Heights of Special Points on Quaternionic Shimura Varieties

Let $E$ be a CM field, and $F$ be its totally real subfield, so that $[E:F]=2$. Set $g\coloneqq[F:\mathbb{Q}]$. Let $\phi\subset\operatorname{Hom}(E,\mathbb{C})$ be a partial CM-type, meaning that $\phi\cap\overline{\phi}=\varnothing$. Write $\Sigma\subset\operatorname{Hom}(F,\mathbb{R})$ for the restriction of $\phi$ to $F$. Suppose that $B/F$ is a quaternion algebra with the following properties:

1. (1)

   There exists an embedding $E\hookrightarrow B$;

2. (2)

   The ramification set of $B$ at infinity is $\Sigma^{c}$;

3. (3)

   If $B$ is ramified at a finite prime $\mathfrak{p}$ of $F$, then $E$ is also ramified over $\mathfrak{p}$.

We define the algebraic group $G$ over $\mathbb{Q}$ as

$$G\coloneqq\operatorname{Res}_{F/\mathbb{Q}}B^{\times}.$$

For each compact open subgroup $U\subset G(\mathbb{A}_{f})$, we obtain a (quaternionic) Shimura variety $X_{U}$, defined over a number field $E_{X}$, with complex uniformization given by:

$$X_{U}(\mathbb{C})=G(\mathbb{Q})\backslash(\mathcal{H}^{\pm})^{\Sigma}\times G(\mathbb{A}_{f})/U,$$

where $\mathcal{H}^{\pm}$ is the upper and lower complex half-planes. This is a Shimura variety of abelian type, and by utilizing ideas from [Car83] and [PZ13], we can construct a regular integral model $\mathcal{X}_{U}$ for $X_{U}$ over $\operatorname{Spec}\mathcal{O}_{E_{X}}$.

Let $\widehat{\mathcal{L}_{U}}$ be the arithmetic Hodge bundle of $\mathcal{X}_{U}$, which consists of a line bundle $\mathcal{L}_{U}$ on $\mathcal{X}_{U}$ and a Hermitian metric given by:

$$\left\lVert\bigwedge_{\sigma\in\Sigma}dz_{\sigma}\right\rVert\coloneqq\prod_{\sigma\in\Sigma}2\mathrm{Im}(z_{\sigma}),$$

where the $z_{\sigma}$ are given by the complex uniformization of $X_{U}$. When $U$ is sufficiently small, the Hodge bundle is simply the canonical bundle $\mathcal{L}_{U}=\omega_{\mathcal{X}_{U}/\mathcal{O}_{E_{X}}}$. The precise definition of $\widehat{\mathcal{L}_{U}}$ is given in Section 6.

Let $P_{U}\subset X_{U}(\overline{\mathbb{Q}})$ be a special point arising from the embedding $E\hookrightarrow B$, and let $\overline{P_{U}}$ be the closure of this point in $\mathcal{X}_{U}$, which we will also denote by $P_{U}$ by abuse of notation. The height of this point relative to $\widehat{\mathcal{L}_{U}}$ is the Arakelov height:

$$h_{\widehat{\mathcal{L}_{U}}}(P_{U})\coloneqq\frac{1}{[\mathbb{Q}(P_{U}):\mathbb{Q}]}\widehat{\deg}(\widehat{\mathcal{L}_{U}}|_{P_{U}}).$$

Finally, let $\Phi$ be a full CM-type, and let $h(\Phi)$ be the Faltings height of an abelian variety with complex multiplication by $(\mathcal{O}_{E},\Phi)$. Let $d_{\phi},d_{\overline{\phi}}$ and $d_{\Sigma}\coloneqq d_{\phi\sqcup\overline{\phi}}$ be certain absolute discriminants of $\phi,\overline{\phi}$, and $\phi\sqcup\overline{\phi}$. These are defined in detail in Section 2.

Let $\mathfrak{d}_{E/F}$ denote the relative discriminant of the extension $E/F$. There is a reflex norm $N_{F/E_{X}}\colon F\to E_{X}$ defined by $N_{F/E_{X}}(x)=\prod_{\sigma\in\Sigma}\sigma(x)$. Let $d_{E/F,\Sigma}\in\mathbb{Z}$ be the positive generator of $N_{E_{X}/\mathbb{Q}}(N_{F/E_{X}}(\mathfrak{d}_{E/F}))$. Let $d_{E/F}\in\mathbb{Z}$ be the positive generator of $N_{F/\mathbb{Q}}(\mathfrak{d}_{E/F})$ and let $d_{F}$ be the absolute discriminant of $F$. Let $d_{B}$ be the positive generator of norm from $F$ to $\mathbb{Q}$ of the product of all the finite places of $\mathcal{O}_{F}$ over which $B$ ramifies.

With these definitions in place, we can now state our main theorem.

Additionally, if $\lvert\phi\rvert=1$, then $E_{X}=F$, and we have that $d_{\Sigma}=d_{E/F}=d_{E/F,\Sigma}$ and $d_{\phi}=d_{\overline{\phi}}=1$. As a result, the expression for $\frac{1}{2}h_{\widehat{\mathcal{L}_{U}}}(P_{U})$ simplifies, and we recover [YZ18, Thm. 1.6], where the factor of $g$ is due to different normalizing factors of $h_{\widehat{\mathcal{L}_{U}}}(P_{U})$.

In the Pila–Zannier method, an essential step involves bounding the height of a CM point in terms of the discriminant of its splitting field. For general Shimura varieties, a CM point $P$ is associated with a partial CM-type $\phi$ of a CM-field $E$. In [PST⁺22], a canonical height $h(\phi)$ is introduced for such $\phi$, and this height $h(\phi)$ is shown to be equal to the height $h(P)$ of the associated CM point $P$, up to $O(\log d_{E})$. Then, the height $h(\phi)$ is bounded in terms of the discriminant $d_{E}$, showing that $h(P)$ is as well. The precise definition of $h(\phi)$ can be found in Section 8. Our second main result establishes the compatibility between $h_{\widehat{\mathcal{L}_{U}}}(P_{U})$ and $h(\phi)$.

With the combination of Theorem 1.1 and Theorem 1.2, we obtain the following corollary. This is an ingredient used in the Pila–Zannier method, and serves as one of the results of [PST⁺22].

This result differs from that of [PST⁺22] because in our case, we are able to express $h(\phi)$ in terms of CM-types of $E$, whereas in [PST⁺22], they express $h(\phi)$ in terms of CM-types $\Phi^{\sharp}$ of CM-fields $E^{\sharp}$ containing $E$, whose relative discriminant over $E$ can be bounded.

The André–Oort conjecture states that if $S$ is a Shimura variety and $V\subset S$ is a subvariety with a dense subset of special points, then $V$ itself must be a special subvariety. Definitions and properties for Shimura varieties can be found in [Mil05, Sec. 4] and for special subvarieties in [DR18, Sec. 2]. The André–Oort conjecture was originally proven by André in the case when $S$ is the product of two modular curves [And98] and later for arbitrary products of modular curves by Pila (see [Pil11]). Pila and Tsimerman (see [PT14]) were able to extend this strategy to prove the conjecture unconditionally for $\mathcal{A}_{g}$, the coarse moduli space of principally polarized abelian varieties of dimension $g$, when $g\leq 6$. The averaged Colmez Conjecture, proven by [AGHMP18] and [YZ18], allowed Tsimerman to give an unconditional proof (see [Tsi18]) of the conjecture for $\mathcal{A}_{g}$ for all $g$. Moreover, following results of [BSY23], Pila, Shankar, and Tsimerman recently announced an unconditional proof of the conjecture for all Shimura varieties in [PST⁺22]. There is also a version of the André–Oort conjecture for mixed Shimura varieties that was reduced to the pure Shimura varieties case by Gao (see [Gao16]).

Many of the recent results on the André–Oort conjecture were proven using a strategy that was initially proposed by Zannier, by using the theory of o-minimality and a point counting theorem of Pila and Wilkie from [PW06], combined with estimates on the sizes of certain Galois orbits. This strategy was shown to be viable when Pila and Zannier used it to reprove the Manin–Mumford conjecture in [PZ08]. And it is using this Pila–Zannier strategy that the conjecture was proven for products of the modular curve (see [Pil11]), the moduli space $\mathcal{A}_{g}$ (see [PT14, Tsi18]), and for all Shimura varieties (see [PST⁺22]). The strategy can be split up into three main ingredients.

1. (1)

   The first ingredient is the Pila–Wilkie point counting theorem from [PW06]. It gives an upper bound subpolynomial in height on the rational points of the transcendental component of definable sets. The fundamental domains of the universal covering map of a Shimura variety are definable in an o-minimal structure. This was shown for $\mathcal{A}_{g}$ by Peterzil and Starchenko (see [PS13]), and for arbitrary Shimura varieties by Klinger, Ullmo, and Yafaev (see [KUY16]). A sharpened version proven by Binyamini (see [Bin22]) was necessary for general Shimura varieties.

2. (2)

   The second ingredient is a lower bound polynomial in height for the size of Galois orbits of special points. This is done in two steps. The first step to provide a lower bound polynomial in height for the discriminant of certain endomorphism algebras. This was done by Pila and Tsimerman (see [PT13]) for $\mathcal{A}_{g}$, and by Binyamini, Schmidt, and Yafaev (see [BSY23]) for arbitrary Shimura varieties. The second step is to provide a lower bound subpolynomial in terms of discriminant of those same algebras. Tsimerman proves this for $\mathcal{A}_{g}$ (see [Tsi18]) by combining a result of Masser and Wüstholz (see [MW95]), and the averaged Colmez conjecture, which was proven independently by Andreatta, Goren, Howard, and Madapusi-Pera (see [AGHMP18]), and Yuan and Zhang (see [YZ18]). For general Shimura varieties, Binyamini, Schmidt, and Yafaev (see [BSY23]) reduce this to proving the existence of a canonical height on Shimura varieties and that the height of special points is bounded subpolynomially in terms of the discriminant of certain number fields. A proof of this result was recently announced by Pila, Shankar, and Tsimerman (see [PST⁺22]).

3. (3)

   The third ingredient is the Ax–Lindemann theorem. The first two ingredients combine to show that Galois orbits of special points must lie in the algebraic part of the fundamental domain. Then, the Ax–Lindemann theorem tells us that these algebraic parts are precisely special subvarieties. This was proven for $\mathcal{A}_{g}$ by Pila and Tsimerman (see [PT14]), and by Mok, Pila, and Tsimerman for all Shimura varieties (see [MPT19]).

For this article, we are interested in the contribution of [PST⁺22] in showing that their canonical height for CM points on Shimura varieties is bounded in terms of discriminants of their splitting fields. The first step is to systematically define a Weil height function for any arbitrary Shimura variety. Given a Shimura variety $\mathrm{Sh}_{K}(G,X)$, take a $\mathbb{Q}$-representation $G\to\operatorname{GL}(V)$ and lattice $\Lambda\subset V$. By the Riemann–Hilbert correspondence over $p$-adic local fields, given by [DLLZ23], we get a filtered automorphic vector bundle with connection $(_{\operatorname{dR}}V,\operatorname{Fil}^{\bullet},\nabla)$, which is defined over the reflex field of $\mathrm{Sh}_{K}(G,X)$ and all of its $p$-adic places. Then the plan is to define an adelic norm on $\operatorname{Gr}^{\bullet}_{\operatorname{dR}}V$, which would give rise to an Arakelov height function on $\mathrm{Sh}_{K}(G,X)$. At the archimedean places, this representation admits a polarization $\psi\colon V\times V\to\mathbb{Q}$ and the norm can be defined as the Hodge norm $\psi(v,h(i)v)$. Over the finite places, the crystalline norm is used when the representation is crystalline and an alternative intrinsic norm is used at the finitely many other places. This height is compatible in the sense that if $(G_{1},X_{1})\to(G_{2},X_{2})$ is a map of Shimura datum with $\rho_{i}$ a representation of $G_{i}$ compatible with this morphism, then the height of a point of $\mathrm{Sh}_{K_{2}}(G_{2},X_{2})$ with respect to $\rho_{2}$ is equal to the height of a point of $\mathrm{Sh}_{K_{1}}(G_{1},X_{1})$ with respect to $\rho_{1}$. This height also coincides with the Faltings height for $\mathcal{A}_{g}$.

With this height defined, the next step is to bound the height of special points in terms of discriminants of certain number fields. If $(T,x)\subset(G,X)$ is a Shimura sub-datum of a special point, then $T$ splits over a CM field $E/F$. Given a representation $\rho\colon G\to\operatorname{GL}(V)$ and restricted representation $\rho_{x}\colon T\to G\to\operatorname{GL}(V)$, we can map $(T,x,\rho_{x})\to(\operatorname{Res}_{F/\mathbb{Q}}E^{\times}/F^{\times},x,\rho_{\phi})$ to another Shimura datum where $\phi\subset\operatorname{Hom}(E,\mathbb{C})$ is a partial CM-type and $\rho_{\phi}$ is a representation of $\operatorname{Res}_{F/\mathbb{Q}}E^{\times}/F^{\times}$ in terms of $\phi$. The height $h(\phi)$ is defined as the height of $(\operatorname{Res}_{F/\mathbb{Q}}E^{\times}/F^{\times},x,\rho_{\phi})$. By the compatibility of heights, the problem is reduced to bounding the height $h(\phi)$. Given a set of disjoint CM-fields and partial CM-types $\{(E_{i},\phi_{i})\}_{i=1}^{t}$ they are able to express a linear combination of $h(\phi_{i})$ in terms of a linear combination of heights of full CM-types $\Phi^{S}$ of $E^{S}=\prod_{i\in S}E_{i}$ for various subsets $S\subset\{1,2,\dots,t\}$. By ranging over all subsets $S$, we can express the height of an individual $h(\phi_{i})$ as a linear combination of Faltings heights of CM-types $\Phi^{S}$ of $E^{S}$, where $S$ varies over all possible subsets of $\{1,2,\dots,t\}$. The height of the full CM-type $h(\Phi^{S})$ is bounded subpolynomially in terms of the discriminant $d_{E^{S}}$. Choosing $E_{i}$ carefully, we can bound the relative discriminant $\mathfrak{d}_{E^{S}/E_{i}}$, completing the proof.

Instead of expressing the height $h(\phi)$ in terms of heights of CM-types over the many $E^{S}$, we give an expression in terms of CM-types of $E$ only.

The idea is similar to that of [YZ18]. We use their decomposition of Faltings heights of a CM abelian variety $h(\Phi)$ into constituent parts $h(\Phi,\tau)$, one for each archimedean place $\tau\in\Phi$. The constituent parts are related to the full CM-type by the formula

$$h(\Phi)-\sum_{\tau\in\Phi}h(\Phi,\tau)=\frac{-1}{4[E_{\Phi}:\mathbb{Q}]}\log(d_{\Phi}d_{\overline{\Phi}}),$$

where $d_{\Phi},d_{\overline{\Phi}}$ are discriminants associated with $\Phi,\overline{\Phi}$ respectively, and $E_{\Phi}$ is the reflex field of $\Phi$. Moreover, if $(\Phi_{1},\Phi_{2})$ are nearby CM-types of $E$ in that they differ only at a single place $\tau_{i}=\Phi_{i}\backslash(\Phi_{1}\cap\Phi_{2})$, then [YZ18] proves that the quantity

$$h(\Phi_{1},\tau_{1})+h(\Phi_{2},\tau_{2})$$

is the same across any choice of nearby CM-types.

We define the group

$$G^{\prime\prime}\coloneqq\operatorname{Res}_{F/\mathbb{Q}}(B^{\times}\times E^{\times})/F^{\times},$$

where $F$ embeds diagonally as $a\mapsto(a,a^{-1})$. We can construct a norm $N\colon G^{\prime\prime}\to\operatorname{Res}_{F/\mathbb{Q}}\mathbb{G}_{m}$ and define the group

$$G^{\prime}\coloneqq G^{\prime\prime}\times_{\mathbb{G}_{m}}\operatorname{Res}_{F/\mathbb{Q}}\mathbb{G}_{m}$$

consisting of elements $G^{\prime\prime}$ with norm lying in $\mathbb{Q}^{\times}$. If $\phi$ is a partial CM-type and $\phi^{\prime}$ is a complementary partial CM-type in that $\phi\sqcup\phi^{\prime}$ constitute a full CM-type, then we can construct morphisms $h^{\prime}\colon\mathbb{C}^{\times}\to G^{\prime}(\mathbb{R})$ and $h^{\prime\prime}\colon\mathbb{C}^{\times}\to G^{\prime\prime}(\mathbb{R})$. They give rise to Shimura datum and Shimura varieties $X^{\prime}_{U^{\prime}}$ and $X^{\prime\prime}_{U^{\prime\prime}}$ for compact open subgroups $U^{\prime}\subset G^{\prime}(\mathbb{A}_{f})$ and $U^{\prime\prime}\subset G^{\prime\prime}(\mathbb{A}_{f})$ with complex uniformizations

$$X^{\prime}_{U^{\prime}}(\mathbb{C})=G^{\prime}(\mathbb{Q})\backslash(\mathcal{H}^{\pm})^{\Sigma}\times G^{\prime}(\mathbb{A}_{f})/U^{\prime}$$

and

$$X^{\prime\prime}_{U^{\prime\prime}}(\mathbb{C})=G^{\prime\prime}(\mathbb{Q})\backslash(\mathcal{H}^{\pm})^{\Sigma}\times G^{\prime\prime}(\mathbb{A}_{f})/U^{\prime\prime}$$

They have canonical models defined over the same reflex field $E_{X^{\prime}}=E_{X^{\prime\prime}}$.

The Shimura variety $X^{\prime}_{U^{\prime}}$ is of PEL type and has an integral model $\mathcal{X}^{\prime}_{U^{\prime}}$ by [Car83] and [PZ13]. The pair $(\phi,\phi^{\prime})$ gives rise to a point $P^{\prime}_{U^{\prime}}\in X^{\prime}_{U^{\prime}}$ which parametrizes an abelian variety isogenous to a product $A_{1}\times A_{2}$ of abelian varieties, one with complex multiplication of type $\phi\sqcup\phi^{\prime}$ and the other with complex multiplication of type $\overline{\phi}\sqcup\phi^{\prime}$. After defining a suitable metric on $\omega_{\mathcal{X}^{\prime}_{U^{\prime}}/\mathcal{O}_{E_{X^{\prime}}}}$, the Kodaira–Spencer isomorphism on $X^{\prime}_{U^{\prime}}$ gives us an equality of heights

$$h_{\widehat{\omega_{\mathcal{X}^{\prime}_{U^{\prime}}/\mathcal{O}_{E_{X^{\prime}}}}}}(P^{\prime}_{U^{\prime}})=\sum_{\tau\in\phi}\left(h(\phi\sqcup\phi^{\prime},\tau)+h(\overline{\phi}\sqcup\phi^{\prime},\overline{\tau})\right).$$

Now the idea is to relate $\omega_{\mathcal{X}_{U}/\mathcal{O}_{E_{X}}}$ and $\omega_{\mathcal{X}^{\prime}_{U^{\prime}}/\mathcal{O}_{E_{X^{\prime}}}}$. We do this by mapping both $X_{U}$ and $X^{\prime}_{U^{\prime}}$ into the third Shimura variety $X^{\prime\prime}_{U^{\prime\prime}}$ so that the points $P_{U}$ and $P^{\prime}_{U^{\prime}}$ have the same image $P^{\prime\prime}_{U^{\prime\prime}}\in X^{\prime\prime}_{U^{\prime\prime}}(\overline{\mathbb{Q}})$. We represent both canonical bundles in terms of deformations of $p$-divisible groups $\mathcal{H}_{U}$ and $\mathcal{H}^{\prime}_{U^{\prime}}$ over $\mathcal{X}_{U}$ and $\mathcal{X}^{\prime}_{U^{\prime}}$ respectively, and then relate those $p$-divisible groups to a $p$-divisible group $\mathcal{H}^{\prime\prime}_{U^{\prime\prime}}$ over $\mathcal{X^{\prime\prime}}_{U^{\prime\prime}}$. After showing all this, we get that

$$h_{\widehat{\mathcal{L}_{U}}}(P_{U})=h_{\widehat{\omega_{\mathcal{X}^{\prime}_{U^{\prime}}/\mathcal{O}_{E_{X^{\prime}}}}}}(P^{\prime}_{U^{\prime}})=\sum_{\tau\in\phi}\left(h(\phi\sqcup\phi^{\prime},\tau)+h(\overline{\phi}\sqcup\phi^{\prime},\overline{\tau})\right).$$

Of note is that this formula does not depend on the choice of complementary partial CM-type $\phi^{\prime}$, because the Shimura variety $X_{U}$ was defined independently of the choice of $\phi^{\prime}$. We utilize this by summing over all possible complementary CM-type, which will express $h_{\widehat{\mathcal{L}_{U}}}(P_{U})$ in terms of heights of full CM-types containing $\phi$ as well as nearby CM-types. The sum of heights of nearby CM-types is shown to be constant in [YZ18], and equal to the averaged height of all CM-types of $E$. Combining these two, we are able to express the height in terms of CM-types containing $\phi$ and an average of all possible CM-types.

We first recall from [YZ18] the decomposition of a Faltings height in Section 2. We then describe three Shimura varieties that can be constructed from a quaternion algebra following [TX16] by describing their generic fiber in Section 3 and integral models in Section 4. We then describe some line bundles on these Shimura varieties in terms of Lie algebras of certain $p$-divisible groups described in Section 5 and relate these Lie algebras, and finally define the Hodge bundle, in Section 6. Finally, we prove our theorem for the height of partial CM-types in Section 7 and compare our height with those introduced in [PST⁺22] in Section 8.

We wish to thank Sebastian Eterović, Ananth Shankar, and Xinyi Yuan for their helpful discussions with us.

We first define the Faltings height of an abelian variety. It will be defined as the degree of a metrized line bundle. Let $A$ be an abelian variety of dimension $g$ defined over a number field $K$ and let $\mathcal{A}$ be the Néron model over $\mathcal{O}_{K}$ and let the identity section be $s\colon\operatorname{Spec}\mathcal{O}_{K}\to\mathcal{A}$. Let $\Omega_{\mathcal{A}/\mathcal{O}_{K}}$ be the sheaf of relative differentials. The Hodge bundle of $A$ is the vector bundle $\Omega(\mathcal{A})\coloneqq s^{*}\Omega_{\mathcal{A}/\mathcal{O}_{K}}$ over $\mathcal{O}_{K}$. This is canonically isomorphic to the pushforward $\pi_{*}\Omega_{\mathcal{A}/\mathcal{O}_{K}}$, where $\pi\colon\mathcal{A}\to\mathcal{O}_{K}$ is the structure sheaf morphism.

The Hodge bundle $\Omega(\mathcal{A})$ is a vector bundle over $\mathcal{O}_{K}$ of rank $g$ and taking the determinant $\omega(\mathcal{A})\coloneqq\Omega(\mathcal{A})^{\wedge g}$ gives a line bundle over $\mathcal{O}_{K}$. To make this into a metrized line bundle, we need to define a norm for each archimedean place of $K$. We have that

$$\omega(\mathcal{A})\otimes_{\mathcal{O}_{K}}K\cong s^{*}\omega_{A/K}=H^{0}(A,\omega_{A/K}).$$

For each archimedean place $v$ of $K$, we put the norm as

$$\lVert\alpha\rVert_{v}\coloneqq\left\lvert\frac{1}{(2\pi)^{g}}\int_{A_{v}(\mathbb{C})}\alpha\wedge\overline{\alpha}\right\rvert^{\frac{1}{2}}$$

for each $\alpha\in\omega(\mathcal{A})\otimes_{\mathcal{O}_{K}}K_{v}\cong H^{0}(A_{v},\omega_{A_{v}/K_{v}})$. In this way, we get a metrized line bundle $\widehat{\omega(\mathcal{A})}\coloneqq(\omega(\mathcal{A}),\lVert\cdot\rVert_{v})$.

If $A$ has semistable reduction over $K$, then the Faltings height is invariant under finite field extensions. In general, we define the stable Faltings height as the height after base change to a finite extension $K^{\prime}/K$ such that $A$ has semistable reduction over $K^{\prime}$. Such a $K^{\prime}$ always exists.

A CM-field extension is an extension $E/F$ of number fields such that $F/\mathbb{Q}$ is a totally real field and $E/F$ is a quadratic totally imaginary extension. We say $E$ is a CM-field and $F$ is its totally real subfield.

A (full) CM-type is a subset $\Phi\subset\operatorname{Hom}(E,\mathbb{C})$ such that $\Phi\sqcup\overline{\Phi}=\operatorname{Hom}(E,\mathbb{C})$, where $\overline{\Phi}=\{\overline{\sigma}:\sigma\in\Phi\}$. A partial CM-type is a subset $\phi\subset\operatorname{Hom}(E,\mathbb{C})$ such that $\phi\cap\overline{\phi}=\varnothing$. We say that $\phi^{\prime}$ is a complementary partial CM-type to $\phi$ if $\phi\sqcup\phi^{\prime}$ is a CM-type.

We say that a complex abelian variety $A$ has complex multiplication of type $(\mathcal{O}_{E},\Phi)$ if there exists an embedding $\iota\colon\mathcal{O}_{E}\to\operatorname{End}(A)$ and an isomorphism $\operatorname{Lie}(A)\cong\mathbb{C}^{g}\overset{\Phi}{\cong}E\otimes_{\mathbb{Q}}\mathbb{R}$ of $\mathcal{O}_{E}$ modules.

Let $E$ be a CM-field with degree $[E:\mathbb{Q}]=2g$ and let $\Phi\subset\operatorname{Hom}(E,\mathbb{C})$ be a CM-type. Let $A_{\Phi}$ be an abelian variety of CM-type $(\mathcal{O}_{E},\Phi)$. Then, there is a number field $K$ over which $A_{\Phi}$ is defined and has a smooth projective integral model $\mathcal{A}/\mathcal{O}_{K}$. Colmez proved the following theorem

We write $h(\Phi)\coloneqq h(A_{\Phi})$. Colmez conjectured a formula about $h(\Phi)$ in terms of logarithmic derivatives of Artin L-functions related to $\Phi$. This conjecture has been proven when $E/\mathbb{Q}$ is an abelian extension by Obus and Colmez (see [Obu13]) and when $F$ is a real quadratic field by Yang (see [Yan10]). An averaged version was proven in [YZ18], and independently in [AGHMP18].

We recall the results of [YZ18] decomposing the Faltings height of a CM-type $\Phi$ into its constituent embeddings $\tau\in\Phi$. To decompose the height, we first decompose the Hodge bundle into its eigenspaces.

Let $A$ be an abelian variety with complex multiplication of type $(\mathcal{O}_{E},\Phi)$. We define

$$\Omega(A)_{\tau}\coloneqq\Omega(A)\otimes_{E_{\tau}}\mathbb{C},$$

where $E_{\tau}$ acts on $\mathbb{C}$ through the projection $E\otimes_{\mathbb{Q}}\mathbb{C}\cong\prod_{\sigma\in\operatorname{Hom}(E,\mathbb{C})}\mathbb{C}_{\sigma}\to\mathbb{C}_{\tau}$. This gives us a decomposition of the Hodge bundle as

$$\Omega(A)\cong\bigoplus_{\tau\colon E\to\mathbb{C}}\Omega(A)_{\tau}\cong\bigoplus_{\tau\in\Phi}\Omega(A)_{\tau}.$$

The latter isomorphism holds because $\Omega(A)_{\tau}=0$ for $\tau\not\in\Phi$.

Let $A^{t}$ be the dual abelian variety of $A$. Then, we have canonical isomorphisms

$$\Omega(A^{t})=\operatorname{Lie}(A^{t})^{\vee}\cong H^{1}(A,\mathcal{O}_{A})^{\vee}\cong H^{0,1}(A)^{\vee}=\overline{\Omega(A)}^{\vee},$$

so that if $A$ is of CM-type $(\mathcal{O}_{E},\Phi)$, then $A^{t}$ is of CM-type $(\mathcal{O}_{E},\overline{\Phi})$. From this isomorphism, we also get a perfect Hermitian pairing $\Omega(A^{t})\otimes\Omega(A)\to\mathbb{C}$.

Just as before, we can decompose

$$\Omega(A^{t})\cong\bigoplus_{\tau\in\overline{\Phi}}\Omega(A^{t})_{\tau}.$$

The Hermitian pairing from before decomposes into a sum of orthogonal pairings $\Omega(A)_{\tau}\otimes\Omega(A^{t})_{\overline{\tau}}\to\mathbb{C}$. Taking the determinant gives a Hermitian norm on the line bundle

$$N(A,\tau)\coloneqq\det\Omega(A)_{\tau}\otimes\det\Omega(A^{t})_{\overline{\tau}}.$$

We can extend $N(A,\tau)$ to an integral model of $A$. If $\mathcal{A}$ is the Néron model over $\mathcal{O}_{K}$ as before, with $K$ including all embeddings of $E\to\overline{\mathbb{Q}}$, define

$$\Omega(\mathcal{A})_{\tau}\coloneqq\Omega(\mathcal{A})\otimes_{\mathcal{O}_{K}\otimes\mathcal{O}_{E},\tau}\mathcal{O}_{K}$$

for each $\tau\colon E\to K$. We define $\Omega(\mathcal{A}^{t})_{\tau}$ analogously. For each archimedean place of $K$, we use the aforementioned Hermitian norm $\lVert\cdot\rVert$ on the generic fiber of $\det\Omega(\mathcal{A})_{\tau}\otimes\det\Omega(\mathcal{A}^{t})_{\overline{\tau}}$, and thus we get a metrized line bundle

$$\widehat{\mathcal{N}(\mathcal{A},\tau)}\coloneqq(\det\Omega(\mathcal{A})_{\tau}\otimes\det\Omega(\mathcal{A}^{t})_{\overline{\tau}},\lVert\cdot\rVert).$$

Note that if $\tau\not\in\Phi$, then $\mathcal{N}(\mathcal{A},\tau)=0$ and so the height contribution is $0$ as well.

Just as with the Faltings height, this $\tau$-component is independent of the abelian variety itself. Thus, we will write $h(\Phi,\tau)$ for $h(A,\tau)$.

We call a pair of CM-types $(\Phi_{1},\Phi_{2})$ nearby if $\lvert\Phi_{1}\cap\Phi_{2}\rvert=g-1$. Let $\tau_{i}=\Phi_{i}\backslash(\Phi_{1}\cap\Phi_{2})$ be the place where they differ. Then, the sum of the $\tau_{i}$-components of $h(\Phi_{i})$ is independent of the choice of nearby CM-type.

Finally, we compare $h(\Phi)$ with its constituents $h(\Phi,\tau)$.