2.1 Tuning stacks and tuning sheaves

Throughout we let K, C and $\mathcal {X}$ be as at the start of Section 2. Motivated by the tuning module of Yasuda–Wood [Reference Machett Wood and Yasuda74, Definition 3.3], we begin by defining the central object of this subsection.

Definition 2.1. Given $x\in \mathcal {X}(K)$ , we say that $(\mathcal {C},\overline {x},\pi )$ is a tuning stack for x if $\mathcal {C}$ is a normal Artin stack with finite diagonal, $\pi \colon \mathcal {C} \to C$ is a birational coarse space map, and the diagram

commutes. A morphism $(\mathcal {C}',\overline {x}',\pi ')\to (\mathcal {C},\overline {x},\pi )$ of tuning stacks for x is a map $f\colon \mathcal {C}'\to \mathcal {C}$ such that $\pi \circ f=\pi '$ and $\overline {x}\circ f=\overline {x}'$ . Finally, if $(\mathcal {C},\overline {x},\pi )$ is terminal among all tuning stacks, we say $\mathcal {C}$ is a universal tuning stack for x.

We show the existence of a universal tuning stack after some preliminaries.

The next result makes use of relative normalization for morphisms of stacks. We refer the reader to [Reference Meier and Ozornova52, Definition 5.3].

Proof. By definition of the relative normalization, f factors as $\mathcal {Y}\to \mathcal {Y}' := \underline {\operatorname {\mathrm {Spec}}}_{\mathcal {Z}} \mathcal {O}' \to \mathcal {Z}$ , where the sheaf $\mathcal {O}'$ is the integral closure of $\mathcal {O}_{\mathcal {Z}}$ in $f_* \mathcal {O}_{\mathcal {Y}}$ (i.e., the integral closure relative to the morphism of sheaves $\mathcal {O}_{\mathcal {Z}} \to f_* \mathcal {O}_{\mathcal {Y}}$ induced by the map f). Letting $Z\to \mathcal {Z}$ be a smooth cover by a scheme, we have a Cartesian diagram

where $\mathcal W$ may not be a scheme since we have not assumed f is representable. Since relative normalization commutes with smooth base change, $W'\to Z$ is the relative normalization of $\mathcal W\to Z$ . Since $W'\to \mathcal {Y}'$ is a smooth cover, to show normality of $\mathcal {Y}'$ it suffices to prove $W'$ is normal. We have therefore reduced to the case where $\mathcal {Z}$ is a scheme, which we will denote by Z.

We are now in the situation where $f\colon \mathcal {Y}\to Z$ and Z is a scheme. Notice that $\mathcal {Y}'\to Z$ is affine, and so $\mathcal {Y}'=Y'$ is a scheme. Since Z is a scheme, we know that f factors as $\mathcal {Y}\stackrel {\pi }{\longrightarrow } Y\stackrel {g}{\longrightarrow } Z$ , where $\pi $ is a coarse space map (which exists since $\mathcal {Y}$ has finite diagonal). By definition, $\mathcal {O}'$ is the integral closure of $\mathcal {O}_{Z}$ in $f_* \mathcal {O}_{\mathcal {Y}}=g_*\pi _*\mathcal {O}_{\mathcal {Y}}=g_*\mathcal {O}_Y$ where the last equality holds because $\pi $ is Stein. Thus, $Y'\to Z$ is the relative normalization of $Y\to Z$ . Since $\mathcal {Y}$ is normal, Y is as well so $Y'$ is normal by [66, Tag 035L].

We are now ready to show the existence of universal tuning stacks. We thank Martin Olsson for suggesting this construction.

Proof. We abusively refer to the extended map $U\to \mathcal {X}$ as x. By definition of the relative normalization, x factors as

$$\begin{align*}U\longrightarrow \mathcal{C} := \underline{\operatorname{\mathrm{Spec}}}_{\mathcal{X}} \mathcal{O}' \stackrel{\overline{x}}{\longrightarrow} \mathcal{X}, \end{align*}$$

where the sheaf $\mathcal {O}'$ is the integral closure relative to the morphism of sheaves $\mathcal {O}_{\mathcal {X}} \to x_* \mathcal {O}_U$ induced by the map x. Lemma 2.4 shows that $\mathcal {C}$ is normal. Since $\overline {x}$ is representable, integral and of finite type, it follows from [66, Tag 01WJ] that it is finite. Then finiteness of the diagonal for $\mathcal {C}$ follows from finiteness of the diagonal for $\mathcal {X}$ . Thus, $\mathcal {C}$ has a coarse space map $\pi \colon \mathcal {C}\to C'$ . Since $\mathcal {C}$ is normal, $C'$ is as well. The morphism $\overline {x}$ induces a map $q\colon C'\to C$ .

We next show that $\mathcal {C}\to C$ is an isomorphism over U. Consider the Cartesian diagram

Since relative normalization commutes with smooth base change, $\beta \colon \mathcal {C}_U\to \mathcal {X}_U$ is the relative normalization of $\beta \circ \alpha \colon U\to \mathcal {X}_U$ . Note that $\gamma \circ \beta \circ \alpha =\mathrm {id}_U$ is proper quasi-finite and $\gamma \colon \mathcal {X}_U\to U$ is separated, so $\beta \circ \alpha $ is proper quasi-finite, hence finite as it is representable. Thus, $\beta \alpha $ is integral so its relative normalization $\alpha \colon U\to \mathcal {C}_U$ is an isomorphism. As a result, $\gamma \circ \beta \colon \mathcal {C}_U\to U$ is an isomorphism.

Now that we have established $\mathcal {C}\to C$ is an isomorphism over U, it follows that $q\colon C'\to C$ is an isomorphism over U. So, q is a birational map of normal curves (or Dedekind schemes), hence an isomorphism. This shows that $\pi \colon \mathcal {C}\to C'\simeq C$ is a birational coarse space map, and hence $\mathcal {C}$ is a tuning stack.

Before turning to the claim concerning universality, we show that $\overline {x}\colon \mathcal {C}\to \mathcal {X}$ is independent of the choice of open subset U and extension $U\to \mathcal {X}$ of x. To see this, it suffices to show that if $i\colon V\to U$ is the inclusion of a nonempty open subset, then the relative normalizations of $x\colon U\to \mathcal {X}$ and $x\circ i\colon V\to \mathcal {X}$ are the same. Letting $\overline {x}\colon \mathcal {C}\to \mathcal {X}$ be the former normalization and $\overline {x}'\colon \mathcal {C}'\to \mathcal {X}$ be the latter one, by functoriality of the relative normalization we obtain a morphism $f\colon \mathcal {C}'\to \mathcal {C}$ of tuning stacks. Lemma 2.3 (3) shows f is an isomorphism.

To prove universality, let $(\mathcal {C}',\overline {x}',\pi ')$ be another tuning stack. By Lemma 2.3 (1), we need only show the existence of a map $f\colon \mathcal {C}'\to \mathcal {C}$ of tuning stacks. We let $\mathcal {C}'\longrightarrow \mathcal {C}"\stackrel {\overline {x}"}{\longrightarrow } \mathcal {X}$ be the relative normalization of $\overline {x}'$ . Since $\pi $ and $\pi '$ are birational, we can choose a nonempty open subset $U\subseteq C$ over which $\pi $ and $\pi '$ are isomorphisms. We have just showed that $\mathcal {C}$ is independent of the choice of U, so we have a commutative diagram

where we obtain the morphism $g\colon \mathcal {C}\to \mathcal {C}"$ (shown as a dotted arrow above) from the universal property of the relative normalization of $x\colon U\to \mathcal {X}$ . By Lemma 2.4, we know $\mathcal {C}"$ is normal. We also know that $\overline {x}"$ is representable, integral and of finite type, hence finite by [66, Tag 01WJ]. Then $\mathcal {C}"$ has finite diagonal, so it has a coarse space. Since $\pi '$ is an isomorphism over U, we see $\mathcal {C}"\to C$ is a coarse space which is an isomorphism over U; this follows from the same argument used to establish this fact for $\mathcal {C}\to C$ . So, $\mathcal {C}"$ is a tuning stack for x. Finally, Lemma 2.3 (3) shows that g is an isomorphism, and so $\mathcal {C}'\longrightarrow \mathcal {C}"\stackrel {g^{-1}}{\longrightarrow }\mathcal {C}$ is our desired map of tuning stacks.

2.2 Heights

We are now ready to give the definition of the height of a rational point on a stack (with respect to a given vector bundle). We define the height to be the degree of any associated tuning sheaf. The tuning sheaf is, in general, a vector bundle, so by degree we mean the degree of the top wedge power, which is now a line bundle (metrized in the arithmetic case) on C. We show that this is well defined in Proposition 2.13.

Definition 2.11. Let $\mathcal {X}$ be a stack over C, and let $K = K(C)$ . Let $\mathcal {V}$ be a vector bundle on $\mathcal {X}$ and $x\in \mathcal {X}(K)$ be a rational section. If $\mathcal {C}$ is any tuning stack for x and $\mathcal T_{x,\mathcal {V},\mathcal {C}}$ is the associated tuning sheaf, we let $\operatorname {\mathrm {ht}}_{\mathcal {V}}(x)=-\deg (\mathcal T_{x,\mathcal {V},\mathcal {C}})$ . In other words, the height of the rational point $x\in \mathcal {X}(K)$ with respect to $\mathcal {V}$ is

$$\begin{align*}\operatorname{\mathrm{ht}}_{\mathcal{V}}(x):=-\deg(\pi_*\overline{x}^*\mathcal{V}^{\vee}), \end{align*}$$

where $(\mathcal {C},\overline {x},\pi )$ is any choice of tuning stack for x.

If L is a finite extension of K, we can define the height of a point of $\mathcal {X}(L)$ by letting $C'$ be $\operatorname {\mathrm {Spec}} \mathcal {O}_L$ (if K is a number field) or the smooth projective curve with function field L (if K is a function field), and consider $\mathcal {X}' = \mathcal {X} \times _C C'$ , which carries a vector bundle obtained by pulling back $\mathcal {V}$ . Then we define the height of a point of $\mathcal {X}(L)$ to be the height of the corresponding point of $\mathcal {X}'(L)$ .

At this point, we need to comment on a piece of notation. When C is a curve over a finite field k, the degree of a divisor $D= P_1 + \cdots + P_r$ on C is understood to be $\sum _i \log |k_{P_i}|$ , where $k_{P_i}$ is the residue field of the closed point $P_i$ . In particular, $\deg D$ does not lie in $\mathbf {Z}$ but in $(\log q)\mathbf {Z}$ , where $q = |k|$ . This choice of notation is most natural in a context, as here, where we want to write down theorem statements and arguments which treat the case of number fields and function fields at once. The reader who wants to work in the context where C is a curve over a fixed finite field k and avoid the number field case is free to take heights to be integers, which just modifies everything in this paper by a multiplicative factor of $\log q$ .

The reader may wonder why the height is defined as the negative of the degree of a bundle obtained from $\mathcal {V}^\vee $ , rather than as the degree of a bundle obtained from $\mathcal {V}$ itself. The answer is that, in cases arising naturally, the heights as defined here will typically be bounded below (Northcott property) while a height defined to be $\deg (\pi _*\overline {x}^*\mathcal {V})$ will often take values unbounded both above and below, or only bounded above (Southcott property).

Another natural question: Why do we not define the height of x as $\deg _{\mathcal {C}} \overline {x}^* \mathcal {V}$ (where degree is defined in Definition B.5), which would be more similar to the usual definition? The main reason is that, as we shall see, $\deg _{\mathcal {C}} \overline {x}^* \mathcal {V}$ is identically zero for many choices of $\mathcal {X}$ and nontrivial $\mathcal {V}$ (e.g., for any line bundle on $BG$ ). Nonetheless, this function will play a key role for us (it will differ from $\operatorname {\mathrm {ht}}_{\mathcal {V}}(x)$ by local terms supported on the stacky locus of $\mathcal {C}$ , as we will see in § 2.3), so we give it a name here.

Definition 2.12. Let $\mathcal {X}$ , $\mathcal {V}$ and K be as in Definition 2.11. Then stable height $\operatorname {\mathrm {ht}}^{\operatorname {\mathrm {st}}}_{\mathcal {V}}(x)$ is defined by

$$ \begin{align*} \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\mathcal{V}}(x) = -\deg_{\mathcal{C}} \overline{x}^* \mathcal{V}^{\vee} \end{align*} $$

for any choice of tuning stack $\mathcal {C}$ .

We justify the name ‘stable height’ in Proposition 2.14 below. When x is an integral point of $\mathcal {X}$ , we may take C itself to be the tuning stack; in this case, $\pi $ is the identity and $\operatorname {\mathrm {ht}}(x)$ and $\operatorname {\mathrm {ht}}^{\operatorname {\mathrm {st}}}(x)$ agree.

In fact we show more: Not only the height, but the isomorphism class of the tuning sheaf is independent of the choice of tuning stack.

Proof. Let $(\mathcal {C},\overline {x},\pi )$ be the universal tuning stack for x whose existence we have shown in Proposition 2.5. By the universal property, there exist unique morphisms $f_i\colon \mathcal {C}_i\to \mathcal {C}$ of tuning stacks. Thus, we reduce immediately to the case where $\mathcal {C}_1$ is universal and $f\colon \mathcal {C}_2 \to \mathcal {C}_1$ is a map of tuning stacks. Now, let

$$\begin{align*}\mathcal{C}_2 \to \underline{\operatorname{\mathrm{Spec}}}\, f_*\mathcal{O}_{\mathcal{C}_2} \to \mathcal{C}_1 \end{align*}$$

be the Stein factorization. Then $\operatorname {\mathrm {Spec}} f_*\mathcal {O}_{\mathcal {C}_2} \to \mathcal {C}_1$ is a birational, finite, representable map with normal codomain and hence an isomorphism by Zariski’s main theorem.

In particular f is Stein (i.e. the map $\mathcal {O}_{\mathcal {C}_1} \to f_*\mathcal {O}_{\mathcal {C}_2}$ is an isomorphism). Then for any vector bundle $\mathcal {W}$ on $\mathcal {C}_1$ ,

$$\begin{align*}\mathcal{W} \simeq \mathcal{O}_{\mathcal{C}_1} \otimes_{\mathcal{O}_{\mathcal{C}_1}} \mathcal{W}\simeq f_*\mathcal{O}_{\mathcal{C}_2} \otimes_{\mathcal{O}_{\mathcal{C}_1}} \mathcal{W}\simeq f_*f^*\mathcal{W}, \end{align*}$$

where the third isomorphism is the projection formula. Applying $\pi _{1*}$ to the above isomorphism with $\mathcal {W} = \overline {x}_1^*\mathcal {V}^{\vee }$ , we see $\pi _{1*}\overline {x}_1^*\mathcal {V}^{\vee }\simeq \pi _{2*}\overline {x}_2^*\mathcal {V}^{\vee }$ and so height is independent of the choice of tuning stack. (In the Arakelov case, we note that the tuning stacks are all birational so that the metric does not change.) Independence of the stable height follows from Lemma B.9 applied to $f_i$ .

The justification for the name ‘stable height’ is as follows. As we shall see, the height $\operatorname {\mathrm {ht}}_{\mathcal {V}}(x)$ does not behave well under ramified base change. That is: If $L/K$ is a finite extension, and $x_L$ the point of $\mathcal {X}(L)$ obtained by composing $x\colon \operatorname {\mathrm {Spec}} K {\rightarrow } \mathcal {X}$ with the structure map $p\colon \operatorname {\mathrm {Spec}} L {\rightarrow } \operatorname {\mathrm {Spec}} K$ , the relationship between $\operatorname {\mathrm {ht}}_{\mathcal {V}}(x)$ and $\operatorname {\mathrm {ht}}_{\mathcal {V}}(x_L)$ is not in general very transparent. For example, if $\mathcal {X} = BG$ and $x \in \mathcal {X}(K)$ corresponds to a Galois extension $L/K$ with Galois group G, then $\operatorname {\mathrm {ht}}_{\mathcal {V}}(x_L) = 0$ , but $\operatorname {\mathrm {ht}}_{\mathcal {V}}(x) \neq 0$ in general. For stable height, by contrast, the situation is much as we are used to from heights on schemes.

Proposition 2.14 (Stable height is stable under base change).

With $\mathcal {X}$ , $\mathcal {V}$ , x and $x_L$ as above, and $L/K$ is a separable extension, then

$$ \begin{align*} \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\mathcal{V}}(x_L) = [L:K] \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\mathcal{V}}(x). \end{align*} $$

Proof. If L is a number field, then let $C'=\operatorname {\mathrm {Spec}}\mathcal O_L$ ; if L is a function field, then let $C'$ be the projective normal curve with function field L. Let $\mathcal {C}$ be a tuning stack for $x_K$ . Then the normalization $\mathcal {C}'$ of the fiber product $\mathcal {C} \times _{C} C'$ is a tuning stack for $x_L$ , and we compute that

$$ \begin{align*} \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\mathcal{V}}(x_L) := \deg \overline{x}_L^*\mathcal{V} = \deg g \cdot \deg \overline{x}^*\mathcal{V}= [L:K] \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\mathcal{V}}(x), \end{align*} $$

where g is the projection $\mathcal {C}' \to \mathcal {C}$ and the middle inequality is Lemma B.9.

When $\mathcal {X}$ is a scheme, we can take $\mathcal {C} = C$ and so stable height and height are the same. More generally, height agrees with stable height whenever the vector bundle $\mathcal {V}$ is pulled back from a vector bundle on a scheme.

Proposition 2.15. Suppose $f\colon \mathcal {X} {\rightarrow } Y$ is a morphism over C, where Y is a scheme. Let V be a vector bundle on Y. Then, for all $x \in X(K)$ ,

$$ \begin{align*} \operatorname{\mathrm{ht}}_{f^* V}(x) = \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{f^* V}(x). \end{align*} $$

Proof. Let $\mathcal {C}$ be a tuning stack for x, and let $\overline {x}\colon \mathcal {C} {\rightarrow } \mathcal {X}$ be an extension of x to $\mathcal {C}$ . The map $f \circ \overline {x}\colon \mathcal {C} {\rightarrow } Y$ factors as $g \circ \pi $ for some $g\colon C {\rightarrow } Y$ , by the universal property of the coarse space. So the vector bundle $\overline {x}^* f^* V$ can be written as $\pi ^* g^* V$ . Noting that duality commutes with pullback, we now have

$$ \begin{align*} \operatorname{\mathrm{ht}}_{f^* V}(x) = - \deg_C \pi_* \pi^* g^* V^\vee \end{align*} $$

and

$$ \begin{align*} \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{f^*V}(x) = -\deg_{\mathcal{C}} \pi^* g^* V^\vee = -\deg_{C} g^* V^\vee \end{align*} $$

(where the last equality follows from Lemma B.9 since $\deg \pi = 1$ ). The result now follows from the fact that for any bundle W on C,

$$\begin{align*}W \cong \mathcal{O}_C \otimes_{\mathcal{O}_C} W \cong \pi_*\mathcal{O}_{\mathcal{C}} \otimes_{\mathcal{O}_C} W \cong \pi_*\pi^*W; \end{align*}$$

the last isomorphism is the projection formula, and the middle follows since the coarse map is Stein [Reference Rydh61, Theorem 6.12].

Remark 2.16. Similarly, if $f\colon \mathcal {X} \to \mathcal {Y}$ is a morphism of stacks and $\mathcal {V}$ is a vector bundle on $\mathcal {Y}$ , then for any $x \in \mathcal {X}(K)$ ,

$$\begin{align*}\operatorname{\mathrm{ht}}_{f^* \mathcal{V}}(x) = \operatorname{\mathrm{ht}}_{\mathcal{V}}(f \circ x) \end{align*}$$

since a tuning stack for x is also a tuning stack for $f \circ x$ .

Definition 2.17. We say that a vector bundle $\mathcal {V}$ on $\mathcal {X}/K$ satisfies the Northcott property if for every finite extension $L/K$ and every integer B,

$$\begin{align*}\{x \in \mathcal{X}(L) \colon \operatorname{\mathrm{ht}}_{\mathcal{V}}(x) \leq B\} \end{align*}$$

is finite.

This definition is slightly unsatisfactory because it will be too lenient for some choices of $\mathcal {X}$ . For instance, if $\mathcal {X}$ is a curve of genus at least $2$ , it has finitely many points over every global field, so under this definition the Northcott property will be satisfied by every vector bundle. In the present paper, however, we will almost always be considering stacks $\mathcal {X}/K$ which have infinitely many K-rational points. Under such circumstances we expect $\mathcal {V}$ to satisfy the Northcott property if $\mathcal {V}$ is ‘positive enough’, which we demonstrate through several examples; see Section 3. (Be warned, however, that the Northcott vector bundles do not form a cone in any sense. For instance, it is possible that a line bundle $\mathcal {L}$ is Northcott but positive multiples $\mathcal {L}^{\otimes n}$ of it are not; the nontrivial line bundle on $B{\mu _2}$ has this property.) It is in order to ensure that natural examples exhibit the Northcott property that we use $\mathcal {V}^{\vee }$ rather than $\mathcal {V}$ when defining height.

Definition 2.18. Let $\mathcal {X}$ , $\mathcal {V}$ and K be as in Definition 2.11, with $\mathcal {V}$ Northcott. We define the counting function associated to $\mathcal {V}$ and K to be

$$\begin{align*}N_{\mathcal{V},K}(B) := \#\{x \in \mathcal{X}(K) \colon \operatorname{\mathrm{Ht}}_{\mathcal{V}}(x) \leq B\}. \end{align*}$$

2.3 Computing heights: local discrepancies

We now turn to the problem of practical computation of heights of points on stacks.

As above, let C be the spectrum of the ring of integers of a number field or a smooth curve over a finite field, let K be the fraction field of C and let $\mathcal {X}$ a normal proper Artin stack over C with finite diagonal. Let $\mathcal {V}$ be a vector bundle on $\mathcal {X}$ , where we recall once again that, if K is a number field, $\mathcal {V}$ is a metrized vector bundle, as defined in § A.4.

Let $x\colon \operatorname {\mathrm {Spec}} K {\rightarrow } \mathcal {X}$ be a rational point of $\mathcal {X}$ , let $\mathcal {C}$ be a tuning stack, $\pi \colon \mathcal {C} {\rightarrow } C$ the coarse moduli map and $\overline {x}\colon \mathcal {C} {\rightarrow }\mathcal {X}$ an integral extension of x.

By Definition 2.11, the height of x is

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{V}}(x) = -\deg \pi_* \overline{x}^* \mathcal{V}^{\vee}, \end{align*} $$

and by Definition 2.12 we have

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{V}}^{\operatorname{\mathrm{st}}}(x) = -\deg \overline{x}^* \mathcal{V}^{\vee}. \end{align*} $$

Our goal in this section is to study the difference between height and stable height. To this end, we recall the natural map of vector bundles on $\mathcal {C}$

(2.20) $$ \begin{align} \pi^* \pi_* \overline{x}^* \mathcal{V}^{\vee} {\rightarrow} \overline{x}^* \mathcal{V}^{\vee} \end{align} $$

whose cokernel is a sheaf $M(\overline {x}^* \mathcal {V}^{\vee })$ on $\mathcal {C}$ with trivial generic fiber. This map is the counit of adjunction and we claim that it is injective. Indeed, we can check injectivity locally and assume that C is affine, in which case $\pi _* \overline {x}^* \mathcal {V}^{\vee } = \Gamma (\overline {x}^* \mathcal {V}^{\vee })$ , and the map (2.20) is thus the inclusion

$$\begin{align*}\Gamma(\overline{x}^* \mathcal{V}^{\vee}) \otimes_{\mathcal{O}_C}\mathcal{O}_{\mathcal{C}} \to \overline{x}^* \mathcal{V}^{\vee} \end{align*}$$

of global sections.

Let $C'$ be a smooth proper curve (or in the arithmetic case, $\operatorname {\mathrm {Spec}} \mathcal {O}_L$ for some étale algebra $L/K$ ) endowed with a finite flat surjection $p\colon C' {\rightarrow } \mathcal {C}$ whose degree we denote by m; such a $C'$ exists by Proposition B.3. The sheaf $p^* M(\overline {x}^* \mathcal {V}^{\vee })$ is now a generically trivial and finitely generated sheaf on $C'$ , which is to say it is a finite abelian group with the structure of an $\mathcal {O}_{C'}$ -module. It follows from Proposition B.10 and exactness of $p^*$ that

$$ \begin{align*} \log |p^* M(\overline{x}^* \mathcal{V}^{\vee})| & = \deg p^* \overline{x}^* \mathcal{V}^{\vee} - \deg p^* \pi^* \pi_* \overline{x}^* \mathcal{V}^{\vee} \\ & = m(\deg \overline{x}^* \mathcal{V}^{\vee} - \deg \pi^* \pi_* \overline{x}^* \mathcal{V}^{\vee}) \\ & = m(\operatorname{\mathrm{ht}}_{\mathcal{V}}(x) - \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\mathcal{V}}(x)). \end{align*} $$

Now, $p^* M(\overline {x}^* \mathcal {V}^{\vee })$ is a finite $\mathcal {O}_{C'}$ -module and as such has a canonical decomposition as a finite direct sum $\oplus _v p^* M(\overline {x}^* \mathcal {V}^{\vee })_v$ , where v varies over non-Archimedean places of $C'$ .

Definition 2.21. With all notation as above, the local discrepancy $\delta _{\mathcal {V};v}$ is defined as

$$ \begin{align*} \delta_{\mathcal{V};v}(x) = \frac{1}{m} \log |p^* M(\overline{x}^* \mathcal{V}^{\vee})_v|. \end{align*} $$

We thus arrive at the formula

(2.22) $$ \begin{align} \operatorname{\mathrm{ht}}_{\mathcal{V}}(x) = \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\mathcal{V}}(x) + \sum_v \delta_{\mathcal{V};v}(x). \end{align} $$

One can think of the structural information imparted by equation (2.22) as follows. The height $\operatorname {\mathrm {ht}}_{\mathcal {V}}$ is a nonadditive function which changes under field extensions and lacks a canonical decomposition into local terms. However, it canonically decomposes into two pieces; one of which, $\operatorname {\mathrm {ht}}^{\operatorname {\mathrm {st}}}_{\mathcal {V}}$ , is additive and stable under field extensions, while the other, $\sum _v \delta _{\mathcal {V};v}$ , canonically decomposes into local terms. These good features of the summands often make it manageable to compute them individually.

Concretely, we may think of local discrepancy as follows. Write $K_v$ for the completion of K at v. Define $L_v = K_v \otimes _C C'$ so that $L_v$ is an étale algebra over $K_v$ . We can thus write $C^{\prime }_v = \operatorname {\mathrm {Spec}} \mathcal {O}_{L_v}$ . Choose an identification of $\overline {x}^* \mathcal {V}^\vee |_{\operatorname {\mathrm {Spec}} K_v}$ with $K_v^r$ . Then the generic stalk of $p^* \overline {x}^* \mathcal {V}^\vee $ is identified with $L_v^r$ , and we can think of $p^* \overline {x}^* \mathcal {V}^\vee $ as a $C^{\prime }_v$ -lattice $\Lambda $ in the vector space $L_v^r$ . Then the $\mathcal {O}_{K_v}$ -module $\pi _* \overline {x}^* \mathcal {V}^{\vee }$ is $\Lambda \cap K_v^r$ , and so $p^* \pi ^* \pi _* \overline {x}^* \mathcal {V}^{\vee }$ is

$$ \begin{align*} (\Lambda \cap K_v^r) \otimes_{\mathcal{O}_{K_v}} \mathcal{O}_{L_v} \subset \Lambda \end{align*} $$

and

$$ \begin{align*} \delta_{\mathcal{V};v}(x) = \frac{1}{m} \log \left| \frac{\Lambda}{(\Lambda \cap K_v^r) \otimes_{\mathcal{O}_{K_v}} \mathcal{O}_{L_v}} \right|. \end{align*} $$

Proposition 2.24. Let $E_v$ be an unramified extension of $K_v$ of degree d, let x be a point of $\mathcal {X}(K_v)$ and let $x_E$ be the corresponding point of $\mathcal {X}(E_v)$ . Then

$$ \begin{align*} \delta_{\mathcal{V};v}(x_E) = d \delta_{\mathcal{V};v}(x). \end{align*} $$

Proof. (This proof is essentially the same as that of the ‘geometric’ part of [Reference Machett Wood and Yasuda74, Lemma 3.4].)

Write $\Lambda _E$ for $(\Lambda \otimes _{\mathcal {O}_{K_v}} \mathcal {O}_{E_v})$ . Observe first that

$$ \begin{align*} \Lambda_E \cap E_v^r = (\Lambda \cap K_v^r) \otimes_{\mathcal{O}_{K_v}} \mathcal{O}_{E_v} \end{align*} $$

since the condition of being in $K^r$ is cut out by K-linear conditions on $L^r$ considered as a K-module; the same linear conditions applied to $(L \otimes _K E)^r$ cut out $E^r$ . We then get an equality

$$ \begin{align*} d \delta_{\mathcal{V};v}(x) = d \frac{1}{m} \log \left| \frac{\Lambda}{(\Lambda \cap K_v^r) \otimes_{\mathcal{O}_{K_v}} \mathcal{O}_{L_v}} \right| = \frac{1}{m} \log \left| \frac{\Lambda_E}{(\Lambda \cap K_v^r) \otimes_{\mathcal{O}_{K_v}} \mathcal{O}_{L_v} \otimes_{\mathcal{O}_{K_v}} \mathcal{O}_{E_v}} \right|. \end{align*} $$

On the other hand, writing $F_v$ for the etale algebra $E_v \otimes _{K_v} L_v$ , we have

$$ \begin{align*} \delta_{\mathcal{V};v}(x_E) = \frac{1}{m} \log \left| \frac{\Lambda_E}{(\Lambda_E \cap E_v^r) \otimes_{\mathcal{O}_{E_v}} \mathcal{O}_{F_v}}\right| = \frac{1}{m} \log \left| \frac{\Lambda_E}{(\Lambda \cap K_v^r) \otimes_{\mathcal{O}_{K_v}} \mathcal{O}_{F_v}} \right|. \end{align*} $$

The desired equality now follows from the fact that, since $E_v / K_v$ is unramified, we have $\mathcal {O}_{F_v} = \mathcal {O}_{L_v} \otimes _{\mathcal {O}_{K_v}} \mathcal {O}_{E_v}$ .

There will be some cases where our life is simpler if we can ignore a specified finite set of primes. The following proposition is useful when we need to show this negligence does not perturb our height functions by very much.

Proposition 2.25. Suppose K is a number field. There is a constant $C(\mathcal {X},\mathcal {V},v)$ such that

$$ \begin{align*} \delta_{\mathcal{V};v}(x) \leq C(\mathcal{X},\mathcal{V},v) \end{align*} $$

for all x in $\mathcal {X}(K_v)$ .

Proof. (The following proof is adapted from a nice proof of Hilbert 90 that we learned from [Reference Berhuy8, Lemma 3.3].)

There is some constant B such that every point $x \in \mathcal {X}(K_v)$ extends to an integral point of $\mathcal {X}(L_v)$ for some finite Galois extension L of K of degree at most B; this follows from the fact that $\mathcal {X}$ has a finite cover by a scheme; see [Reference Rydh62, Theorem B]. Since K is a number field, there are only finitely many isomorphism classes of extensions of $K_v$ of degree at most B. We may thus prove the required bound for a single choice of $L_v$ .

Write G for $\operatorname {\mathrm {Gal}}(L/K)$ . Write $\alpha _1, \ldots , \alpha _m$ for a subset of $\mathcal {O}_{L_v}$ which freely spans $\mathcal {O}_{L_v}$ as an $\mathcal {O}_{K_v}$ -module. Let $\lambda $ be an element of $\Lambda $ , and for each i in $1,\ldots ,m$ define

$$ \begin{align*} \lambda_i = \sum_{g \in G} (\alpha_i \lambda)^g. \end{align*} $$

The action of G permutes the summands above, so $\lambda _i$ is fixed by G and thus lies in $\Lambda \cap K_v^r$ .

We can also write

(2.26) $$ \begin{align} \lambda_i = \sum_{g \in G} (\alpha_i^g)(\lambda^g). \end{align} $$

Write A for the $m \times m$ matrix in with rows indexed by $\alpha _1, \ldots , \alpha _m$ and columns by the elements of G; by Dedekind’s lemma this matrix lies in $\operatorname {\mathrm {GL}}_m(L_v)$ . Write $\overrightarrow {\lambda }$ for the vector $\lambda _1, \ldots , \lambda _m \in L_v^m$ and $\overrightarrow {\mu }$ for the vector whose entries are $\{\lambda ^g\}_{g \in G}$ . With this notation, equation (2.26) becomes

$$ \begin{align*} \overrightarrow{\lambda} = A \overrightarrow{\mu} \end{align*} $$

which we can rewrite as

$$ \begin{align*} \overrightarrow{\mu} = A^{-1} \overrightarrow{\lambda}. \end{align*} $$

In particular, we can write

(2.27) $$ \begin{align} \lambda = \sum a_i \lambda_i, \end{align} $$

where $a_i$ are entries of $A^{-1}$ . But note that A depends only on the choice of $\alpha _i$ ; in particular, there is some constant C such that the entries of $A^{-1}$ lie in $C^{-1} \mathcal {O}_{L_v}$ . Thus, equation (2.27) expresses an arbitrary $\lambda \in \Lambda $ as a linear combination of the $\lambda _i$ , which lie in $\Lambda \cap K_v^r$ , with coefficients in $C^{-1} \mathcal {O}_{L_v}$ . We conclude that

$$ \begin{align*} \Lambda \subset C^{-1}[(\Lambda \cap K_v^r) \otimes_{\mathcal{O}_{K_v}} \mathcal{O}_{L_v}] \end{align*} $$

which provides a bound for

$$ \begin{align*} \left| \frac{\Lambda}{(\Lambda \cap K_v^r) \otimes_{\mathcal{O}_{K_v}} \mathcal{O}_{L_v}} \right| \end{align*} $$

depending only on $L_v$ , as required.

We note that Proposition 2.25 does not hold in general when K has characteristic p. For instance, we will see that the local discrepancy at v for a point of $BG$ , with $\mathcal {V}$ the regular representation of G, computes the discriminant of the local extension: but we know that the discriminant of a $\mathbf {Z}/p\mathbf {Z}$ -extension of ${\mathbb F}_p((t))$ can be arbitrarily large, by contrast with the discriminant of a $\mathbf {Z}/p\mathbf {Z}$ -extension of $\mathbf {Q}_p$ .

2.4 Computing heights: line bundles on $\mathcal {X}$ with globally generated powers

In this section, we consider the special case where $\mathcal {V}$ is a line bundle $\mathcal {L}$ . It turns out that, speaking loosely, if some tensor power $\mathcal {L}^m$ has “enough sections,” we can use these sections to compute heights of rational points on $\mathcal {X}$ with little explicit reference to stacks. (Whether this is a virtue depends on the reader’s taste.)

Suppose $\mathcal {X}$ is a stack over C, $\mathcal L$ is a metrized line bundle on $\mathcal {X}$ , and $s_1, \ldots , s_k$ are sections of $\mathcal L$ . We say $\mathcal L$ is generically globally generated by $s_1, \ldots , s_k$ if the cokernel $\mathcal F$ of the corresponding morphism of sheaves

$$ \begin{align*} \mathcal{O}_{\mathcal{X}}^{\oplus k} {\rightarrow} \mathcal{L} \end{align*} $$

vanishes over the generic point of C. In particular, this implies that $\mathcal F$ is supported at finitely many places v of C. More specifically: For each non-Archimedean v with uniformizer $\pi _v \in \mathcal {O}_{C_v}$ , there is an integer $m_v$ such that the restriction of $\mathcal F$ to $\mathcal {X} \times _C \mathcal {O}_{C_v}$ is killed by $\pi _v^{m_v}$ (since $\mathcal {X}$ is finite type, it suffices to check this on a finite flat cover). In the case where C has no Archimedean places, we say $\mathcal L$ is globally generated by $s_1, \ldots , s_k$ when the map from $\mathcal {O}_{\mathcal {X}}^{\oplus k}$ to $\mathcal {L}$ is surjective. We write $q_v$ for the order of the residue field at v, if v is a non-Archimedean place; when v is Archimedean, we can take $q_v = e$ .

Proposition 2.28. Suppose $\mathcal {X}$ is a stack, and suppose $\mathcal L$ is a metrized line bundle on $\mathcal {X}$ such that some power $\mathcal L^n$ is generically globally generated by sections $s_1, \ldots , s_k$ . Let K be a global field, and let $x\colon \operatorname {\mathrm {Spec}} K {\rightarrow } \mathcal {X}$ be a point of $\mathcal {X}(K)$ . Choose an identification of $x^*\mathcal {L}$ (whence also $x^* \mathcal {L}^n$ ) with K, and write $x_1, \ldots , x_k$ for the pullbacks of $s_1, \ldots , s_k$ by x. Then

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal L}(x) = \sum_{v} \left\lceil (1/n) \log_{q_v} \max(|x_1|_v, \ldots, |x_k|_v) \right\rceil \log q_v + E(x), \end{align*} $$

where $E(x)$ is a function bounded above and below on $\mathcal {X}(K)$ . When C has no Archimedean places and $\mathcal L^n$ is globally generated by $s_1, \ldots , s_k$ we have

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal L}(x) = \sum_{v} \left\lceil (1/n) \log_{q_v} \max(|x_1|_v, \ldots, |x_k|_v) \right\rceil \log q_v \end{align*} $$

exactly.

From now on, we denote a bounded function on $\mathcal {X}(K)$ by $O_{\mathcal {X}(K)}(1)$ . Note that, when $K=\mathbf {Q}$ , we may take $x_1, \ldots , x_k$ to be integers, with the property that, for every p, there is some $x_i$ which is not a multiple of $p^n$ . We say such a tuple $(x_1, \ldots , x_k) \in \mathbf {Z}^k$ is in minimal form. Suppose $(x_1, \ldots , x_k)$ corresponds to a point x of $\mathcal {X}(\mathbf {Q})$ as in Proposition 2.28. The hypothesis of minimal form implies that the non-Archimedean contributions all vanish, and we are left with

(2.29) $$ \begin{align} \operatorname{\mathrm{ht}}_{\mathcal L}(x) = (1/n) \log \max_i |x_i|_{\mathbf{R}} + O_{\mathcal{X}(K)}(1) \end{align} $$

up to a function bounded on $\mathcal {X}(\mathbf {Q})$ . (The ceiling function can now be neglected since, having restricted to a single summand, the difference between a number and its floor is bounded and can be absorbed into the error term.)

We now prove Proposition 2.28.

Proof. We note, first of all, that we have not specified the choice of metric on $\mathcal {L}$ at Archimedean places, but this choice can be absorbed in the error term; if $\mathcal {L}$ and $\mathcal {L}'$ are line bundles which differ only with respect to the Archimedean metric, it is easy to see from the proof below that $\operatorname {\mathrm {ht}}_{\mathcal {L}}- \operatorname {\mathrm {ht}}_{\mathcal {L}'} = O_{\mathcal {X}(K)}(1)$ . (At the moment when we say ‘Fubini–Study metric on $\mathcal {O}(1)$ on complex projective space’, just insert your own favorite metric, which differs from Fubini–Study by a bounded function.)

We begin by computing the degree of $\overline {x}^* \mathcal {L}^n$ on $\mathcal {C}$ . Let $L/K$ be a finite extension of some degree m such that the pullback of x to $\operatorname {\mathrm {Spec}} L$ extends to a morphism $y\colon C' {\rightarrow } \mathcal {X}$ , where $C'$ is the curve (or Dedekind domain) with fraction field L. We then have a commutative diagram:

Now, $\deg _{\mathcal {C}} \overline {x}^* \mathcal {L}^n = (1/m) \deg _{C'} p^* \overline {x}^* \mathcal L^n$ . The latter is a metrized line bundle on $\mathcal {O}_L$ whose degree we can compute by means of a section. For ease of notation, write $\Lambda $ for $p^* \overline {x}^* \mathcal L^n$ .

$$ \begin{align*} \deg_{C'} p^* \overline{x}^* \mathcal L^n = \log |\Lambda/ s_1 \mathcal{O}_L| - \sum_{\sigma \colon L {\rightarrow} \mathbf{C}} |\sigma^* s_1|_\sigma. \end{align*} $$

Write $\Lambda '$ for the submodule of $\Lambda $ spanned by $s_1, \ldots , s_k$ . By hypothesis, there is a bound independent of x for the size of $\Lambda / \Lambda '$ . Thus, we may replace $\Lambda $ with $\Lambda '$ and get

$$ \begin{align*} \deg_{C'} p^* \overline{x}^* \mathcal L^n = \log |\Lambda'/ s_1 \mathcal{O}_L| - \sum_{\sigma \colon L {\rightarrow} \mathbf{C}} |\sigma^* s_1|_\sigma + O_{\mathcal{X}(K)}(1). \end{align*} $$

Now, the torsion $\mathcal {O}_L$ -module $\Lambda '/ s_1 \mathcal {O}_L$ can be broken up into v-adic components $T_v$ , one for each non-Archimedean place v of K, and by the explicit description of $\Lambda '$ we have

$$ \begin{align*} \log |T_v| = m (\log \max_i |x_i|_v - \log |x_1|_v). \end{align*} $$

Thus, we have

$$ \begin{align*} \log |\Lambda'/ s_1 \mathcal{O}_L| = \sum_{v \nmid \infty} m(\log \max_i |x_i|_v - \log |x_1|_v). \end{align*} $$

We now turn to the Archimedean places, which requires us to specify the metric on $\mathcal {L}^n$ . The sections $s_1, \ldots , s_n$ provide a map of complex manifolds $f\colon \mathcal {X}(\mathbf {C}) {\rightarrow } \mathbb {P}^{k-1}(\mathbf {C})$ , and $\mathcal {L}^n|\mathcal {X}(\mathbf {C})$ is pulled back from $\mathcal {O}(1)$ under f. So we may choose for our metric on $\mathcal {L}^n|\mathcal {X}(\mathbf {C})$ the pullback of the Fubini–Study metric on $\mathcal {O}(1)$ . Having done so, we have

$$ \begin{align*} \sum_{\sigma\colon L {\rightarrow} \mathbf{C}} |\sigma^* s_1|_\sigma = \sum_{v | \infty} m (\log|x_1|_v - \log \max_i |x_i|_v ) + O_{\mathcal{X}(K)}(1). \end{align*} $$

To sum up, we have computed that

$$ \begin{align*} \log |\Lambda'/ s_1 \mathcal{O}_L| - \!\sum_{\sigma\colon L {\rightarrow} C} \!|\sigma^* s_1|_\sigma = - \sum_v m |x_i|_v + \sum_v m \log \max_i |x_i|_v = \sum_v \log \max_i |x_i|_v + O_{\mathcal{X}(K)}(1), \end{align*} $$

whence

$$ \begin{align*} \deg_{C'} p^* \overline{x}^* \mathcal L^n = (\sum_v m\log \max_i |x_i|_v) + O_{\mathcal{X}(K)}(1), \end{align*} $$

whence

$$ \begin{align*} \deg_{C} \overline{x}^* \mathcal L^\vee = -(1/n) (\sum_v \log \max_i |x_i|_v) + O_{\mathcal{X}(K)}(1). \end{align*} $$

We note that, in the case where K is a function field and $s_1,\ldots ,s_k$ globally generate $\mathcal {L}^n$ , the expression $(\sum _v m\log \max _i |x_i|_v)$ is just the usual expression for the degree of a line bundle pulled back from $\mathcal {O}(1)$ on $\mathbb {P}^{k-1}$ by a morphism with coordinates $(x_1: \ldots : x_k)$ .

Having computed this degree, which is the negative of the stable height $\operatorname {\mathrm {ht}}^{\operatorname {\mathrm {st}}}_{\mathcal {L}}(x)$ , we can compute $\operatorname {\mathrm {ht}}_{\mathcal {L}}(x)$ by adding local discrepancies as in the previous section. First of all, if v is one of the finitely many non-Archimedean places where $\mathcal {L}^n$ is not generated by $s_1, \ldots , s_k$ , we observe that $\delta _{\mathcal {L};v}(x)$ is $O_{\mathcal {X}(K)}(1)$ by Proposition 2.25, and since the number of such places is bounded independently of x, we can absorb the contribution of those local discrepancies $\delta _{\mathcal {L};v}(x)$ into the error term.

So let v be a non-Archimedean place where $\mathcal {L}^n$ is generated by $s_1, \ldots , s_k$ . Then, given our choice of identification of $x^* \mathcal {L}^n$ with K, and writing $L_v$ for the etale algebra $L \otimes _K K_v$ , we can write $\overline {x}^* \mathcal {L}^n$ as the Galois-stable lattice I in $L_v$ spanned as an $\mathcal {O}_{L_v}$ -module by $x_1, \ldots , x_k$ . Then $\overline {x}^* \mathcal {L}^\vee $ is the submodule $I^{-1/n}$ of $L_v$ consisting of all $\alpha \in L_v$ such that $\alpha ^n I \subset \mathcal {O}{L_v}$ . The pushforward $\pi _* \overline {x}^* \mathcal {L}^\vee $ is then the submodule $I^{-1/n} \cap K_v$ of $K_v$ consisting of all $\beta \in K_v$ with $\beta ^n x_i \subset \mathcal {O}_{K_v}$ for all i, which is to say it is the fractional ideal $m_v^{c_v}$ , where

$$ \begin{align*} c_v = \lceil -(1/n)\min_i \operatorname{\mathrm{ord}}_v x_i \rceil = \lceil (1/n) \log_{q_v} \max_i |x_i|_v \rceil. \end{align*} $$

So

$$ \begin{align*} \delta_{\mathcal{L};v}(x) = (1/m) \log |I^{-1/n} / I^{-1/n} \cap K| = (\log q_v) \lceil (1/n) \log_{q_v} \max_i |x_i|_v \rceil - (1/n) \log \max_i |x_i|_v. \end{align*} $$

Recalling from above that

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{L}}^{\operatorname{\mathrm{st}}}(x) = (1/n) \sum_v \log \max_i |x_i|_v + O_{\mathcal{X}(K)}(1), \end{align*} $$

we conclude that

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{L}}(x) = \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}(x) + \sum_v \delta_{\mathcal{L};v}(x) = \sum_v \lceil (1/n) \log_{q_v} \max_i |x_i|_v \rceil \log q_v+ O_{\mathcal{X}(K)}(1) \end{align*} $$

which was the desired result.

3.1 Heights on $BG$

Let G be a constant finite group scheme over C, let $\mathcal {X}$ be the classifying stack $BG/C$ , and let $q\colon C {\rightarrow } BG$ be the universal G-cover. Let $x\colon \operatorname {\mathrm {Spec}} K {\rightarrow } \mathcal {X}$ be a rational point, and let $\overline {x}\colon \mathcal {C} {\rightarrow } BG$ be the extension of x to a tuning stack. Then we have a commutative diagram

where $C'$ is a smooth proper curve (not necessarily irreducible) whose fiber over $\operatorname {\mathrm {Spec}} K$ is an étale G-algebra $L/K$ .

Let $\mathcal {V}$ be a vector bundle of rank r on $BG$ ; in other words, $\mathcal {V}$ is an r-dimensional representation V of G over C. Then, by equation (2.22), we have

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{V}}(x) = \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\mathcal{V}}(x) + \sum_v \delta_{\mathcal{V};v}(x). \end{align*} $$

First of all, note that $p^* \overline {x}^* \mathcal {V}^\vee = x_{C'}^* q^* \mathcal {V}$ is a vector bundle on $C'$ pulled back from the trivial bundle on C, and thus has degree $0$ . So

$$ \begin{align*} \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\mathcal{V}}(x) = -\deg \overline{x}^* \mathcal{V}^{\vee} = - (\deg p)^{-1} \deg p^* \overline{x}^* \mathcal{V}^{\vee} = 0. \end{align*} $$

We have thus reduced ourselves to the local problem of computing $\delta _{\mathcal {V};v}(x)$ at the finite set of non-Archimedean places v of K where $L/K$ is ramified. Let v be such a place.

The pullback of $\mathcal {V}^\vee $ along $x_{C'}^*$ from C to $C'$ is $\mathcal {O}_{C'} \otimes _{\mathcal {O}_C} V^{\vee }$ .

Thus, locally, the G-stable lattice $\Lambda _v \subset L_v^r$ we use to compute the local discrepancy can be written as

$$ \begin{align*} \mathcal{O}_{L_v} \otimes_{\mathcal{O}_{K_v}} V^\vee. \end{align*} $$

We note that this is precisely the G-module studied by Yasuda and Wood in section 3 of [Reference Machett Wood and Yasuda74]. (The free rank $r \mathcal {O}_{L_v}$ -module we call $\Lambda _v$ is identified with $\mathcal {O}_{L_v}^r$ in their notation.) In particular, the free rank $r \mathcal {O}_{K_v}$ -module $\Lambda ^G$ is precisely the tuning submodule in [Reference Machett Wood and Yasuda74, Def 3.1], and the local discrepancy $\delta _{\mathcal {V};v}(x)$ is exactly the quantity denoted $\mathbf {v}_\tau (\rho )$ in [Reference Machett Wood and Yasuda74, Def 3.3]. Thus, we can make use of their results to compute the local discrepancies explicitly.

The case where $\mathcal {V}$ is a permutation representation is an important example; in this case, we find that the discriminant of a field extension can be understood as a height on $BG$ in the sense of this paper. In particular: When $\mathcal {V}$ is a degree-n permutation representation of G, and x is a point of $BG(K)$ , we can associate to x a map

$$ \begin{align*} \rho_x\colon \operatorname{\mathrm{Gal}}(K) {\rightarrow} G {\rightarrow} S_n \end{align*} $$

which in turn specifies a degree-n étale algebra $L/K$ .

Proposition 3.1. Let $\mathcal {V}$ be a vector bundle on $BG$ corresponding to a degree-n permutation representation $\rho $ of G, let x be a point in $BG(K)$ and let $L/K$ be the algebra corresponding to x as described above. Then

(3.2) $$ \begin{align} \operatorname{\mathrm{ht}}_{\mathcal{V}}(x) = (1/2) \log |\Delta_{L/K}|. \end{align} $$

Proof. It follows from [Reference Machett Wood and Yasuda74, Theorem 4.8] that

$$ \begin{align*} \delta_{\mathcal{V};v}(x) = (1/2) a_v(\rho_x), \end{align*} $$

where $a_v$ is the Artin conductor of $\rho _x|_{K_v}$ , which is precisely the local component at v of $\Delta _{L/K}$ . Thus,

(3.3) $$ \begin{align} \operatorname{\mathrm{ht}}_{\mathcal{V}}(x) = \sum_v \delta_{\mathcal{V};v}(x) = (1/2) \log |\Delta_{L/K}|, \end{align} $$

where by $|\Delta _{L/K}|$ we mean the absolute norm of the discriminant, that is, the order of the finite group $\mathcal {O}_C / \Delta _{L/K}$ .

In other words, the general definition of height introduced here, when applied to a G-extension (thought of as a point of $BS_n$ ), recovers the discriminant. Of course, a point of $BS_n$ can be thought of as a G-extension in different ways; one might have in mind a degree-n extension, the Galois $S_n$ -extension obtained by applying Galois closure or some other number field with the same Galois closure. Each such field corresponds to a permutation representation of $S_n$ (in the first and second case above, the standard representation and the regular representation) and the discriminant of the field is computed by the height with respect to the vector bundle $\mathcal {V}$ specified by the corresponding permutation representation.

The case $\mathcal {X} = BG$ demonstrates the necessity of computing heights with respect to vector bundles of arbitrary rank, not only line bundles. Line bundles on $BG$ correspond to one-dimensional representations of G. If, for example, G is a finite group with trivial abelianization, there are no nontrivial line bundles at all. In order to have a theory of heights rich enough to capture the invariants of G-extensions, we have no alternative than to consider vector bundles of higher rank on $BG$ .

The work of Yasuda and Wood is not limited to permutation representations. For example, Wood and Yasuda work out in [Reference Machett Wood and Yasuda74, Example 4.10] the example where $G = \mathbf {Z}/p\mathbf {Z}$ , K is a function field of characteristic p and $\mathcal {V}$ is the two-dimensional nonsemisimple representation of $\mathbf {Z}/p\mathbf {Z}$ over K. A rational point of $BG$ corresponds to a $\mathbf {Z}/p\mathbf {Z}$ -extension $L/K$ . If v is a place of K, we denote by $j_v$ the largest integer i such that the higher ramification group $G_i$ at v surjects onto $\mathbf {Z}/p\mathbf {Z}$ . Then Yasuda and Wood’s computation shows

(3.4) $$ \begin{align} \operatorname{\mathrm{ht}}_v(x) = 1 + \left\lfloor \frac{j_v}{p} \right\rfloor. \end{align} $$

When $K = {\mathbb F}_q(t)$ with q a power of p, the points of $B(\mathbf {Z}/p\mathbf {Z})(K)$ correspond to Artin–Schreier curves, and the height of an Artin–Schreier curve with respect to this $\mathcal {V}$ is the sum of the local terms (3.4) over all places v of ${\mathbb F}_q(t)$ which are ramified in the Artin–Schreier cover. We do not know if this notion of height of an Artin–Schreier curve corresponds to anything that has appeared in previous literature, but we note that the expression above is closely related to that appearing in the computation of dimensions of irreducible components of moduli space for Artin–Schreier curves of specified p-rank in the work of Pries and Zhu [Reference Pries and Zhu60, Theorem 1.1].

This example also illustrates the important point that the height function $\operatorname {\mathrm {ht}}_{\mathcal {V}}$ is not determined by the class of $\mathcal {V}$ in $K_0$ of the category of vector bundles; the vector bundle above is an extension of the trivial line bundle by the trivial line bundle, but its associated height function is not zero.Footnote ²

3.2 Heights on $B\mu _n$

Suppose $\mathcal {X} = B\mu _n$ , and $\mathcal {L}$ is the line bundle on $B\mu _n$ corresponding to the standard one-dimensional representation $\mu _n {\rightarrow } \mathbb G_m$ . In that case, $\mathcal {L}^n$ is the trivial bundle on $\mathcal {X}$ and thus admits a generating section s. On the other hand, if x is a K-point of $B\mu _n$ , the pullback $x^* \mathcal {L}$ is isomorphic to K. The obstruction to $x^*s \in \Gamma (\operatorname {\mathrm {Spec}} K, x^* \mathcal {L}^n)$ being an nth power of an nonzero section of $x^* \mathcal {L}$ now yields a class in $K^* / (K^*)^n$ . Put another way: Choosing an identification of $x^* \mathcal {L}$ with K induces an identification of $x^* \mathcal {L}^n$ with K, under which $x^* s$ is identified with an element $x_0 \in K^*$ , which represents the class in $K^* / (K^*)^n$ corresponding to x. Note that a change in the choice of s will apply a translation to the identification $B\mu _n(K) \cong K^* / (K^*)^n$ , but such a change will modify heights by a bounded quantity, and if K is a function field over a finite field k and we require s to globally generate $\mathcal {L}^n$ , the ambiguity in s imposes translation by $k^*$ , which will not change the heights we compute at all. (If we want to remove this ambiguity entirely, we can fix for all time a choice of universal $\mu _n$ -torsor $q\colon \operatorname {\mathrm {Spec}} K {\rightarrow } B\mu _n/K$ and an identification of $q^* \mathcal {L}$ with K; having done so, we can require that s pull back under q to an element of $(K^*)^n$ .)

We note that the above setup applies even when $\text {char}\, K$ divides n.

In particular: Proposition 2.28 yields

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal L}(x) = \sum_v \left\lceil \frac{1}{n} \log_{q_v} |x_0|_v \right\rceil \log q_v. \end{align*} $$

We note that our formula for $\operatorname {\mathrm {ht}}_{\mathcal L}(x)$ is unchanged, as it must be, when $x_0$ is modified by an element of $(K^*)^n$ .

By the computation above, when $K = \mathbf {Q}$ we see that the height of a point x of $B \mu _n (\mathbf {Q}) = \mathbf {Q}^\times / (\mathbf {Q}^\times )^n$ is obtained as follows: The class of $\mathbf {Q}^\times / (\mathbf {Q}^\times )^n$ corresponding to x is represented uniquely by an integer N with no nth power divisor, and as in equation (2.29) we have

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal L}(x) = \log |N|^{1/n}. \end{align*} $$

(In the examples, we will often suppress the $O_{\mathcal {X}(K)}(1)$ error term when no confusion is likely.)

Once again, the height recovers the measure of complexity most frequently used in practice; when enumerating the elements of $\mathbf {Q}^*/(\mathbf {Q}^*)^n$ , one typically identifies the elements of the group with nth power-free integers and lists in order of absolute value.

Of course, this choice $\mathcal L$ is not the only option. Suppose, for instance, $K = \mathbf {Q}$ and $n=3$ ; then there are two equally good choices of nontrivial line bundle on $\mathcal {X}$ , namely $\mathcal L$ and $\mathcal L^2$ . Suppose $x \in B\mu _3(\mathbf {Q})$ corresponds to $N M^2 \in \mathbf {Q}^\times / (\mathbf {Q}^\times )^3$ , with N and M coprime and squarefree. Then, as we have already observed above,

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal L}(x) = \log |NM^2|^{1/3} = (1/3) \log N + (2/3) \log M. \end{align*} $$

On the other hand, consider $\mathcal L' = \mathcal L^2$ . Then, having chosen s as above, $s^2$ is a generating section of $(\mathcal L')^3$ , so we can take $x_1$ to be $x^* s^2$ , which corresponds to $N^2 M^4 \in \mathbf {Q}^\times $ . Putting this integer in minimal form modifies it to $N^2 M$ , and another application of equation (2.29) shows that

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal L'}(x) = (2/3) \log N + (1/3) \log M. \end{align*} $$

As a final illustration, we can see how the above two computations combine to yield Proposition 3.1 for $B\mu _3$ . Let $\mathcal {V}$ be the vector bundle $\mathcal {L} \oplus \mathcal {L}^2 \oplus \mathcal {O}_{\mathcal {X}}$ . Then

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{V}}(x) = \log N + \log M \end{align*} $$

which we note is also $(1/2) \Delta _{L/\mathbf {Q}}$ , where $L = \mathbf {Q}((NM^2)^{1/3}) = \mathbf {Q}((N^2M)^{1/3}) $ is the cubic extension of $\mathbf {Q}$ arising from x. This is as it must be, as we now explain. First, note that $\operatorname {\mathrm {ht}}_{\mathcal {V}}^{\operatorname {\mathrm {st}}}(x) = 0$ for all x just as in the case $\mathcal {X} = BG$ , because $\mathcal {V}$ pulls back to a trivial bundle on a finite cover of $B\mu _3$ . So

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{V}}(x) = \sum_v \delta_{\mathcal{V};v}(x). \end{align*} $$

Now, the size of $\delta _{\mathcal {V};3}(x)$ is bounded by Proposition 2.25, so at the expense of a bounded error term we can write

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{V}}(x) = \sum_{v \neq 3} \delta_{\mathcal{V};v}(x). \end{align*} $$

Let $x'$ be the point of $B(\mu _3)(\mathbf {Q}(\zeta _3))$ obtained by base change from x. Since every prime other than $3$ is unramified in $\mathbf {Q}(\zeta _3)/\mathbf {Q}$ , Proposition 2.24 tells us that

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{V}}(x') = 2 \sum_{v \neq 3} \delta_{\mathcal{V};v}(x) = 2\operatorname{\mathrm{ht}}_{\mathcal{V}}(x). \end{align*} $$

On the other hand, over $\mathbf {Q}(\zeta _3)$ , there is an isomorphism between $B(\mu _3)$ and $B(\mathbf {Z}/3\mathbf {Z})$ , which carries $\mathcal {V}$ to the reduced permutation representation of $\mathbf {Z}/3\mathbf {Z}$ , which we denote by $\mathcal {W}$ . In fact, this isomorphism extends to $\mathbf {Z}[\zeta _3][1/3]$ . Let y be the point of $B(\mathbf {Z}/3\mathbf {Z})(\mathbf {Q}(\zeta _3))$ corresponding to $x'$ under this isomorphism, which we can also think of as the point associated to the Galois $\mathbf {Z}/3\mathbf {Z}$ -extension $L(\zeta _3)/\mathbf {Q}(\zeta _3)$ . Then

$$ \begin{align*} \delta_{\mathcal{W};v}(y) = \delta_{\mathcal{V}';v}(x') \end{align*} $$

for all places v of $\mathbf {Q}(\zeta _3)$ not dividing $3$ . We conclude that (as always, up to bounded error)

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{W}}(y) = \sum_{v \neq 3} \delta_{\mathcal{W};v}(y) = \sum_{v \neq 3} \delta_{\mathcal{V}';v}(x') = 2\operatorname{\mathrm{ht}}_{\mathcal{V}}(x). \end{align*} $$

On the other hand, by equation (3.3) we have

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{W}}(y) = (1/2) \log |\Delta_{L(\zeta_3)/\mathbf{Q}(\zeta_3)}| = \log \Delta_{L/\mathbf{Q}} \end{align*} $$

which shows that $\operatorname {\mathrm {ht}}_{\mathcal {V}}(x) = (1/2) \log |\Delta _{L/\mathbf {Q}}|$ .

3.3 Heights on weighted projective space and weighted projective stacks

In this section, we consider rational points on the weighted projective space $\mathcal {X} = \mathbb {P}(a_0, \ldots , a_k)$ . This stack is, by definition, the quotient $[\mathbb {A}^{k+1}\smallsetminus 0/\mathbb G_m]$ where $\mathbb G_m$ acts by the rule

$$ \begin{align*} \lambda\cdot(X_0, \ldots, X_k) = (\lambda^{a_0} X_0, \ldots, \lambda^{a_k} X_k). \end{align*} $$

Then $\mathbb {P}(a_0, \ldots , a_k)$ is a smooth proper stack, and $\mathbb {A}^{k+1} \smallsetminus 0$ is the total space of a line bundle on $\mathcal {X}$ , whose dual is the tautological bundle $\mathcal {O}_{\mathbb {P}(a_0, \ldots , a_k)}(1)$ ; for simplicity of notation, we denote the tautological bundle by $\mathcal {L}$ for the rest of this section. The coordinate function $X_i$ is a section of $\mathcal {L}^{a_i}$ . Writing A for the least common multiple of the $a_i$ , the $k+1$ sections $X_i^{A/a_i}$ of $\mathcal {L}^A$ generate $\mathcal {L}^A$ . So we can compute heights of points in $\mathbb {P}(a_0, \ldots , a_k)(K)$ by applying Proposition 2.28, as we now explain.

Let x be a point of $\mathbb {P}(a_0, \ldots , a_k)(K)$ . As in Proposition 2.28, we choose an identification of $x^* \mathcal {L}$ with K; this assigns a value in K to each of the $k+1$ coordinates, which values we denote $x_0, \ldots , x_k$ . Changing the identification of $x^*\mathcal {L}$ with K modifies this tuple by elementwise multiplication by tuples of the form $\lambda ^{a_0}, \ldots , \lambda ^{a_k}$ , and we say that two tuples $x_0, \ldots , x_k$ are equivalent if they differ by such a transformation. Then Proposition 2.28 tells us that

(3.5) $$ \begin{align} \operatorname{\mathrm{ht}}_{\mathcal{L}}(x) = \sum_v \lceil \log_{q_v} \max_i |x_i|_v^{1/{a_i}} \rceil \log q_v. \end{align} $$

In particular, when $K = \mathbf {Q}$ , a rational point x of $\mathbb {P}^1(a_0, \ldots , a_k)(\mathbf {Q})$ can be identified with a tuple of integers $(M_0: \ldots :M_k)$ such that there is no prime p with $p^{a_i} | M_i$ for all i. Given a tuple which is in minimal form in this sense, the non-Archimedean primes contribute nothing to equation (3.5), and we get

(3.6) $$ \begin{align} \operatorname{\mathrm{ht}}_{\mathcal{L}}(x) = \log \max_i |M_i|^{1/a_i}. \end{align} $$

We note that this definition recovers the notion called ‘naive height’ for points of weighted projective space in [Reference Beshaj, Gutierrez and Shaska9].

Here is another means by which it is often practical to compute heights on weighted projective space when K is a global function field. Let F be a section of $\mathcal {L}^A$ —for instance, it might be $X_i^{A/a_i}$ for some i—and let y be the pullback of F along x to $x^*\mathcal {L}^A$ , which we have identified with K. We define the minimal valuation of F at a place v of K as follows. Let $\pi _v \in K^*$ be an element which is a uniformizer at v, and define

$$ \begin{align*} c_v = \min \lfloor (1/a_i) \operatorname{\mathrm{ord}}_v x_i \rfloor. \end{align*} $$

Note that $c_v = 0$ if and only if all the $x_i$ are integral at v and there is some i such that $\operatorname {\mathrm {ord}}_v x_i < a_i$ . In this case, we say that $(x_0, \ldots , x_k)$ is in minimal form. If $(x_0, \ldots , x_k)$ is not in minimal form, we find an equivalent tuple in minimal form by modifying each $x_i$ by $\pi _v^{-a_i c_v}$ ; the effect of this transformation on y is multiplication by $\pi _v^{-A c_v}$ . We therefore define the minimal valuation of F to be

$$ \begin{align*} \operatorname{\mathrm{ord}}^{\min}_v F = \operatorname{\mathrm{ord}}_v y - A c_v = \operatorname{\mathrm{ord}}_v y - A \min \lfloor (1/a_i) \operatorname{\mathrm{ord}}_v x_i \rfloor. \end{align*} $$

We note that this quantity does not depend on the identification of $x^* \mathcal {L}$ with K, but only on F and v. Furthermore, we have

$$ \begin{align*} \sum_v \operatorname{\mathrm{ord}}^{\min}_v F = \sum_v \operatorname{\mathrm{ord}}_v y - \sum_v A \min \lfloor (1/a_i) \operatorname{\mathrm{ord}}_v x_i. \rfloor = A \sum_v \max \lceil (1/a_i) \log_{q_v} \max |x_i|_v \rceil \log q_v \end{align*} $$

and, by Proposition 2.28, this last quantity, taking $X_i^{A/a_i}$ to be the sections generating $\mathcal {L}^A$ , is exactly $A \operatorname {\mathrm {ht}}_{\mathcal {L}} x$ . We conclude that

(3.7) $$ \begin{align} \operatorname{\mathrm{ht}}_{\mathcal{L}} x = (1/A) \sum_v \operatorname{\mathrm{ord}}^{\min}_v F \log q_v. \end{align} $$

The classical theory of Weil heights is often set up by defining heights on projective spaces, and then defining a height $\operatorname {\mathrm {ht}}_{\mathcal {O}(1)}$ on $X(K)$ for other projective schemes X by restriction. In a similar manner, one can define height functions on weighted projective stacks $\mathbb {P}(a_0,\dots ,a_n)$ and obtain a height function $\operatorname {\mathrm {ht}}_{\mathcal {L}}$ on $\mathcal {X}(K)$ whenever $\mathcal {L}$ is a generically globally generated power as in Section 2.4. However, we stress that this naive approach does not apply to all stacks of interest. Indeed, if $\mathcal {X}$ is any stack with a nonabelian stabilizer group, it does not embed into a weighted projective stack, hence the necessity of our construction of heights given in Section 2.2.

One example of weighted projective stacks which is of great interest is the moduli stack of elliptic curves $\overline {\mathcal {M}}_{1,1}$ . If K is a field of characteristic not equal to 2 or 3, this stack is isomorphic over K to the weighted projective line $\mathbb {P}(4,6)$ : concretely, given an elliptic curve $E/K$ , we can write it in Weierstrass form $y^2 = x^3 + Ax + B$ with $A,B$ in K. This Weierstrass form is unique up to transformations $(A,B) {\rightarrow } (\lambda ^4 A, \lambda ^6 B)$ . So $(A:B)$ is a well-defined point on $\mathbb {P}(4,6)$ . Moreover, the isomorphism takes the line bundle $\mathcal {O}(1)$ on $\mathbb {P}(4,6)$ to the Hodge bundle $\mathcal {L}$ on $\overline {\mathcal {M}}_{1,1}$ (the bundle whose kth powers have weight $2k$ modular forms as sections). We conclude that, if $E/K$ is an elliptic curve over a global field of characteristic at least $5$ , with Weierstrass equation $y^2 = x^3 + Ax+B$ , thought of as a K-point of $\overline {\mathcal {M}}_{1,1}$ , we have

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{L}} E = \log \max (|A|^{1/4},|B|^{1/6}). \end{align*} $$

In other words, the familiar ‘naive height’ of an elliptic curve is indeed a height in the sense of this paper.

When K is a number field, the identification of $\overline {\mathcal {M}}_{1,1}/\mathbf {Q}$ with $\mathbb {P}(4,6)/\mathbf {Q}$ does not extend to $\operatorname {\mathrm {Spec}} \mathbf {Z}$ but only to $\operatorname {\mathrm {Spec}} \mathbf {Z}[1/6]$ . However, this is enough to ensure that $\mathcal {L}^{12}$ is still generically globally generated by $A^3$ and $B^2$ in the sense of Proposition 2.28, and so equation (3.3) still holds up to a bounded error term.

When K is a global function field of characteristic at least $5$ , we can also apply equation (3.7); here, $A = \operatorname {\mathrm {lcm}}(4,6) = 12$ and the discriminant $\Delta $ is a natural section of $\mathcal {L}^{12}$ to use. So we find

(3.8) $$ \begin{align} \operatorname{\mathrm{ht}}_{\mathcal{L}} E = \frac{1}{12}\sum_v \operatorname{\mathrm{ord}}_v^{\min} \Delta, \end{align} $$

where $\operatorname {\mathrm {ord}}_v^{\min } \Delta $ is the valuation of the discriminant of a Weierstrass equation for E which is minimal at v.

We will return to the interesting case where K is a global function field of characteristic $2$ or $3$ in Section 3.4.

More generally, the moduli space of hyperelliptic curves over K with a marked Weierstrass point can be thought of as a weighted projective space as long as the characteristic of K is large enough: If $Y {\rightarrow } \mathbb {P}^1$ is the hyperelliptic map, we can move the image of the marked Weierstrass point to $\infty $ and (assuming the characteristic of K is not $2$ ) complete the square in y so that the curve has affine equation

$$ \begin{align*} y^2 = x^{2g+1} + a_1 x^{2g} + \cdots + a_{2g+1} \end{align*} $$

then (again throwing out a finite set of characteristics for K) modify by the automorphism $x {\rightarrow } x+\frac {a_1}{2g+1}$ of $\mathbb {P}^1$ in order to make $a_1 = 0$ . We now have an equation for Y of the form

(3.9) $$ \begin{align} y^2 = x^{2g+1} + a_2 x^{2g-1} + \cdots + a_{2g+1} \end{align} $$

which is unique up to the operation of multiplying $a_i$ by $\lambda ^{2i}$ for $\lambda \in K^*$ . In other words, the moduli stack of hyperelliptic curves with marked Weierstrass point is isomorphic over K to the weighted projective $(2g-1)$ -space $\mathbb {P}(4,6,8,\ldots ,4g+2)$ . So a hyperelliptic curve over K can be thought of as a point x on $\mathbb {P}(4,6,8,\ldots , 4g+2)$ , whose height with respect to $\mathcal {O}(1)$ we have computed above. In particular, if Y is a hyperelliptic curve over $\mathbf {Q}$ with Weierstrass equation (3.9), where the $a_i$ are chosen to be integers so that there is no prime p with $p^{2i}|a_i$ , the height of Y is $\log \max |a_i|^{1/2i}$ , which again is equivalent to the notion of height typically used for hyperelliptic curves with a specified Weierstrass point as in, for example, the work of Bhargava and Gross [Reference Bhargava and Gross11].

3.4 Heights of abelian varieties

We have established above in equation (3.3) that, when K is a global field of characteristic at least $5$ , the height of an elliptic curve with respect to the Hodge bundle on $\overline {\mathcal {M}}_{1,1}$ is the same as the customary naive height. There is another natural height on an elliptic curve over a global field: the Faltings height $\operatorname {\mathrm {ht}}_{\operatorname {\mathrm {Fal}}}(E)$ . In this section, we study the extent to which Faltings height can be seen as a height in the sense of the present paper.

We note first that Faltings height satisfies some of the same formal properties as the heights defined in this paper do. For example: If $L/K$ is a field extension, it is not necessarily the case that $\operatorname {\mathrm {ht}}_{\operatorname {\mathrm {Fal}}}(E/L)$ is $[L:K]\operatorname {\mathrm {ht}}_{\operatorname {\mathrm {Fal}}}(E/K)$ ; however, this equality does hold if $E/K$ has everywhere semistable reduction, so we can define a stable Faltings height $\operatorname {\mathrm {ht}}_s(E/K)$ to be $[L:K]^{-1}\operatorname {\mathrm {ht}}_{\operatorname {\mathrm {Fal}}}(E/L)$ for any $L/K$ such that $E/L$ has everywhere semistable reduction. The height $\operatorname {\mathrm {ht}}_{\mathcal {V}}$ for any vector bundle on $\overline {\mathcal {M}}_{1,1}$ has the same properties, since an elliptic curve over $L = K(C')$ with everywhere semistable reduction is an integral point of $\overline {\mathcal {M}}_{1,1}$ , that is, a morphism from $C'$ to $\overline {\mathcal {M}}_{1,1}$ . Lastly, $\operatorname {\mathrm {ht}}_{\operatorname {\mathrm {Fal}}}(E/K) - \operatorname {\mathrm {ht}}_s(E/K)$ has a canonical local decomposition, just as does $\operatorname {\mathrm {ht}}_{\mathcal {V}}(E/K) - \operatorname {\mathrm {ht}}^{\operatorname {\mathrm {st}}}_{\mathcal {V}}(E/K)$ ; see (2.22).

It is thus natural to ask whether Faltings height is $\operatorname {\mathrm {ht}}_{\mathcal {V}}$ for some vector bundle $\mathcal {V}$ or at least whether the two heights differ by a bounded function. One can even guess which vector bundle one might use; for everywhere semistable $E/K$ , or in other words morphisms $f\colon C {\rightarrow } \overline {\mathcal {M}}_{1,1}$ , we have

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\operatorname{\mathrm{Fal}}}(E) = \deg f^* \mathcal{L}, \end{align*} $$

where $\mathcal {L}$ is the Hodge bundle $\Omega ^1_{\mathcal {E}/\overline {\mathcal {M}}_{1,1}}$ and $\mathcal {E}$ the universal semielliptic curve over the moduli stack.

So does $\operatorname {\mathrm {ht}}_{\operatorname {\mathrm {Fal}}}$ differ from $\operatorname {\mathrm {ht}}_{\mathcal {L}}$ by a bounded function? Unfortunately, the answer is in general no—remember, in the number field case, $\operatorname {\mathrm {ht}}_{\mathcal {L}}$ is naive height, and the difference between the naive height and the Faltings height of an elliptic curve over a number field K is not bounded, as one can see, for instance, in the proof of [Reference Pazuki55, Lemma 3.2].

The reason for this is the following. When K is a number field, the specification of the degree above requires a choice of metrization on $\mathcal {L}$ at the Archimedean places; for Faltings height, the appropriate Hermitian norm actually has a singularity at the cusp of moduli space, and in the present paper we have not considered metrized line bundles in this level of generality; rather, we have assumed that our choice of metrization on $\mathcal {L}$ is defined on all of $\overline {\mathcal {M}}_{1,1}(\mathbf {C})$ , including the cusp.

However, when K is a global function field, this Archimedean issue is absent, and we find the following.

Proposition 3.11. Let K be a global function field of characteristic at least $5$ , and let $\mathcal {L}$ be the Hodge bundle on $\overline {\mathcal {M}}_{1,1}$ as above. Then

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\operatorname{\mathrm{Fal}}}(E) = \operatorname{\mathrm{ht}}_{\mathcal{L}}(E) \end{align*} $$

for all elliptic curves $E/K$ .

The case of small characteristic is a different story. Let K be the function field of a curve C in characteristic p. Then the Faltings height of an elliptic curve over K is still the valuation of a minimal discriminant divisor on C, even if the characteristic of K is $2$ or $3$ , and the Faltings height has the Northcott property.Footnote ³

On the other hand, $\operatorname {\mathrm {ht}}_{\mathcal {L}}$ is not Northcott in this setting, Note for instance that $\overline {\mathcal {M}}_{1,1}/\mathbf {F}_3$ contains as a closed substack a copy of $BG$ lying over the coarse point $j=0=1728$ , where G is the automorphism group scheme of an elliptic curve with j-invariant $0$ in characteristic $3$ . The group scheme G has order $12$ and sits in an exact sequence

$$ \begin{align*} 1 {\rightarrow} A {\rightarrow} G {\rightarrow} \mu_4 {\rightarrow} 1, \end{align*} $$

where $A \cong \mathbf {Z}/3\mathbf {Z}$ (see, for instance, [Reference Silverman65, Exercise A.1]) and $\lambda \in \mu _4$ acts on A by multiplication by $\lambda ^2$ . The pullback of $\mathcal {L}$ to $BG$ is a line bundle on $BG$ , which is necessarily trivial on the commutator subgroup A. So $\mathcal {L}$ pulls back to the zero bundle under the composition $BA {\rightarrow } BG {\rightarrow } \overline {\mathcal {M}}_{1,1}$ , which means that any point x in the image of $BA(K) {\rightarrow } \overline {\mathcal {M}}_{1,1}(K)$ has $\operatorname {\mathrm {ht}}_{\mathcal {L}}(x) = 0$ . There are infinitely many such points, corresponding to the $\mathbf {Z}/3\mathbf {Z}$ -extensions of K. Concretely, elliptic curves given by Weierstrass equations of the form

(3.12) $$ \begin{align} y^2 = x^3 - x - f(t) \end{align} $$

all have height $0$ with respect to $\mathcal {L}$ . Another way to see this is to observe that the space of sections of $\mathcal {L}^{12}$ —that is, of weight- $12$ modular forms of level $1$ in characteristic $3$ —is two-dimensional and is spanned by $\Delta $ and $b_2$ , where $b_2$ is the Hasse invariant. [Reference Deligne22, Prop 6.2]. Any Weierstrass equation of type (3.12) has $b_2(E) = 0$ and $\Delta (E) = 1$ ([Reference Silverman65, Appendix A, Prop 1.1. b)]). So by Proposition 2.28, using the fact that $\Delta $ is constant, we see again that $\operatorname {\mathrm {ht}}_{\mathcal {L}}(E) = 0$ for any such E.

This does not mean, however, that Faltings height is a different kind of height from those discussed in this paper; it only means it does not agree with the height arising from the Hodge bundle or any of its powers. But, as explained in a paper of Meier [Reference Meier51], there are other vector bundles! When K is a field of characteristic greater than $3$ , every vector bundle on $\overline {\mathcal {M}}_{1,1}$ is isomorphic to a direct sum of line bundles, which can only be powers of the Hodge bundle [Reference Meier51, Cor 3.6]), essentially because $\overline {\mathcal {M}}_{1,1}$ is a weighted projective line in this case. But in characteristic $2$ and $3$ , Meier constructs indecomposable higher-rank vector bundles on $\overline {\mathcal {M}}_{1,1}/K$ .Footnote ⁴ Thus, the following question still makes sense.

We originally asked this question with $c=1$ ; that is, is the Faltings height itself a height in our sense? Landesman showed in his thesis [Reference Landesman45] that this is too much to hope for; in characteristic $3$ , there is no vector bundle $\mathcal {V}$ on $\overline {\mathcal {M}}_{1,1}/\mathbf {F}_3$ with $\operatorname {\mathrm {ht}}_{\mathcal {V}} = \operatorname {\mathrm {ht}}_{\operatorname {\mathrm {Fal}}}$ . However, Landesman then raises the question stated above, which remains open: Is there a vector bundle which computes some integer multiple of the Faltings height in characteristic $3$ ? For that matter, is there even a vector bundle whose associated height is Northcott?

Furthermore, one may ask the same question about abelian varieties of higher dimension. The Faltings height is usually thought of as being related to the Hodge bundle on the moduli stack $\bar {\mathcal {A}}_g$ . But the stacky height associated to this line bundle, or any line bundle, will not be Northcott on $\bar {\mathcal {A}}_g$ , for the same reason it failed to be Northcott for $\overline {\mathcal {M}}_{1,1}$ in low characteristic; there are abelian varieties of dimension d with nonabelian automorphism group, which give rise to maps $BG \hookrightarrow \bar {\mathcal {A}}_g$ for nonabelian G, and no line bundle on $BG$ can be Northcott. This problem can be avoided by computing heights on $\bar {\mathcal {A}}_g$ with respect to the rank-g vector bundle $\mathcal {V} = e^* \Omega ^1_{A/\bar {\mathcal {A}}_g}$ , where A is the universal principally polarized abelian variety over the moduli stack, rather than with respect to its determinant, the Hodge bundle. There will still be problems in low characteristic, as we have seen from the case of elliptic curves. One way of understanding the difficulty with curves of the form (3.12) is that a wildly ramified extension of K is necessary in order to arrive at a curve with semistable reduction; this cannot be the case for elliptic curves over fields of characteristic $5$ or greater. The following question thus seems reasonable.

Question 3.14. When K is a global function field, $\mathcal {V}$ is the vector bundle $e^* \Omega ^1_{A/\overline {\mathcal {A}}_g}$ on $\overline {\mathcal {A}}_g$ , and $A/K$ is an abelian variety that becomes semistable over an everywhere tamely ramified extension of K, is it the case that

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{V}}(A) = \operatorname{\mathrm{ht}}_{\operatorname{\mathrm{Fal}}}(A)? \end{align*} $$

If Questions 3.13 and 3.14 both have a positive answer, one might well ask the common descendant of both questions: are there “exotic” vector bundles on $\mathcal {A}_g$ in small (relative to g) characteristic which compute the Faltings height of abelian varieties that require a wild extension to become semistable?

Finally, we return for a moment to the number field case. Because of the singularity at the boundary of $\overline {\mathcal {A}}_g$ of the Faltings metric, we cannot expect $\operatorname {\mathrm {ht}}_{\mathcal {V}}$ to match $\operatorname {\mathrm {ht}}_{\operatorname {\mathrm {Fal}}}$ exactly. But there is a way to ask whether the two heights agree ‘apart from the Archimedean place.’ Namely, we can ask the following.

This is a purely local question which has to do with the behavior of the tangent space to the Néron model of A under ramified base change. A positive answer to Question 3.15 would imply a positive answer to Question 3.14, as follows. The stable Faltings height $\operatorname {\mathrm {ht}}_s(A)$ agrees with $\operatorname {\mathrm {ht}}^{\operatorname {\mathrm {st}}}_{\mathcal {V}}$ in this setting, because both are given by the degree of the pullback of the Hodge bundle to an integral point $C' {\rightarrow } \overline {\mathcal {A}}_g$ , where $C'$ is a cover of C. And since there are no Archimedean places, the positive answer to Question 3.15 shows that

$$ \begin{align*} \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{Fal}}}(A) - \operatorname{\mathrm{ht}}_s(A) = \sum_v \delta_{\mathcal{V};v}(A) = \operatorname{\mathrm{ht}}_{\mathcal{V}}(A) - \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\mathcal{V}}(A). \end{align*} $$

3.5 Heights on footballs

A footballFootnote ⁵ $\mathcal {F}(a,b)$ is a $\mathbb {P}^1$ rooted at $0$ and $\infty $ , with residual gerbes $\mu _a$ and $\mu _b$ , respectively. Let K be a global field; we emphasize that K is allowed to have any characteristic, including characteristics dividing a or b. (When K has one of these characteristics, $\mathcal {F}(a,b)$ is a tame Artin stack but not a Deligne–Mumford stack.) As an illustration of the (moderate) subtlety of the Northcott condition in the stacky case, we will work out which line bundles on $\mathcal {F}(a,b)$ are Northcott.

There are three kinds of K-points of $\mathcal {F}(a,b)$ , which may be treated separately.

• • The points supported at $0$ ; these are naturally identified with K-points of $B(\mu _a)$ , which are in turn identified with the set $K^* / (K^*)^a$ ;

• • The points supported at $\infty $ ; these are naturally identified with K-points of $B(\mu _b)$ , which are in turn identified with the set $K^* / (K^*)^b$ ;

• • The rest of the points, which are naturally identified with the points on $\mathbb {P}^1(K)$ other than $0$ and $\infty $ ; that is, these points are in bijection with $K^*$ .

Any divisor on $\mathcal {F}(a,b)$ is linearly equivalent to one of the form $d[P] + n[0] + m[\infty ]$ , where P is some point on $\mathbb {G}_m$ ; such a divisor has degree $d + n/a + m/b$ . This expression is not unique but is subject to the relations $a[0] \sim b[\infty ] \sim [P]$ . Take $\mathcal {L}$ to be the line bundle on $\mathcal {F}(a,b)$ corresponding to $d[P] + n[0] + m[\infty ]$ . We now explain how to compute $\operatorname {\mathrm {ht}}_{\mathcal {L}}(x)$ for $x \in \mathcal {F}(a,b)(K)$ .

For the first two types of points, this computation of height has already been carried out in Section 2.4. For a point x of the first type, d and m are irrelevant. The class in $K^* / (K^*)^a$ associated to x is represented by a function $f \in K^*$ , and the height of x is a sum over places of K:

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal L}(x) = \sum_v \left\lceil \frac{n}{a} \operatorname{\mathrm{ord}}_v(f) \right\rceil. \end{align*} $$

Similarly, for a point of the second type, represented by the class of g in $K^* / (K^*)^b$ the height is

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal L}(x) = \sum_v \left\lceil \frac{m}{b} \operatorname{\mathrm{ord}}_v(g) \right\rceil. \end{align*} $$

We now treat points of the third, or generic type. For simplicity of description, take K to be the function field of a smooth proper curve $C/{\mathbb F}_q$ . Then x affords a rational map $\phi $ from C to $\mathcal {F}(a,b)$ . Write $\phi _c\colon C {\rightarrow } \mathbb {P}^1$ for the composition of $\phi $ with the coarse moduli map, denote $\deg \phi _c = \deg \phi $ by e, and write $\sum e_i P_i$ for the divisor $\phi _c^* [0]$ and $\sum e_i' Q_i$ for the divisor $\phi _c^* [\infty ]$ . Then $\sum e_i \deg P_i = \sum e^{\prime }_i \deg Q_i = e$ .

We may take $\mathcal {C}$ to be a root stack with residual gerbe $\mu _a$ at the $P_i$ and $\mu _b$ at the $Q_i$ . Then $\overline {x}^* \mathcal {L}^\vee $ is the divisor

$$ \begin{align*} -\left(d\phi^{-1}(P) + \sum_i \frac{e_i n}{a} P_i + \sum_i \frac{e^{\prime}_i m}{b} Q_i\right) \end{align*} $$

whose degree, as it must be, is $-e \deg \mathcal {L}$ .

We then have

(3.16) $$ \begin{align} \operatorname{\mathrm{ht}}_{\mathcal{L}}(x) & = - \deg \pi_* \overline{x}^* \mathcal{L} = -\left(-ed + \sum_i \left\lfloor -\frac{e_i n}{a} \right\rfloor \deg P_i + \sum_i \left\lfloor -\frac{e^{\prime}_i m}{b} \right\rfloor \deg Q_i\right) \log q \end{align} $$

(3.17) $$ \begin{align} & \hspace{-42pt}= \left(ed + \sum_i \left\lceil \frac{e_i n}{a} \right\rceil \deg P_i + \sum_i \left\lceil \frac{e^{\prime}_i m}{b} \right\rceil \deg Q_i \right) \log q.\hspace{42pt}\\[-16pt]\nonumber \end{align} $$

In particular, we note that $\operatorname {\mathrm {ht}}_{\mathcal {L}}(x) \geq e \log q \deg \mathcal {L}$ , with equality holding just when every $e_i$ is a multiple of a and every $e^{\prime }_i$ is a multiple of b, which is to say, when x actually extends to an integral point of $\mathcal {F}(a,b)$ .

This description suffices to tell us which line bundles have the Northcott property. We already see that the set of Northcott line bundles does not form a cone because it is not closed under addition. (Indeed, we could have already seen that from the case $B(\mathbf {Z}/2\mathbf {Z})$ , where the nontrivial line bundle $\mathcal {L}$ is Northcott and $\mathcal {L}^{\otimes 2}$ , which is trivial, is not Northcott.)

Proof. Suppose $(n,a) = r> 1$ . Then any point of $\mathbb {P}(a,b)$ of the first type which corresponds to $f \in (K^*)^{a/r}/(K^*)^a \subset K^* / (K^*)^a$ has height $0$ with respect to L, which contradicts Northcott. The argument is just the same if $(m,b)> 1$ .

We observe that there are infinitely many maps $\mathbb {P}^1 {\rightarrow } \mathcal {F}(a,b)$ ; namely, those whose coarse map $\mathbb {P}^1 {\rightarrow } \mathbb {P}^1$ is of the form $[B(s,t)^b:A(s,t)^a]$ . Any such map, pulled back to C via a map $C {\rightarrow } \mathbb {P}^1$ , gives an integral point $C {\rightarrow } \mathcal {F}(a,b)$ of some coarse degree e, whose height is $e \deg L$ ; we can make e as large as we want, which shows that L cannot be Northcott if $\deg L \leq 0$ .

Suppose, on the other hand, that all three conditions are met. We have already shown that points x of the third type have $\operatorname {\mathrm {ht}}_L(x) \geq e \log q \deg L$ ; since $\deg L$ is positive, $\operatorname {\mathrm {ht}}_L(x)$ gives an upper bound for e, which makes the set of possible x finite. For points of the first type represented by $f \in (K^*)/(K^*)^a$ , we observe that

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{L}}(x) = \sum_v \left\lceil \frac{n}{a} \operatorname{\mathrm{ord}}_v(f) \right\rceil = \sum_v \left\{\frac{n}{a} \operatorname{\mathrm{ord}}_v(f) \right\},\\[-16pt] \end{align*} $$

the latter equality following from $\sum _v \operatorname {\mathrm {ord}}_v(f) = 0$ . So a bound on the height of x yields a bound on the number of places, where $\frac {n}{a} \operatorname {\mathrm {ord}}_v(f)$ is not an integer; since $(n,a) = 1$ , this bounds the number of places where (more precisely: the degree of the divisor where) $\operatorname {\mathrm {ord}}_v(f)$ is not a multiple of a. Bounding this quantity places f within a finite set of cosets of $(K^*)^a$ , so we are done. The case of points of the second type is exactly the same.

3.5.1 Consistency check: footballs and weighted projective lines

When a and b are relatively prime, the football $\mathcal {F}(a,b)$ is isomorphic to the weighted projective line $\mathbb {P}(a,b)$ ; on K-points, the isomorphism $\psi $ from $\mathbb {P}(a,b)$ to $\mathcal {F}(a,b)$ sends $(s:t)$ to the point $t^a/s^b$ when $st \neq 0$ . Let $m,n$ be integers such that $ma+nb = 1$ ; then the line bundle $\mathcal {L} = n[0] + m[\infty ]$ on ${\mathbb F}(a,b)$ has degree $1/ab$ , and its pullback to $\mathbb {P}(a,b)$ is the tautological bundle $\mathcal {O}_{\mathbb {P}(a,b)}(1)$ . If x is a point of $\mathbb {P}(a,b)(K)$ , we have

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{O}_{\mathbb{P}(a,b)}(1)}(x) = \operatorname{\mathrm{ht}}_{\mathcal{L}}(\psi(x)).\\[-16pt] \end{align*} $$

This provides an opportunity to check consistency between the formulas we have given for the height of a point on weighted projective space and the height of a point on a football. Let $x = (s:t)$ be a point of $\mathbb {P}(a,b)$ . Then by equation (3.5), we have

(3.19) $$ \begin{align} \operatorname{\mathrm{ht}}_{\mathcal{O}_{\mathbb{P}(a,b)}(1)}(x) = \sum_v \lceil \log_{q_v} \max (|s|_v^{1/a},|t|_v^{1/b}) \rceil \log q_v. \end{align} $$

We now compute $\operatorname {\mathrm {ht}}_{\mathcal {L}}(\psi (x))$ . Recall that $\psi (x)$ is the point on ${\mathbb F}(a,b)$ corresponding to the point $t^a/s^b$ of $\mathbb {P}^1(K)$ . In the notation of the above section, the points $P_i$ correspond to those places v of K, where $a \operatorname {\mathrm {ord}}_v t - b \operatorname {\mathrm {ord}}_v s> 0$ , and the points $Q_i$ to those places where $a \operatorname {\mathrm {ord}}_v t - b \operatorname {\mathrm {ord}}_v s < 0$ . When v is a prime with $a \operatorname {\mathrm {ord}}_v t - b \operatorname {\mathrm {ord}}_v s> 0$ , we have, again maintaining the notation of equation (3.17),

$$ \begin{align*} e_i = a \operatorname{\mathrm{ord}}_v t - b \operatorname{\mathrm{ord}}_v s\\[-15pt] \end{align*} $$

and

$$ \begin{align*} \deg P_i = \log q_v / \log q.\\[-15pt] \end{align*} $$

So the contribution of v to equation (3.17) is

$$ \begin{align*} \left\lceil \frac{(a \operatorname{\mathrm{ord}}_v t - b \operatorname{\mathrm{ord}}_v s)n}{a} \right\rceil \log q_v &= \left( n \operatorname{\mathrm{ord}}_v t - \left\lceil \frac{nb \operatorname{\mathrm{ord}}_v s}{a} \right\rceil \right)\log q_v\\ &= \left( n \operatorname{\mathrm{ord}}_v t + m \operatorname{\mathrm{ord}}_v s - \left\lceil \frac{\operatorname{\mathrm{ord}}_v s}{a} \right\rceil \right) \log q_v.\\[-15pt] \end{align*} $$

By a similar argument, one shows that when $a \operatorname {\mathrm {ord}}_v t - b \operatorname {\mathrm {ord}}_v s < 0$ one gets a contribution of

$$ \begin{align*} \left( n \operatorname{\mathrm{ord}}_v t + m \operatorname{\mathrm{ord}}_v s - \left\lceil \frac{\operatorname{\mathrm{ord}}_v t}{b} \right\rceil \right) \log q_v.\\[-15pt] \end{align*} $$

Since the first case obtains exactly when $\operatorname {\mathrm {ord}}_v s / a < \operatorname {\mathrm {ord}}_v t / b$ , we can express the contribution of v uniformly as

$$ \begin{align*} \left( n \operatorname{\mathrm{ord}}_v t + m \operatorname{\mathrm{ord}}_v s - \left\lceil \min \left(\frac{\operatorname{\mathrm{ord}}_v s}{a} , \frac{\operatorname{\mathrm{ord}}_v t}{b} \right) \right\rceil \right) \log q_v.\\[-15pt] \end{align*} $$

Summing over v, the first two terms vanish by the product formula, and we are left with

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\mathcal{L}}(\psi(x)) = -\sum_v \min \left(\left\lceil \frac{\operatorname{\mathrm{ord}}_v s}{a} \right\rceil, \left\lceil \frac{\operatorname{\mathrm{ord}}_v t}{b} \right\rceil \right) \log q_v\\[-15pt] \end{align*} $$

which is just equation (3.19) in another form.

3.6 Heights on symmetric powers of varieties

There is a substantial literature about points on varieties of bounded algebraic degree. We explain how these questions look through the lens of heights on stacks. Let X be a smooth proper scheme of dimension n over K. A point on X of algebraic degree m over K can be thought of as a K-point on the stack $\operatorname {\mathrm {Sym}}^m X = [X^m / S_m]$ . In this section, we explain how to compute the height of such a point. Slightly more generally, let G be a subgroup of $S_m$ , and let $\mathcal {X}$ be the quotient $[X^m / G]$ ; when $G = S_m$ , our stack $\mathcal {X}$ is $\operatorname {\mathrm {Sym}}^m X$ .

In order to talk about height, we need to choose a vector bundle $\mathcal {V}$ on $\mathcal {X}$ ; this is the same thing as an G-equivariant vector bundle on $X^m$ . The choice we make is as follows: Let $V_0$ be some vector bundle of rank r on X, and let $\pi _1,\ldots , \pi _m\colon X^m {\rightarrow } X$ be the m projections. Then $\widetilde {V} = \oplus _i \pi _i^* V_0$ is an G-equivariant vector bundle of rank $mr$ , which descends to a vector bundle $\mathcal {V}$ of rank $mr$ on $\mathcal {X}$ .

Let x be a point of $\mathcal {X}(K)$ . We begin by computing the stable height $\operatorname {\mathrm {ht}}^{\operatorname {\mathrm {st}}}_{\mathcal {V}}(x)$ . The Cartesian square

provides an étale algebra L over K which carries an $S_m$ -action and a rational point $x_L$ which extends to an integral point $C {\rightarrow } X^m$ . By Proposition 2.14,

$$ \begin{align*} \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\mathcal{V}}(x) = [L:K]^{-1}\operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\widetilde{V}}(x_L). \end{align*} $$

(Should L be an étale algebra which is not a field but rather a direct sum $\oplus _i F_i$ , our convention is that the height of a point of $X^m(L)$ is $\sum _i \operatorname {\mathrm {ht}}(P_i)$ , where $P_i \in X^m(F_i)$ are the points corresponding to the restriction of $x_L\colon \operatorname {\mathrm {Spec}} L {\rightarrow } X^m$ to connected components of $\operatorname {\mathrm {Spec}} L$ .)

Since $X^m$ is a scheme, we have

$$ \begin{align*} \operatorname{\mathrm{ht}}^{\operatorname{\mathrm{st}}}_{\widetilde{V}}(x_L) = \operatorname{\mathrm{ht}}_{\widetilde{V}}(x_L). \end{align*} $$

The latter quantity is a very natural one, what you might call the ‘absolute height’ of x. Suppose, for instance, that $L/K$ is a field extension, necessarily Galois with Galois group G. Then $x_L$ is a point of $X^m(L)$ on which $\operatorname {\mathrm {Gal}}(L/K)$ acts by permutations; in other words, it is an element $(\alpha _1, \ldots , \alpha _m)$ , where the $\alpha _i$ are conjugate and each $\alpha _i$ is contained in a degree-m extension $L_i/K$ whose Galois closure is L. The (unordered) set $\alpha _1, \ldots , \alpha _m$ can be thought of as a K-rational Galois orbit of points on X, and the height of $x_L$ is then given by the usual Weil height on X:

$$ \begin{align*} \operatorname{\mathrm{ht}}_{\widetilde{V}}(x_L) = \sum_i \operatorname{\mathrm{ht}}_{V_0;L} \alpha_i = m \operatorname{\mathrm{ht}}_{V_0;L} \alpha_1, \end{align*} $$