However, the mutual energy pairing and our inner product do agree on adelic measures of total mass zero. We may then consider our inner product to be an extension of the mutual energy pairing from the space of adelic measures of total mass zero to $\mathcal{A}_{K}$. It therefore follows from Theorem 9 of [Fil17] and Theorem 1 of [PST12] that if $\mu_{f}$ and $\mu_{g}$ are the canonical adelic measures of two rational maps $f$ and $g$ then

It is a straightforward consequence of Theorem 1.3 and (1.1) that, when $g$ is fixed, the Arakelov–Zhang pairing is commensurate with the height of $f$.

Bounds on the Arakelov–Zhang pairing of certain families of rational maps with the power map $z^{2}$ are studied by Petsche, Szpiro, and Tucker; one such bound, which is generalized by Corollary 1.4, is $(z^{2},z^{2}+c)_{\mathrm{AZ}}=\frac{1}{2}h(c)+O(1)$ [PST12]. The lower bound $(z^{2},f)_{\mathrm{AZ}}\geq h_{f}(0)$, where $h_{f}$ is the Call–Silverman canonical height of a rational map $f$, has been shown by Bridy and Larson [BL21]. Perhaps the strongest known comparisons between the dynamical height pairing and a height on rational maps have been shown by DeMarco, Krieger, and Ye. In [DKY20], they show that $(f_{t_{1}},f_{t_{2}})_{\mathrm{AZ}}\asymp h(t_{1},t_{2})$ for a certain family of Latès maps $f_{t}$ associated to an elliptic curve written in Legendre form with parameter $t$. In a similar spirit, they show that $(z^{2}+c_{1},z^{2}+c_{2})_{\mathrm{AZ}}\asymp h(c_{1},c_{2})$ [DKY22]. These estimates are stronger than those of Corollary 1.4, though they hold in specific families of rational maps. It is immediate from Corollary 1.4 and the Northcott property of heights that $(f,g)_{\mathrm{AZ}}$ can be bounded from below by constant depending on one of the rational maps.

A uniform lower bound on $(z^{2}+c_{1},z^{2}+c_{2})_{\mathrm{AZ}}$ with $c_{1}\neq c_{2}$ and $c_{1},c_{2}\in\bar{{\mathbb{Q}}}$, is given by DeMarco, Krieger, and Ye ([DKY22], Theorem 1.6). They use this lower bound on the Arakelov–Zhang pairing to prove a uniform upper bound on the number of preperiodic points shared by $z^{2}+c_{1}$ and $z^{2}+c_{2}$. It is to be hoped that the lower bound of Corollary 1.5 has analogous implications for the number of common preperiodic of two rational maps of degree $d$, but we do not explore this possibility here.

Fix a number field $K$ and let $M_{K}$ be the set of places of $K$. Let ${\mathbb{C}}_{v}$ be the completion of a fixed algebraic closure of $K_{v}$. We denote by $|\cdot|_{v}$ the absolute value on ${\mathbb{C}}_{v}$ whose restriction to ${\mathbb{Q}}$ coincides with the usual real or $p$-adic absolute value. Let $r_{v}=[K_{v}:{\mathbb{Q}}_{v}]/[K:{\mathbb{Q}}]$ be the ratio of local-to-global degrees. It it well known that $K$ satisfies the product formula, which states that $\prod_{v\in M_{K}}|x|_{v}^{r_{v}}=1$ for all $x\in K^{\times}$. Moreover the extension formula reads $\sum_{\begin{subarray}{c}w\in M_{L}\\ w\mid v\end{subarray}}r_{w}=r_{v}$, where $L$ is a finite extension of $K$, and where $w\mid v$ means that $w$ lies above $v$ (so $|\alpha|_{w}=|\alpha|_{v}$ for all $\alpha\in K$). We write $M_{K}^{0}$ for the set of non-Archimedean places of $K$ and $M_{K}^{\infty}$ for the Archimedean places.

Of central importance to this article is the v-adic chordal metric, $||\cdot,\cdot||_{v}:{\mathbb{P}}^{1}({\mathbb{C}}_{v})\times{\mathbb{P}}^{1}({\mathbb{C}}_{v})\to[0,1]$. In homogeneous coordinates $x=[x_{1}:x_{2}]$ and $y=[y_{1}:y_{2}]$ the chordal metric is given by

It is well known that $||\cdot,\cdot||_{v}$ is a metric on ${\mathbb{P}}^{1}({\mathbb{C}}_{v})$, and that when $v$ is not Archimedean, then $||\cdot,\cdot||_{v}$ satisfies the strong triangle inequality (see, for example, chapters 1 and 3 of [Sil07]). For non-Archimedean $v$, there is a continuous extension of $||\cdot,\cdot||_{v}$ to $\mathbf{P}_{v}^{1}$ called the spherical kernel which we denote again by $||\cdot,\cdot||_{v}$. The spherical kernel is not a metric on $\mathbf{P}_{v}^{1}$; indeed, the Gauss point $\zeta_{\mathrm{Gauss}}$ satisfies $||z,\zeta_{\mathrm{Gauss}}||_{v}=1$ for all $z\in\mathbf{P}_{v}^{1}$. See Proposition 4.7 of [BR10] for more properties of the spherical kernel.

There is an analogous extension of the absolute value $|x-y|_{v}$ on ${\mathbb{C}}_{v}$ to the Berkovich affine line, $\mathbf{A}_{v}^{1}$, which, following [BR10], we refer to as the Hsia Kernel, which we will write as $\delta_{v}(x,y)$. (In [BR10], this is written $\delta(x,y)_{\infty}$ to distinguish the Hsia kernel from the generalized Hsia kernel. In the work of Favre and Rivera-Letelier, this extension is written $\sup\{S,S^{\prime}\}$.) Similar to the classical relationship between the affine distance on ${\mathbb{C}}$ and the chordal metric, we have

for any two points $x,y\in\mathbf{A}_{v}^{1}\cong\mathbf{P}_{v}^{1}\setminus\{\infty\}$. We will also write $\delta_{v}(x,y)=|x-y|_{v}$ for the affine distance on ${\mathbb{C}}_{v}$ when $v$ is Archimedean.

We come to the main object of study in this article. We adopt the convention that complex or real valued measures have finite total variation.

Observe that if $\mu=(\mu_{v})_{v\in M_{K}}$ is an adelic measure with $\mu_{v}=\mu_{r,v}+i\mu_{i,v}$ for signed measures $\mu_{r,v},\mu_{i,v}$ at each place, then $(\mu_{r,v})_{v\in M_{K}}$ and $(\mu_{i,v})_{v\in M_{K}}$ are again (signed) adelic measures. Indeed, $\mathrm{Re}\,U_{\mu,v}=U_{\mu_{r},v}$ so that $\mu_{r,v}$ has continuous potentials at each place $v$; moreover $\mathrm{Re}\,c_{\mu}=\mathrm{Re}\,\mu_{v}(\mathbf{P}_{v}^{1})=\mu_{r,v}(\mathbf{P}_{v}^{1})$ so that $\mu_{r,v}(\mathbf{P}_{v}^{1})$ is a constant independent of $v$. So $\mu_{r}=(\mu_{r,v})$ is a signed adelic measure. Similar remarks apply to $(\mu_{i,v})_{v\in M_{K}}$.

The primary examples of adelic probability measures in $\mathcal{A}_{K}$ relevant to arithmetic dynamics are the adelic canonical measures associated to a rational map, which can be defined as the weak (really, weak-$*$) limit of the sequence $(d^{-1}f^{*})^{n}\lambda_{\mathrm{Ar},v}$ of normalized pull-backs of the Arakalov measure (or, any other measure with continuous potentials). It follows from this definition that $f^{*}\mu_{f,v}=d\mu_{f,v}$; in fact, $\mu_{f,v}$ is the unique log-continuous probability measure satisfying this equation (Theorem (d) in [FLM83], Théorème A in [FR10]). As shown in [FR04], $\mu_{f,v}=\delta_{\zeta_{\mathrm{Gauss}}}$ precisely when $f$ has good reduction, and so $\mu_{f,v}=\delta_{\zeta_{\mathrm{Gauss}}}$ for all but finitely many $v$. That $\mu_{f}$ has continuous potentials (in fact, Holder continuous) is also due to Favre and Rivera-Letelier for non-Archimedean $v$, and due to Mañé for Archimedean $v$ [Mañ06]. So $\mu_{f}=(\mu_{f,v})_{v\in M_{K}}$ is indeed an element of $\mathcal{A}_{K}$.