# Just-likely intersections on Hilbert modular surfaces

Asvin G.111email<EMAIL_ADDRESS>Qiao He, Ananth N. Shankar222All
authors affiliated with the University of Wisconsin-Madison.

###### Abstract

In this paper, we prove an intersection-theoretic result pertaining to curves
in certain Hilbert modular surfaces in positive characteristic. Specifically,
we show that given two appropriate curves $C,D$ parameterizing abelian
surfaces with real multiplication, the set of points in $(x,y)\in C\times D$
with surfaces parameterized by $x$ and $y$ isogenous to each other is Zariski
dense in $C\times D$, thereby proving a case of a just-likely intersection
conjecture. We also compute the change in Faltings height under appropriate
$p$-power isogenies of abelian surfaces with real multiplication over
characteristic $p$ global fields.

##  1 Introduction

The study of intersection theory in the context of Shimura varieties has
produced several breakthroughs in arithmetic. For instance, Gross-Zagier
[GZ86] prove their celebrated theorem on ranks of elliptic curves by
extensively studying the arithmetic intersection theory of Heegner points on
modular curves. This program has been extended to more complicated Shimura
varieties in various directions (see [KRY06], [YZZ13], [YZ17], also [LL21]),
and arithmetic intersection theory has been crucial to the development of this
program.

In a different direction, the proof of the Andre-Oort conjecture heavily
relies on the Average Colmez conjecture (interesting in its own right), whose
proof ([AGHMP18], [YZ18]) relies on the Arakelov intersection theory of
special cycles in orthogonal Shimura varieties.

Most relevant to our paper is the work of Charles [Cha18] proving that two
elliptic curves over a number field are isogenous modulo infinitely many
primes. This is achieved by studying the arithmetic intersection theory of the
modular curve by an approach inspired by work of Chai-Oort, who prove the
analogous result over function fields [CO06, Proposition 7.3]. This work has
been generalized to the splitting of abelian surfaces over global fields
[ST20], [SSTT22], [MST22b], (where the authors prove that abelian surfaces
over global fields are isogenous to a product of elliptic curves modulo
infinitely many primes) and to Picard-rank jumping results [SSTT22] [MST22a].
These theorems are obtained by proving that any fixed arithemtic curve
intersects the Hecke translates of an appropriate “special divisor” at
infinitely many points. These special divisors solve moduli problems, and
(roughly speaking) parameterize abelian varieties with extra endomorphisms.
This moduli-theoretic interpretation of special divisors is crucial to
understanding and computing local and global intersections.

Despite the proofs relying heavily on this moduli theory of special divisors,
these problems can be naturally phrased in the setting of arbitrary
subvarieties of a Shimura variety. Specifically, let $\mathcal{S}$ denote a
Shimura variety, and let $C,D\subset\mathcal{S}$ be subschemes having
complementary dimension. How does the set of points on $C$ isogenous to some
point of $D$ distribute in $C$? In a recent AIM workshop, the participants
developed a framework that makes the following prediction.

######  Conjecture 1.

Let $\mathcal{S}$ denote a simple Shimura variety over $\mathbb{F}_{q}$, and
suppose that $C,D\subset\mathbb{F}_{q}$ are generically ordinary subvarieties
having complementary dimension. Then, the set of points in $C$ isogenous to
some point of $D$ is Zariski-dense in $C$. Further, the subset of $C\times D$
consisting of pairs of isogenous points is dense in $C\times D$.

This conjecture is inspired both by the work in [Cha18, CO06, ST20, SSTT22,
MST22b] as well as the Hecke orbit conjecture. There is also an arithmetic
(i.e., number-field) analogue of this conjecture. These conjectures have
several applications. For instance, they would imply that any abelian fourfold
over a global field should be isogenous to a Jacobian modulo infinitely many
primes! This is all the more striking as the existence of abelian fourfolds
over global fields _not_ isogenous to Jacobians has been established in [ST21]
and [Tsi12].

In [G22], the first-named author proves this conjecture in the setting of
$X(1)^{n}$. In this paper, we establish this conjecture for Hilbert modular
surfaces (with some mild constraints).

######  Theorem 1.1.

Let $F$ denote a real quadratic field, and let $p$ denote a rational prime
that splits in $F$. Let $\mathcal{H}$ denote the mod $p$ Hilbert modular
surface associated to $\mathcal{O}_{F}$, and let $C,D\subset\mathcal{H}$
denote two generically ordinary, proper, curves in $\mathcal{H}$, at least one
of which is ample. Then Conjecture 1 holds for $C,D\subset\mathcal{H}$.

###  1.1 Outline of proof

Our proof is inspired by Chai-Oort’s work. Roughly speaking, we consider a
sequence of Hecke operators $T_{n}$, and consider the intersections
$(T_{n}(C).D)$. In order to prove our theorem, it suffices to prove that the
local contributions of any finite set of points $x_{1}\ldots x_{m}\in D$ to
the intersection $(T_{n}(C).D)$ is smaller than the global intersection
numnber $(T_{n}(C).D)$ for some $n$. However, we encounter several additional
difficulties that aren’t present in their work.

Firstly, their ambient space is a product of curves, and their proof makes
crucial use of the product structure. In the hitherto solved cases where the
ambient space is _not_ a product variety, either $C$ or $D$ is special and the
local intersection numbers at any fixed point is bounded using the moduli
interpretation of the special divisor (see [SSTT22, Lemma 7.2], and [MST22b,
Lemma 7.2.1]). We overcome this first difficulty by constructing a _local_
product structure at every closed point (see Proposition 2.2) which is adapted
to a very specific set of Hecke operators that exist only in positive
characteristic. In fact, these Hecke operators endow $\mathcal{H}$ with a
partial Frobenius structure (see 2). This construction allows us to control
the local intersection multiplicites (see Theorem 3.1).

The second difficulty is in estimating the global intersection number. The
previously known cases (in the setting of special divisors) use earlier
results establishing the modularity of sequences of special divisors ([Bor99])
to compute the global intersection numbers. In our setting, we have no such
modularity results and we instead compute the intersections of $T_{n}(C)$ with
the non-ordinary locus of $\mathcal{H}$, and then use the ampleness of $D$ to
control $(T_{n}(C).D)$. While doing so, we establish a result (Theorem 1.2)
pertaining to the change of Faltings height under $p$-power isogeny which is
of independent interest.

###  1.2 Change of Faltings height under $p$-power isogeny

The Faltings height of an abelian variety over a number field is defined to be
the Arakelov degree of the Hodge bundle. The change of the Faltings height of
an Abelian variety under isogeny was a crucial ingredient in the proof of the
Mordell Conjecture by Faltings.

In the function field case, the definition of Faltings height for a proper
curve is even simpler. Suppose $\mathcal{A}/C$ is a family of abelian
varieties over a proper curve $C$. Then we define the Faltings height of
$\mathcal{A}/C$ to just be the degree of the Hodge bundle. In our setting,
this height only depends on the image of $C$ in $\mathcal{H}$. The prime-
to-$p$ Hecke operators on $\mathcal{H}$ are etale, and this makes it easy to
compute the change in Faltings height under prime-to-$p$ isogeny. The non-
etaleness of $p$-power hecke operators makes computing the change-in-height a
formidable prospect. Further, $p$-power Hecke correspondences aren’t even
separable!

Indeed, the only prior work pertaining to this that we are aware of is [GP20].
They consider an isogeny $\varphi\mathrel{\mathop{\mathchar 58\relax}}E_{1}\to
E_{2}$ between two elliptic curves over $k(C)$ for $C$ a curve over
$\mathbb{F}_{q}$) and [GP20, Theorem A] proves that

$h(j(E_{2}))=\frac{\deg_{\mathrm{ins}}(\varphi)}{\deg_{\mathrm{ins}}(\hat{\varphi})}h(j(E_{1}))$

where $h$ denotes the Weil height on $X(1)$ and $\deg_{\mathrm{ins}}(\varphi)$
denotes the inseparable degree of $\varphi$. The proof there proceeds by an
identification of the Weil height with an intersection number and since the
Picard group of $X(1)$ is simply $\mathbb{Z}$, and the only $p$-power Hecke
operators (mod $p$) are iterates of Frobenius and Verschibung. This lets one
calculate the effect of a Hecke translation on the Picard group simply as
multiplication by an integer.

In our case, the Picard group of $\operatorname{\mathcal{H}}$ is complicated
and the action of $p$-power Hecke correspondences (which are not étale!) on
the Picard group is not a-priori easy to compute. Nevertheless we fully
compute the change in Faltings height of an abelian surface with real
multiplication by $\mathcal{O}_{F}$ under $p$-power isogenies that respect the
$\mathcal{O}_{F}$-action.

Suppose $C$ is a proper curve and $\mathcal{A}/C$ is a generically ordinary
abelian surface with endomorphisms by $\mathcal{O}_{F}$ (with $F$ a real
quadratic field as in Theorem 1.1) so that the image of $C$ under the
corresponding map $C\to\mathcal{H}$ is an ample divisor of $\mathcal{H}$.
Suppose moreover that $\mathcal{A}_{n}/C$ is another abelian surface, also
having endomorphisms by $\mathcal{O}_{F}$, isogenous to $\mathcal{A}/C$ by a
purely inseparable isogeny $\varphi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{A}^{\prime}\to\mathcal{A}$ which respects the
$\mathcal{O}_{F}$-actions, whose kernel is cyclic of size $p^{n}$ (indeed,
this is the hardest case to treat, and the general case can be deduced from
this case). Let $h_{F}(\mathcal{A}/C)$ denote the Faltings height of the
abelian variety $\mathcal{A}/C$, which is defined to be the degree of the
Hodge bundle $\omega$ of $\mathcal{H}$ restricted to $C$.

######  Theorem 1.2.

The height of $\mathcal{A}_{n}/C$ grows exponentially in $n$.

In fact, we prove an exact formula for the change of height under inseparable
isogenies (Theorem 4.2) but for simplicity of notation, we will be content
with the version above for the introduction. We prove this result by comparing
the Hodge bundle to the class of the Hasse invariant, and using our
computation of the intersection of Hecke orbits of $C$ with the non-ordinary
locus, and the product structure that we construct.

### Acknowledgements

We’re very grateful to Jordan Ellenberg, Eyal Goren, Ruofan Jiang, Yunqing
Tang, Yifan Wei and Tonghai Yang for useful discussions. We are also grateful
to the AIM workshop on Arithmetic Intersection Theory on Shimura Varieties, in
particular Shou-Wu Zhang’s group, where the broader framework extending
Theorem 1.1 was discussed and formalized.

Q.H. is partially supported by a graduate school grant of UW-Madison. A.N.S.
is partially supported by the NSF grant DMS-2100436.

##  2 Background material

We provide some background (and standardize notation) on Hilbert modular
surfaces in this section, define the notion of a partial Frobenius structure
and prove a theorem about curves parametrizing abelian varieties without extra
endomorphisms.

Let $F$ be a real quadratic field, $\mathcal{O}_{F}$ its ring of integers,
$p=\mathfrak{P}_{1}\mathfrak{P}_{2}$ a prime that splits in $\mathcal{O}_{F}$
and $\mathfrak{a}$ a fractional ideal of $\mathcal{O}_{F}$. It is a general
result that there exists a scheme parametrizing Abelian surfaces with
endomorphisms by $\mathcal{O}_{F}$. More precisely:

######  Definition 1.

Let $\mathcal{H}_{n}^{\mathfrak{a}}$ be the moduli functor that associates to
a $\mathbb{F}_{p}$ scheme $S$ the groupoid of tuples $(A,\iota,\lambda,\eta)$
where:

  1. 1.

$A\to S$ is an abelian surface;

  2. 2.

$\iota\mathrel{\mathop{\mathchar
58\relax}}\mathcal{O}_{F}\to\operatorname{End}_{S}(A)$ is a ring homomorphism
;

  3. 3.

$\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{a}\to\operatorname{Hom}^{\operatorname{Sym}}_{\mathcal{O}_{F}}(A,A^{V})$
is an $\mathcal{O}_{F}$-linear homomorphism such that $\lambda(a)$ is an
$\mathcal{O}_{F}$-linear polarization of $A$ for every totally positive
$a\in\mathfrak{a}$, and the homomorphism
$A\otimes_{\mathcal{O}_{F}}\mathfrak{a}\stackrel{{\scriptstyle\sim}}{{\to}}A^{\vee}$
induced by $\lambda$ is an isomorphism of abelian surfaces;

  4. 4.

$\eta\mathrel{\mathop{\mathchar
58\relax}}(\mathcal{O}_{F}/n\mathcal{O}_{F})^{2}_{S}\to A[n]$ is an
$\mathcal{O}_{F}$-linear isomorphism from the constant group scheme
$(\mathcal{O}_{F}/n\mathcal{O}_{F})^{2}_{S}$ to $A[n]$.

It is known that $\mathcal{H}_{n}^{\mathfrak{a}}$ is represented by a Deligne-
Mumford stack over $\mathbb{F}_{p}$ and when $n\geq 3$, it is even represented
by a scheme (see [Gor02]). From now on, we fix some $n\geq 3$ that is prime to
$p$, a polarization $\mathfrak{a}$ and denote the representing scheme by
$\mathcal{H}$.

We now set up some notation. Let $\mathcal{A}/\mathcal{H}$ denote denote the
universal Abelian surface over $\mathcal{H}$, and let $\mathcal{G}$ denote its
$p$-divisible group. We have that
$\operatorname{End}(\mathcal{A})=\mathcal{O}_{F}$, and as $p$ splits in $F$,
we have that
$\operatorname{End}(\mathcal{G})=\mathcal{O}_{F}\otimes\mathbb{Z}_{p}=\mathbb{Z}_{p}\times\mathbb{Z}_{p}$.
This implies that $\mathcal{G}=\mathcal{G}_{1}\times\mathcal{G}_{2}$. At any
point $x\in\mathcal{H}$, let $\mathcal{A}_{x}$ and $\mathcal{G}_{i,x}$ denote
the pullback of $\mathcal{A}$ and $\mathcal{G}_{i}$ to $x$. Note that both
factors $\mathcal{G}_{i}$ are one-dimensional and have height 2.

######  Lemma 2.1.

Let $x\in\mathcal{H}(\overline{\mathbb{F}}_{p})$ and let the deformation space
of $\mathcal{G}_{i}$ be
$\operatorname{Spf}\overline{\mathbb{F}}_{p}[\\![t_{i}]\\!]$333Implicit here
is that the deformation space of a $p$-divisible group having height 2 and
dimension 1 is one-dimensional and smooth.. Then $\widehat{\mathcal{H}}_{x}$
is canonically isomorphic to
$\operatorname{Spf}\overline{\mathbb{F}}_{p}[\\![t_{1}]\\!]\times\operatorname{Spf}\overline{\mathbb{F}}_{p}[\\![t_{2}]\\!]$.

###### Proof.

The formal neighbourhood $\widehat{\mathcal{H}}_{x}$ parameterizes formal
deformations of $\mathcal{A}_{x}$ which admit an action of $\mathcal{O}_{F}$
(compatible with the action of $\mathcal{O}_{F}$ on $\mathcal{A}_{x}$). By the
Serre-Tate lifting theorem, we have that this is the same as formal
deformations of $\mathcal{G}_{x}$ which admit an action of
$\mathcal{O}_{F}\otimes\mathbb{Z}_{p}$ – but this is the same as pairs of
$p$-divisible groups which lift the pair
$(\mathcal{G}_{1,x},\mathcal{G}_{2,x})$. The lemma follows. ∎

Henceforth, we will refer to the decomposition above as the product structure
on $\widehat{\mathcal{H}}_{x}$.

###  2.1 Partial Frobenius structure and coordinates

For ease of exposition, we henceforth assume that $\mathfrak{P}_{1}$ (and
therefore $\mathfrak{P_{2}}$) are trivial in the narrow class group.
Otherwise, the analysis below will go through identically, except we work with
$p^{a}$ (and $\operatorname{Frob}_{p^{a}}$) in place of $p$ (and
$\operatorname{Frob}_{p}$), where $a$ is the order of $\mathfrak{P}_{i}$ in
the narrow class group.

######  Definition 2.

We say a surface $X$ over $\mathbb{F}_{p}$ has partial Frobenius structure if
there is a factorization

$\operatorname{Frob}_{p}=\pi_{1}\pi_{2},$

where $\pi_{i}\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow X$ are maps
which satisfy the following condition:

For any $q$ a power of $p$, locally around any $x\in X(\mathbb{F}_{q})$, we
can find coordinates $t_{1},t_{2}$ such that:

$(\pi_{i}^{q*})(t_{j})=\begin{cases}t_{j}^{q}&\text{ if }i=j,\\\ t_{j}&\text{
otherwise }.\end{cases}$ (1)

A product of curves over $\mathbb{F}_{p}$ is one example of a variety that has
a partial Frobenius structure (where the $\pi_{1},\pi_{2}$ are induced from
each factor independently), and there is an obvious choice of coordinates as
in (1). In particular, the product of modular curves will be a very pertinent
example.

The key assumption that $p$ splits in $\mathcal{O}_{F}$ results in a partial
Frobenius structure on $\mathcal{H}$. In this case, we may define the maps
$\pi_{i}$ using our moduli interpretation. Suppose $S$ is an arbitrary
$\mathbb{F}_{p}$ scheme and $\mathcal{A}/S$ is an abelian surface
corresponding to a point $x\in\mathcal{H}(S)$. Define $\pi_{1}(x)$ to be the
point $y$ corresponding to the abelian scheme $\mathcal{A}/G_{1}$, where
$G_{1}\subset\mathcal{G}_{1,x}$ is the kernel of Frobenius. According to
Section 2.2 of [Pap95], $\mathcal{A}\mapsto\mathcal{A}/G_{1}$ defines a
morphism
$\mathcal{H}^{\mathfrak{a}}\to\mathcal{H}^{\mathfrak{P}_{1}\mathfrak{a}}$. As
$\mathfrak{P}_{1}$ is trivial in the narrow class group, we have indeed obtain
a morphism $\pi_{1}^{a}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{H}^{\mathfrak{a}}\to\mathcal{H}^{\mathfrak{a}}$.

We define $\pi_{2}(x)$ analogously. Clearly,
$\operatorname{Frob}_{p}=\pi_{1}\pi_{2}$. Given this description, we have for
any $x\in\mathcal{H}(\mathbb{F}_{q})$, $\pi_{1}$ induces the
$q^{\textrm{th}}$-power map on the deformation space of $\mathcal{G}_{1,x}$
and leaves the deformation space of $\mathcal{G}_{2,x}$ unchanged. We
therefore have the following proposition:

######  Proposition 2.2.

$\mathcal{H}$ has a partial frobenius structure, with
$\operatorname{Frob}=\pi_{1}\pi_{2}$. Further, at every point
$x\in\mathcal{H}(\overline{\mathbb{F}}_{p})$, the coordinates $t_{1},t_{2}$
are just the coordinates induced by the product structure as in Lemma 2.1.

We will use the following lemma in the proof of the theorem immediately after.

######  Lemma 2.3.

Assume $K$ is a global function field, and $K_{v}$ is a completion of $K$ at
some place $v$. Let $\mathcal{G}_{/K}$ be an ordinary $p$-divisible group over
$K$ with height 2 and dimension 1, and $\mathcal{G}_{K_{v}}$ be its base
change to $K_{v}$. If the connected-étale exact sequence for
$\mathcal{G}_{K_{v}}$ splits then the connected-étale exact sequence for $G$
splits over $K$.

###### Proof.

Jiang proves a more general result in [Jia], but we include a different proof
here for completeness. For any $n$ the connected-étale exact sequence of
$\mathcal{G}_{/K_{v}}[p^{n}]$ splits over some finite extension $L_{n}/K$. We
will prove that $L_{n}$ can be chosen to be a subfield of $K_{v}$.

In order to show that we can choose $L_{n}\subset K_{v}$, we use the
representability of $\operatorname{Hom}_{\text{gp sch}}(H,G)$ for any finite
flat group schemes $H,G$. In general, $\operatorname{Hom}_{S}(X,Y)$ is
representable if $X,Y$ are finite type schemes over $S$ and moreover, $X$ is
flat, projective and $Y$ is quasi-projective over $S$ [§4c, [Gro60]] and the
conditions for being a homomorphism of groups defines a closed subscheme.
Splittings of the connected étale sequence

$0\to G_{n,\text{conn}}\to
G_{n}\stackrel{{\scriptstyle\pi}}{{\to}}G_{n,\text{\'{e}t}}\to 0$

can be identified with the closed subset of points of
$f\in\operatorname{Hom}_{\text{gp sch}}(G_{n,\text{\'{e}t}},G_{n})$ such that
$f\circ\pi=\mathrm{id}$. In particular, they are parametrized by a scheme
which we call $\operatorname{Split}/\operatorname{Spec}K$.

We therefore have a splitting over $K_{v},L_{n}$, i.e., $K_{v},L_{n}$ valued
points of $\operatorname{Split}$. We now consider $K_{v},L_{n}\subset M$ for
some algebraically closed field $M$. Let $\alpha\in\operatorname{Split}(M)$ be
defined over some subfield $L$. Since (upto etale-twists)
$G_{n,\text{conn}}\cong\mu_{p^{n}},G_{n,\text{\'{e}t}}\cong
p^{-n}\mathbb{Z}_{p}/\mathbb{Z}_{p}$, their automorphisms are defined over the
ground field, we have that every element of $\operatorname{Split}(M)$ is
defined over $L$. Therefore,

$\operatorname{Split}(K_{v})=\operatorname{Split}(M)=\operatorname{Split}(L_{n})\implies\operatorname{Split}(K_{v})=\operatorname{Split}(L_{n}\cap
K_{v}).$

Therefore, we can replace $L_{n}$ by $L_{n}\cap K_{v}$ to assume that
$L_{n}\subset K_{v}$.

We now claim that $L_{n}/K$ is separable. Indeed, being a one-parameter
function field, $K$ admits a unique degree $p$ inseparable extension
necessarily containing all $p$th roots of $K$, and whicih therefore contains
the pth root of a uniformizer of $K$ at $v$, which therefore can’t be
contained in $K_{v}$. It follows that $L_{n}/K$ is indeed separable.

The splitting behaviour of the connected-etale exact sequence for any
$p$-divisible group is insensitive to separable extensions, and therefore the
connected-etale exact sequenc for $\mathcal{G}[p^{n}]$ splits over $K$ for all
$n$. The result follows.

∎

######  Theorem 2.4.

Let $D\subset\mathcal{H}$ be a generically ordinary, reduced, irreducible
curve such that the generic abelian variety $A_{/K(D)}$ does not have an extra
endomorphism444i.e., the ring of endomorphisms is $\mathcal{O}_{F}$.. We
identify $\widehat{\mathcal{O}}_{\mathcal{H},x}$ with
$\overline{\mathbb{F}}_{p}[\\![t_{1},t_{2}]\\!]$ as in Lemma 2.1. Let
$\widehat{\mathcal{O}}_{\mathcal{H},x}/(f_{D,x})$ be the formal completion of
the local ring of $D$ at $x$. Then $t_{i}$ does not divide $f_{D,x}$ for any
$i$.

######  Remark 1.

As an analogous situation to the above lemma, consider $X=X(1)\times X(1)$,
then locally around $x$, we may choose $t_{i}$ to be the corresponding
coordinate of the $i$-th $X(1)\cong\mathbb{P}^{1}$.

###### Proof.

Without loss of generality, we may replace $D$ with its normalization, and
pull back the universal abelian surface to $D$. Therefore, we may assume that
$D$ is smooth - however, the map $D\rightarrow\mathcal{H}$ need no longer be
an embedding. Nevertheless, given any point $x\in D$, we obtain a map of
formal schemes $\psi\mathrel{\mathop{\mathchar
58\relax}}\widehat{D}_{x}\rightarrow\widehat{\mathcal{H}}_{x}$ (we abuse
notation by letting $x$ denote both the point of $D$ and its image in
$\mathcal{H}$). Let $f_{D,x}\in\widehat{\mathcal{O}}_{\mathcal{H}_{x}}$ denote
the defining equation of the image of $\widehat{D}_{x}$.

It suffices to prove that if $t_{i}$ divides $f_{D,x}$ for $i=1$ or $2$, then
either $D$ is not generically ordinary, or $\mathcal{A}/K(D)$ has extra
endomorphisms. Without loss of generality, assume $t_{1}\mid f_{D,x}$. If
$\mathcal{G}_{x,1}$ is supersingular, then by the description of the non-
ordinary locus in Section 4.1, we see that $\mathcal{A}/D$ must non-ordinary.
Therefore, we suppose that $\mathcal{G}_{x,1}$ is ordinary. The inclusion of
the divisor corresponds to

$\displaystyle\widehat{D}_{x}\cong\operatorname{Spec}\overline{\mathbb{F}}_{p}[\\![u]\\!]$
$\displaystyle\to\widehat{\mathcal{H}}_{d,x}\cong\operatorname{Spec}\widehat{\mathcal{O}}_{\mathcal{H},x};$
$\displaystyle t_{1}$ $\displaystyle\stackrel{{\scriptstyle\psi}}{{\to}}0,$
$\displaystyle\quad t_{2}$
$\displaystyle\stackrel{{\scriptstyle\psi}}{{\to}}\bar{t}_{2}\in\overline{\mathbb{F}}_{p}[\\![u]\\!].$

Here, $\widehat{D}_{x}$ is the formal completion of $D$ at $x$. As in Lemma
2.1, we know $\widehat{\mathcal{O}}_{\mathcal{H},x}$ is the product of the
deformation spaces of $\mathcal{G}_{i,x}$. Notice that $\psi(t_{1})=0$ implies
that $\mathcal{G}_{x,1}$ remains constant along $\widehat{D}_{x}$. In other
words,

$\mathcal{G}_{\widehat{D}_{x}}=(\mathcal{G}_{x,1}\times\widehat{D}_{x})\times\mathcal{G}_{\widehat{D}_{x},2}.$

The connected-étale exact sequence of $\mathcal{G}_{x,1}$ splits since it is
over a perfect field and hence, so does the connected-étale exact sequence of
$\mathcal{A}_{\hat{D}_{x}}[p^{\infty}]_{1}$. By Lemma 2.3, this implies that
the connected-étale exact sequence for $\mathcal{G}_{1}$ already splits over
$K(D)$. Consequently,
$\operatorname{End}(\mathcal{G}_{D,1})=\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$,
whence $\operatorname{End}(\mathcal{G}_{D})$ is strictly larger than
$\mathbb{Z}_{p}\otimes\mathcal{O}_{F}$. Applying the $p$-adic Tate conjecture
for endomorphisms of abelian varieties implies $\mathcal{A}_{D}$ has
endomorphism ring larger than $\mathcal{O}_{F}$, as required.

∎

##  3 Local intersection on varieties with a split frobenius

In this section, we assume that $X$ is a surface over $\mathbb{F}_{p}$ with a
partial Frobenius structure. Throughout this section, we fix such a system of
coordinates around each point as in (1). As in Section 2, Hilbert modular
surfaces where $p=\mathfrak{P}_{1}\mathfrak{P}_{2}$ with $\mathfrak{P}_{i}$
trivial in the narrow class group of $\mathcal{O}_{F}$ are examples of such
surfaces. As noted in Section 2, as long as $p$ splits completely in
$\mathcal{O}_{F}$, $\mathcal{H}$ will admit a partial $p^{a}$-Frobenius
structure were $a$ is the order of $\mathfrak{P}_{1}$ in the narrow class
group.

Note that the Frobenius (and hence the $\pi_{i}$) are universal
homeomorphisms, i.e., they induce homeomorphisms on the underlying topological
space. In particular, the preimage of any point under the $\pi_{i}$ is also
exactly one point.

The results in this section will be purely local around a point $x\in
X(\mathbb{F}_{q})$. Note that $\pi_{1}$ need not fix $x$ but some power of it
will. so we may assume that $X=\operatorname{Spec}R$ with
$R=\mathbb{F}_{q}[\\![t_{1},t_{2}]\\!]$ and $x$ is the origin given by the
vanishing of $t_{1},t_{2}$. In local co-ordinates (a power of) $\pi_{1}$
corresponds to (a power of) Frobenius along one of the co-ordinates.

Now, let $C,D\subset X$ be two curves and $x\in C\cap D$ and define
$C_{n}=(\pi_{1}^{n})^{-1}(C)$ to be the pullback of $C$ under $\pi_{1}^{n}$,
where $n$ satisfies $\pi_{1}^{n}(x)=x$. In this section, we prove results
about the intersection numbers $C_{n}\cdot D$. As is clear, Theorem 3.1 (1) is
inspired by [CO06, Proposition 7.3].

######  Theorem 3.1.

Let $C=V(f),D=V(g)$ locally around $x\in\operatorname{\mathcal{H}}$ and
$n=mr(x)\to\infty$ be part of a sequence of integers divisible by $r(x)$ and
increasing without bound. Suppose moreover that $t_{1},t_{2}\nmid f$ and
$t_{1}\nmid g$.

  1. 1.

Suppose $t_{2}\nmid g$. Then, the local intersection number $(C_{n}\cdot
D)_{x}$ is bounded as $n\to\infty$.

  2. 2.

Suppose $t_{2}\mid g$. Then, the local intersection number $(C_{n}\cdot
D)_{x}\to\infty$ as $n\to\infty$.

The analogous result holds with the roles of $\pi_{1},\pi_{2}$ (and
$t_{1},t_{2}$) reversed.

###### Proof.

In order to compute intersection numbers, we can replace $D$ by its
normalization and consider each component separately [Ful13, Example A.3.1].
We therefore have
$\widehat{\mathcal{O}}_{D,x}=\overline{\mathbb{F}}_{p}[\\![u]\\!]$ and we
suppose that the morphism $D\to\operatorname{\mathcal{H}}$ is locally around
$x$ given by

$\displaystyle\overline{\mathbb{F}}_{p}[\\![t_{1},t_{2}]\\!]$
$\displaystyle\to\overline{\mathbb{F}}_{p}[\\![u]\\!]$ $\displaystyle t_{i}$
$\displaystyle\to\alpha_{i}.$

For the first part of the theorem, $\alpha_{1},\alpha_{2}\neq 0$ since
$t_{1},t_{2}\nmid g$. We therefore define $\alpha_{i}=a_{i}u^{k_{i}}$ with
$a_{i}$ a unit and $k_{i}\geq 1$. Since $x\in C$ and $r(x)|n$, we have $x\in
C_{n}$. As $C$ is defined by $f(t_{1},t_{2})$, $C_{n}$ is defined by
$f(t_{1}^{p^{n}},t_{2})$ so that

$(C_{n}.D)_{x}=\dim_{\overline{\mathbb{F}}_{p}}\frac{\overline{\mathbb{F}}_{p}[\\![u]\\!]}{f(\alpha_{1}^{p^{n}},\alpha_{2})}.$

By assumption, $t_{1}\nmid f$ so that we can write
$f(t_{1},t_{2})=t_{2}^{e}\tilde{f}(t_{2})+t_{1}h(t_{1},t_{2})$ for $e\geq 1$
and $\tilde{f}(t_{2})$ a unit. Therefore,

$\displaystyle f(\alpha_{1}^{p^{n}},\alpha_{2})$
$\displaystyle=\alpha_{2}^{e}\tilde{f}(\alpha_{2})+\alpha_{1}^{p^{n}}h(\alpha_{1}^{p^{n}},\alpha_{2})$
$\displaystyle=a_{2}^{e}u^{k_{2}e}\tilde{f}(a_{2}u^{k_{2}})+a_{1}^{p^{n}}u^{p^{n}k_{1}}h(a_{1}^{p^{n}}u^{p^{n}k_{1}},a_{2}u^{k_{2}}).$

For $n$ large enough, the $u$-adic valuation of the second term is larger than
the $u$-adic valuation of the first term since $\tilde{f}$ is a unit.
Consequently (for $n$ large enough), $f(\alpha_{1}^{p^{n}},\alpha_{2})$ is
divisible by exactly $u^{ek_{2}}$ which proves that $(C_{n}.D)_{x}=ek_{2}$
which is independent of $n$.

In the second case, $\alpha_{2}=0$. Let $f(t_{1},0)=t_{1}^{d}w(t_{1})$ with
$d\geq 1$ and $w(t_{1})$ a unit. We then have (with $k_{1}$ as in the first
part)

$(C_{n}.D)_{x}=\dim_{\overline{\mathbb{F}}_{p}}\frac{\overline{\mathbb{F}}_{p}[\\![u]\\!]}{f(\alpha_{1}^{p^{n}},0)}=dk_{1}p^{n}$
(2)

which proves that $(C_{n}.D)_{x}\to\infty$ as $n\to\infty$.

∎

We have the following corollary.

######  Corollary 3.2.

Let $C$ be as above, and suppose now that
$Z_{i}\subset\widehat{\mathcal{H}}_{x}$ are the formal curves defined by
$t_{i}=0$ for $i=1,2$. Then, $(C_{n}.Z_{1})_{x}=(C.Z_{1})_{x}$ and
$(C_{n}.Z_{2})_{x}=p^{n}(C.Z_{2})_{x}$.

###### Proof.

The first equality follows by inspection and the second from Equation (2). ∎

##  4 Change of height under $p$-power isogenies

In this section, we will describe some instances in which we can describe how
the height of a generically ordinary proper curve $C\subset\mathcal{H}$
changes under isogenies induced by the $\pi_{i}$. The idea is to use the fact
that the Faltings height equals (up to a scaling factor of $\frac{1}{p-1}$)
the intersection of $C$ with the non-ordinary locus.

###  4.1 The non-ordinary locus.

Let $x\in Z(\mathbb{F}_{q})$ be a point, where $Z\subset\mathcal{H}$ is the
non-ordinary locus. As in Section 2, the formal completion
$\widehat{\operatorname{\mathcal{H}}}_{x}=\operatorname{Spf}\mathbb{F}_{q}[\\![t_{1},t_{2}]\\!]$,
where $t_{i}$ controls the deformation theory of $\mathcal{G}_{x,i}$ and
leaves $\mathcal{G}_{x,i+1}$ constant (here, the indices are read modulo 2).
Therefore, if $\mathcal{G}_{x,1}$ is ordinary and $\mathcal{G}_{x,2}$ is
supersingular, the local equation of $Z$ at $x$ is just $t_{1}=0$; if
$\mathcal{G}_{2,x}$ is ordinary and $\mathcal{G}_{1,x}$ is supersingular, then
the local equation is $t_{2}=0$; and if $x$ itself is supersingular, then the
local equation is $t_{1}t_{2}=0$.

We now recall the global geometry of $Z\subset\operatorname{\mathcal{H}}$,
following [BG99, §4, Theorem 4.2]. The non-ordinary locus is a union of smooth
irreducible curves which intersect transversally at supersingular points.
There are two ‘types’ of curves – curves of type 1 and curves of type 2. Every
non-supersingular point $x$ on curves of type 1 has the property that
$\mathcal{G}_{1,x}$ is ordinary while every non-supersingular point $y$ on
curves of type 2 has the property that $\mathcal{G}_{2,y}$ is ordinary. Curves
of type 1 never intersect (and neither do curves of type 2) while every
supersingular point has exactly one curve of type 1 and one curve of type 2
passing through it. Finally, the local equations of these curves are precisely
as described in the paragraph just above.

For brevity, we write $Z=Z_{1}\cup Z_{2}$, where $Z_{i}$ is the union of
curves of type $i$. The main result of this section is the following:

######  Theorem 4.1.

Let $C/\mathbb{F}_{q^{\prime}}\subset\mathcal{H}$ denote a proper generically
ordinary curve, and suppose that all the non-ordinary points of $C$ are
contained in $C(\mathbb{F}_{q})$ for some $q=p^{n_{0}}$. Then as $n$ goes to
infinity, we have $((\pi_{1}^{n_{0}n})^{-1}(C)\cdot
Z)=q^{n}(C.Z_{2})+(C.Z_{1})$, and $((\pi_{2}^{n_{0}n})^{-1}(C).Z)=q^{n}(C\cdot
Z_{1})+(C.Z_{2})$.

###### Proof.

This follows directly from the description of the non-ordinary locus as well
as Corollary 3.2. ∎

We keep the notation in the previous result. Recall that we define the
Faltings height of the abelian variety $\mathcal{A}_{C}$ to be the degree of
the Hodge bundle $\omega$ restricted to $C$. As in [MST22a] and [MST22b], the
class of the Hodge bundle can be expressed in terms of the non-ordinary locus.
To be precise, we have that $Z$ is the vanishing locus of a section of
$\omega^{p-1}$ (see [Box15, Sections 1.4,1.5, and Theorem 6.23]). This,
together with Theorem 4.1 gives the following result.

######  Theorem 4.2.

Notation as above. The Faltings height of
$\mathcal{A}_{(\pi_{1}^{n_{0}n})^{-1}(C)}$ is

$h_{F}(\mathcal{A}_{(\pi_{1}^{n_{0}n})^{-1}(C)})=\frac{q^{n}(C\cdot
Z_{2})+(C.Z_{1})}{p-1},$

and $\mathcal{A}_{(\pi_{2}^{n_{0}n})^{-1}(C)}$ is

$h_{F}(\mathcal{A}_{(\pi_{2}^{n_{0}n})^{-1}(C)})=\frac{q^{n}(C.Z_{1})+(C.Z_{2})}{p-1}.$

In particular, if $C$ is also ample555This would imply that $(C.Z_{1})$ and
$(C.Z_{2})$ are both positive. Even if $C$ is not ample, the curve $Z_{1}\cup
Z_{2}$ is ample, and so either $(C.Z_{1})$ or $(C.Z_{2})$ must be positive,
whence we obtain the same asymptotic formula for the Faltings heights of
either $\mathcal{A}_{(\pi_{1}^{n_{0}n})^{-1}(C)}$ or
$\mathcal{A}_{(\pi_{2}^{n_{0}n})^{-1}(C)}$., we have
$h_{F}(\mathcal{A}_{(\pi_{1}^{n_{0}n})^{-1}(C)})\asymp q^{n}\asymp
h_{F}(\mathcal{A}_{(\pi_{2}^{n_{0}n})^{-1}(C)})$.

##  5 Just-likely intersections on Hilbert Modular Surfaces

######  Definition 3.

Given two proper curves $C,D\subset\operatorname{\mathcal{H}}$ defined over
$\mathbb{F}_{q}$ parametrizing Abelian surfaces
$\mathcal{A}_{C},\mathcal{A}_{D}$, we let

$\mathscr{I}(C,D)=\\{(x,y)\in\operatorname{\mathcal{H}}^{2}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{A}_{C,x}\text{ is isogenous to }\mathcal{A}_{D,y}\text{
through a power of }\pi_{i}^{n}\\}.$

Moreover, let

$\widetilde{\mathscr{I}}(C,D)=\\{(x,y)\in\operatorname{\mathcal{H}}^{2}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{A}_{C,x}\text{ is isogenous to }\mathcal{A}_{D,y}\text{
through a }p\text{-power isogeny }\\}.$

Note that $\mathscr{I}(C,D)\subset\widetilde{\mathscr{I}}(C,D)$.

We use the results of Section 3 to prove:

######  Theorem 5.1.

Let $C,D$ be as above and suppose that $\mathcal{A}_{C},\mathcal{A}_{D}$
parametrized respectively by $C,D$ have no extra endomorphisms generically and
moreover, suppose that $D$ is ample.

Then $\mathscr{I}(C,D)$ has infinitely many points not contained in a finite
union of axes of the form $\bigcup_{i}\\{x_{i}\\}\times
D\cup\bigcup_{j}C\times\\{y_{j}\\}$.

######  Remark 2.

If one of the curves generically does have extra endomorphisms, then the
theorem is still true by the results of [MST22b].

######  Theorem 5.2.

Suppose that $C,D\subset\operatorname{\mathcal{H}}$ are two curves such that
$\mathscr{I}(C,D)$ has infinitely many points not contained in a finite union
of the axes as in the previous theorem. Then, $\widetilde{\mathscr{I}}(C,D)$
is dense in $C\times D$.

The strategy for the proof of Theorem 5.1 is as follows. We first prove that
the local intersection numbers $d_{n}=((\pi_{1}^{n})^{-1}(C).D)_{x}$ are
bounded for any $x\in D$. Next, we prove that the global intersection numbers
$((\pi_{1}^{n})^{-1}(C).D)$ are unbounded, by comparing these numbers with the
quantities $((\pi_{1}^{n})^{-1}(C).Z)$, using Theorem 4.1 and the ampleness of
$D$. Finally, Theorem 5.2 follows by a ”soft” argument. Note that Theorem 5.2
always holds for two curves which satisfy the conclusions of Theorem 5.1.

Throughout, we use the local coordinates $t_{1},t_{2}$ of Lemma 2.1. Let $C,D$
be as in Theorem 5.1. We prove Theorem 5.1 through a sequence of lemmas.
Recall that for a point $x\in X$, $r(x)$ is the smallest value so that
$\pi_{1}^{r(x)}(x)=\pi_{2}^{r(x)}(x)=x$.

######  Lemma 5.3.

For any point $x\in C$, the intersection number of $(\pi_{i}^{n})^{-1}(C)$
with $D$ at $(\pi_{i}^{n})^{-1}(x)=x$ is bounded absolutely as $n$ ranges
through the multiples of $r(x)$.

###### Proof.

Around any point $x\in C,D$, the curves $C,D$ don’t contain the divisors
$t_{i}=0$ since otherwise they would either be generically non-ordinary or
generically have extra endomorphisms (by Lemma 2.4).

Therefore, part (1) of Theorem 3.1 applies and shows that the intersection
number of $((\pi_{i}^{n})^{-1}(C)\cdot D)$ at $(\pi_{1}^{n})^{-1}(x)$ is
absolutely bounded as $n=mr(x)\to\infty$. ∎

The next lemma deals with the global intersection number. We have that
$(C.Z)=(C.Z_{1})+(C.Z_{2})$ is positive, so we assume without loss of
generality that $(C.Z_{2})$ is positive and define
$C_{n}=(\pi_{i}^{n})^{-1}(C)$.

######  Lemma 5.4.

The global intersection number $C_{n}\cdot Z\to\infty$ as $n\to\infty$ through
the multiples of $r(x)$.

###### Proof.

This follows immediately from Theorem 4.1. ∎

######  Lemma 5.5.

Let $D$ now be an arbitrary ample, proper divisor. Then, the global
intersection number $C_{n}\cdot D\to\infty$ as $n\to\infty$.

###### Proof.

We will reduce to the case where our divisor is $Z$, the Hasse locus.

Since $D$ is ample, we can find a large enough $m$ such that $mD-Z$ is also
ample. Therefore:

$C_{n}.(mD-Z)\geq 0\iff mC_{n}.D\geq C_{n}.Z$

and since $C_{n}.Z\to\infty$ by the previous Lemma 5.4, $C_{n}.D$ is also
unbounded. ∎

We are now ready to prove the main results of this paper.

###### Proof of Theorem 5.1.

Let $S_{n}\subset C_{n}=(\pi_{1}^{n})^{-1}(C)$ be the set of points $x_{i,n}$
on $C_{n}$ isogenous to some point $y_{i,n}$ on $D$. Note that there is a
unique $x_{i}\in C$ so that $\pi_{1}^{n}(x_{i,n})=x_{i}$ since $\pi_{1}$ is a
universal homeomorphism. We first show that $|S_{n}|\to\infty$ as
$n\to\infty$. For contradiction, suppose that $|S_{n}|\leq N$. Then, for $r\gg
0$ and $n$ ranging through the multiples of the lcm of
$r(x_{1}),\dots,r(x_{N})$:

$(C_{rn}\cdot D)=\sum_{i=1}^{N}(C_{rn}\cdot D)_{x_{i,n}}\leq NC_{0}$

where the bound on the right hand side follows from Lemma 5.3 and $C_{0}$ is
some constant. On the other hand, the left hand side goes to infinity by Lemma
5.5 which provides us with our contradiction.

We have thus shown that there is an infinite set of points
$x_{1},x_{2},\dots\subset C$ isogenous to some point on $D$. We would like to
show that the corresponding points $y_{1},y_{2},\dots$ on $D$ also form an
infinite set. Suppose otherwise for contradiction.

Thus, we can find an infinite subset $x_{i_{1}},x_{i_{2}},\dots$ isogenous to
the same point $y$ on $D$ through the isogenies corresponding to
$\pi_{1}^{n_{1}},\pi_{2}^{n_{2}},\dots$ with $n_{i}\to\infty$. That is:

$\pi_{1}^{n_{j}}(y)=x_{i_{j}}.$

On the other, the orbit of $\pi_{1}$ on any $\overline{\mathbb{F}}_{q}$ point
of $\operatorname{\mathcal{H}}$ is finite since, in local co-ordinates,
$\pi_{1}$ just corresponds to the Frobenius on one of the co-ordinates which
certainly has a finite orbit. This forces us to identify some of the
$x_{i_{j}}$ which is contradictory to our assumption that the $x_{i}$ are
distinct.

We have thus shown that we can find two sequences $x_{i},y_{i}\in C,D$ such
that $x_{i}\neq x_{j},y_{i}\neq y_{j}$ for $i\neq j$ and the $x_{i}$ are
isogenous to the $y_{i}$ as required. ∎

We now prove Theorem 5.2, and therefore finish the proof of Theorem 1.1.

###### Proof of Theorem 5.2.

Suppose for contradiction that the closure of $\widetilde{\mathscr{I}}(C,D)$
is a proper subset of $C\times D\subset\operatorname{\mathcal{H}}^{2}$. Let
$W$ be the complement of the axes inside the closure of $\mathscr{I}(C,D)$. By
Theorem 5.1, $W$ has positive dimension. We can therefore find an infinite
sequence of points $(x_{1},y_{1}),(x_{2},y_{2}),\dots\in W$ such that the
fields of definition of $y_{i}$ individually go to infinity. We will prove
that $N_{i}=\\#\\{y\in D\mathrel{\mathop{\mathchar 58\relax}}(x_{i},y)\in
W\\}\to\infty$ as $i\to\infty$ which contradicts the fact that $W$ is a closed
subset of $C\times D$ and therefore has finite degree projections onto the
first factor.

Indeed, if $y_{i}$ has field of definition $\mathbb{F}_{q^{m_{i}}}$, then the
size of the orbit of $y_{i}$ under $\operatorname{Frob}_{q}$ has size $m_{i}$
and moreover, each point in this orbit is $p$-power isogenous to $x_{i}$ and
lies on $D$ (since $D$ is defined over $\mathbb{F}_{q}$). Since $W$ is defined
over $\mathbb{F}_{q}$ too, it is fixed by any Frobenius and the entire orbit
is contained inside $W$ proving that $N_{i}\geq m_{i}\to\infty$ as
$i\to\infty$. ∎

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finitely many classes of causal teams of signature $\sigma$. Thus, any set of
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$-formulas
is equivalent to a single
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$-formula,
and it then suffices to prove the statement for $\Gamma=\\{\varphi\\}$.

Now suppose $\varphi\models\psi$. Then by Lemma 5.9 and soundness we have that
$\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{R}(\varphi)\models\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{R}(\psi).$
Thus, for every $\gamma\in\mathcal{R}(\varphi)$,
$\gamma\models^{g}\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{R}(\psi),$
which further implies, by Lemma 3.8, that there is an
$\alpha_{\gamma}\in\mathcal{R}(\psi)$ such that
$\gamma\models\alpha_{\gamma}$. Since $\gamma,\alpha_{\gamma}$ are
$\mathcal{CO}[\sigma]$-formulas, and the system for
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$
extends that for $\mathcal{CO}[\sigma]$, we obtain by the completeness theorem
of $\mathcal{CO}[\sigma]$ (Theorem 4.14) that $\gamma\vdash\alpha_{\gamma}$.
Applying $\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$I
and Lemma 5.9, we obtain
$\gamma\vdash\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{R}(\psi)\vdash\psi$
for each $\gamma\in\mathcal{R}(\varphi)$. Thus, by Lemma 5.9 and
$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$E, we
conclude that
$\varphi\vdash\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{R}(\varphi)\vdash\psi.$
$\square$

###  5.3 Axiomatizing $\mathcal{COD}$ over generalized causal teams

In this section we provide calculi for the language $\mathcal{COD}[\sigma]$,
as interpreted over the generalized semantics. The resulting system is a
variant of the deduction system for propositional dependence logic given in
[30].

Given a subformula $\theta$ of $\varphi$, we use the notation
$\varphi(\psi/[\theta,k])$ for the formula obtained by replacing _the $k$th
occurrence_ of the formula $\theta$ in $\varphi$ with $\psi$. For instance, if
$\varphi=\,\,=({X})\vee(Y=y\boxright=({X}))$, then
$\varphi(X=x/[=({X}),2])=\,\,=({X})\vee(Y=y\boxright X=x)$.

######  Definition 5.11.

The system for $\mathcal{COD}[\sigma]$ over generalized causal teams consists
of all the rules of the system for $\mathcal{CO}[\sigma]$ (Definition 4.12)
plus the additional rules for $\vee$ from Definition 5.6 and the following
rules for dependence atoms:

$X=x$ ConI $=({X})$ |  $=({X_{1},\dots,X_{n}};{Y})$ $=({X_{1}})\,\dots\,=({X_{n}})$ DepE $=({Y})$
---|---
$\varphi$ $\forall x\in Ran(X)$ [$\varphi(X=x/[=({X}),k])$] $\vdots$ $\psi$ ConE $\psi$ |  [$=({X_{1}})$] $\dots$ [$=({X_{n}})$] $=({Y})$ $\mathsf{DepI}$ $=({X_{1},\dots,X_{n}};{Y})$

Again, for the rules from Definition 4.12 we assume that $\alpha$ ranges over
$\mathcal{CO}[\sigma]$ formulas only. The rule ConE states that if $\psi$ can
be derived from the formula $\varphi(X=x/[=({X}),k])$ for every
$x\in\mathrm{Ran}(X)$, then we conclude $\psi$ from $\varphi$. The rule
$\mathsf{DepI}$ is read as: if the constancy atom $=({Y})$ can be derived from
the constancy atoms $=({X_{1}}),\dots,=({X_{n}})$ (possibly together with
other assumptions $\Gamma$), then the dependence atom
$=({X_{1},\dots,X_{n}};{Y})$ follows (from $\Gamma$).

Since the above system contains all applicable rules from the system for
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$,
Proposition 4.13 and Proposition 5.7(iii) are still derivable in the system
for $\mathcal{COD}[\sigma]$. It is easy to show that the entailment relation
$\vdash$ is monotone with respect to the substitution of positive subformula
occurrences.

######  Lemma 5.12.

Let $\varphi,\theta,\theta^{\prime}$ be $\mathcal{COD}[\sigma]$-formulas.
Suppose $\theta$ is a subformula of $\varphi$ that is not in the scope of
negation $\neg$, nor in the antecedent of a counterfactual implication
$\boxright$. Then, $\theta\vdash\theta^{\prime}$ implies
$\varphi\vdash\varphi(\theta^{\prime}/[\theta,k])$.

#### Proof.

A routine inductive proof, using $\boxright\\!\\!\textsf{Rpl}_{A}$ and
$\boxright\\!\\!\textsf{Rpl}_{C}$ in the $\boxright$ case. $\square$

In the next proposition we derive some other useful clauses involving
dependence atoms.

######  Proposition 5.13.

  1. (i)

$=({\mathbf{X}};{Y}),\mathbf{X}=\mathbf{x}\vdash=({Y})$

  2. (ii)

$=({\mathbf{X}};{Y})\dashv\vdash\bigvee_{\mathbf{x}\in
Ran(\mathbf{X})}(\mathbf{X}=\mathbf{x}\ \wedge=({Y}))$

#### Proof.

Item (i) follows from ConI and DepE. We now prove item (ii). We will
implicitly use Lemma 5.12 in many of the steps. For the left to right
direction, we first have by ValDef that $\vdash\bigvee_{\mathbf{x}\in
Ran(\mathbf{X})}\mathbf{X}=\mathbf{x}$. Thus,

$\displaystyle=({\mathbf{X}};{Y})$
$\displaystyle\vdash\,=({\mathbf{X}};{Y})\wedge(\bigvee_{\mathbf{x}\in
Ran(\mathbf{X})}\mathbf{X}=\mathbf{x})$ ($\land\textsf{I}$)
$\displaystyle\vdash\bigvee_{\mathbf{x}\in
Ran(\mathbf{X})}(\mathbf{X}=\mathbf{x}\wedge=({\mathbf{X}};{Y}))$ (Proposition
5.7(iii), $\lor$Com) $\displaystyle\vdash\bigvee_{\mathbf{x}\in
Ran(\mathbf{X})}(\mathbf{X}=\mathbf{x}\wedge=({Y})).$ (item (i),
$\vee\textsf{Rpl}$)

For the right to left direction, putting $\mathbf{X}=\langle
X_{1},\dots,X_{n}\rangle$, by DepI, it suffices to derive
$=({X_{1}}),\dots,=({X_{n}}),\bigvee_{\mathbf{x}\in
Ran(\mathbf{X})}(\mathbf{X}=\mathbf{x}\wedge=({Y}))\vdash=({Y})$, which, by
ConE, is further reduced to deriving
$\mathbf{X}=\mathbf{x}^{\prime},\bigvee_{\mathbf{x}\in
Ran(\mathbf{X})}(\mathbf{X}=\mathbf{x}\wedge Y=y)\vdash=({Y})$ for all
$\mathbf{x}^{\prime}\in\mathrm{Ran}(\mathbf{X})$ and all
$y\in\mathrm{Ran}(Y)$. Now, we have that

$\displaystyle\mathbf{X}=\mathbf{x}^{\prime},\bigvee_{\mathbf{x}\in
Ran(\mathbf{X})}(\mathbf{X}=\mathbf{x}\wedge Y=y)$
$\displaystyle\vdash\bigvee_{\mathbf{x}^{\prime}\in
Ran(\mathbf{X})}(\mathbf{X}=\mathbf{x}\wedge\mathbf{X}=\mathbf{x}^{\prime}\wedge
Y=y)$ ($\land\textsf{I}$, Proposition 5.7(iii))
$\displaystyle\vdash(\mathbf{X}=\mathbf{x}\wedge
Y=y)\vee\bigvee_{\begin{subarray}{c}\mathbf{x}^{\prime}\neq\mathbf{x}\\\
\mathbf{x}^{\prime}\in Ran(\mathbf{X})\end{subarray}}\bot$ (ValUnq)
$\displaystyle\vdash\mathbf{X}=\mathbf{x}\wedge Y=y$ ($\neg\textsf{E}$,
$\lor\textsf{E}$) $\displaystyle\vdash=({Y}).$ ($\land\textsf{E}$, ConI)

$\square$

Since dependence atoms never occur in the scope of $\neg$ or in the antecedent
of $\boxright$, by Lemma 5.12 and Proposition 5.13(ii), we can transform every
$\mathcal{COD}[\sigma]$-formula into an equivalent one with constancy
dependence atoms only.

######  Corollary 5.14.

For any $\mathcal{COD}[\sigma]$-formula $\varphi$, let $\varphi^{\ast}$ be the
formula obtained from $\varphi$ by replacing every occurrence of
$=({\mathbf{X}};{Y})$ by $\bigvee_{\mathbf{x}\in
Ran(\mathbf{X})}(\mathbf{X}=\mathbf{x}\wedge=({Y}))$. Then
$\varphi\dashv\vdash\varphi^{\ast}$.

The soundness of DepI and DepE follows from the easily verified fact that

$\Gamma,=({X_{1}}),\dots,=({X_{n}})\models\,=({Y})\iff\Gamma\models\,=({X_{1},\dots,X_{n}};{Y}).$

The rule ConI is clearly sound, since
$=({X})\equiv\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{x\in\mathrm{Ran}(X)}X=x$.
The soundness of the more complex rule ConE follows from the equivalence

$\varphi\equiv\varphi(\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{x\in\mathrm{Ran}(X)}X=x/[=({X}),k])\equiv\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{x\in\mathrm{Ran}(X)}\varphi(X=x/[=({X}),k]),$
(4)

which is a special case of a more general equivalence we shall prove in Lemma
5.15 below. In order to state the more general fact, let us first introduce
some terminologies.

Let $\mathbf{d}=\langle[=({X_{1}}),k_{1}],\dots,[=({X_{n}}),k_{n}]\rangle$ be
a sequence of occurrences of constancy dependence atoms in a formula
$\varphi$. A function $f:\\{1,\dots,n\\}\to\bigcup_{1\leq i\leq
n}\mathrm{Ran}(X_{i})$ is called an instantiating function over $\mathbf{d}$
if $f(i)\in\mathrm{Ran}(X_{i})$ for each $i$. We define an $f$-instantiation
of $\varphi$, denoted by $\varphi_{f}$, as the formula

$\varphi(X_{1}=f(1)/\\![=({X_{1}}),k_{1}]\,\dots,X_{n}=f(n)/\\![=({X_{n}}),k_{n}]).$

If $\mathbf{d}$ lists all occurrences of constancy dependence atoms in
$\varphi$, then we call the formula $\varphi_{f}$ a full instantiation of
$\varphi$.

######  Lemma 5.15.

Let $\mathbf{d}$ be a sequence of occurrences of constancy atoms in a
$\mathcal{COD}[\sigma]$-formula $\varphi$, and $F$ denote the set of all
instantiating functions over $\mathbf{d}$. Then
$\varphi\equiv\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{f\in
F}\varphi_{f}$.

#### Proof.

We prove the proposition by induction on $\varphi$. If $\varphi$ is a
constancy dependence atom $=({X_{1}})$, it is easy to verify that
$=({X_{1}})\equiv(\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{x\in\mathrm{Ran}(X_{i})}X_{1}=x)=(\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{f\in
F}X_{1}=f(1))$. If $\varphi$ is a $\mathcal{CO}[\sigma]$-formula, then the
claim is trivial.

If $\varphi=\psi\ast\chi$ for $\ast\in\\{\wedge,\vee\\}$, then the claim
follows from the induction hypothesis and the fact that
$(\eta\raisebox{1.0pt}{$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$}\theta)\ast\chi\equiv(\eta\ast\chi)\raisebox{1.0pt}{$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$}(\theta\ast\chi)$.
If $\varphi=\mathbf{X}=\mathbf{x}\boxright\psi$, then by induction hypothesis
we have that
$\psi\equiv\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{f\in
F}\psi_{f}$. We then have that

$\mathbf{X}=\mathbf{x}\boxright\psi\,\equiv\,\mathbf{X}=\mathbf{x}\boxright\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{f\in
F}\psi_{f}\,\equiv\,\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{f\in
F}(\mathbf{X}=\mathbf{x}\boxright\psi_{f})\,\equiv\,\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{f\in
F}(\mathbf{X}=\mathbf{x}\boxright\psi)_{f}.$

$\square$

Now, we know that the equivalence (4) holds, and thus our system for
$\mathcal{COD}[\sigma]$ is sound.

For any $\mathcal{COD}[\sigma]$-formula $\varphi$, consider the equivalent
formula $\varphi^{\ast}$ given by Corollary 5.14. Denote the set of all full
instantiations of $\varphi^{\ast}$ by $\mathcal{I}(\varphi)$. Since all
dependence atoms occurring in $\varphi^{\ast}$ are constancy atoms,
$\mathcal{I}(\varphi)$ is a finite set of $\mathcal{CO}[\sigma]$-formulas.

######  Corollary 5.16.

$\varphi\equiv\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{I}(\varphi)$.

#### Proof.

By Corollary 5.14 and the soundness of the rules, we have
$\varphi\equiv\varphi^{\ast}$. Then by Lemma 5.15,
$\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{I}(\varphi)\equiv\varphi^{\ast}\equiv\varphi$.
$\square$

As a further corollary, we have that each validity of $\mathcal{COD}$ is
“witnessed” by a $\mathcal{CO}$ validity, one of its instatiations; or, more
generally, we have the following analogous of the disjunction property:

######  Corollary 5.17 (Instantiation property).

Let $\Delta$ be a set of $\mathcal{CO}[\sigma]$-formulas, and $\varphi$ a
$\mathcal{COD}[\sigma]$-formula. Then $\Delta\models\varphi$ iff
$\Delta\models\varphi_{f}^{\ast}$ for some full instantiation
$\varphi_{f}^{\ast}\in\mathcal{I}(\varphi)$ of $\varphi^{\ast}$ via an
instantiating function $f$.

#### Proof.

By Corollary 5.16 and the disjunction property (Theorem 3.8). $\square$

To prove the completeness, we will make use of the disjunctive normal form
$\varphi\equiv\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{I}(\varphi)$.
While the global disjunction
$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$ in the
equivalence is not in the vocabulary of $\mathcal{COD}[\sigma]$, we now show
that in our system $\varphi$ behaves proof-theoretically as
$\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{I}(\varphi)$,
in the sense that it proves the same consequences and it is derivable from the
same assumptions.

######  Lemma 5.18.

Let $\Gamma\cup\\{\varphi,\psi\\}$ be a set of
$\mathcal{COD}[\sigma]$-formulas. Let
$\mathbf{d}=\langle=({X_{1}}),\dots,=({X_{n}})\rangle$ be a sequence of
occurrences of constancy atoms in $\varphi$.

  1. (i)

For any instantiating function $f$ over $\mathbf{d}$, we have that
$\varphi_{f}\vdash\varphi$.

  2. (ii)

If $\Gamma,\varphi_{f}\vdash\psi$ for all instantiating functions $f$ over
$\mathbf{d}$, then $\Gamma,\varphi\vdash\psi$.

#### Proof.

(i) By ConI, we have $X_{i}=f(i)\vdash=({X_{i}})$ for each $1\leq i\leq n$.
Then we obtain
$\varphi(X_{1}=f(1)/[=({X_{1}}),k_{1}],\dots,X_{n}=f(n)/[=({X_{n}}),k_{n}])\vdash\varphi$
by repeatedly applying Lemma 5.12.

(ii) Let $f_{1}$ be an arbitrary instantiating function over $\mathbf{d}$. For
any arbitrary $x\in\mathrm{Ran}(X_{1})$, let $g_{x}$ be the instantiating
function over $\mathbf{d}$ such that $g_{x}(1)=x$ and $g_{x}(i)=f_{1}(i)$ for
all $2\leq i\leq n$. By the assumption, we have then
$\Gamma,\varphi_{g_{x}}\vdash\psi$. Since we can obtain this clause for each
$x\in\mathrm{Ran}(X_{1})$, we conclude by ConE that

$\Gamma,\varphi(X_{2}=f_{1}(2)/[=({X_{2}}),k_{2}],\dots,X_{n}=f_{1}(n)/[=({X_{n}}),k_{n}])\vdash\psi$
(5)

for an arbitrary instantiating function $f_{1}$ over $\mathbf{d}$. By the same
argument, we can show, by using (5) and ConE, that

$\Gamma,\varphi(X_{3}=f_{2}(3)/[=({X_{3}}),k_{3}],\dots,X_{n}=f_{2}(n)/[=({X_{n}}),k_{n}])\vdash\psi$

for arbitrary instantiating function $f_{2}$ over
$\langle[=({X_{2}}),k_{2}],\dots,[=({X_{n}}),k_{n}]\rangle$. Proceeding in the
same way, in the end we obtain that $\Gamma,\varphi\vdash\psi$, as required.
$\square$

Now, we can prove the completeness theorem by essentially the same argument as
in the proof of Theorem 5.10.

######  Theorem 5.19 (Completeness).

For any set $\Gamma\cup\\{\psi\\}$ of $\mathcal{COD}[\sigma]$-formulas, we
have that $\Gamma\models^{g}\psi\iff\Gamma\vdash\psi.$

#### Proof.

We prove the direction “$\Longrightarrow$”. As in Theorem 5.10, it suffices to
prove it for $\Gamma=\\{\varphi\\}$. Suppose $\varphi\models\psi$. By
Corollary 5.16, we have that
$\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{I}(\varphi)\models\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{I}(\psi).$
It then follows that for every $\gamma\in\mathcal{I}(\varphi)$, we have that
$\gamma\models\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}\mathcal{I}(\psi).$
Since each $\gamma$ is a $\mathcal{CO}[\sigma]$-formula, by Theorem 3.8, there
is a $\mathcal{CO}[\sigma]$-formula $\beta\in\mathcal{I}(\psi)$ such that
$\gamma\models\beta.$ Now, by applying Theorem 4.14 and Lemma 5.18(i), we
obtain that $\gamma\vdash\beta\vdash\psi$ for all
$\gamma\in\mathcal{I}(\varphi)$, which implies by Lemma 5.18(ii) that
$\Gamma\vdash\psi$. $\square$

###  5.4 Axiomatizing
$\mathcal{CO}_{\footnotesize{\setminus}\mspace{-5.0mu}{\setminus}/}$ and
$\mathcal{COD}$ over causal teams

In this section, we axiomatize
$\mathcal{CO}_{\footnotesize{\setminus}\mspace{-5.0mu}{\setminus}/}$ and
$\mathcal{COD}$ over causal teams. It can be verified that all the rules in
the systems of the two logics over generalized causal teams are also sound
over causal teams; in the case of rules without discharging of assumptions,
this immediately follows from the fact that causal teams can be regarded as
generalized causal teams with uniform function components. To obtain complete
systems over causal teams, we shall then add new axioms or rules to
characterize the property of having uniform function components (and thus
being “indistinguishable” from a causal team). Recall from Corollary 3.5 that
the property is defined by the formula
$\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{\mathcal{F}\in\mathbb{F}_{\sigma}}\Phi^{\mathcal{F}}$,
which is, however, not a formula of $\mathcal{COD}$. To give a unified
axiomatization for both
$\mathcal{CO}_{\footnotesize{\setminus}\mspace{-5.0mu}{\setminus}/}$ and
$\mathcal{COD}$, we will instead extend the corresponding system with a new
rule. Let us now present our systems.

######  Definition 5.20.

The system for
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$
over causal teams consists of all rules of
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$
over generalized causal teams (Definition 5.6) plus the following rule:

$\forall\mathcal{F}\in\mathbb{F}_{\sigma}$ [$\Phi^{\mathcal{F}}$] $\vdots$
$\psi$ FunE $\psi$
---

Similarly, the system for $\mathcal{COD}[\sigma]$ over causal teams consists
of all rules of $\mathcal{COD}[\sigma]$ over generalized causal teams
(Definition 5.11) plus the above rule FunE.

The new rule FunE states that if $\psi$ can be derived from
$\Phi^{\mathcal{F}}$ for every $\mathcal{F}\in\mathbb{F}_{\sigma}$, then
$\psi$ can be derived. The soundness of this rule follows immediately from
Corollary 3.6. We write $\vdash^{c}$ for the entailment relation associated
with the system of
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$
or $\mathcal{COD}[\sigma]$ over causal teams (which system is meant will
always be clear from the context). Obviously, $\vdash^{c}$ is an extension of
the entailment relation $\vdash^{g}$ for the corresponding system over
generalized causal teams.

######  Lemma 5.21.

For any set $\Gamma\cup\\{\psi\\}$ of
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$-formulas
(or $\mathcal{COD}[\sigma]$-formulas), we have that

$\Gamma,\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{\mathcal{F}\in\mathbb{F}_{\sigma}}\Phi^{\mathcal{F}}\models^{g}\psi\iff\Gamma,\Phi^{\mathcal{F}}\models^{g}\psi\text{
for all }\mathcal{F}\in\mathbb{F}_{\sigma}\iff\Gamma\vdash^{c}\psi.$

#### Proof.

The first “$\iff$” is easy to verify. For the second “$\iff$”, suppose first
that $\Gamma,\Phi^{\mathcal{F}}\models^{g}\psi$ for all
$\mathcal{F}\in\mathbb{F}_{\sigma}$. Then we have
$\Gamma,\Phi^{\mathcal{F}}\vdash^{g}\psi$ for all
$\mathcal{F}\in\mathbb{F}_{\sigma}$ by the completeness theorem of the
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$-system
(or $\mathcal{COD}[\sigma]$-system) over generalized causal teams. This then
gives that $\Gamma,\Phi^{\mathcal{F}}\vdash^{c}\psi$ for all
$\mathcal{F}\in\mathbb{F}_{\sigma}$. Finally we derive $\Gamma\vdash^{c}\psi$
by applying the rule FunE. Conversely, if $\Gamma\vdash^{c}\psi$, then we have
$\Gamma\models^{c}\psi$ by the soundness theorem. But then we conclude by
Lemma 3.7 that
$\Gamma,\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{\mathcal{F}\in\mathbb{F}_{\sigma}}\Phi^{\mathcal{F}}\models^{g}\psi$,
which completes the proof. $\square$

######  Theorem 5.22 (Completeness).

For any set $\Gamma\cup\\{\psi\\}$ of
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$-formulas
(or $\mathcal{COD}[\sigma]$-formulas), we have that
$\Gamma\models^{c}\psi\iff\Gamma\vdash^{c}\psi.$

#### Proof.

It suffices to show the direction “$\Longrightarrow$”. Suppose
$\Gamma\models^{c}\psi$. By Lemma 3.7, we have that
$\Gamma,\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{\mathcal{F}\in\mathbb{F}_{\sigma}}\Phi^{\mathcal{F}}\models^{g}\psi$,
which then gives $\Gamma\vdash^{c}\psi$ by Lemma 5.21. $\square$

Now that we have axiomatized the two logics
$\mathcal{CO}_{\footnotesize{\setminus}\mspace{-5.0mu}{\setminus}/}$ and
$\mathcal{COD}$ over causal teams, let us devote the rest of the section to
alternative complete systems for these logics, or equivalently, alternative
ways to axiomatize the property of a generalized causal team having a uniform
function component.

As mentioned already, such a property can be axiomatized in
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$
using the following axiom instead of the rule FunE:

$\textsf{Unf}_{\footnotesize\raisebox{1.0pt}{$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$}}$
$\mathop{{{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}}_{\mathcal{F}\in\mathbb{F}_{\sigma}}\Phi^{\mathcal{F}}$
---

It is straightforward to show that the rule FunE and the above axiom
$\textsf{Unf}_{\footnotesize\raisebox{1.0pt}{$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$}}$
are inter-derivable from the rules
$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$I and
$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$E. Therefore,
the system consisting of all rules of
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$
over generalized causal teams together with the axiom
$\textsf{Unf}_{\footnotesize\raisebox{1.0pt}{$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$}}$
is also sound and complete for
$\mathcal{CO}_{\footnotesize{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}}[\sigma]$
over causal teams.

We now turn to the logic $\mathcal{COD}$. Consider the following axiom schema:
For $s,t\in\mathbb{A}_{\sigma}$ and
$\mathcal{F},\mathcal{G}\in\mathbb{F}_{\sigma}$,

$\textsf{Unf}_{\textsf{D}}$ $\Xi^{\\{(s,\mathcal{F}),(t,\mathcal{G})\\}}$
---
where $\mathcal{F}\not\sim\mathcal{G}$

We now show that the above axiom schema $\textsf{Unf}_{\textsf{D}}$ also
defines uniformity of generalized causal teams.

######  Lemma 5.23.

For any generalized causal team $T$ over some signature $\sigma$,

$T\models\Xi^{\\{(s,\mathcal{F}),(t,\mathcal{G})\\}}\text{ for all
}s,t\in\mathbb{A}_{\sigma}\text{ and
}\mathcal{F},\mathcal{G}\in\mathbb{F}_{\sigma}\text{ with
}\mathcal{F}\not\sim\mathcal{G}\iff T\text{ is uniform}.$

#### Proof.

Suppose $T$ is uniform. Then $T=T^{\mathcal{H}}$ for some
$\mathcal{H}\in\mathbb{F}_{\sigma}$. For any
$\mathcal{F},\mathcal{G}\in\mathbb{F}_{\sigma}$ with
$\mathcal{F}\not\sim\mathcal{G}$, it must be that either
$\mathcal{F}\not\sim\mathcal{H}$ or $\mathcal{G}\not\sim\mathcal{H}$. Thus,
for any $s,t\in\mathbb{A}_{\sigma}$, we have
$\\{(s,\mathcal{F}),(t,\mathcal{G})\\}\not\preccurlyeq T$, which by Lemma 5.5
implies that $T\models\Xi^{\\{(s,\mathcal{F}),(t,\mathcal{G})\\}}$.

Conversely, suppose $T$ is not uniform. Then, there exist
$(s,\mathcal{F}),(t,\mathcal{G})\in T$ such that
$\mathcal{F}\not\sim\mathcal{G}$. Clearly
$\\{(s,\mathcal{F}),(t,\mathcal{G})\\}\preccurlyeq T$. But this means, by
Lemma 5.5, that $T\not\models\Xi^{\\{(s,\mathcal{F}),(t,\mathcal{G})\\}}$.
$\square$

The above lemma implies (by Corollary 3.5) that the axiom schema
$\textsf{Unf}_{\textsf{D}}$ is equivalent to the axiom
$\textsf{Unf}_{\footnotesize\raisebox{1.0pt}{$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$}}$,
and thus to the rule FunE, given
$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$I and
$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$E. From these
it follows that the system consisting of all rules of $\mathcal{COD}[\sigma]$
over generalized causal teams together with the axiom schema
$\textsf{Unf}_{\textsf{D}}$ is sound and complete for $\mathcal{COD}[\sigma]$
over causal teams.

##  6 Conclusion

We have answered the main questions concerning the expressive power and the
existence of complete deduction calculi for the languages that were proposed
in [3] and [4], and which involve both interventionist counterfactuals and
(contingent) dependencies. We have introduced a generalized causal team
semantics that extends the causal discourse to cases in which there may be
uncertainty about the causal laws; and we have analyzed the properties of the
languages from [3, 4] also when they are evaluated in this generalized
semantics. We believe the present work may provide guidelines for the
investigation of further notions of dependence and causation in causal team
semantics and its variants.

Our work shows that many of the methodologies developed in the literature on
team semantics can be adapted to the (generalized) causal team semantics. At
the same time, a number of peculiarities emerged that set apart these semantic
frameworks from the usual team semantics. An interesting anomaly (as noted in
[4]) is the existence of formulas whose interpretation only depends on the
causal structure; this is the case for the “direct cause” formula mentioned in
the introduction (call it $\beta_{\mathrm{DC}}(X,V)$), or for the following
formula:

$\beta_{\mathrm{En}}(V):=\bigvee_{X\in\mathbf{W}_{V}}\beta_{\mathrm{DC}}(X,V),$

which can be shown to characterize the property of $V$ being endogenous (via a
non-constant function). One unusual aspect of such formulas is that not only
the class of causal teams they define is downward closed, but also its
complement (taken together with the empty team) is. Some more peculiarities
have emerged that only concern the original causal team semantics: first, the
failure of the disjunction property for
$\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$, and of
analogous instantiation properties for the dependence atoms; and secondly, the
collapse of $\mathop{\,{\mathbin{{\setminus}\mspace{-5.0mu}{\setminus}}/}\,}$
(global disjunction) to $\lor$ (tensor disjunction) for some special classes
of disjuncts. This kind of “collapse” is possible in pure team semantics (and
over generalized causal teams) only if one of the disjuncts entails the other,
which is however not the case in some of the examples we isolated.

We point out that our calculi are sound only for _recursive_ systems, i.e.,
when the causal graph is acyclic. The general case (and special cases such as
the “Lewisian” systems considered in [32]) will require a separate study. We
point out, however, that each of our deduction systems can be adapted to the
case of _unique-solution_ (possibly generalized) causal teams by replacing the
Recur rule with an inference rule that expresses the _Reversibility_ axiom
from [12]:

$\Large\frac{(\mathbf{X}=\mathbf{x}\land Y=y)\boxright W=w\hskip
25.0pt(\mathbf{X}=\mathbf{x}\land W=w)\boxright
Y=y}{\mathbf{X}=\mathbf{x}\boxright Y=y}\normalsize\hskip 5.0pt\text{ (for
$Y\neq W$) }.$

See [4] for the details on how the semantics should be adapted for this case.
The analysis of causal languages beyond the unique-solution case leads
naturally to the consideration of operators (such as the _might_
-counterfactuals) that do not preserve downward closure. It is difficult to
guess, at this stage, to what extent the methodologies of team semantics can
be adapted to the study of causal languages that are not downward closed. When
this issue is solved, it might be interesting to investigate languages that
also include non-downward closed contingent (in)dependencies, such as the
_independence atoms_ and _inclusion_ atoms considered in the literature on
pure team semantics ([15, 13]). Finally, it would be interesting to consider
counterfactuals with more complex antecedents – e.g., antecedents containing
disjunctions or negations. There have been a number of (often conflicting)
proposals in the literature for the semantics of such interventionist
counterfactuals [7, 1, 10, 22, 21]; it is our hope that causal team semantics
can shed some light on this issue. For example, a natural interpretation of a
counterfactual $(X=1\lor X=2)\boxright\psi$ is that $\psi$ certainly holds,
even if we are not sure which of the interventions $do(X=1)$ or $do(X=2)$ has
been performed. This uncertainty may be modeled by taking the union of the two
(generalized) causal teams resulting from applying each intervention.

## Acknowledgments

We would like to thank Johan van Benthem for an interesting discussion with
the first author, which inspired us to further generalize causal team
semantics towards the direction taken in this paper. We also thank two
anonymous referees for their helpful comments.

The first author was supported by grant 316460 of the Academy of Finland. The
second author was supported by grant 330525 of the Academy of Finland, and
Research Funds of the University of Helsinki.

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