We assume that $X$ is smooth and relatively minimal, i.e. there are no $(-1)$ curves in the fibers of the morphism to $B$, and that the fibration is not isotrivial (this implies that it has at least one singular fiber, see e.g. <cit.>).

We additionally assume that for each singular fiber, at least one irreducible component has multiplicity one (other components can have higher multiplicity).
In the analytic topology near the singular fiber, this is equivalent to the fibration having locally a section.
This assumption is needed to avoid some torsion phenomena, and is implicit in the literature that considers elliptic curves, since those have a section.
Note also that this assumption is always satisfied when $X$ is a K3, by the classification of singular fibers of elliptic fibrations on K3s <cit.>.

§.§.§ Vertical divisors

A Weil or Cartier divisor on $X$, will be called vertical if it is vertical in the sense of <cit.>.
See also <cit.> where such divisors are called fibral.
Note that all fibers of the elliptic fibration have arithmetic genus one, so when a fiber is not smooth, its irreducible components must have geometric genus $0$.

§.§.§ Jacobian surface

It will be technically convenient to refer to the Jacobian surface $J_X\to B$.
See <cit.> which describes the construction (the fact that the original $X$ is a K3 and $B\isom \bP^1$ is not used; it is the case, however, that if $X$ is a K3 then $J_X$ is also a K3).

One definition of $J_X$ is as follows.
View $X$ as a genus one curve over the function field $k(B)$.
It has an associated Jacobian elliptic curve $J_{X,k(B)}$ and take $J_X$ to be the surface over $k$ which is relatively minimal over $B$.

We now proceed to study the Picard group of $X$ and some of its subgroups.

Let $\pi\colon X\to B$ be a non-isotrivial relatively minimal elliptic surface over $\mathbb{C}$ (we do not assume $X$ projective or that $\pi$ admits a section).
Then $\pi^*\colon\Pic^0(B)\to \Pic^0(X)$ is an isomorphism.

As in <cit.>, the Leray spectral sequence gives the exact sequence

\[0\to H^1(B,\pi_*\mathcal{O}_X)\to H^1(X,\mathcal{O}_X)\to H^0(B, R^1 \pi_* \mathcal{O}_X)\to H^2(B,\pi_* \mathcal{O}_X),\]

but $\pi_*\mathcal{O}_X\cong\mathcal{O}_B$ since $\pi$ has connected fibers, hence

\[0\to H^1(B,\mathcal{O}_B)\to H^1(X,\mathcal{O}_X)\to H^0(B, R^1 \pi_* \mathcal{O}_X)\to 0,\]

is exact, and the dimensions of the first two terms are $g(B)$ and $q(X)$ respectively. As for the third term, <cit.> (see also their Remark after Thm V.12.1 on p.213) shows that the degree of the dual of $R^1 \pi_* \mathcal{O}_X$ is strictly positive, unless all smooth fibers of $\pi$ are isomorphic (i.e. $\pi$ is isotrivial) and the only singular fibers are multiples of a smooth fiber.  So by our assumption it follows that the degree of the line bundle $R^1 \pi_* \mathcal{O}_X$ is strictly negative, so it has no sections, and the exact sequence above gives $q(X)=g(B)$.

We can now conclude as in <cit.>: first, the map $\pi:\Pic(B)\to\Pic(X)$ is injective since the projection formula gives $\pi_*\pi^*L\cong L\otimes\pi_*\mathcal{O}_X\cong L$ for every $L\in \Pic(B)$. And then since $q(X)=g(B)$, it follows that the tori $\Pic^0(B)$ and $\Pic^0(X)$ have the same dimension, hence $\pi^*:\Pic^0(B)\to\Pic^0(X)$ is an isomorphism.

§.§.§ Line bundles and automorphisms

Let $\Aut_\pi(X)$ be the group of parabolic automorphisms of the elliptic surface $X\xrightarrow{\pi}B$, i.e. those preserving the fibers of $\pi$.
Denote by $\Aut_\pi^\circ(X)$ those automorphisms that act trivially on the set of irreducible components of fibers of $\pi$, and such that the induced automorphism of the genus one curve $X_{k(B)}$ over the function field $k(B)$ is given by a translation.
Then $\Aut_{\pi}^{\circ}(X)$ is of finite index in $\Aut_{\pi}(X)$, since there are only finitely many fibers with distinct irreducible components, and because the translation automorphisms are of finite index in the group of all automorphisms of an elliptic curve (by composing by a translation, we can assume a point is fixed, and the group of automorphisms preserving a polarization is finite).

To handle more general elliptic fibrations compared to <ref>, we will replace the group $[E]^\perp/[E]$ from previous sections by a different one.
Specifically, let $\Pic^0_{\pi}(X)$ denote the subgroup of line bundles on $X$ which restrict to have degree $0$ on each irreducible component of a fiber of the fibration.
Consider also the larger group $\Pic^0_{\pi}(X)^{sm}$ of line bundles that have degree $0$ on the generic smooth fiber.
Finally let $\pi^*\Pic(B)\subset \Pic(X)$ be the group of line bundles pulled back from $B$.
We have the filtration

\begin{align}
\label{eqn:filtration_general_pic}
\Pic(X)\supset
\Pic^0_\pi(X)^{sm}\supset
\Pic^0_\pi(X)\supset
\pi^* \Pic(B).
\end{align}

Define now $\Pic^{rel}_\pi(X):=\Pic^0_{\pi}(X)/\pi^* \Pic(B)$ which will be the analogue of $[E]^\perp/[E]$.
Note that this is a discrete and finitely generated group by <ref>.
We will now consider the action of parabolic automorphisms on <ref>.

Suppose that $\gamma\in \Aut^\circ_\pi(X)$ is a parabolic automorphism.

If $L^0\in \Pic^0_{\pi}(X)^{sm}$, then we have that $\gamma^*L^0-L^0\in \pi^*\Pic(B)$.

If $L\in \Pic(X)$ then $\gamma^*L-L\in \Pic^0_\pi(X)$.

For part (i), note that adding a vertical divisor to $L^0$ doesn't affect the desired conclusion, and by <cit.> we can adjust $L^0$ by vertical divisors to assume that it is in fact in $\Pic^0_\pi(X)$, i.e. it has degree $0$ on each irreducible component of each vertical divisor.

For each fiber $X_b$ we have the Jacobian variety of degree $0$ line bundles $\Jac(X_b)$.
The restriction of the parabolic automorphism $\gamma$ to each fiber induces a group automorphism of $\Jac(X_b)$.
In the case of smooth fibers, and also the generic fiber, the induced automorphism on $\Jac(X_{k(B)})$ is the identity by the definition of $\Aut^\circ_{\pi}(X)$ in <ref>.
Therefore over the open locus $X^{\circ}\to B^{\circ}$ where the fibration is smooth, the line bundle is pulled back from the base.
So there exists some line bundle $L_B$ on $B$ (which we identify with a Cartier divisors) such that $\gamma^*L^0-L^0 - \pi^*L_B$ is linearly equivalent to a Cartier divisor supported on the singular vertical fibers (see also <ref>).
However, this divisor must have zero intersection number with any irreducible component of any fiber, since this holds for $\gamma^*L^0 - L^0$.
It follows that this integral divisor is a rational multiple of full singular fibers (recall that some singular fibers could have multiplicity) by <cit.>.
However, since we assumed that each singular fiber has at least one component of multiplicity $1$, it follows that the rational number is an integer and so the divisor is pulled back from $B$.

Part (ii) follows because the restriction of $L$ and $\gamma^*L$ to each irreducible component have the same degree, since $\gamma$ acts trivially on the set of irreducible components.

§.§.§ Assigning line bundles to automorphisms

Returning to <ref>, the quotient $\Pic(X)/\Pic_\pi^0(X)^{sm}$ is a cyclic subgroup of $\bZ$.
There is also a canonical choice among the two possible generators, given by the one whose multiple is in the image of an ample class from $\Pic(X)$.
Fix therefore $L\in \Pic(X)$ to be an ample line bundle projecting to the generator.
For an element $\gamma\in \Aut^\circ_\pi(X)$ we then set

\[
\xi(\gamma):=\gamma^* L - L \in \Pic^{rel}_\pi(X).
\]

where by <ref> <ref> the line bundle $\gamma^*L-L$ belongs to $\Pic^{0}_\pi(X)$, and we further consider its image under the quotient by $\pi^*\Pic(B)$.

The map $\xi$ in <ref> is independent of the choice of representative line bundle $L$ mapping to the generator of $\Pic(X)/\Pic^0_{\pi}(X)^{sm}$ and gives a group homomorphism:

\[
\xi\colon \Aut^{\circ}_\pi(X) \to \Pic^{rel}_\pi(X).
\]

Suppose that $L'$ is another such line bundle.
Then $L-L'\in \Pic^0_\pi(X)^{sm}$ and so by <ref><ref> it follows that $\gamma^*(L-L')-(L-L')\in \pi^* \Pic(B)$ so the map $\xi$ is independent of the choice of $L$.
To check the group homomorphism property, we must show that for two parabolic automorphisms $\gamma_1,\gamma_2$, the line bundles

\[
(\gamma_1^*L-L) + (\gamma_2^*L-L) \quad \text{ and }\quad
(\gamma_1^*\gamma_2^*)L - L
\]

give the same element after quotienting by $\pi^* \Pic (B)$.
We rewrite the second one as

\[
(\gamma_1^*\gamma_2^*)L - L = \gamma_1^*(\gamma_2^*L - L) + (\gamma_1^*L-L)
\]

and observe that since $\gamma_2^*L-L$ is an element of $\Pic^0_\pi(X)$, by <ref><ref> its image under $\gamma_1^*$ is equal to itself, modulo an element of $\pi^*\Pic(B)$.

§.§.§ A pairing valued in line bundles

The next construction appears in <cit.> in a slightly different notation.
We have a bilinear map

\[
\Aut^\circ_\pi(X)\times \Pic^{rel}_\pi(X) \xrightarrow{\eta_{\Pic}} \Pic(B)
\]

given by

\[
\pi^*\eta_{\Pic}(\gamma,L^0)= \gamma^* L^0 - L^0  \quad \text{ as line bundles on }X.
\]

In other words, the line bundle $\gamma^* L^0 - L^0$ is the pullback of a line bundle $\eta_{\Pic}(\gamma,L^0)$ from $B$.
Furthermore, if we change $L^0$ by an element of $\pi^* \Pic(B)$ the resulting line bundle on $B$ doesn't change.

§.§.§ Recollections on heights

We work with projective varieties which are not necessarily smooth.
Height functions are naturally associated to elements of the Picard group, i.e. line bundles.
For instance Serre <cit.> has a discussion of heights and their functorial properties for not necessarily smooth projective varieties and line bundles.

For a projective variety $X$, denote by $\Pic(X)$ its Picard group and by $\cH(X)$ the (additive) group of height functions on $X(\ov{k})$, so that we have a natural extension

\[
0 \to L^\infty\left(X(\ov{k});\bR\right)\into \cH(X) \onto \Pic(X)\to 0
\]

Our interest will be to find preferred choices of height functions for various subgroups of the Picard group.

§.§.§ Variation of canonical height

We recall briefly the theory developed by Silverman [64, 66, 65, 62]
and Tate [71] for the variation of canonical height in a family.
Assuming that the elliptic fibration has a section $\sigma\colon B\to X$, there are canonical –Tate heights on individual fibers $X_b$ and denoted $h^{can}_{X_b,\sigma}$.
Then the basic result is that for another section $\sigma'$ we have that

\[
h^{can}_{X_b,\sigma}(\sigma'(b)) =: h^{vcan}_{\sigma,\sigma'}(b)
\]

is a height function on $B$, which is associated to the line bundle $\eta_{\Pic}(\sigma'-\sigma,\sigma'-\sigma)$.
Note that the difference of the two sections induces a parabolic automorphism of the surface, and is also a line bundle on the surface which has degree $0$ on the vertical fiber.

§.§.§ Variation of canonical height as a pairing

The original statements of Tate and Silverman are concerned with elliptic fibrations with a section.
But this assumption can be relaxed to view the pairing as one between parabolic automorphisms and relative degree $0$ line bundles, without the need to introduce a section.
Indeed, fix a $\bQ$-line bundle $L$ which has degree $1$ when restricted to a generic smooth fiber of the fibration (this is the replacement of the section).
Fix some basepoint $x_b\in X_b$ in each fiber, so as to obtain a canonical height $h^{can}_{L\vert_{X_b},x_b}$.
Note that the canonical height thus defined depends on the choice of $x_b$ up to a constant.
Then we simply define

\begin{align}
\label{eqn:vcan_definition}
h^{vcan}_{\gamma,\xi(\gamma)} (b) :=
\lim_{n\to +\infty}
\frac{1}{n^2} h^{can}_{L\vert_{X_b},x_b}(\gamma^n x_b)
\end{align}

Note that this limit is independent of $x_b$ (we could independently replace each occurrence of $x_b$ by another $x_b'$ in the above limit without changing it).
Furthermore, this definition feeds directly into the telescoping argument of Tate to establish the existence of the limit and its properties.

Alternatively, we can work on the Jacobian surface $J_X\to B$.
The results of Silverman and Tate apply to it directly, and we do have fiberwise isomorphisms $J_{X,b}\isom X_{b}$ after some choice of basepoint in each fiber.
In particular the definition in <ref> of $h^{vcan}$ on $X$ translates directly to the Silverman–Tate definition on $J_X$.

The next result is the analogue of the current-valued pairing constructed in <ref> but in the setting of heights.
It follows from the previous discussion by “polarizing” the variation of canonical height for a single parabolic automorphism to a bilinear pairing between parabolic automorphisms.
For the proof, see <cit.>.

There exists a map

\[
\eta_h\colon
\Aut^\circ_\pi(X)
\times
\Pic^{rel}_\pi(X)
\to
\cH(B)
\]

which is linear in each coordinate and with the following properties.

* The map is compatible with the corresponding pairing on line bundles, namely $\eta_h(\gamma,L^0)$ is a height on $B$ associated to the line bundle $\eta_{\Pic}(\gamma, L^0)$.

* For an automorphism $\gamma\in \Aut^\circ_\pi(X)$ we have that $\eta_h(\gamma,\xi(\gamma))$ agrees with Silverman's variation of canonical height on $B$ and

\[
\eta_h(\gamma,\xi(\gamma)) \geq 0 \quad \text{ as a function on }B(\ov{k}).
\]

* The map is symmetric, i.e. if $\xi_i=\xi(\gamma_i)$

\[
\eta_h(\gamma_1,\xi_2) = \eta_h(\gamma_2,\xi_1)
\]

Essential for our construction will be the analogue of the existence, for a $(1,1)$-form $\alpha$ of integral $0$ on fibers, of a potential $\phi$ which makes it trivial on fibers, and in particular satisfies the identity:

\[
\gamma_{*}(\alpha + dd^c\phi)
(\alpha + dd^c \phi) + \pi^{*}\eta_h(\gamma,[\alpha])
\]

as in <ref>.
This analogue is the content of <ref>.

§.§ Relative dynamics in the elliptic fibration

§.§.§ Setup

For a smooth genus one curve $E$ over a field, we denote by $\sim$ or $\sim_E$ linear equivalence of divisors.
This will apply to both $X$ viewed as a genus one curve over the function field $k(B)$ and to the fibers $X_b$ for $b\in B(\ov{k})$ viewed as genus one curves over $k(b)$ (the residue field at $b$).

Since our genus one curves do not have a preferred basepoint, we will distinguish between addition for a group law and formal addition of divisors.
For a divisor $p^0$ of degree $0$ on $E$, and for any divisor $q$ on $E$ of degree $1$, the degree $1$ divisors $q\oplus p^0$ and $q\ominus p^0$ are well-defined and equal to the unique degree $1$ divisors linearly equivalent to $q\pm p^0$ where $\pm$ is used now as formal addition of divisors.
Throughout, the symbols $+$ and $-$ for divisors will mean formal addition of divisors, whereas $\oplus$ and $\ominus$ will mean translation by corresponding degree $0$ divisors.

§.§.§ Hyperbolic dynamics

Let $D\subset X$ be an effective divisor, with no vertical components (see <ref>), and such that $[D].[X_b]=d$ i.e. it intersects the fibers in degree $d\geq 1$.
Associated to $D$ we have the “multiplication by $(d+1)$ map” in the fibers, defined on the open set of smooth fibers $X^\circ$ as follows:

\begin{align}
\begin{split}
\label{eqn:delta_circ_def}
\delta_D^\circ \colon X^{\circ} &\to X^{\circ}\\
\delta_D^\circ(x) & \sim_{X_b} (d+1)\cdot x - (D\cap X_b) \qquad b = \pi(x)
\end{split}
\end{align}

In other words $\delta_D^\circ(x)$ is the unique point in the same fiber which is linearly equivalent to the degree $1$ divisor $(d+1)\cdot x - (D\cap X_b)$.
Note that if $D\cap X_b\sim_{X_b} d\cdot r_D(b)$, where $r_D(b)\in X_b(\ov{k})$ is well-defined up to $d$-torsion, then the map can also be expressed as $\delta_D^\circ(x)=x \oplus d(x-r_D(b))$, i.e. corresponds to multiplication by $d+1$ if we took $r_D(b)$ as the origin on $X_b$.
In particular, the map $\delta_D^\circ$ has degree $(d+1)^2$ on each fiber.
We will also consider the rational map

\[
\delta_D \colon X \dashrightarrow X
\]

induced by $\delta_D^\circ$ on the open locus.

Let us note that the map $\delta_D^\circ$ only depends on the class of $D$ in the Picard group of $X$ (hence the –Severi group for a K3 surface $X$), and furthermore does not change if we add a divisor pulled back from $B$ to the linear equivalence class of $D$ on $X$.

§.§.§ Parabolic dynamics

Let now $\gamma\colon X\to X$ be a parabolic automorphism in the sense of <ref>.
Our goal will be to understand the interaction between $\gamma$ and $\delta_D$.
On the smooth locus $X^\circ$ we have the identity

\[
\delta_D^\circ \left(\gamma(x)\right) = \gamma^{d+1}\left(\delta_D^\circ (x)\right)
\]

which it suffices to verify on each closed fiber $X_b$.
Let $\gamma_b$ be the degree $0$ divisor on $X_b$ such that $\gamma(x)=x \oplus \gamma_b$ in the fiber, for $b=\pi(x)$.
Then we have, using the formula for $\delta_D^\circ$, that

\begin{align*}
\delta_D^\circ \left(\gamma(x)\right)& \sim_{X_b} (d+1)(x+\gamma_b)- (D\cap X_b)\\
\gamma^{d+1}\left(\delta_D^\circ(x)\right)&
\sim_{X_b}
\big[(d+1)x- (D\cap X_b)\big] \oplus (d+1)\gamma_b
\end{align*}

and these divisors are clearly linearly equivalent on $X_b$.

§.§.§ A common space for the dynamics

In order to analyze the behavior of heights under the rational map $\delta_D$ on $X$ we need to pass to a completion on which to compare line bundles.
We therefore consider the graphs:

\begin{align*}
\Gamma(\delta^\circ_D)& :=
\{\left(x,\delta_D^\circ(x)\right)\colon x\in X^\circ\} \subset X^{\circ}\times X^\circ\\
\Gamma(\delta_D)& = \ov{\Gamma(\delta^\circ_D)} \subset X\times X
\end{align*}

with projections to each factor denoted $\pi_1$ and $\pi_2$ respectively.
Consider the map

\begin{align*}
\gamma_2\colon X\times X & \to X\times X\\
\gamma_2(x,y) & := \left(\gamma(x),\gamma^{d+1}(y)\right)
\end{align*}

The map $\gamma_2$ preserves $\Gamma(\delta_D)$ and induces an automorphism on it.

We already know that $\gamma_2$ preserves $\Gamma(\delta_D^\circ)$, since this is equivalent to the identity

\[
\left(\gamma(x),\delta_D^\circ(\gamma(x))\right) =
\left(\gamma(x),\gamma^{d+1}(\delta_D^\circ(x))\right)
\]

established in <ref>.
But $\gamma_2$ is a regular (invertible) map on $X\times X$ and hence preserves the closure $\Gamma(\delta_D)$ of $\Gamma(\delta_D^\circ)$.

§.§.§ Normalization of the graph

Denote by $\wtilde{X}$ the normalization of the graph $\Gamma(\delta_D)=:\ov{X}$.
The automorphism $\gamma_2$ extends to $\wtilde{X}$ since the normalization is functorial and we will denote it by $\tilde{\gamma}_2$.
The same applies to the projection $\pi_1,\pi_2$ on $\ov{X}$, but we will also denote the projection $\tilde{\pi}_2\colon \wtilde{X}\to X$ by $\tilde{\delta}_D$.

§.§.§ Singularities of $\wtilde{X}$

The normal surface $\wtilde{X}$ has a birational morphism to the smooth surface $X$, and since the singularities of $\wtilde{X}$ can be resolved by a birational morphism to $\wtilde{X}$, the singularities of $\wtilde{X}$ are called “sandwiched” (see [68] for more on sandwiched singularities).

Sandwiched (normal) singularities are rational (see [3] for the definition of rational singularities).
Indeed <cit.> says that singularities mapping birationally to rational ones are also rational, and smooth points are rational singularities. Furthermore, rational singularities on normal surfaces are in particular $\mathbb{Q}$-factorial <cit.>, so every Weil divisor on $\wtilde{X}$ has some multiple which is Cartier.

§.§.§ Action on line bundles

We now consider the dynamics of the two maps on line bundles.
Consider $L:=\cO(D)$ the line bundle on $X$ associated to $D$.
We also denote $L^0:=\gamma^*L - L$.

With notation as above:

* We have the isomorphism of line bundles on $\wtilde{X}$:

\[
\tilde{\delta}_D^{*}L = (d+1)^2 \pi_1^* L + E^{vert}
\]

where $E^{vert}$ is a vertical line bundle.

* After passing to a finite index subgroup of $\Aut^\circ_\pi(X)$, for any $\gamma$ and associated $\tilde{\gamma}_2$, the action of $\tilde\gamma_2$ on vertical divisors is trivial.

* For any $\gamma$ in the finite index subgroup of $\Aut^\circ_\pi(X)$ as in the previous part, we also have the isomorphism of line bundles on $\wtilde{X}$:

\[
\tilde{\delta}_D^{*}\left((\gamma^{d+1})^* L\right) = (d+1)^2 \pi_1^* \left(\gamma^*L\right) + E^{vert}
\]

where $E^{vert}$ is the same line bundle as in part (i).

Subtracting the result in part (i) from that in part (iii) yields:

There exists a finite index subgroup of $\Aut_\pi^\circ(X)$ such that any element $\gamma$ in it satisfies the identity of line bundles on $\wtilde{X}$:

\begin{align*}
\tilde{\delta}_D^{*}\left((\gamma^{d+1})^* L - L\right) & = (d+1)^2 \pi_1^* \left(\gamma^*L - L\right)\\
\intertext{
\quad or using the notation $L^0 = \gamma^*L-L$
\tilde{\delta}_D^{*}\left(L^0 + \gamma^*L^0 + \cdots + (\gamma^d)^* L^0\right)
& = (d+1)^2 \pi_1^* L^0
\end{align*}

Note that the second form of the expression telescopes to the first one.

For part (i), let $X_{k(B)}$ denote the (scheme-theoretic) generic fiber of $X$ over $k(B)$, in other words the associated elliptic curve over the function field $k(B)$.
We can restrict line bundles from $X$ to $X_{k(B)}$ and will do so without additional notation.
Note also that we can do the same on $\wtilde{X}$ and the associated elliptic curves are isomorphic (via the induced map): $\wtilde{X}_{k(B)}\toisom X_{k(B)}$.
Now the line bundle $\tilde\delta_D^* L - (d+1)^2\tilde\pi_1 L$ is trivial on $X_{k(B)}$, since $\tilde\delta_D$ induces a multiplication by $(d+1)$ map on the elliptic curve $X_{k(B)}$.
In particular the bundle has a trivializing section over $k(B)$, which gives a rational section on $\wtilde{X}$ which has only vertical poles and zeros (see also <cit.>).

For part (ii), observe that vertical divisors are permuted by the lifts of parabolic automorphisms $\tilde\gamma_2$, so they are fixed by a finite index subgroup of $\Aut^\circ_\pi(X)$.

For part (iii), apply the automorphism $\tilde\gamma_2$ to the identity in part (i).
This gives:

\[
\tilde\gamma_2^*\tilde\delta_D^{*}L = (d+1)^2 \tilde\gamma_2^*\pi_1^* L + \tilde\gamma_2^*E^{vert}
\]

But as maps we have the identities $\tilde\delta_D\circ \tilde\gamma_2 = \gamma^d \circ \tilde\delta_D$ (using that $\tilde\delta_D=\tilde\pi_2$ and $\tilde\gamma_2$ acts on $\wtilde{X}$) and $\pi_1\circ \tilde\gamma_2 = \gamma\circ \pi_1$.
Combined with the triviality of the action of $\tilde{\gamma}_2$ on vertical divisors, the claimed identity follows.

§.§ Preferred heights on elliptic fibrations

§.§.§ Setup

The heights we construct are not quite canonical, but they have some preferred properties.
In particular, they restrict to canonical heights on smooth fibers.

For the next result, recall that canonical height functions on elliptic curves depend on the choice of basepoint.
However, changing the basepoint only changes the canonical height function by a constant.
Therefore, we can speak of “a canonical height function up to a constant”.

Consider a line bundle $L^0\in \Pic^0_\pi(X)$, i.e. one that restricts to have degree $0$ on each irreducible component of the fibers of the fibration.
Then there exists a “preferred” height function $h^{pf}_{L^0}$ in its class, with the property that, when restricted to any smooth fiber $X_b$, the height function agrees, up to a constant depending on $b$, with a canonical height function of $L^0$ restricted to that fiber.

In particular, the restricted height function is affine for the “group law” on the genus one smooth fibers.
On the fibers where $\pi$ is not a smooth map, the height $h^{pf}_{L^0}$ can be taken to be identically $0$.

Note that preferred height functions are only unique up to pullback of a bounded function from $B$.
This is in analogy to the uniqueness of the potential $\phi$ from <ref> up to pullbacks of potentials from $B$.

Fix an ample divisor $D$ and associated line bundle $L$ as in <ref>.
Any line bundle $L^0$ as in the statement can be written as a linear combination of bundles of the form $\gamma^* L - L$, as $\gamma$ ranges over generators of $\Aut_\pi^\circ(X)$.
The desired conclusion is also preserved by taking linear combinations, so it suffices to prove the statement assuming that $L^0 = \gamma^*L - L$.
Furthermore, we can assume that $\gamma$ belongs to the finite index subgroup to which <ref> applies, so we have on $\wtilde{X}$ the identity of line bundles:

\begin{align*}
\tilde{\delta}_D^{*}\left(L^0 + \gamma^*L^0 + \cdots + (\gamma^d)^* L^0\right)
& = (d+1)^2 \pi_1^* L^0.
\end{align*}

Recall that we also have the identity

\[
\gamma^* L^0 = L^0 + \eta_{\Pic}(\gamma,L^0)
\]

where $\eta_{\Pic}$ is the pairing valued in line bundles from <ref>.
In particular $(\gamma^*)^i L^0 = L^0 + i \cdot \eta_{\Pic}(\gamma,L^0) $ so we can rewrite the above identity on $\wtilde{X}$ as

\begin{align*}
\tilde{\delta}_D^{*}\left((d+1)\cdot L^0
\frac{d(d+1)}{2}\eta_{\Pic}\left(\gamma,L^0\right)
\right)
& = (d+1)^2 \pi_1^* L^0\\
\intertext{and after cancelling a factor of $d+1$:}
\tilde{\delta}_D^{*}\left(L^0
\frac{d}{2}\eta_{\Pic}\left(\gamma,L^0\right)
\right)
& = (d+1) \pi_1^* L^0.
\end{align*}

Let now $h_{L^0}$ be an arbitrary height function on $X$ associated to $L^0$.
Recall also that we had a pairing valued in heights, compatible with the pairing valued in line bundles, denoted $\eta_h(\gamma,L^0)$.
Then on $\wtilde{X}$, using functoriality of heights, we obtain the identity of height functions

\begin{align*}
\tilde{\delta}_D^{*}
\left(
h_{L^0} +
\frac{d}{2}
\eta_h(\gamma,L^0)
\right)
& = (d+1) \pi_1^* h_{L^0} + O(1)\\
\end{align*}

where the error term $O(1)$ only depends on the choice of initial height function $h_{L^0}$, $\gamma$, and $D$.
Restrict this identity on the open locus $\wtilde X^\circ$, which can then be rewritten on $X^\circ$ to yield:

\[
h_{L^0} = \frac{1}{d+1}\left(
(\delta_D^{\circ})^* h_{L^0}
+ \frac{d}{2} \eta_h\left(\gamma,L^0\right)
\right)
+ O(1)
\]

This identity shows that if we consider the transformation which changes any height function associated to $L^0$ on $X$ by the formula

\begin{align}
\label{eqn:fixed_point_equation}
h_{L^0} \mapsto \frac{1}{d+1}\left(
(\delta_D^{\circ})^* h_{L^0}
+ \frac{d}{2} \eta_h\left(\gamma,L^0\right)
\right)
\end{align}

on the open locus $X^\circ$, then the resulting function on $X(\ov{k})$ is again a height function associated to $L^0$.
Note that the transformation does not do anything to the values of the height on the singular fibers.
This transformation is also visibly a contraction for the $L^\infty$-norm on heights on $X^\circ(\ov{k})$, by a factor of $\frac{1}{d+1}$.
Using the standard converging geometric series argument, it follows that if we start from some height function $h_{L^0}$, there is a corresponding height function $h^{pf}_{L^0}$ which is fixed by the above transformation.

It remains to observe that if we restrict $h^{pf}_{L^0}$ to any smooth fiber $X_b$, it is still a fixed point of the transformation in <ref>.
Note also that the value $\eta_h(\gamma,L^0)(b)$ is independent of the point on the fiber.
Consider then the function on $X_b(\ov{k})$ defined by $h_b:= h^{pf}_{L^0} - \frac{1}{2}\eta_h(\gamma,L^0)(b)$, which differs from $h^{pf}_{L^0}$ by a constant.
Observe that $h_b$ satisfies exactly the equation for the canonical height associated to the line bundle $L^0\vert_{X_b}$ and basepoint $r_D(b)$, where $r_D(b)\in X_b(\ov{k})$ is a point such that $d \cdot r_D(b)\sim D\cap X_b$.
Therefore $h^{pf}_{L^0}$ is (up to a constant) a canonical height function on each smooth fiber and thus satisfies the conclusion of the theorem.

For irreducible components of fibers where $\pi$ is not a smooth map, observe that they are all birational to $\bP^1$.
By assumption $L^0$ restricts to a bundle of degree $0$ on each such irreducible component, hence the induced height function is a bounded distance from the trivial height function $0$.

In the proof of <ref> above, we could have used any height in the class $\eta_{\Pic}(\gamma,L^0)$, not necessarily $\eta_{h}(\gamma,L^0)$.
This would have changed the preferred height $h^{pf}_{L^0}$ by a (bounded) function pulled back from $B$.
But the variation of canonical height does emerge dynamically as follows.

The statement of <ref> does not involve any parabolic automorphism, but we did use one to construct the preferred height.
Suppose now $\gamma'$ is another parabolic automorphism.
Then we have the formula:

\begin{align}
\label{eqn:preferred_heights_dynamics}
\left(\gamma'\right)^* h^{pf}_{L^{0}}	= h^{pf}_{L^{0}} + \eta_h(\gamma',L^{0})
\end{align}

and when $L^0 = \xi(\gamma)$ we will write $\eta_h(\gamma',\gamma)$ instead.
Indeed, it suffices to check <ref> pointwise, and for this we can use the fact that the preferred height is affine, for the group law on fibers.
Then, after picking in each fiber a basepoint, the result follows from <ref>.

§ BOUNDARY HEIGHTS, DIRECT CONSTRUCTION

§.§.§ Outline of section

We now use the same arguments that established the existence of canonical currents to produce canonical heights for arbitrary boundary classes.
Our main result on the existence of canonical heights is stated in <ref>.

In <ref> we collect the needed estimates needed to handle elliptic fibrations.
Then in <ref> we make everything “explicit” by fixing non-canonical representatives for the heights, analogously to choosing smooth differential forms to represent every cohomology class.
Then any other height function differs from the non-canonical representative by a bounded “potential” function on $X(\ov k)$.

§.§ Preliminary parabolic estimates

Using the preferred representatives of height functions for relative degree $0$ lines bundles constructed in <ref>, we can now establish some estimates for heights on general ample line bundles.

§.§.§ Preparations with line bundles

Fix a ($\bQ$-)line bundle $L$ on $X$ of degree $1$ on the fibers of the elliptic fibration, let $h_{L}$ be a height function for $L$, and let $\gamma$ be a parabolic automorphism.
Set $L^0 = \gamma^* L - L$, let $h_{L^0}:=\gamma^*h_L- h_L$, and let $h^{pf}_{L^0}$ be a preferred height function provided by <ref>.
The function $\phi\colon X\left(\ov k\right)\to \bR$ defined by $\phi:=h_{L^0}-h^{pf}_{L^0}$ is uniformly bounded.
We will make use of the following immediate formulas:

\begin{align*}
\gamma^* h_L & = h_L + h_{L_0}		&	\gamma^* h^{pf}_{L^0} & = h^{pf}_{L^0} + \eta_h(\gamma,L^0)\\
h_{L^0} & = h_{L^0}^{pf} + \phi 	&	(\gamma^i)^* h^{pf}_{L^0} & = h^{pf}_{L^0} + i\cdot \eta_h(\gamma,L^0)
\end{align*}

where $\eta_h$ is the height-valued pairing from <ref>.
Note that the formula for the pullback $\gamma^* h^{pf}_{L^0}$ follows from <ref>.
We abuse notation and view $\eta_h$ as giving heights not only on $B$, but also on $X$ by pullback along $\pi$.
The next result is a direct analogue of <ref>.

With notation as above, we have the identity of functions on $X(\ov{k})$:

\begin{align*}
\left(\gamma^i\right)^* h_{L^0} & = h_{L^0} + i\cdot \eta_h(\gamma,L^0) +
(\gamma^i)^* \phi - \phi\\
\left(\gamma^{n}\right)^* h_L & = h_L + n\cdot h_{L^0} + \tfrac{n(n-1)}{2}\eta_h(\gamma,L^0) + \\
& + \left[S_n(\gamma,\phi) - n\phi\right]
\end{align*}

where $S_n(\gamma,\phi):=\phi + \gamma^*\phi + \cdots + \gamma^{n-1}\phi$ is a Birkhoff sum associated to $\phi$ for the dynamics of $\gamma$.

The identity for $(\gamma^i)^*h_{L^0}$ follows by applying $(\gamma^i)^*$ to the identity $h_{L^0}=h_{L^0}^{pf}+\phi$ and using the simple iteration property for $h_{L^0}^{pf}$ established above.
To obtain the identity for $\left(\gamma^n\right)^* h_{L}$, we simply observe that (by induction, say)

\[
\left(\gamma^n\right)^* h_L = h_L + h_{L^0} + \cdots + \left(\gamma^{n-1}\right)^* h_{L^0}
\]

and sum the identity for $(\gamma^i)^* h_{L^0}$ to conclude.

We also need to establish an estimate combining several generators, in direct analogy with <ref>.

Suppose that $\gamma_1,\ldots,\gamma_k$ are parabolic automorphisms, $L$ is a fixed line bundle as before, we denote by $L_i=\gamma_i^*L - L$ and associated height functions $h_L, h_{L_i}$, and preferred height functions $h^{pf}_{L_i}$ and functions $\phi_i=h_{L_i}-h_{L_i}^{pf}$.

Then we have the identity

\begin{align*}
\left(\gamma_k^{n_k}\right)^*
\cdots
\left(\gamma_1^{n_1}\right)^* h_{L}
& =
h_L + \sum_{i=1}^k n_i h_{L_i} +
\\
& + \left[\sum_{i=1}^k S_{n_i}\left(
\gamma_i,
\cdots
(\gamma_{i+1}^{n_{i+1}})^* \phi_i
\right)
- n_i \phi_i\right]
\\
& + \left[
\sum_{i=1}^k \tfrac{n_i(n_i-1)}{2}\eta_h(\gamma_i,\gamma_i)
\sum_{1\leq i<j\leq k} n_i n_j \cdot \eta_h(\gamma_i,\gamma_j)
\right]
\end{align*}

where by abuse of notation we denoted by $\eta_h(\gamma_i,\gamma_j)$ the pullback to $X$ of the height function $\eta_h(\gamma_i,\xi(\gamma_j))$ on $B$.

We apply the earlier formula from <ref> for a single automorphism successively, but also taking into account that preferred heights satisfy the formula in <ref>.
In particular, for the heights $h_{L_i}$ and automorphisms $\gamma_j$, using the intermediate preferred heights (which disappear from the final statement), we see that:

\[
\left(\gamma_j^{n_j}\right)^* h_{L_i} =
h_{L_i} + n_j \cdot \eta_h(\gamma_j,\gamma_i) +
\left[
\left(\gamma_j^{n_j}\right)^{*}\phi_i - \phi_i
\right].
\]

Then the desired formula follows by induction on $k$.
The case $k=1$ is <ref>.
Now apply $(\gamma_k^{n_k})^*$ to each term in the formula for $k-1$ automorphisms to find:

\begin{align*}
\left(\gamma_k^{n_k}\right)^*
h_L  & = h_L + n_k h_{L_k} + \tfrac{n_k(n_k-1)}{2}\eta_h(\gamma_k,L_k) + \left[S_n(\gamma_k,\phi_k) - n_k\phi_k\right] \\
\left(\gamma_k^{n_k}\right)^* n_i h_{L_i} & =
n_i h_{L_i} + n_i n_k \cdot \eta_h(\gamma_k,\gamma_i) +
n_i \left[
\left(\gamma_k^{n_k}\right)^{*}\phi_i - \phi_i
\right]
\\
\left(\gamma_k^{n_k}\right)^* n_i \phi_i & = n_i \left(\gamma_k^{n_k}\right)^* \phi_i\\
\end{align*}

and the identity

\[
\left(\gamma_k^{n_k}\right)^* S_{n_i}\left(
\gamma_i,
\cdots
(\gamma_{i+1}^{n_{i+1}})^* \phi_i
\right) =
\gamma_i,
\cdots
(\gamma_{i+1}^{n_{i+1}})^* \phi_i
\right)
\]

which follows because the automorphisms $\gamma_\bullet$ commute with each other.
Note also that parabolic automorphisms preserve any height functions pulled back from $B$, so the desired identity follows.

Let us set $\gamma=\gamma_k^{n_k}\cdots \gamma_1^{n_1}$ so that we can write a bit more concisely the expression in <ref> as:

\begin{align}
\label{eqn:large_iterate_grouped_terms}
\begin{split}
\gamma^* h_{L}
& =
h_L + \sum_{i=1}^k n_i \left(h_{L_i} + \tfrac{1}{2}\eta_h(\gamma_i,\gamma_i)\right) + \frac{1}{2}\eta_h(\gamma,\gamma)+
\\
& + \left[\sum_{i=1}^k S_{n_i}\left(
\gamma_i,
\cdots
(\gamma_{i+1}^{n_{i+1}})^* \phi_i
\right)
- n_i \phi_i\right]
\end{split}
\end{align}

where, in comparison to similar expressions in <ref>, we have already normalized $L$ to have degree $1$ on fibers.

§.§ Heights from a basis

§.§.§ Setup

In order to establish the existence of canonical heights associated to every point on the boundary of the ample cone, we fix a basis for the –Severi group of $X$ and choose once and for all representatives for the height functions of the basis elements.
Extended by linearity, we have for every $\bR$-line bundle $L$ a height function $h_{L}^{lin}$.
For any other height function $h_L$ associated to the same line bundle there is a unique function $\phi(h_L)$ in $L^\infty\left(X(\ov{k})\right)$ such that we have

\begin{align}
\label{eqn:height_noncanonical_potential}
h_L = h_{L}^{lin} + \phi(h_L)
\end{align}

directly analogous to <ref>.

§.§.§ Change of height under automorphism

For an element $\gamma\in \Aut(X)$ we then have the basic formula

\[
\gamma^* h_L^{lin} = h^{lin}_{\gamma^*L} + \phi(\gamma^*h^{lin}_{L})
\]

which relates to height functions associated to the line bundle $\gamma^*L$, namely $h_{\gamma^*L}^{lin}$ and $\gamma^* h_{L}^{lin}$.
Let us now iterate this formula when applying several automorphisms:

\begin{align*}
(\gamma_n\cdots \gamma_1)^* h^{lin}_L & =
\gamma_1^*\cdots \gamma_n^* h^{lin}_L =
\gamma_1^*\cdots \gamma_{n-1}^* \left(h^{lin}_{\gamma_{n}^*L} + \phi\left(\gamma_{n}^*h^{lin}_{L}\right)\right)\\
& = \gamma_1^*\cdots \gamma_{n-2}^*
\left(h^{lin}_{\gamma_{n-1}^*\gamma_{n}^*L} + \phi\left(\gamma_{n-1}^*h^{lin}_{\gamma_n^* L}\right) + \gamma_{n-1}^*\phi\left(\gamma_{n}^*h^{lin}_{L}\right)\right)\\
& = \cdots\\
& = h^{lin}_{\gamma_1^* \cdots \gamma_n^* L} + \sum_{i=1}^{n} \gamma_1^*\cdots \gamma_{i-1}^{*}\phi\left(\gamma_{i}^* h^{lin}_{\gamma_{i+1}^*\cdots\gamma_n^* L}\right)
\end{align*}

In other words, we have that

\begin{align*}
\phi\left(\gamma_1^*\cdots \gamma_n^* h_L^{lin}\right) & =
\sum_{i=1}^{n} \gamma_1^*\cdots \gamma_{i-1}^{*}\phi\left(\gamma_{i}^* h^{lin}_{\gamma_{i+1}^*\cdots\gamma_n^* L}\right)
\end{align*}

which is the direct analogue of <ref>, except that we now pull back line bundles, instead of pushing forward metrics.

We record the estimates that we will need.
The next proposition simply replaces in <ref> the class $[\omega]$ by $[L]$ and the representative $(1,1)$-form $A([\omega])$ by the representative height $h_{L}^{lin}$.
It uses the normed space $L^\infty(X(\ov{k}))$ instead of $C^0(X(\bC))$.

Consider a parabolic automorphism $\gamma=\gamma_k^{n_k}\cdots \gamma_1^{n_1}$ written as a product of generators and $[L]$ a parabolically normalized class.

There exists a constant $C_2=C_2(E)$ giving the estimate on potentials:

\[
\norm{
\frac{\phi\left(\gamma^* h_L^{lin}\right)}{M(\gamma^* [L])
\frac{\phi\left(\eta_h(\gamma,\gamma)\right)}{M\left(\eta(\gamma,\gamma)\right)}
\leq
\frac{C_2}{1 + \norm{\gamma}_{\NS}}
\]

We also have the estimate

\[
\norm{
\phi\left(\gamma^* h_L^{lin}\right)
\leq C_3 \left(1+\norm{\gamma}_{\NS}^2\right)\norm{[L]}
\]

which holds for arbitrary $[L]$.

The proof is word for word the same as for <ref>, with the replacements indicated above.

§.§.§ Adapted generators

We will use the generators for $\Aut(X)$ fixed in <ref>, and keep the notation $\leqapprox$ to denote inequalities that hold up to constants that only depend on the fixed background geometric objects.
For any such generator $\gamma$, we have the basic inequality

\begin{align}
\label{eqn:operator_norm_bound_heights}
\norm{\phi(\gamma^* h_L^{lin})}_{L^\infty}\leqapprox \norm{\gamma}_{op}  \cdot M(L)
\end{align}

where as before $M(L)$ is the mass of $L$ relative to a fixed once and for all ample class, see <ref>.
For parabolic automorphisms this is <ref><ref>, and for the finitely many remaining automorphisms we pick a sufficiently large constant in $\leqapprox$ so that the bound continues to hold.

§.§ Existence of boundary heights

Here is the main result of this section, in analogy with <ref>.
Recall that $\partial^\circ \Amp_c(X)$ is the blown-up boundary of the ample cone, as in <ref>, and $\cH(X)$ denotes the space of heights on $X$.

There exists an assignment of height functions

\[
h^{can}\colon \partial^\circ \Amp_c(X) \to \cH(X)
\]

compatible with the map from $\partial^\circ \Amp_c(X)$ to $\partial \Amp(X)\subset \NS(X)$, and from heights $\cH(X)$ to $\NS(X)$.
The map has the following additional properties:

The map $h^{can}$ is $\Aut(X)$-equivariant.

For a fixed $p\in X\left(\ov k\right)$, the function

\begin{align*}
\partial^\circ \Amp_c(X) & \xrightarrow{\alpha\mapsto h^{can}_\alpha(p)}\bR_{\geq 0}
\end{align*}

is continuous.

On the real projective $(\rho-2)$-spaces in $\partial^\circ \Amp_c(X)$ associated to rational boundary rays corresponding to elliptic fibrations, the height function $h^{can}$ coincides with Silverman's variation of canonical height pairing:

\[
h^{can}(\lambda[E]\times [\xi])=\lambda\cdot \frac{\eta_{[E],h}([\xi],[\xi])}{\norm{\xi}_{\NS}^2} \text{ for }[\xi]\in \bP([E]^{\perp}/[E]).
\]

where $\eta_{[E],h}$ is the height-valued pairing from <ref> associated to the genus one fibration given by $[E]$.

For boundary classes $\alpha_\pm$ expanded/contracted by hyperbolic automorphisms of $X$, the heights $h^{can}_{\alpha_\pm}$ coincide with Silverman's canonical heights [61].

Fix an ample line bundle $L$, height function $h_L$, and mass function (on cohomology classes) $M$ as in <ref>.
For $p\in X(\ov{k})$ and a sequence of automorphisms $\gamma_i\in \Aut(X)$ as in <ref>, such that $L_i:=\gamma_1^*\cdots \gamma_i^* L$ satisfies $\frac{1}{M(L_i)}L_i \to \alpha\in \partial^\circ \Amp_c(X)$, we have that

\[
h^{can}_\alpha(p) = \lim_{i\to \infty} \frac{1}{M(L_i)}\gamma_1^* \cdots \gamma_i^* h_L(p)
\]

and the limit exists.
In particular $h^{can}_\alpha(p)\geq 0$ for any $p\in X\left(\ov k\right)$ and any $\alpha \in \partial^\circ \Amp_c(X)$.

Furthermore, property <ref> characterizes $h^{can}$, as does the combination of properties <ref>, <ref>, together with either <ref> or <ref>.

It would be interesting and useful to have a criterion that uniquely characterizes $h^{can}$ in terms of its equivariance and positivity properties.
It is reasonable to expect that the uniqueness of positive currents in <ref>, due to Sibony–Verbitsky in the archimedean case, also holds in the non-archimedean case.
Then the canonical height functions $h^{can}_{\alpha}$ for irrational $\alpha$ would be the unique adelic height function with subharmonic potentials at each place.

Let us note that if a mapping $h^{can}$ satisfies the continuity property in <ref>, and say <ref> or <ref>, then it is unique and satisfies <ref> automatically.
Indeed, the expanded/contracted rays of hyperbolic automorphisms are dense on the boundary, as are the parabolic classes, so together with continuity they uniquely determine the values of other heights.
Together with uniqueness of canonical heights for hyperbolic automorphisms, the equivariance property from <ref> then follows.
So it suffices to construct $h^{can}$ by the process described in <ref> and check that it yields, for a fixed $p\in X(\ov{k})$, continuous functions on $\partial^\circ \Amp_c(X)$.

The existence of the limit

\[
h^{can}_\alpha(p) =
\lim_{i\to \infty} \frac{1}{M(L_i)}\gamma_1^* \cdots \gamma_i^* h_L(p) =
\lim_{i\to \infty} \frac{1}{M(L_i)}h_L(\gamma_i\cdots \gamma_1 p)
\]

follows from arguments word for word identical to those in <ref>, with the following replacements.
Instead of $C^0(X(\bC))$-bounds, use $L^\infty\left(X(\ov{k})\right)$, and instead of $A([\omega])$ use $h^{lin}_{L}$.
Continuity, for fixed $p$ and varying point on the boundary, follows analogously.

To be more specific, the bounds used in the proof of <ref> for currents are listed in <ref>.
Besides the ones that involve masses of cohomology classes in <ref>, which continue to hold in the present case for the same reason, the bound in <ref> holds in the case of heights by <ref>.

The uniform boundedness of the sequence $\frac{\phi\left(h_{L}^{lin}(\gamma_i\cdots \gamma_1 p)\right)}{M(\gamma_1^*\cdots \gamma_i^* L)}$ (with bounds independent of $p\in X(\ov{k})$) follows just like in <ref> for potentials.
Similarly, the Cauchy property for such sequences is proved just like in <ref> and the same estimates imply continuity of $h_{[\alpha]}^{can}(p)$ at an irrational $[\alpha]$ for fixed $p$.

The continuity of the height function at irrational points uses the bound in <ref><ref>, just like in <ref>.

§ SUSPENSION SPACE CONSTRUCTION

§.§.§ Outline of section

It is natural to wonder if the constructions of canonical currents, and heights, could be formulated in terms of the geodesic flow for an appropriate hyperbolic manifold, instead of the more direct, combinatorial construction in <ref> and <ref>.
We describe below the candidate space and some of its properties, but remark that because of the need to consider the blown up boundary of the ample cone, and non-uniqueness of the associated currents and heights for boundary classes corresponding to elliptic fibrations, this more intrinsic approach to the construction is not directly applicable.

We establish some results from ergodic theory and formulate a number of questions.
Some of the constructions in this section will be taken up in future work.

§.§ Suspension space construction

§.§.§ Setup

We return to the constructions in <ref> and discuss how they interact with the geometry of the K3 surface.
Specifically, let $X$ be a K3 surface satisfying the standing assumptions in <ref>.
Let $N:=\NS(X)$ be its –Severi group, and set $\bbG:=\SO(N)$ to be the orthogonal group preserving the intersection pairing on $N$.
We will denote by $G:=\bbG(\bR)^+$ the connected component of the identity, and by $\Gamma:=\Aut(X)\cap G$ the corresponding subgroup.
By our standing assumptions $\Gamma$ is a lattice in $G$.

Denote by $\wtilde\cM:=\Amp^1(X)$ the ample classes of volume $1$, which we view as the space of Ricci-flat metrics on $X$ normalized to volume $1$ (and in the –Severi part).
The quotient $\cM:=\leftquot{\Gamma}{\wtilde{\cM}}$ is then naturally the moduli space, and its frame bundle is $\cQ := \leftquot{\Gamma}{G} $ after we choose a basepoint in $\wtilde\cM$ with a fixed frame in its tangent space.
We have a right action of $G$ on $\cQ$, a natural object in homogeneous dynamics.

§.§.§ Suspended space

Consider now the space $\wtilde{\cX}:=G\times X$, equipped with an action of $G$ on the right by only acting on the first factor by right multiplication, and a left (diagonal) action of $\Gamma=\Aut(X)$ on both factors.
We can then form

\[
\leftquot{\Gamma}{\wtilde{\cX}}
=: \cX \to \cQ =
\leftquot{\Gamma}{G}
\]

which is $G$-equivariant on the right.
This is a nonlinear analogue of constructing a flat vector bundle from a representation of the fundamental group.

§.§.§ Basic ergodic theory

If a group $H$ acts on a space $Y$, an $H$-invariant measure $\mu$ is ergodic if there does not exist a decomposition $Y=Y_1\coprod Y_2$ into measurable $H$-invariant sets with $\mu(Y_1)\neq 0 \neq \mu(Y_2)$.
We will be interested in the situation when $Y$ is a manifold and carries a topology, in which case orbit closures $\ov{H\cdot y}$ for $y\in Y$ are to be understood in the ordinary, Euclidean topology (even if $Y$ also has a Zariski topology).

§.§.§ The groups of interest

Besides the full semisimple group $G$, we will be interested in the dynamics of the following subgroups:

* A parabolic subgroup denoted $P\subset G$.

* A diagonal (Cartan) subgroup $A\subset P$ whose elements will be denoted $g_t$, and we view as generating the geodesic flow on $\cQ$.

* The maximal unipotent subgroup $U\subset P$, or subgroups $U'\subset U$.

We fix but do not specify the necessary data (a Cartan subalgebra and Weyl chamber) needed to define these groups.
For future reference, we also a maximal compact subgroup $K\subset G$, determined by a fixed basepoint in the ample cone, and $M:=K\cap Z_K(A)$ the centralizer of $A$ inside $K$.

§.§ G-dynamics

In this section, we begin by recalling some elementary facts from dynamics.
Then we proceed to formulate some questions that we find natural and of interest.

§.§.§ Correspondence principle

It is immediate to check from the definitions that the following objects give equivalent data:

* A finite $\Gamma$-invariant measure on $X$.

* A finite $G$-invariant measure on $\cX$.

* A locally finite measure on $G\times X$ which is invariant by $G$ on the right, and by $\Gamma$ on the left.

An analogous discussion applies to establish an equivalence between closed $\Gamma$-invariant sets on $X\times G/H$ and closed $H$-invariant sets in $\cX$, for general closed subgroups $H\subseteq G$.
The next result is an immediate corollary of Cantat's <cit.> (see also Wang [75]):

Suppose that the K3 surface, with assumptions as in <ref>, has an elliptic fibration.

Then any ergodic $G$-invariant measure on $\cX$, or closed $G$-invariant set, is obtained by suspension as in <ref> from $X$ and one of the following:

* A finite $\Aut(X)$-invariant set, with uniform measure.

* A totally real $\Aut(X)$-invariant submanifold in $X$, equipped with a smooth measure in the Lebesgue class of the submanifold.

* All of $X$ equipped with $\dVol_X$.

§.§.§ Consequences of Howe–Moore theorem

Recall that the Howe–Moore theorem <cit.> says that if the semisimple Lie group $G$ has a unitary representation on a Hilbert space, and there are no $G$-invariant vectors, then there are no $H$-invariant vectors for any unbounded (i.e. with noncompact closure) subgroup of $G$.
As a consequence, if $G$ acts ergodically on a measure space, then so does any of its unbounded subgroups.
Since the measures in <ref> are $G$-ergodic, we obtain:

Fix the assumptions as in <ref> and let $\mu$ be one of the ergodic $G$-invariant measures.

Then for any noncompact Lie subgroup $H\subset G$ (e.g. $U,A$ or $P$) and $\mu$-a.e. $x\in \cX$ the orbit $H\cdot x$ is dense in the support of $\mu$.

§.§.§ Relation to $P$-dynamics

We recall here some results of Furstenberg <cit.>.
Fix a probability measure $\nu$ on $G$ which is “admissible”, i.e. some convolution power of it is absolutely continuous for Haar measure on $G$.
Suppose $Y$ is locally compact and second countable, with a continuous $G$-action.
Then results of Furstenberg, see <cit.> for precise statements, imply that there is a one-to-one correspondence between $\nu$-stationary measures on $Y$ and $P$-invariant measures on $Y$.
Recall that a $\nu$-stationary measure on $Y$ is a (probability) measure $\mu$ satisfying $\int_{G}\left(g_*\mu\right) d\nu(g)=\mu$.
One says that the action of $G$, or of $\Gamma$, on $Y$ is stiff if any stationary measure is actually invariant, i.e. $g_*\mu = \mu,\forall g\in G$, or for all $g\in \Gamma$ respectively.
Stiffness for $\Gamma$ is expected to hold in situations such as $\Aut(X)$ acting on $X$, see e.g. conjectures of Furstenberg [35].
In the homogeneous setting these were resolved by Benoist–Quint [9] quite generally and by Bourgain–Furman–Lindenstrauss–Mozes in the case of tori [5].
Analogous results in the nonlinear setting, closer to the one considered here, have been obtained by Brown–Rodriguez-Hertz [10] for group actions on smooth real surfaces, and by Eskin, Mirzakhani, and Mohammadi in the setting of dynamics [29, 28].
It is therefore natural to pose:

[$P$-invariant implies $G$-invariant]

If a measure $\mu$ on $\cX$ is $P$-invariant, must it be also $G$-invariant?

This is close to, but not equivalent to stiffness for the action of $\Gamma=\Aut(X)$ on $X$.
This stiffness has been established under some assumptions by Cantat–Dujardin [15].
Let us also note that an affirmative answer to <ref> implies stiffness for AC measures in the sense of <cit.>, by the same method as the proof of Thm. 4.2 in loc. cit. which identifies the boundaries of $\Gamma$ and $G$ when the measure on $\Gamma$ is AC.

§.§.§ $U$-dynamics

The discussion in <ref> illustrates that there is little essential difference between random walks on $\Aut(X)$ and the action of $P$ on $\cX$.
In the setting of the suspended space $\cX$, it is also meaningful to ask about the action of unipotent subgroups.
This cannot be, as far as we know, directly translated into properties of the random walk.
The most interesting challenge to us seems:
[$U$-invariant implies $P$-invariant]

Can one classify all $U'$-invariant measures, for $U'\subset G$ a unipotent subgroup?

Specifically, suppose that $\mu$ is a measure on $\cX$, invariant under $U$, and that projects to Haar measure $\dVol_\cQ$ on $\cQ$.
Is it true that $\mu$ must also be $P$-invariant?

On the homogeneous space $\cQ$, the classification of orbit closures and invariant measures for $U$, or one of its subgroups $U'$, follows from Ratner's work [56, 55], see also earlier work of Margulis [50].
Using this classification of invariant measures and sets for $U'$ on $\cQ$,
<ref> is most interesting when the invariant measure, resp. set on $\cX$, when projected to $\cQ$ is $\dVol_{\cQ}$, resp. all of $\cQ$.
The possibilities for $U'$-invariant measures on $\cQ$ are homogeneous ones supported on closed $U''$-orbits, for $U''$ an intermediate unipotent subgroup between $U'$ and $U$, or homogeneous measures on $\leftquot{\Gamma\cap H}{H}$ where $H\subset G$ is a semisimple subgroup (yielding a totally geodesic hyperbolic submanifold in $\cM$).

The ergodic invariant measures on $\cX$ projecting to homogeneous closed $U''$-orbits on $\cQ$, for $U''$ a unipotent subgroup, are immediately classified.
They must be homogeneous measures on tori of dimension $0$, $1$, or $2$, contained in a single fiber of the corresponding elliptic fibration.

§.§ A-dynamics

§.§.§ Setup

We now look in more detail at the dynamics of the subgroup $A$ generating the geodesic flow on $\cM$.
To introduce some notation, let $\pi\colon \cX\to \cQ$ denote the projection.
Given a probability measure $\mu$ on $\cQ$, we will be interested in probability measures on $\cX$ that project to $\mu$.
The space of all such measures will be denoted $\pi_{*}^{-1}(\mu)$, it is compact in the weak topology.

§.§.§ The centralizer of $A$

Recall from <ref> that $K$ is a maximal compact and $M$ is the centralizer in $K$ of the $A$-action.
For this section only, all measures on $\cX$ and $\cQ$ will be assumed to be $M$-invariant, or equivalently we work on the spaces $\cX/M$ and $\cQ/M$ and only consider the right action of $A$.
For example, $\cQ/M=\leftrightquot{\Gamma}{G}{M}$ is the unit tangent bundle of $\cM$, denoted from now on $T^1\cM$, and the right action of $A$ gives the geodesic flow.

§.§.§ The space of geodesics

The stabilizer of an oriented hyperbolic geodesic in $\wtilde{\cM}$ is the group $MA$, with $A$ acting as translation along the geodesic and $M$ acting as a rotation along the axis of the geodesic.
Therefore $G/MA$ is naturally identified with the space of geodesics in $\wtilde{\cM}$.
Furthermore, by the correspondence principle in <ref>, a $\Gamma$-invariant locally finite measure on $G/MA$ is equivalent to an $A$-invariant locally finite measure on $\leftrightquot{\Gamma}{G}{M}=:T^1\cM$.
We will only work with finite $A$-invariant measures on $T^1\cM$.

§.§.§ Bruhat cell and boundaries

If we denote by $U^{\pm}$ the stable and unstable unipotent subgroups associated to $A$, then the embedding

\[
\rightquot{G}{MA} \into \rightquot{G}{MAU^+}\times \rightquot{G}{MAU^-}
\]

identifies the space of oriented geodesics with the space of points on the boundary of $\wtilde{\cM}$ that are transverse.
Equivalently, this is the open Bruhat cell on the right-hand side <cit.>.

Let us denote the image by $\partial^{(2)}\wtilde{\cM}$ and observe that it consists of pairs of transverse points on the boundary:

\[
\partial^{(2)}\wtilde{\cM} =
\Big\lbrace
(l_+, l_-)\in \bP(N)^2 \colon
l_+^2 = 0 =  l_-^2 \text{ and } l_+\wedge l_-\neq 0
\Big\rbrace
\]

where $l_{\pm}$ are isotropic lines in the –Severi group.
We can also make the identification with pairs of actual cohomology classes $([\eta_+],[\eta_-])\in N\times N$ satisfying the condition $[\eta_+]\wedge[\eta_-]=1$ modulo the equivalence $([\eta_+],[\eta_-])\sim \left(e^\lambda[\eta_+],e^{-\lambda}[\eta_-]\right)$ for $\lambda\in \bR$ and $[\eta_\pm]$ in the boundary of the ample cone.

Finally, let $\partial^{(2)}_{irr}\wtilde{\cM}$ denote the subset of $\partial^{(2)}\wtilde{\cM}$ consisting of pairs of rays both of which are irrational.

Suppose that $\mu$ is a finite $A$-invariant measure $T^1\cM$.
Then $\Gamma$-invariant measure on $G/MA$ associated to $\mu$ by the correspondence principle and denoted $\mu_{G/MA}$, satisfies:

\[
\mu_{G/MA}\left(\partial^{(2)}\wtilde{\cM}\setminus \partial^{(2)}_{irr}\wtilde{\cM}\right) = 0
\]

or in other words, it is carried by the irrational directions.

The set excluded from $\partial^{(2)}_{irr}\wtilde{\cM}$ consists of pairs of the form $[E]\times (\bP(N)\setminus [E])$ and those obtained by flipping the two factors, where $E$ is an integral isotropic vector.
Since there are countably many such sets, arguing by contradiction it would follow that one such set has positive $\mu_{G/MA}$-measure.

But then going back through the correspondence principle construction, it would follow that $\mu$ has infinite mass in the cusp associated to $[E]$.

Note that there are locally finite $A$-invariant but infinite measures on $T^1\cM$, such as those supported on a geodesic that goes between two cusps of $\cM$.

§.§.§ Canonical lifts

Let now $\mu$ be an ergodic, $A$-invariant probability measure on $T^1\cM$.
Using the equivariant family of currents on the boundary of the ample cone, we construct a lift $\mu_{\cX}$ of $\mu$ to an $A$-invariant measure on $\cX$.
First, observe that as a consequence of <ref>, which equivariantly assigns closed positive currents to each boundary class, we have:

There exists a $\Gamma$-equivariant measurable map

\[
\mu_{mme}\colon \partial^{(2)}_{irr}\wtilde{\cM}
\to
\cP(X)
\]

were $\cP(X)$ denotes the probability measures on $X$.

We can associate to a pair of cohomology classes $[\eta_\pm]$ the currents $\eta_{\pm}$ provided by <ref>, and define

\[
\mu_{mme}([\eta_+],[\eta_-]):=\eta_+\wedge \eta_-
\]

where the wedge-product is well-defined since each of $\eta_\pm$ has continuous potentials and is positive, while the normalization $[\eta_+]\wedge[\eta_-]=1$ ensures we get a probability measure, and the scaling equivariance of the canonical currents implies the measure is independent of the choice of representatives for the pair of cohomology classes.

The map $\mu_{mee}$ is measurable since it is the restriction to a measurable subset of a continuous map.

Let $\mu$ be an $A$-invariant probability measure on $T^1\cM$.
Then there exists a measure $\mu_\cX$ on $\cX/M$ which pushes forward to $\mu$ and which is also $A$-invariant (and $M$-invariant), with fiberwise conditional measures given by <ref>.

Indeed, $\mu$ gives rise to the $\Gamma$-invariant measure $\mu_{G/MA}$ on $G/MA$, and this space is identified with $\partial^{(2)}\wtilde{\cM}$.
Furthermore, by <ref> the measure $\mu_{G/MA}$ is supported on the irrational points, and we can form

\[
\mu_{G/MA}\otimes \mu_{mme} \text{ on }\partial^{(2)}\wtilde{\cM}\times X
\]

which is $\Gamma$-equivariant, and by the correspondence principle again yields an $A$-invariant measure on $\cX/M$.

A subsequent work will establish that $\mu_{\cX}$ has relative entropy $1$ for $A$, i.e. that the difference between the entropy of $\mu$ and that of $\mu_{\cX}$ is $1$.
Furthermore $1$ is the maximum relative entropy among lifts of $\mu$, and $\mu_{\cX}$ is unique with this property.

This puts together and generalizes the Gromov–Yomdin theorem [37, 78] which says that the entropy of a hyperbolic automorphism $\gamma$ is equal to the spectral radius of $\gamma^*$ acting on $H^{1,1}$.
Indeed, we can take $\mu$ to be the probability measure supported by the closed geodesic corresponding to $\gamma$ and observe that for a flow such as $g_t$ (generating $A$) the relation between entropies of different elements is $h(g_t)=|t|h(g_1)$, and the spectral radius of $\gamma^*$ is precisely the length of the associated geodesic.

§.§ A consequence of homogeneous dynamics

§.§.§ Setup

We keep the notation for Lie groups from the previous sections.
Our goal is to explain how rigidity theorems in homogeneous dynamics imply the following statement:

Let $[\alpha]\in \partial \Amp(X)$ be a point in the boundary of the ample cone.
Then either its orbit $\Aut(X)\cdot[\alpha]$ is dense in the ample cone, or $[\alpha]$ is proportional to an integral vector $[E]$ and its orbit is discrete.

By the correspondence principle <ref> and passing to finite-index subgroups, it suffices to show the same dichotomy for $H$-orbits on $\leftquot{\Gamma}{G}$, where $H$ is the stabilizer of an isotropic vector $[\alpha]\in \partial \Amp(X)$.
In our earlier notation, the subgroup is $H=MU^+$, where $M\subset K$ is the centralizer of $A$ and $U^+$ is a horospherical subgroup.
Note that $M$ also normalizes $U^+$, in particular it acts on $U^+$-orbits.

Now $U^+$ is a unipotent group, in fact a maximal horospherical subgroup, so Dani's theorem <cit.> (see also <cit.> where incidentally Raghunathan's conjectures are stated, that eventually became Ratner's theorems) applies to show that $U^+$-orbit closures are homogeneous under some connected Lie subgroup $L$ with $U^+\subseteq L\subseteq G$.
there exists $U'\subset U^+$ which is normal in $L$ such that $L/U'$ is reductive, and
$L$ is unimodular (so the $L$-orbit can admit a finite invariant measure).

If $L\subset MU^+$ with the closed orbit $\Gamma gL\subset \leftquot{\Gamma}{G}$, then since $gLg^{-1}\cap\Gamma$ in a lattice in $gLg^{-1}$, and by the structure of lattices in compact-by-unipotent groups, it follows that also $gU^+g^{-1}\cap\Gamma$ is a lattice in $gU^+g^{-1}$.
Therefore $gU^+g^{-1}$ stabilizes an integral isotropic vector $[E]$.
It follows that $gMg^{-1}$ also stabilizes $[E]$ and hence the orbit $\Gamma gH=\Gamma gU^+M$ is closed, and $\Gamma gL$ is contained in this closed orbit.

Because $G$ has real rank $1$, we now check that the only other possibility is $L=G$.
It suffices to look at the Lie algebra decomposition

\[
\frakg = \fraku^+ \oplus (\frakm \oplus \fraka) \oplus \fraku^-
\]

Suppose that the Lie algebra of $L$, denoted $\frakl$, is not entirely contained in $\fraku^+ \oplus \frakm$, but contains $\fraku^+$.
Considering the Lie algebra $\fraku'\subset \fraku^+$ such that $\frakl/\fraku$ is reductive, we see that we must have $\frakl/\fraku'=\fraku^{+}{}''\oplus \frakm'' \oplus \fraka''\oplus \fraku^{-}{}''$, where the $''$ parts come from the corresponding subspaces in the decomposition.
By the assumption that $\frakl$ is not contained in $\fraku^+\oplus \frakm$ it follows that $\fraka''\neq 0$ (if $\frakl$ only had something in $\fraku^-$, we could still obtain elements in $\fraka$ by commutators with $\fraku^+$).
Thus in fact $\fraka\subset \frakl$, since $\dim \fraka = 1$.
Furthermore $\fraku^{-}{}''\neq 0$ since otherwise $\frakl$ would not be unimodular (it would be almost a parabolic, up to some compact parts in $\frakm$).
Finally, an explicit calculation with commutators of matrices shows that by taking commutators of a vector in $\fraku^{-}$ with arbitrary vectors in $\fraku^+$, and then subsequent commutators, we can generate first all of $\frakm$ and then all of $\fraku^-$.
So it must be that $\frakl=\frakg$ as claimed.

§ APPLICATIONS OF HEIGHTS

§.§.§ Outline of section

We now consider some elementary applications of the canonical heights constructed in <ref>.
In <ref> we construct an invariant of $\Aut(X)$-orbits of rational points.
This can be viewed as analogous to the invariant constructed by Silverman <cit.> for a single hyperbolic automorphism, which in his case is the product $\hat{h}^+(p)\cdot \hat{h}^-(p)$ of the canonical heights.

In <ref> we analyze the situation when a point has vanishing canonical height, for an irrational direction on the boundary.
It is established in <ref> that this is equivalent to the point having finite orbit under an appropriate sequence of automorphisms.
This can be seen as analogous to results of Silverman for a single automorphism, and to results about torsion points in elliptic fibrations for parabolic automorphisms.

§.§ Invariants associated to canonical heights

§.§.§ Setup

We maintain the conventions from <ref>.
Let $h^{can}\colon \partial^\circ \Amp_c(X)\to \cH(X)$ be the assignment of canonical heights constructed in <ref>.
We will denote by $[\alpha]$ an element of $\partial^\circ \Amp_c(X)$ and by $p$ an element of $X(\ov{k})$, and we will denote by $h^{can}([\alpha],p)$ the associated height.
This is $\Aut(X)$-equivariant, and also scales as $h^{can}(\lambda[\alpha],p)=\lambda\cdot h^{can}([\alpha],p)$.

We will also keep the Lie-theoretic notions introduced in <ref> and particularly the groups defined in <ref>.

§.§.§ The space of horocycles

The stabilizer in $G$ of a null-vector in $\NS(X)$ is the group $MN$, therefore we can identify in a $\Gamma$-equivariant manner $\partial \Amp(X)$ (the usual boundary of the ample cone) with $G/MN$.
Note that there is a well-defined right action of $A$, since $A$ normalizes $MN$, which commutes with the left action of $\Gamma$.
Under the identification with null-vectors, this is nothing but the scaling action of $\bR_+^\times$.
Our goal will be to associate invariants (for the left $\Gamma$-action) to functions that are homogeneous for the $A$-action.

Both $G$ and $MN$ are unimodular groups, therefore there is a $G$-invariant measure on $G/MN$, unique up to scaling.
Here is an explicit description in coordinates.
Take the quadratic form to be $x_0^2-x_1^2-\cdots - x_{\rho-1}^{2}$ and identify the boundary of the ample cone with $\bR^{\rho-1}\setminus 0$ via

\[
(x_1,\ldots,x_{\rho-1})\mapsto \left(\left(x_1^2+\cdots + x_{\rho-1}^2\right)^{\tfrac 12},x_1,\ldots,x_{\rho-1}\right)
\]

and the square root is the positive one.
Then the differential form

\begin{align}
\label{eqn:invariant_measure}
\frac{dx_1\wedge\cdots\wedge dx_{\rho-1}}{\left(x_1^2+\cdots +x_{\rho-1}^2\right)^{\tfrac 12}}
\end{align}

is clearly $\SO_{\rho-1}(\bR)$-invariant, and if we check that it is invariant also under $A$ which only acts on the first two coordinates by

\[
\begin{bmatrix}
\end{bmatrix}
\mapsto
\begin{bmatrix}
\cosh(t)x_0 + \sinh(t)x_1\\
\sinh(t)x_0 + \cosh(t)x_1
\end{bmatrix}
\]

the invariance under the full group $G$ follows (since $G$ admits the Iwasawa, or $KAK$ decomposition).
However, identifying the denominator in <ref> with $x_0$, an immediate calculation gives that $\frac{dx_1\wedge \cdots \wedge dx_{\rho-1}}{x_0}$ is invariant by the above transformation and the result follows.
To check the invariance one needs to use that $x_0 dx_0 = \sum_{i=1}^{\rho-1} x_i dx_i$ and the above differential form can also be obtained as the residue of $\frac{dx_0\wedge\cdots\wedge dx_{\rho-1}}{x_0^2-x_1^2 - \cdots - x_{\rho-1}^2}$ along the null hypersurface.

§.§.§ The null quadric

We can also view the boundary of $\Amp^1(X)$ as the null quadric $Q\subset \bP(\NS(X))$ in the projectivized –Severi group.
Now the projective space $\bP:=\bP\left(\NS(X)\right)$ carries the tautological bundle $\cO(-1)$.
Then the null vectors are the total space of $\cO(-1)\vert_Q \to Q$, and the boundary of the positive cone is one of its two connected components (all over $\bR$).
The quadric itself is given as the vanishing locus of a section of $\cO_{\bP}(2)$ so its canonical bundle, by the adjunction formula, is $\cO_{\bP}(-\rho+2)\vert_{Q}$.

§.§.§ An invariant of $\Gamma$-orbits

Let us now identify the irrational rays on $Q$ with the irrational part of the boundary $\partial^{\circ}_{irr}\Amp(X)$ and denote this set, and its rays, by $Q_{irr}$ and $\cO(-1)\vert_{Q_{irr}}$.
The case $\rho=3$ is special and we don't need to restrict to irrational rays, since then the boundaries of interest agree, see the discussion in <ref>.
In general, for the constructions we make below, ignoring a set of Lebesgue measure zero, such as that of irrational directions, will be irrelevant.

For a point $p\in X(\ov{k})$ the height function $h^{can}(-,p)$ gives a $1$-homogeneous function on $\cO(-1)\vert_{Q_{irr}}$, i.e. it provides us with a measurable section of $\cO(1)\vert_Q$, let us call this section $s_p$.
Because the canonical bundle of $Q$ is $\cO(-\rho+2)$, if we raise $s_p$ to the power $-\rho+2$ we can then integrate it.
Recall that by our standing assumption $\rho\geq 3$.

For a point $p\in X\left(\ov k\right)$, the integral

\[
\int_{Q}s_p^{-\rho+2}=:h^{tot}(p)
\]

is independent of the choice of representative in the $\Gamma$-orbit, i.e. we have

\[
h^{tot}(p)=h^{tot}(\gamma p) \quad \forall \gamma \in \Gamma.
\]

In fact this holds for any $\gamma\in \Aut(X)$.

Furthermore, the invariant can be alternatively described as follows.
Consider the subset

\[
S_{p}:=\{[\alpha]\in \cO(-1)\vert_{Q_{irr}}\colon h^{can}([\alpha],p)\leq 1\}
\]

Then up to a constant $c(\rho)$ that only depends on $\rho$, we have that

\[
\Vol(S_p) = c(\rho)\cdot \frac{1}{h^{tot}(p)}
\]

where $\Vol$ is a fixed $G$-invariant volume on the boundary of the ample cone, e.g. the one in <ref>.

The boundaries of the sets $S_p$ are depicted in <ref> in blue, using computer simulations.

The $\Gamma$-invariance of $h^{tot}(p)$ follows from the $G$-invariance of the operation of integration of differential forms, i.e. $\int_{Q}s_p^{-\rho+2}$.
In fact this is further invariant under the larger group of orthogonal transformations preserving the positive cone (this group has two connected components) but it contains the entire automorphism group $\Aut(X)$.

For the relation between the volume of $S_p$ and $h^{tot}(p)$, it suffices to remember only the measurable section $s_p$.
Then if we change $s_p$ to $\lambda s_p$, then $\left(\int_Q s_p^{-\rho+2}\right)$ rescales by $\lambda^{-\rho+2}$.
Similarly, the set $S_p$ changes to $\frac{1}{\lambda}S_p$ and its volume changes by $\frac{1}{\lambda^{-\rho+2}}$.
Both assignments are $G$-invariant, and clearly given locally on $Q$ by integration, so they must be the same up to some fixed constant that depends on the $G$-invariant measure fixed on $G/MN$.

Recall that in [61], starting from a hyperbolic automorphism $\gamma$ with expanded/contracted directions $[\alpha_{\pm}]$, and given a point $p\in X(\ov{k})$, Silverman defined the quantity $h^{can}([\alpha_+],p)\cdot h^{can}([\alpha_-],p)$ and verified that it is an invariant associated to the $\gamma$-orbit of $p$.
His construction corresponds to the critical case $\rho=2$ in the above constructions.
The space of sections of $\cO(1)\vert_{Q}$ is $2$-dimensional and identified naturally with $\NS(X)$, and it admits a $G$-invariant quadratic form of signature $(1,1)$.
Silverman's invariant is the quadratic form applied to the section $s_p$.

§.§ Small points and bounded orbits

§.§.§ Setup

Suppose that $\gamma\in \Aut(X)$ is a hyperbolic automorphism and $[\alpha_+]$ is the expanded direction.
Silverman in [61] proved that a point $p\in X\left(\ov k\right)$ satisfies $h^{can}([\alpha_+],p)=0$ if and only if $p$ is periodic for $\gamma$ (and hence also $h^{can}([\alpha_-],p)=0$ if $[\alpha_-]$ denotes the contracted direction).
We will now develop some analogous corollaries, both for the irrational boundary points, and for the ones contained in the rational directions associated to elliptic fibrations.

§.§.§ The irrational boundary points

Suppose that $[\alpha]\in \partial^\circ\Amp_c(X)$ is an irrational direction.
Using the fixed generating set in <ref>, there is therefore a sequence of elements $\gamma_1,\gamma_2,\ldots$ among them such that the appropriately normalized basepoint ample class $[L]$ converges to $[\alpha]$ under the iterations $\gamma_1^*\cdots \gamma_n^*[L]$.
To a point $p\in X(\ov{k})$ we can associate the sequence of points $p_n:=\gamma_n\cdots \gamma_1 p$.
Observe that this sequence is, up to finite ambiguity, independent of the choice of generators.
Namely, if we picked a different set of generators $\gamma_{\bullet}'$, considered the new sequence $\gamma_i'$ associated to $[\alpha]$, and hence sequence of points $p_n':=\gamma_n'\cdots \gamma_1'p$, then there exists a finite set of automorphisms $\beta_1,\ldots,\beta_K$, depending only on the two fixed sets of generators, such that $p_n'=\beta_{\psi(n)}p_{\phi(n)},\forall n$ for appropriate $\phi(n),\psi(n)$.
Indeed this already holds at the level of group elements, i.e. $\gamma_n'\cdots \gamma_1' = \beta_{\psi(n)}\cdot \gamma_{\phi(n)}\cdots \gamma_1$.

Suppose that $[\alpha]\in \partial^\circ\Amp_c(X)$ is an irrational class and $p\in X\left(\ov k\right)$ is a point.
Then the canonical height satisfies $h^{can}([\alpha],p)=0$ if and only if the set of points $\{\gamma_n\cdots \gamma_1p\}_{n\geq 1}$ is finite, where $\gamma_i$ is the sequence of generators associated to $[\alpha]$ by <ref>.

Fix an ample class $[L]$ with $[L]^2=1$ and a height function $h_{[L]}$ in this class.
Then the converse direction is immediate: if the set of points is finite, then the set of heights $h_{[L]}(p_n)$ is finite, so the limit of $\frac{1}{M(\gamma_1^*\cdots \gamma_n^*[L])}h_L(p_n)$, which defines the canonical height, is clearly zero.
To show that the vanishing of the height implies the set of points is finite, we will show that the set of heights $h_{[L]}(p_n)$ is bounded, with a bound independent of $n$, so the result will follow by the Northcott property of heights associated to ample line bundles.

$[L]$-normalized representatives.
Suppose that $[\alpha_+]$ satisfies $[\alpha_+]^2=0$ and $[\alpha_+].[L]=\tfrac 12$; call such classes $[L]$-normalized isotropic.
Then there exists a unique $[L]$-normalized and isotropic class $[\alpha_-]$, which we will also denote by $op_{[L]}([\alpha_+])$ such that $[L]=[\alpha_+]+[\alpha_-]$.
We can view $[\alpha_-]$ as the “opposite” point on the boundary to $[\alpha_+]$, with respect to the interior basepoint $[L]$.
We then have the following family of heights associated to $[L]$, namely:

\[
h^{can}_{[\alpha_+]} + h^{can}_{op_{[L]}([\alpha_+])} \text{ as }[\alpha_+] \text{ ranges over $[L]$-normalized isotropic classes.}
\]

When $[\alpha_+]$ or its opposite land at a rational point, we can take an arbitrary height $h^{can}$ corresponding to that elliptic fibration.
Then by the analogue for heights of <ref>, it follows that all of these heights are within a uniform constant of the fixed height $h_{[L]}$.
So it suffices to find for each $p_n$ an $[L]$-normalized isotropic class $[\alpha_{+,n}]$ (with $[\alpha_{-,n}]:=op_{[L]}([\alpha_{+,n}])$) such that $h^{can}([\alpha_{+,n}],p_n)+h^{can}([\alpha_{-,n}],p_n)$ is bounded by a constant independently of $n$.

Dynamics and choice of classes.
Set $g_n:=\gamma_1^*\cdots \gamma_n^*$ viewed as an orthogonal transformation in $\SO(\NS(X))$.
Then we have, using the equivariance property of canonical heights:

\begin{multline*}
h^{can}([\alpha_{+,n}],p_n)+h^{can}([\alpha_{-,n}],p_n)  = \\
= h^{can}([\alpha_{+,n}],\gamma_n\cdots\gamma_1p)+h^{can}([\alpha_{-,n}],\gamma_n\cdots \gamma_1 p)\\
= h^{can}(g_n[\alpha_{+,n}],p)+h^{can}(g_n[\alpha_{-,n}],p)
\end{multline*}

Let us now choose $[\alpha_{+,n}]$ such that $g_n[\alpha_{+,n}]$ is proportional to the class $[\alpha]$ for which we know that $h^{can}([\alpha];p)=0$.
Note that the constant of proportionality is going to be large (increasing with $n$), but the vanishing of the height still holds.
The boundedness of heights will then follow if we show that $g_n[\alpha_{-,n}]$ is a vector whose norm (relative the fixed basepoint $[L]$) is bounded independently of $n$.
The uniform boundedness of $g_n[\alpha_{-,n}]$ and ampleness of $[L]$ will imply $h_{L}\geqapprox h_{g_n[\alpha_{+,n}]}^{can}+O(1)$ and the result will follow, since $h(L;p)$ is a fixed constant.

Estimating the opposite vector.
The above argument has reduced the claim to an estimate in linear algebra.
Specifically, we have that

\begin{align}
\label{eqn:L_gnL}
\begin{split}
[L] & = [\alpha_{+,n}] + [\alpha_{-,n}]\\
g_n[L] & = e^{A_n}[\alpha] + g_n[\alpha_{-,n}] \quad \text{ so }g_n[\alpha_{n,+}]=e^{A_n}[\alpha]
\end{split}
\end{align}

by choice of classes $[\alpha_{\pm,n}]$.
The construction of the group element $g_n$ is such that, in hyperbolic space, the point $g_n[L]$ is a uniformly bounded distance away from the geodesic connecting $[L]$ with the boundary point determined by $[\alpha]$.
Let us call $[\alpha_+]:=[\alpha]$ and $[\alpha_-]:=op_{[L]}([\alpha])$.
Then the last geometric statement in hyperbolic space translates as follows.

Let $m_n:=\norm{g_n}$, for the norm relative to the fixed basepoint $[L]$.
Let $a_n\in \SO(\NS(X))$ be the element which acts by $[\alpha_\pm]\mapsto e^{\pm m_n}[\alpha_{\pm}]$ and the identity on the orthogonal complement.
Then the property that $g_n[L]$ is a bounded distance from the geodesic connecting $[L]$ and $[\alpha_+]$ is equivalent to the existence of $h_n\in \SO(\NS(X))$, of uniformly bounded norm, such that

\[
g_n = a_n h_n \quad \text{ as elements of }\SO(\NS(X)).
\]

Indeed, there exists an $h_n'$ of uniformly bounded size such that $g_nh_n'[L]=a_n[L]$ for the action on hyperbolic space (this is the statement about uniform distance to the geodesic) and so $a_n^{-1}g_nh_n'$ stabilizes $[L]$, hence must be in the corresponding compact subgroup:

\[
a_n^{-1} g_n h_n' = k_n \in \Stab_{[L]}(\SO(\NS(X)))
\]

Thus $g_n = a_n k_n \left(h_n'\right)^{-1}$ as claimed.

Consider now the action of $h_n$ on $[L]=[\alpha_+]+[\alpha_-]$:

\[
h_n[L] = c_n[\alpha_+] + d_n[\alpha_-] + v_n
\]

where, by boundedness of $h_n$, we see that $c_n, d_n$ and $\norm{v_n}$ are uniformly bounded above, and $v_n$ is orthogonal to the span of $[\alpha_{\pm}]$.
It follows that:

\[
g_n[L]=a_n h_n[L] = c_ne^{m_n}[\alpha_+] + d_n e^{-m_n}[\alpha_-]+v_n
\]

so it must be that $g_n[\alpha_{-,n}] = d_n e^{-m_{n}}[\alpha_-]+v_n$, which is of bounded norm, as required.

§.§.§ Heights along elliptic fibrations

Let now $[E]$ be the class of a fiber in a genus one fibration $X\xrightarrow{\pi}B$ and $p\in X\left(\ov k\right)$.
Set also $V({[E]}):=[E]^{\perp}/[E]$ viewed as a real vector space, with a $\bZ$-lattice inside, which we can identify with a finite-index subgroup of $\Aut_\pi(X)$ as we now do.
By <ref>, we have a quadratic form:

\[
h^{vcan}(-,p)\colon V([E])\to \bR
\]

which only depends on the projection $b:=\pi(p)\in B$, so we will write it as $h^{vcan}(-,b)$.
In fact, if $\Jac(X)\to B$ denotes the Jacobian fibration associated to the original genus one fibration, there is a natural specialization map $\Aut_{\pi}^{\circ}(X)\to \Jac(X)_{b}(k)$ (recall that we assume all automorphisms are already defined over $k$) and the above quadratic form is just the canonical height on the elliptic curve $\Jac(X)_{b}$ restricted to the image of $\Aut_{\pi}^{\circ}(X)$.
The next result collects several known facts, due to Silverman and Tate [71, 64].

* A parabolic automorphism $\gamma\in \Aut_\pi^{\circ}(X)\toisom V([E])$ has $h^{vcan}(\xi(\gamma);b)=0$ if and only if it is of finite order when restricted to the fiber $X_b$.

* The kernel of the quadratic form $h^{vcan}(-;b)$ is a rational subspace inside $V([E])$.

If $B\isom \bP^1$ then the kernel of the quadratic form can have dimension $\geq 1$ for only finitely many values of $b\in X(k)$.

Part (i) is immediate from the classical properties of canonical heights on the elliptic curve $\Jac(X)_b$, and part (ii) similarly identifies the kernel of the quadratic form with the kernel of the specialization map $\Aut_\pi^\circ(X)\to \Jac(X)_b(k)$.

Part (iii) follows from the results of Silverman and Tate, see <ref>.
Indeed, the height functions $h^{vcan}(v;-)\colon B\left(\ov{k}\right)\to \bR$, as $v$ varies in $V([E])$ subject to the normalization $[v]^2=-1$, are within uniform constants of each other, and in particular for a fixed height $h_{L_B}$ on $\cO_{B}(1)$, we will have that $h^{vcan}(v;b)+C\geq h_{L_B}(b)$ for a constant $C$ independent of $b$.
Therefore, the set of points $b\in B\left(\ov k\right)$ where $h^{vcan}(v;b)$ vanishes for some $v\in V([E])\setminus 0$ is of bounded height, and hence there are only finitely many such in $B(k)$.

Based on the results of Masser–Zannier [53, 52], it seems plausible also that the kernel of the quadratic form can have dimension $\geq 2$ for only finitely many values of $b\in X\left(\ov k\right)$.
Indeed, what they proved is that any specific $2$-dimensional subspace inside $V([E])$ can be in the kernel at only finitely many $b\in B\left(\ov k\right)$.

Note also that the result in <ref> of <ref> above is frequently only stated for a single section or automorphism of the elliptic fibration, but the proof of the slightly stronger statement, allowing for any linear combinations, is analogous.

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