# Non-simple polarised abelian surfaces

Robert Auffarth and Paweł Borówka  R. Auffarth
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile,
Santiago
Chile<EMAIL_ADDRESS>P. Borówka
Institute of Mathematics, Jagiellonian University in Kraków
Poland<EMAIL_ADDRESS>
###### Abstract.

We study the space of non-simple polarised abelian surfaces. Specifically, we
describe the locus of polarised abelian surfaces of type $(1,d)$ that contain
an elliptic curve of exponent $n$ and characterise its irreducible components.
We find a condition on the complementary exponent $m$ and show that the locus
of surfaces containing complementary elliptic curves of exponents $m,n$ is
irreducible. We also find equations for this locus in the Siegel space.

###### 2020 Mathematics Subject Classification:

14K10,14K12,14H52

##  1\. Introduction

The moduli space of non-simple principally polarised abelian varieties is well
understood. Indeed, the space of all non-simple principally polarised abelian
varieties that contain an abelian subvariety of fixed dimension and exponent
is irreducible and can be described as the quotient of a product space of
polarised abelian varieties (see for instance [Deb88], [Auf16], [Bor16]). In
particular, for principally polarised surfaces, exponents of complementary
elliptic curves coincide and the periods of such surfaces have to satisfy so
called Humbert singular equations (see for example [ALR17, Bor16, vdG88]).

On the other hand, the moduli space of polarised abelian varieties of a given
(non-principal) type is much harder to deal with. The main obstacle to extend
the results of the principally polarised case is the fact that two
complementary abelian subvarieties of a non-principally polarised abelian
variety do not necessarily have the same exponent. This inevitably leads to
the conclusion that the moduli space of non-simple polarised abelian varieties
that contain an abelian subvariety of a fixed dimension is not necessarily
irreducible, and it is not clear a priori how to describe the irreducible
components.

In [vdG88, Ch. IX], G. van der Geer recalls Humbert’s singular equations and
states that they can be generalised to non-principally polarised surfaces in
[vdG88, Th. IX.2.4]. However, in Corollary 5.5 we show that different singular
equations can give the same irreducible component in the moduli. Hence,
[vdG88, Th. IX.2.4] is not enough to describe irreducible components of non-
simple abelian surfaces.

In [G06], L. Guerra constructs moduli spaces of non-simple abelian varieties
and especially their irreducible components. However, due to the fact that the
construction is general and not very explicit, the author concludes his paper
with questions about whether the constructed spaces are actually non-empty
[G06, page 334]. In Corollary 5.6, in the case of polarised abelian surfaces,
we show a necessary and sufficient condition for his spaces to be non-empty
and that they consist of unique irreducible components.

In [ALR17] the authors have found equations for non-simple polarised
varieties. However, since they are defined over $\mathbb{Q}$ they do not
provide any information about irreducible components and possible exponents of
subvarieties.

The aim of this paper is to understand completely the case of non-simple
polarised abelian surfaces. Let $\mathcal{A}_{2}(d)$ be a moduli of $(1,d)$
polarised abelian surfaces. Denote by

$\displaystyle\mathcal{E}_{d}(m,n):=\left\\{(A,\mathcal{L})\in\mathcal{A}_{2}(d):\begin{array}[]{l}A\text{
contains a pair of complementary}\\\ \text{ elliptic curves of exponents
}m,n\end{array}\right\\}.$

The main result of the paper (that follows from Theorem 4.2 and Corollary 5.4)
can be summarised in the following theorem

######  Theorem 1.1.

For $d,m,n\in{\mathbb{Z}}_{+}$, the moduli space $\mathcal{E}_{d}(m,n)$ is a
(non-empty) irreducible subvariety of dimension 2 of $\mathcal{A}_{2}(d)$ if
and only if

$mn\cdot gcd(m,n,d)=gcd(m,n)^{2}d.$

In such a case, we can write $m=\frac{cd}{a},n=\frac{cd}{b}$ for some pairwise
coprime $a,b,c\in{\mathbb{Z}}$. Then, there exist $u,v\in{\mathbb{Z}}$
satisfying $au-bv=c$ and $\mathcal{E}_{d}(m,n)$ is the image of the set of
period matrices

$\\{Z=\left[\begin{array}[]{cc}z_{1}&z_{2}\\\
z_{2}&z_{3}\end{array}\right]:Im(Z)>0,z_{3}=(\frac{du}{b}+\frac{dv}{a})z_{2}-\frac{d^{2}uv}{ab}z_{1}\\}.$

The idea of the proof is to use certain cyclic isogenies from principally
polarised abelian surfaces in order to reduce the problem to working in the
principally polarised case. Since we understand the situation for principally
polarised surfaces, we need to carefully compute the degrees of the induced
restrictions of the isogeny to each elliptic curve. To show the irreducibility
we consider modular curves as moduli spaces of elliptic curves with level
structure and show that our objects of interest are equivalent under the
action of the symplectic group.

Our approach can be seen as a careful combination of ideas in [vdG88] and
[G06]. We reduce the problem of finding exponents to the principally polarised
case with additional structure and use symplectic action to conclude the
results. Therefore, we significantly extend the statement of Theorem IX.2.4
from [vdG88] and solve an open problem stated in [G06, page 334].

The structure of the paper is the following: In Section 2 we go over some
preliminaries of abelian surfaces and find important condition on the
exponents of elliptic curves, in Section 3 we prove crucial results on finite
abelian groups with symplectic action, in Section 4 we prove the
irreducibility of the moduli spaces we are studying, and finally in Section 5
we compute equations and state some concluding remarks.

### Acknowledgements

The work of the second author has been partially supported by the Polish
National Science Center project number 2018/30/E/ST1/00530. He also wishes to
thank his wife Aleksandra for some valuable discussions on proofs in Section
3.

##  2\. Preliminaries on abelian surfaces

Let $(A,{\mathcal{L}})$ be a complex polarised abelian surface of type
$D:=(1,d)$. That is, ${\mathcal{L}}$ is an ample line bundle and the map

$\varphi_{{\mathcal{L}}}:A\to A^{\vee}=:\mathrm{Pic}^{0}(A)$ $x\mapsto
t_{x}^{*}{\mathcal{L}}\otimes{\mathcal{L}}^{-1}$

is an isogeny with kernel isomorphic to $({\mathbb{Z}}/d{\mathbb{Z}})^{2}$.
Moreover let $\mathcal{A}_{2}(d)$ denote the space of polarised abelian
surfaces of type $(1,d)$. If $d=1$ we will simply use the standard notation
$\mathcal{A}_{2}$. We wish to study the moduli space

$\mathcal{E}_{d}(n):=\\{(A,\mathcal{L})\in\mathcal{A}_{2}(d):A\text{ contains
an elliptic curve of exponent }n\\},$

where the exponent of an elliptic curve $E\subseteq A$ is by definition the
degree of ${\mathcal{L}}|_{E}$, and in particular we want to understand what
its irreducible components are.

If $E\subseteq A$ is an elliptic curve, then it corresponds to the image of a
symmetric idempotent $\epsilon_{E}\in\mathrm{End}(A)\otimes{\mathbb{Q}}$
(where symmetric means that it is fixed by the Rosati involution given by
${\mathcal{L}}$). We can define the complementary elliptic curve of $E$ with
respect to ${\mathcal{L}}$ as

$F:=\mathrm{Im}(1-\epsilon_{E}),$

where $\epsilon_{F}:=1-\epsilon_{E}$ is also a symmetric idempotent. We have
that $E\cap F$ is finite and $A=E+F$, and therefore the natural addition map

$E\times F\to A$

is an isogeny.

We define another moduli space

$\displaystyle\mathcal{E}_{d}(m,n):=\left\\{(A,\mathcal{L})\in\mathcal{A}_{2}(d):\begin{array}[]{l}A\text{
contains a pair of complementary}\\\ \text{ elliptic curves of exponents
}m,n\end{array}\right\\}.$

Note that for a general pair $(m,n)$ the space $\mathcal{E}_{d}(m,n)$ will be
empty, and that

$\mathcal{E}_{d}(n)=\bigcup_{m}\mathcal{E}_{d}(m,n).$

######  Remark 2.1.

It is a classical result (see [BL06, Cor 12.1.2]) that if $d=1$ then $m=n$. In
the non–principally polarised case the complementary polarisation types can be
quite unexpected. For $a,b\in\mathbb{N}$, consider a product $E\times F$ with
the product polarisation ${\mathcal{O}}_{E}(a)\boxtimes{\mathcal{O}}_{F}(b)$.
The kernel of the isogeny associated to the polarisation is
$\mathbb{Z}_{a}^{2}\times\mathbb{Z}_{b}^{2}$, and by elementary group theory,
this group is isomorphic to
$\mathbb{Z}_{gcd(a,b)}\times\mathbb{Z}_{lcm(a,b)}$. In particular, the product
polarisation is of type $(gcd(a,b),lcm(a,b))$, and so $E\times F$ can
naturally be seen to belong to the moduli space
$\mathcal{A}_{2}\left(d\right)$ where $d=lcm(a,b)/gcd(a,b)$. Moreover, with
this primitive polarisation $E$ has exponent $a/gcd(a,b)$ and $F$ has exponent
$b/gcd(a,b)$.

Going back to the original problem, we would like to find conditions on the
pair $(n,m)$ for it to correspond to complementary polarisation types on a
polarised abelian surface of type $(1,d)$. In other words, we want to know
when $\mathcal{E}_{d}(m,n)\neq\varnothing$. An obvious necessary condition
that follows from [BL06, Cor 2.4.6.d] is that $d|lcm(m,n)$.

If $\\{\lambda_{1},\lambda_{2},\mu_{1},\mu_{2}\\}$ is a symplectic basis for
$\Lambda$, then we see that the sublattice given by $\langle
d\lambda_{1},\lambda_{2},\mu_{1},\mu_{2}\rangle$ gives an abelian variety $X$
and a cyclic isogeny

$f:X\to A$

of degree $d$. Moreover, we see that $f^{*}{\mathcal{L}}\equiv d\theta$ where
$\theta$ is a principal polarisation and $\equiv$ stands for numerical
equivalence. Let

$E^{\prime}:=f^{-1}(E)^{0},\hskip 14.22636ptF^{\prime}:=f^{-1}(F)^{0}$

be the corresponding elliptic curves on $X$. By [ALR17, Corollary 2.5], we
have that $E^{\prime}$ and $F^{\prime}$ are complementary elliptic curves on
$X$ with respect to $d\theta$, and therefore with respect to $\theta$. In
particular, since $\theta$ is a principal polarisation, we have that
$E^{\prime}$ and $F^{\prime}$ have the same exponent which we will denote by
$c$.

Define

$a:=|E^{\prime}\cap\ker f|=\deg f|_{E^{\prime}},\hskip
14.22636ptb:=|F^{\prime}\cap\ker f|=\deg f|_{F^{\prime}}.$

In order to construct possible pairs of complementary types, we start by
reducing possible cases. We do this with the following lemma.

######  Lemma 2.2.

Let $f:X\to A$, $E^{\prime}$ and $F^{\prime}$ be as before. Assume $\ker f\cap
E^{\prime}\cap F^{\prime}=\langle x\rangle$ for some point $x$. Then there
exists another principally polarised abelian variety $Y$ with elliptic curves
$E^{\prime\prime}=E^{\prime}/\langle x\rangle$ and
$F^{\prime\prime}=F^{\prime}/\langle x\rangle$ and a map $h:Y\to A$ such that
$\ker h\cap E^{\prime\prime}\cap F^{\prime\prime}=\\{0\\}$.

###### Proof.

In the proof, we abuse notation by writing $E^{\prime}$ both as an abstract
curve and its image in $X$. In the same way, $x$ is both a point of $X$ that
is in the image of $E^{\prime}$ and a point of $E^{\prime}$.

For any map $g$, let $\overline{g}$ denote the induced map on the quotient.
Let $q$ denote the quotient map $X\to X/\langle x\rangle$, and let $g$ be the
addition map. We have the following diagram of polarised isogenies:

(1) $\textstyle{E^{\prime}\times
F^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{q}$$\textstyle{A}$$\textstyle{X/\langle
x\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}}$

Since $\ker(g\circ q)$ contains the group generated by $(x,0)$ and $(0,x)$ in
$E^{\prime}\times F^{\prime}$ we get the quotient map and the extended diagram

(2) $\textstyle{E^{\prime}\times
F^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{quotient}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{q}$$\textstyle{A}$$\textstyle{E^{\prime}/\langle
x\rangle\times F^{\prime}/\langle
x\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{g\circ
q}}$$\textstyle{X/\langle
x\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}}$

The polarisation types are as follows, where $l$ is the order of $x$:

(3)
$\textstyle{(cd,cd)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{quotient}$$\textstyle{(d,d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{q}$$\textstyle{(1,d)}$$\textstyle{(\frac{c}{l}d,\frac{c}{l}d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{g\circ
q}}$$\textstyle{(\frac{d}{l},d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}}$

Since $\overline{f}\circ\overline{g\circ q}$ is a polarised isogeny of order
$\frac{c^{2}d}{l^{2}}$, we can factor it via the map $h^{\prime}$ of order
$\frac{c^{2}}{l^{2}}$ to a surface $(Y,d\theta_{Y})$ and $h:Y\to A$ that is a
cyclic isogeny of order $d$. Moreover, the restrictions of $h^{\prime}$ to
$E^{\prime\prime}$ and to $F^{\prime\prime}$ are injective, we get that
$h^{\prime}$ is addition and $E^{\prime\prime}$, $F^{\prime\prime}$ in $Y$ are
elliptic curves that are complementary to each other. ∎

By the previous lemma, we can therefore assume that $E^{\prime}\cap
F^{\prime}\cap\ker f=\\{0\\}$. We are now ready to prove the following
proposition.

######  Proposition 2.3.

Let $A$ be $(1,d)$ polarised abelian surface that is non-simple. Let $E,F$ be
a pair of complementary elliptic curves with exponents $n$ and $m$,
respectively. Then $m$, $n$ and $d$ satisfy the following equation:

(4) $mn\cdot gcd(m,n,d)=gcd(m,n)^{2}d.$

###### Proof.

Assume $E^{\prime}$ is of exponent $c$ in $X$. Then $E^{\prime}\cap
F^{\prime}=E^{\prime}[c]=F^{\prime}[c]$. Now, in order to have $\ker f\cap
E^{\prime}\cap F^{\prime}=\\{0\\}$, we need that $a,b,c$ are pairwise coprime,
i.e. $(a,b)=(a,c)=(b,c)=1$. Moreover, the exponents of $E,\ F$ are
$\frac{cd}{a}$ and $\frac{cd}{b}$. Note that
$gcd(\frac{cd}{a},\frac{cd}{b})=\frac{cd}{ab}$. Hence both sides of Equation 4
equal $\frac{c^{2}d^{3}}{a^{2}b^{2}}$. ∎

We define the function

$\mu:\mathbb{N}^{2}\to\mathbb{N}$

as follows: If $n,d\in\mathbb{N}$ we can write $n=\tilde{n}t$ and
$d=\tilde{d}t$ where $t$ is the maximal integer such that
$gcd(\tilde{n},t)=gcd(\tilde{d},t)=1$. Then we define

$\mu(n,d):=\sigma(t),$

where $\sigma$ is the function that counts the number of positive divisors of
a number.

######  Lemma 2.4.

If $n,d\in\mathbb{N}$, then the number of $m\in\mathbb{N}$ that satisfy
Equation (4) is $\mu(n,d)$.

###### Proof.

Write

$n=\prod_{i=1}^{r}p_{i}^{\alpha_{i}},\hskip
14.22636ptd=\prod_{i=1}^{r}p_{i}^{\beta_{i}},\hskip
14.22636ptm=\prod_{i=1}^{r}p_{i}^{\gamma_{i}}$

where $\alpha_{i},\beta_{i},\gamma_{i}\geq 0$ and
$\beta_{i}\leq\max\\{\alpha_{i},\gamma_{i}\\}$. Now Equation (4) turns into

$\alpha_{i}+\gamma_{i}+\min\\{\alpha_{i},\beta_{i},\gamma_{i}\\}=2\min\\{\alpha_{i},\gamma_{i}\\}+\beta_{i}.$

If $\alpha_{i}=\beta_{i}$ (only when $p\mid t$), we get the equation

$\gamma_{i}+\min\\{\alpha_{i},\gamma_{i}\\}=2\min\\{\alpha_{i},\gamma_{i}\\},$

which is satisfied if and only if $\gamma_{i}\leq\alpha_{i}$. Therefore
$\gamma_{i}$ can take on $\alpha_{i}+1$ possibilities.

If $\alpha_{i}>\beta_{i}$, then we get the equation

$\alpha_{i}+\min\\{\beta_{i},\gamma_{i}\\}=\gamma_{i}+\beta_{i}.$

This can only be satisfied if $\gamma_{i}=\alpha_{i}$ and so there is one
possibility.

If $\alpha_{i}<\beta_{i}$, then we get the equation

$\alpha_{i}+\gamma_{i}=\min\\{\alpha_{i},\gamma_{i}\\}+\beta_{i}$

which has the unique solution $\gamma_{i}=\beta_{i}$. By counting the total
number of possibilities for each $i$, we get the number $\mu(n,d)$. ∎

The aim of the rest of the paper is to show that the restriction of
Proposition 2.3 is in fact sufficient for the existence of elliptic curves
with the given exponents, and that the locus $\mathcal{E}_{d}(m,n)$ where
$m,n$ and $d$ satisfy Equation (4) is irreducible. Finally as a consequence,
Lemma 2.4 computes the number of irreducible components of
$\mathcal{E}_{d}(n)$.

##  3\. Symplectic modules over ${\mathbb{Z}}_{d}$

Now, we need some results from the theory of abelian groups. Let $d>1$ be a
positive integer. Our main object of interest is the group
${\mathbb{Z}}_{d}^{4}$ equipped with a non-degenerate symplectic form $\omega$
over ${\mathbb{Z}}_{d}$ (i.e. $\omega$ is bilinear, alternating and
$\omega(x,\cdot)=0$ if and only if $x=0$). We will use the perspective that
${\mathbb{Z}}_{d}^{4}$ is a free module over ${\mathbb{Z}}_{d}$. The results
can be stated in a more general setting, but it is not necessary for our
purposes.

We start with a useful fact that we will use quite often:

######  Lemma 3.1.

Let $k|d$ and let $x\in{\mathbb{Z}}_{d}^{n}$ be of order $k$. Then there
exists $y\in{\mathbb{Z}}_{d}^{n}$ of order $d$ such that $x=\frac{d}{k}y$.

###### Proof.

Let $x$ be of order $k$ and consider the canonical projection
$\pi\colon{\mathbb{Z}}_{d}^{n}\to{\mathbb{Z}}_{d}^{n}/\langle x\rangle$. Note
that by the fundamental theorem of finite abelian groups
${\mathbb{Z}}_{d}^{n}/\langle
x\rangle\cong{\mathbb{Z}}_{d}^{n-1}\oplus{\mathbb{Z}}_{\frac{d}{k}}$. Now,
$\pi^{-1}(0\oplus{\mathbb{Z}}_{\frac{d}{k}})$ is a cyclic group of order $d$
that contains $x$. Hence, there exists a generator $y$ such that
$x=\frac{d}{k}y$. ∎

As usual, for $k\in\mathbb{N},E\subset{\mathbb{Z}}_{d}^{n}$ we denote
$kE=\\{kx:x\in E\\}$ and $E[k]=\\{x\in E:kx=0\\}$. Then, the lemma yields an
immediate corollary.

######  Corollary 3.2.

If $E\cong{\mathbb{Z}}_{d}^{2}\subset{\mathbb{Z}}_{d}^{4}$ and $k\mid d$, then
$E[k]=\frac{d}{k}E$.

###### Proof.

The inclusion $\frac{d}{k}E\subseteq E[k]$ is obvious. If $x\in E[k]$ is of
order $\ell$ (and so $\ell\mid k$), then the previous lemma tells us that
$x=\frac{d}{\ell}y$ for some $y$ of order $d$. Now

$x=\frac{d}{k}\frac{k}{\ell}y$

which completes the proof. ∎

The first result deals with the case of a direct sum.

######  Lemma 3.3.

For an integer $d>1$, let $E\cong F\cong{\mathbb{Z}}_{d}^{2}$ with the
canonical bases $e_{1},e_{2}$ and $f_{1},f_{2}$ be equipped with the
alternating forms given by
$\omega_{E}(e_{1},e_{2})=\omega_{F}(f_{1},f_{2})=1$. Consider
${\mathbb{Z}}_{d}^{4}=E\oplus F$ with the direct sum of forms
$\omega_{E}+\omega_{F}$. Let $a,b$ be coprime divisors of $d$. Then, there
exists a cyclic subgroup $G$ of order $d$ such that $|G\cap E|=a$ and $|G\cap
F|=b$. Moreover, all such groups are equivalent under the action of a
(symplectic) group $Sp(E,\omega_{E})\times Sp(F,\omega_{F})$.

###### Proof.

Firstly, consider $P=be_{1}+af_{1}$. Since $e_{1},f_{1}$ are independent, for
$k\in{\mathbb{Z}}_{d}$ we have $kP=0$ if and only if $kb=ka=0\ mod\ d$. Since
$a,b$ are coprime, then $lcm(\frac{d}{a},\frac{d}{b})=d$, so $d|k$. Hence $P$
is of order $d$. Now, we need to prove that $G=\langle P\rangle$ satisfies the
assertion. Firstly, note that $kP\in E$ if and only if $\frac{d}{a}|k$. Hence,
we can write $k=\frac{d}{a}l$, for $l=1,\ldots,a$. There are exactly $a$
possible values of $l$ and $\frac{ld}{a}e_{1}=0$ if and only if $l=a$. This
shows $|G\cap E|=a$. Completely analogously, one can show that $|G\cap F|=b$.

Let $H$ be another such group. Let $Q=k_{1}e_{1}+k_{2}e_{2}+zf_{1}+tf_{2}$ be
a generator of $H$. Since $H\cap E$ is of order $a$, we know that
$\frac{d}{a}z=\frac{d}{a}t=0$. Hence $z=az^{\prime},\ t=at^{\prime}$. for some
$z^{\prime},t^{\prime}$. Let $f_{1}^{\prime}=z^{\prime}f_{1}+t^{\prime}f_{2}$.
Since $Q$ is of order $d$ and therefore $zf_{1}+tf_{2}$ is of order
$\frac{d}{a}$, we get that $f_{1}^{\prime}$ is of order $d$. Let
$f_{2}^{\prime}\in F$ be chosen so that
$\omega_{F}(f_{1}^{\prime},f_{2}^{\prime})=1$. Now, for a symplectic basis
$e_{1},e_{2},f^{\prime}_{1},f^{\prime}_{2}$ we have
$Q=k_{1}e_{1}+k_{2}e_{2}+af_{1}^{\prime}$. Analogously, one can find
symplectic $e_{1}^{\prime},e_{2}^{\prime}\in E$ such that
$Q=be_{1}^{\prime}+af_{1}^{\prime}$. Since both bases are symplectic, the
transition matrix belongs to $Sp(E,\omega_{E})\times Sp(F,\omega_{F})$ and
hence $H$ is equivalent to $G$. ∎

######  Remark 3.4.

For a non-prime integer $d$, having
$E\cong{\mathbb{Z}}_{d}^{2},E\subset{\mathbb{Z}}_{d}^{4}$ does not imply the
existence of $F$ such that ${\mathbb{Z}}_{d}^{4}=E\oplus F$. Therefore, we
need a more general and more technical construction that will involve working
over ${\mathbb{Z}}_{cd}$, for some $c>1$.

For an integer $d>1$, let $a,b,c$ be pairwise coprime integers, such that
$a,b$ are divisors of $d$ and $c>1$. Let $E\cong F\cong{\mathbb{Z}}_{cd}^{2}$
with the canonical bases $e_{1},e_{2}$ and $f_{1},f_{2}$ be equipped with the
alternating forms given by
$\omega_{E}(e_{1},e_{2})=\omega_{F}(f_{1},f_{2})=1$. Consider
${\mathbb{Z}}_{d}^{4}=E\oplus F$ with the direct sum of forms
$\omega_{E}+\omega_{F}$. Assume, there is an isotropic subgroup
$K\cong{\mathbb{Z}}_{c}^{2}$ such that $K\cap E=K\cap F=\\{0\\}$ and let
$X=(E\oplus F)/K$ with $\pi:E\oplus F\to X$ be the quotient projection. Such
group is an example of so called allowable isotropic subgroup in [Bor16].

Note that since $K\subset(E\oplus F)[c]$ we can write every element of $K$ as
$dP$ and instead of using restricted symplectic form (that may be trivial), we
use the induced symplectic form that is given by
$\omega_{K}(dP,dQ)=\omega(P,Q)\ mod\ c$ for which $K$ has to be isotropic,
too.

We will start with the following lemma.

######  Lemma 3.5.

For any $K$ satisfying the condition above there exists a symplectic basis
$e_{1},e_{2}\in E,f_{1},f_{2}\in F$ such that $K=\langle
dbe_{1}+daf_{1},dae_{2}-dbf_{2}\rangle$.

###### Proof.

Firstly, note that $K=\langle dbe_{1}+daf_{1},dae_{2}-dbf_{2}\rangle$
satisfies the above properties. Certainly $K\subset(E\oplus F)[c]$. Moreover,
since $a,b$ are coprime, both generators are of order exactly $c$ and they are
linearly independent. $K$ is obviously isotropic since $ab=ba$. Moreover,
since $a,c$ (and $b,c$) are coprime, elements $dbe_{1},dae_{2}$ generates
$E[c]$ and hence $K\cap E=\\{0\\}$. Analogously $K\cap F=\\{0\\}$.

Now, consider the multiplication map
$f_{bd}:{\mathbb{Z}}_{cd}\to{\mathbb{Z}}_{c}$. It is surjective, because $b,c$
are coprime. Moreover, for any generator $x\in{\mathbb{Z}}_{c}$ there exists a
generator $y\in{\mathbb{Z}}_{cd}$ such that $x=bdy$. To see this, one can use
Dirichlet’s theorem on arithmetic progressions to see that $y_{k}=y+kc$ has
infinitely many primes, hence at least one is coprime to $d$ and hence its
class in ${\mathbb{Z}}_{cd}$ is a generator. The map $f_{ad}$ has the same
properties.

Now, let $x,y\in K$ be generators of $K$. By what we have proved, we can write
$x=dbe_{1}+daf_{1}$ and $y=dae_{2}^{\prime}-dbf_{2}^{\prime}$ for some
elements $e_{1},e_{2}^{\prime}\in E,f_{1},f_{2}^{\prime}\in F$ of order $cd$.
By the fact that $K\cap E=\\{0\\}$ we see that $e_{1},e_{2}^{\prime}$ are
independent and hence generate $E$. Therefore
$\omega(e_{1},e_{2}^{\prime})=\alpha$ is a unit in ${\mathbb{Z}}_{cd}$. Note
that $K$ is isotropic so by direct calculation of $\omega_{K}(x,y)$ one gets
that $\omega(f_{1},f_{2}^{\prime})=\alpha\ mod\ c$, too. Hence, by a possible
change of $f_{2}^{\prime}$, we can assume
$\omega(f_{1},f_{2}^{\prime})=\alpha$ in ${\mathbb{Z}}_{cd}$.

Now, let $e_{2}=\alpha^{-1}e_{2}^{\prime},f_{2}=\alpha^{-1}f_{2}^{\prime}$.
Then $e_{1},e_{2},f_{1},f_{2}$ is a symplectic basis and $x,\alpha^{-1}y$ are
generators of $K$ of the desired form. ∎

######  Lemma 3.6.

Let $E^{\prime}=\pi(E\times\\{0\\}),F^{\prime}=\pi(\\{0\\}\times F)$. Then
$E^{\prime}\cap F^{\prime}=\pi(dE\times\\{0\\})=\pi(\\{0\\}\times dF)$. In
particular, we have that $\pi^{-1}(E^{\prime}\cap F^{\prime})=dE+dF$.

###### Proof.

Firstly, note that for every $dx\in dE$ there exists a unique $dy\in dF$ such
that $dx+dy\in K$ and all elements of $K$ are of this form.

Hence, for any $P^{\prime}=Q^{\prime}\in E^{\prime}\cap F^{\prime}$, we have
$P^{\prime}-Q^{\prime}=0\in X$, so there exist $P\in E$ with
$\pi(P)=P^{\prime}$ and $Q\in F$ with $\pi(Q)=Q^{\prime}$ such that $P-Q\in
K$. Hence $P-Q=dx+dy$ so $P-dx=Q+dy=0$. hence
$P^{\prime}=\pi(dx+0)=Q^{\prime}=\pi(0-dy)$.

On the other hand if $P^{\prime}=\pi(dx+0)$ then there exist $dy\in dF$, such
that $dx+dy\in K$, so $P^{\prime}=\pi(dx+0)=\pi(0-dy)=Q^{\prime}\in
E^{\prime}\cap F^{\prime}$.

Note that by comparing the orders, one gets $\pi^{-1}(E^{\prime}\cap
F^{\prime})=dE+dF$. ∎

Let $X[d]$ be the set of $d$-torsion points of $X=(E\oplus F)/K$. Note that
$X[d]\cong{\mathbb{Z}}_{d}^{4}$. Now, we are interested in the set
$\pi^{-1}(X[d])$ that will be denoted by $L$. From the definition, we can
write $L=\\{e+f:e\in E,f\in F,de+df\in K\\}$. By $l$, we denote $l=gcd(c,d)$.
Let $\tilde{c}=\frac{c}{l}$. Note that $cd=\tilde{c}ld$.

######  Lemma 3.7.

The set of cyclic groups $G^{\prime}\subset X[d]$ of order $d$ such that
$G^{\prime}\cap E^{\prime}\cap F^{\prime}=\\{0\\}$ is dominated by the set of
cyclic subgroups $\tilde{G}\subset L$ of order $ld$ (via the map $\pi$).

###### Proof.

If $P\in\tilde{G}$ is a generator of order $ld$ then $kP\in dE+dF$ only for
$k$ such that $d|k$, so $\pi(\tilde{G})$ is a desired group. On the other
hand, if $|\tilde{G}|<ld$, hence $P\in\tilde{G}$ is a generator of order less
than $ld$ then we can write $P=k(e+f)$ for some generators $e\in E$ and $f\in
F$ and $k>1,k|d$. Hence, one gets $\frac{d}{k}P\in dE+dF$. Now, if
$\pi(\tilde{G})$ is of order $d$, then $\frac{d}{k}\pi(P)\neq 0$ and we get
that $\pi(\tilde{G})\cap E^{\prime}\cap F^{\prime}\neq\\{0\\}$. ∎

######  Lemma 3.8.

For any $K$ as above and $G$ that satisfy $|G|=ld,\ |G\cap K|=l,\
G\cap(dE+dF)=\\{0\\}$ and $|G\cap E|=a,|G\cap F|=b$ there exists a symplectic
basis $e_{1},e_{2}\in E,f_{1},f_{2}\in F$ such that
$G=\langle\tilde{c}be_{1}+\tilde{c}af_{1}\rangle,\ K=\langle
dbe_{1}+daf_{1},dae_{2}-dbf_{2}\rangle$.

###### Proof.

Let $u\in K$ be of order $c$, such that $\tilde{c}u\in G$ and let $v\in G$ be
a generator of $G$ (of order $ld$, such that $dv\in K$). We claim that $u+v$
is of order $cd$. To see this, let $k(u+v)=0$ for the smallest positive $k$.
This means that $ku=-kv\in G\cap K$. In order to have $ku\in G$ one needs
$\tilde{c}|k$ and for $kv\in K$ one needs $d|k$. Since $\tilde{c},d$ are
coprime we have $\tilde{c}d|k$. Now, note that $c|\tilde{c}d$, so in fact,
$ku=0$ and therefore $ld|k$. Since $lcm(\tilde{c},ld)=cd$ we get that $cd|k$
and so $u+v$ is of order $cd$.

Note that $d(u+v)\in K$ is of order $c$ and $\tilde{c}(u+v)\in G$ is a
generator of $G$.

By the proof of Lemma 3.3 (applied for order ‘ld’), we can write
$v=\tilde{c}be_{v}+\tilde{c}af_{v}$ and by Lemma 3.5 we can write
$u=dbe_{u}+daf_{u}$ for some $e_{u},e_{v}\in E,f_{u},f_{v}\in F$.

Now, $u+v=be_{1}^{\prime}+af_{1}^{\prime}$ for
$e_{1}^{\prime}=de_{u}+\tilde{c}e_{v},f_{1}^{\prime}=df_{u}+\tilde{c}f_{v}$.
It may happen that $f_{1}^{\prime}$ is not of order $cd$. However, since
$|G\cap E|=a$ and is given by $\frac{cd}{|\langle af_{1}^{\prime}\rangle|}$,
we get that $af_{1}^{\prime}$ is of order $\frac{cd}{a}$. By Lemma 3.1, we get
that $af_{1}^{\prime}=af_{1}$ for some $f_{1}$ of order $cd$. Analogously, we
can find $e_{1}$ of order $cd$ such that $be_{1}^{\prime}=be_{1}$ and
therefore $u+v=be_{1}+af_{1}$ for $e_{1}\in E,\ f_{1}\in F$ of order $cd$. By
the proof of Lemma 3.5 one can extend $e_{1},f_{1}$ symplectically to
$e_{2},f_{2}$ such that $K=\langle d(u+v),dae_{2}-dbf_{2}\rangle$, which
proves the assertion. ∎

######  Remark 3.9.

Note that, a posteriori, Lemma 3.3 can be seen as a special case of Lemma 3.8
when $c=\tilde{c}=1$, $K=\\{0\\}$ and $X=E\oplus F$.

##  4\. Irreducibility of $\mathcal{E}_{d}(m,n)$

We would like to describe the moduli of non-simple abelian surfaces
$\mathcal{E}_{d}$. Our natural starting point is a moduli space of products of
elliptic curves with a chosen $d$-torsion point which in the literature are
sometimes called abelian surfaces with a root (cf. [BL95, HW89]). A crucial
feature that gives us many components of the moduli space of non-simple
surfaces is the fact that a $d$-torsion point on a product of elliptic curves
is not necessarily a sum of $d$-torsion points of elliptic curves. Lemma 3.3,
interpreted correctly, shows that all components of the moduli space are
parametrised by pairs of coprime divisors of $d$. That is why, as before, we
fix coprime $a,b$ that are divisors of $d$.

Since a non-simple principally polarised surface is only isogenous to a
product, we also use a construction from [Bor16] that involves so-called
allowable isotropic subgroups.

In order to make this precise, for $N\in\mathbb{N}$, let $X(N)$ denote the
modular curve associated to the congruence subgroup
$\Gamma(N):=\ker(\mathrm{SL}(2,\mathbb{Z})\to\mathrm{SL}(2,\mathbb{Z}_{N}))$,
i.e. the moduli space of triples $(E,p,q)$ where $p,q\in E$ is a symplectic
basis of torsion points of order $N$.

Now let $a,b,c\in\mathbb{N}$ be coprime integers with $a|d$, $b|d$ and
$\tilde{c}=\frac{c}{\gcd(c,d)}$ and consider the moduli map

$\Phi_{a,b,c}:X(cd)\times X(cd)\to\mathcal{A}_{2}(d)$
$((E,e_{1},e_{2}),(F,f_{1},f_{2}))\mapsto((E\times
F)/\langle\tilde{c}be_{1}+\tilde{c}af_{1},dbe_{1}+daf_{1},dae_{2}-dbf_{2}\rangle),H)$

where $H$ is the polarisation that pulls back to $cd$ times the principal
product polarisation on $E\times F$.

######  Lemma 4.1.

The map $\Phi_{a,b,c}$ is a well-defined morphism and its image lies in
$\mathcal{E}_{d}(\frac{cd}{a},\frac{cd}{b})$.

###### Proof.

We start with $(E\times F,\mathcal{O}_{E}(cd)\boxtimes\mathcal{O}_{F}(cd))$
with a $(cd,cd)$ product polarisation. We define $K=\langle
dbe_{1}+daf_{1},dae_{2}-dbf_{2}\rangle$ and we see that $K$ is an allowable
isotropic subgroup of the set of $c$-torsion points, so we can define a
quotient $\pi_{c}:E\times F\to(E\times F)/K$ and by [BL06, Cor 6.3.5] and
Riemann-Roch we can compute that $X=(E\times F)/K$ has a $(d,d)$ polarisation
and the quotient map $\pi_{c}$ is polarised. Now if
$P=\tilde{c}be_{1}+\tilde{c}af_{1}$ then $\pi_{c}(P)\in X[d]$ and $A=X/\langle
P\rangle$ is indeed (again by [BL06, Cor 6.3.5]) a $(1,d)$-polarised abelian
surface with the quotient map $\pi_{d}$ being a polarised isogeny.

The last assertion follows from the fact that $\pi_{c}|_{E\times\\{0\\}}$ is
injective, whereas $\pi_{d}|_{\pi_{c}(E\times\\{0\\})}$ is of order $a$ and
analogously $(\pi_{d}\circ\pi_{c})|_{\\{0\\}\times F}$ is of order $b$. ∎

Now, we are ready to show the main result of the paper.

######  Theorem 4.2.

Let $m,n,d$ be positive integers that satisfy Equation 4. Then, one can write
$m=\frac{cd}{a},\ n=\frac{cd}{b}$ for some $a,b$ coprime divisors of $d$ and
$c$ coprime to $a,b$ and the map $\Phi_{a,b,c}$ is surjective onto
$\mathcal{E}_{d}(m,n)$. In particular, $\mathcal{E}_{d}(m,n)$ is a non-empty
irreducible subvariety of $\mathcal{A}_{2}(d)$ of dimension 2.

###### Proof.

By dividing Equation 4 by $gcd(m,n)$ one gets $lcm(m,n)|gcd(m,n)d$, hence by
elementary number theory we can write $lcm(m,n)=gcd(m,n)\cdot q$ for some
$q|d$ and so $m=gcd(m,n)\cdot b,\ n=gcd(m,n)\cdot a$ for some $a,b$ that are
coprime and satisfy $ab=q$. In particular $a,b$ are divisors of $q$ and hence
divisors of $d$. Substituting $m,n$ we get $gcd(m,n,d)=\frac{d}{ab}$, so
$m=\frac{cd}{a},\ n=\frac{cd}{b}$ for $c$ coprime to $a,b$. Moreover, the
triple $(a,b,c)$ is uniquely determined by $m,n$ and all triples $(a,b,c)$
appear in this way.

For this reason, we only need to show that a non-simple abelian surface is in
the image of some $\Phi_{a,b,c}$. Fortunately, this is a straightforward
corollary of Lemma 2.2, Poincare’s Irreducibility Theorem and Lemma 3.8.

To be precise, if $A$ is a non-simple $(1,d)$ polarised surface, then one can
find maps $E\times F\to X\to A$, where (up to an isomorphism) $X=(E\times
F)/K$ with $K$ an allowable isotropic subgroup of $c$-torsion points and
$A=X/G$, for some $G$ cyclic of order $d$ with $|G\cap E|=a$ and $|G\cap
F|=b$. Now, by Lemma 3.8 there exists a symplectic basis
$e_{1},e_{2},f_{1},f_{2}$ of the set of $cd$ torsion points of $E\times F$,
such that $K=\langle dbe_{1}+daf_{1},dae_{2}-dbf_{2}\rangle$ and $G$ is
generated by the image of $\tilde{c}be_{1}+\tilde{c}af_{1}$. Hence $A\in
im(\Phi_{a,b,c})$. ∎

######  Corollary 4.3.

The locus of non-simple principally polarised abelian surfaces in
$\mathcal{A}_{2}(d)$ is equal to

$\mathcal{E}_{d}=\bigcup_{m,n\in\mathbb{N}}\mathcal{E}_{d}(m,n)=\bigcup_{\stackrel{{\scriptstyle
a|d,b|d,c}}{{(a,b)=(a,c)=(b,c)=1}}}\mathrm{Im}(\Phi_{a,b,c}),$

where the components are irreducible (or empty).

The following corollary computes possible complementary exponents if we fix a
pair $d,n$.

######  Corollary 4.4.

Let $n,d\in{\mathbb{N}}$. We can write uniquely $n=\frac{cd}{a}$, where
$a=\frac{d}{gcd(n,d)}$ and $c=\frac{n}{gcd(d,n)}=\frac{lcm(n,d)}{d}$. Let $q$
be a divisor of $d$ such that $a|q$ and $a,\frac{q}{a},c$ are pairwise
coprime. Note that $q$ always exists (e.g. $q=a$). Denote by $b=\frac{q}{a}$.

Then $m=\frac{cd}{b}$ satisfies Theorem 4.2 and is a complementary type to
$n$. In particular $\mathcal{E}_{d}(n)$ is non-empty and the number of
irreducible components equals $\mu(n,d)$ (see Lemma 2.4).

######  Remark 4.5.

In extreme cases, we have the following. If $d^{2}|n$ then the only
possibility is $a=b=1$ and both exponents equal $n$. If $(d,n)=1$, then $c=n,\
q=a=d$, so $b=1$ and we have exponents $n,dn$. On the other hand, if $n=cd$
for some $c$ coprime to $d$ then $a=1$ and there are as many complementary
types as divisors of $d$ (of the form $cb$ for $b|d$).

##  5\. Equations for $\mathcal{E}_{d}(m,n)$ and concluding remarks

Using [ALR17, Prop 4.4] and [K94, Prop 5.4] one sees that preimages of
$\mathcal{E}_{d}$ in

$\mathfrak{h}_{2}=\\{Z=\left[\begin{array}[]{cc}z_{1}&z_{2}\\\
z_{2}&z_{3}\end{array}\right]:Im(Z)>0\\}$

satisfy singular relations
$da_{1}z_{1}+a_{2}z_{2}+a_{3}z_{3}+a_{4}(z_{2}^{2}-z_{1}z_{3})+da_{5}=0$ for
some integers $(a_{1},a_{2},a_{3},a_{4},a_{5})$ without a common divisor
satisfying $\Delta=a_{2}^{2}-4da_{1}a_{3}-4da_{4}a_{5}=p^{2}$ for some integer
$p$. Note that [vdG88, Chapter IX] uses slightly different notation that
results in different formulae.

The equations come from the fact that being non-simple is invariant under
isogenies. Hence, one can consider a map that takes a period matrix $[Z\ D]$
to $[Z\ id]$ and use the pullback of a differential form to get equations of
the desired form.

Now we would like to find an equation that induces the locus
$\mathcal{E}_{d}(m,n)$ in $\mathcal{A}_{2}(d)$. We start with the following
lemma.

######  Lemma 5.1.

Fix $x,y\in{\mathbb{Z}}$ satisfying $x\neq y$ and $d|xy$. Let
$A_{x,y}={\mathbb{C}}^{2}/\Lambda$ with $\Lambda$ generated by the columns of

$\left[\begin{array}[]{cccc}z_{1}&z_{2}&1&0\\\
z_{2}&(x+y)z_{2}-xyz_{1}&0&d\end{array}\right].$

Then there exist elliptic curves $E_{x},\ E_{y}$ that are embedded in $A$ via
maps of analytic representations $[1\ x]$ with the primitive lattice vectors
$f_{2}-yf_{1},\frac{lcm(d,x)}{x}e_{1}+lcm(d,x)e_{2}$ and $[1\ y]$ with the
primitive lattice vectors
$f_{2}-xf_{1},\frac{lcm(d,y)}{y}e_{1}+lcm(d,y)e_{2}$. In particular, the
exponents are equal to $|lcm(d,x)-\frac{ylcm(d,x)}{x}|$ and
$|lcm(d,y)-\frac{xlcm(d,y)}{y}|$.

###### Proof.

To shorten the proof, we only show a case of $E_{y}$. A direct computation
shows that $[1\ y]^{t}(z_{2}-xz_{1})=f_{2}-xf_{1}$ which is a primitive vector
in $\Lambda$. A vector $\frac{lcm(d,y)}{y}e_{1}+lcm(d,y)e_{2}$ is indeed
primitive in $\Lambda$ because the coefficients
$\frac{lcm(d,y)}{y},\frac{lcm(d,y)}{d}$ are coprime. The exponent is the
absolute value of a symplectic form on primitive vectors in the image, hence
we get the assertion. ∎

######  Remark 5.2.

Note that for general $z_{1},z_{2}$, the curves are non-isogenous to each
other. In such a case $A$ contains only these two elliptic curves, so they
have to be complementary to each other. One may also explicitly compute the
norm maps and associated idempotents using analytic representations.

If we multiply the exponents of $E_{x},\ E_{y}$ by $gcd(d,x)$ and $gcd(d,y)$
respectively, we get the same number $dx-dy$ (here, we assume $x>y$). The
following lemma is then crucial to find equations.

######  Lemma 5.3.

For any pair $m,n$ of complementary exponents there exist $x,y\in{\mathbb{Z}}$
(with $d|xy$) satisfying

$\begin{cases}dx-dy=m\cdot gcd(d,x)\\\ dx-dy=n\cdot gcd(d,y)\end{cases}$

###### Proof.

Let $m=\frac{cd}{a},\ n=\frac{cd}{b}$. By Bezout’s identity, there exist $u,v$
such that $au-bv=c$. Let $x=\frac{du}{b},\ y=\frac{dv}{a}$. Note that $b,u$
and $a,v$ have to be coprime, so $gcd(d,x)=\frac{d}{b}$ and
$gcd(d,y)=\frac{d}{a}$. Therefore $dx-dy=\frac{d^{2}}{ab}(au-
bv)=\frac{cd^{2}}{ab}=m\frac{d}{b}=n\frac{d}{a}$ and we get the assertion. ∎

######  Corollary 5.4.

For a choice of $x,y$ as in Lemma 5.3, the abelian surface $A_{x,y}$ contains
elliptic curves of exponents $m,n$. This shows that, using irreducibility, we
get that $\mathcal{E}_{d}(m,n)$ is the image of the set of matrices satisfying
$z_{3}=(x+y)z_{2}-xyz_{1}$ with $\Delta=(\frac{cd}{ab})^{2}$. Moreover, having
explicit periods $\tau_{E},\tau_{F}$ of elliptic curves $E,\ F$, one can solve
a system of linear equations to get explicit $z_{1},z_{2}$ and hence explicit
period matrix of a surface $A_{x,y}$ containing $E$ and $F$.

Lemma 5.3 allows us to find two important examples that show that, contrary to
the principally polarised case, the discriminant of a singular equation
carries very little information about irreducible components in
$\mathcal{A}_{2}(d)$.

######  Corollary 5.5.

For $d=6$ the equations $z_{2}=0$ and $6z_{1}-5z_{2}+z_{3}=0$ (both having
$\Delta=1$) yield abelian surfaces containing complementary elliptic curves of
exponents $1,6$ and $2,3$ respectively. Moreover, for $d=90$, the equations
$1620z_{1}-81z_{2}+z_{3}=0$ (with $\Delta=9^{2}$) and
$180z_{1}-27z_{2}+z_{3}=0$ (with $\Delta=3^{2}$) induce the same locus
$\mathcal{E}_{90}(18,45)$ in $\mathcal{A}_{2}(90)$.

We would like to conclude our paper with the remark connecting our results
with the results in [G06]. Guerra’s idea was to use directly Poincare’s
Reducibility Theorem to obtain a map $E\times F\to A$. An advantage of his
approach is that one gets models of irreducible components of the moduli
space, provided it is non-empty. L. Guerra concludes his paper [G06] with some
questions regarding the non-emptiness of the moduli space and irreducible
components that are parametrised by the number of orbits of matrices $[M]$
under the group action. Our paper answers the question stated in [G06, page
334] for surfaces.

######  Corollary 5.6.

Here, we use notation from [G06]. Let $\delta=(E,D,D^{\prime})$, where
$E=(1,d),D=m,D^{\prime}=n$. Then the moduli space $\mathcal{E}_{\delta}[M]$ is
non-empty and all matrices $M$ are equivalent under the action of the group if
and only if $m,n$ satisfy Theorem 4.2. In such a case,
$\mathcal{E}_{\delta}[M]=\mathcal{E}_{d}(m,n)$.

Our result does not generalise well to higher dimensions. A principally
polarised case has already been solved in [Auf16] and [Bor16]. In a non-
principally polarised case, firstly, not every type can be obtained by a
cyclic isogeny from a principally polarised abelian variety. Moreover, you
usually get non-principally polarised components, so you inevitably get the
problem of non-trivial kernel of the polarising isogeny which restrict
possible subgroups to divide by.

## References

  * [Auf16] R. Auffarth On a numerical characterization of non-simple principally polarized abelian varieties. Math. Z. 282 (2016), no. 3-4, 731–746; MR3473641
  * [ALR17] R. Auffarth, H. Lange, A. M. Rojas. A criterion for an abelian variety to be non-simple. J. Pure Appl. Algebra, 221 (2017), 1906–1925; MR3623174
  * [BL06] C. Birkenhake, H. Lange, Complex Abelian Varieties. Second edition. Grundlehren der mathematischen Wissenschaften, 302. Springer-Verlag, Berlin, 2004. xii+635 pp. ISBN: 3-540-20488-1; MR2062673
  * [BL95] C. Birkenhake, H. Lange, Moduli spaces of abelian surfaces with isogeny, Geometry and analysis (Bombay, 1992), 225–243, Tata Inst. Fund. Res., Bombay, 1995; MR1351509
  * [Bor16] P. Borówka, Non-simple principally polarised abelian varieties. Ann. Mat. Pura Appl. (4) 195 (2016), no. 5, 1531–1549; MR3537961
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  * [vdG88] G. van der Geer, Hilbert modular surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 16. Springer-Verlag, Berlin, 1988. x+291 pp. ISBN: 3-540-17601-2; MR0930101
  * [G06] L. Guerra, Siegel coordinates and moduli spaces for morphisms of abelian varieties Ann. Mat. Pura Appl. (4) 185 (2006), no. 3, 319–335; MR2231028
  * [HW89] K. Hulek, S. Weintraub, Bielliptic abelian surfaces. Math. Ann. 283 (1989), no. 3, 411–429; MR0985240
  * [K94] E. Kani Elliptic curves on abelian surfaces. Manuscripta Math. 84 (1994), no. 2, 199–223; MR1285957


# How Good is the Model in Model-in-the-loop Event Coreference Resolution
Annotation?

Shafiuddin Rehan Ahmed1 Abhijnan Nath2 Michael Regan3
Adam Pollins1 Nikhil Krishnaswamy2 James H. Martin1
1University of Colorado, Boulder, CO, USA 3University of Washington, Seattle,
WA, USA
2Colorado State University, Fort Collins, CO, USA
<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Annotating cross-document event coreference links is a time-consuming and
cognitively demanding task that can compromise annotation quality and
efficiency. To address this, we propose a model-in-the-loop annotation
approach for event coreference resolution, where a machine learning model
suggests likely corefering event pairs only. We evaluate the effectiveness of
this approach by first simulating the annotation process and then, using a
novel annotator-centric Recall-Annotation effort trade-off metric, we compare
the results of various underlying models and datasets. We finally present a
method for obtaining 97% recall while substantially reducing the workload
required by a fully manual annotation process.

##  1 Introduction

Event Coreference Resolution (ECR) is the task of identifying mentions of the
same event either within or across documents. Consider the following excerpts
from three related documents:

  * $e_{1}$: 55 year old star will $replace_{m_{1}}$ Matt Smith, who announced in June that he was leaving the sci-fi show.

  * $e_{2}$: Matt Smith, 26, will make his debut in 2010, $replacing_{m_{2}}$ David Tennant, who leaves at the end of this year.

  * $e_{3}$: Peter Capaldi $takes~{}over_{m_{3}}$ Doctor Who … Peter Capaldi $stepped~{}into_{m_{4}}$ Matt Smith’s soon to be vacant Doctor Who shoes.

$e_{1}$, $e_{2}$, and $e_{3}$ are example sentences from three documents where
the event mentions are highlighted and sub-scripted by their respective
identifiers ($m_{1}$ through $m_{4}$). The task of ECR is to automatically
form the two clusters $\\{m_{1},m_{3},m_{4}\\}$, and $\\{m_{2}\\}$. We refer
to any pair between the mentions of a cluster, e.g., $(m_{1},m_{3})$ as an ECR
link. Any pair formed across two clusters, e.g., $(m_{1},m_{2})$ is referred
to as non-ECR link.

Annotating ECR links can be challenging due to the large volume of mention
pairs that must be compared. The annotating task becomes increasingly time-
consuming as the number of events in the corpus increases. As a result, this
task requires a lot of mental effort from the annotator and can lead to poor
quality annotations Song et al. (2018); Wright-Bettner et al. (2019). Indeed,
an annotator has to examine multiple documents simultaneously often relying on
memory to identify all the links which can be an error-prone process.

To reduce the cognitive burden of annotating ECR links, annotation tools can
provide integrated model-in-the-loop for sampling likely coreferent mention
pairs Pianta et al. (2008); Yimam et al. (2014); Klie et al. (2018). These
systems typically store a knowledge base (KB) of annotated documents and then
use this KB to suggest relevant candidates. The annotator can then inspect the
candidates and choose a coreferent event if present.

The model’s querying and ranking operations are typically driven by machine
learning (ML) systems that are trained either actively Pianta et al. (2008);
Klie et al. (2018); Bornstein et al. (2020); Yuan et al. (2022) or by using
batches of annotations Yimam et al. (2014). While there have been advances in
suggestion-based annotations, there is little to no work in evaluating the
effectiveness of these systems, particularly in the use case of ECR.
Specifically, both the overall coverage, or recall, of the annotation process
as well as the degree of annotator effort needed depend on the performance of
the model. In order to address this shortcoming, we offer the following
contributions:

  1. 1.

We introduce a method of model-in-the-loop annotations for ECR111repo:
$\mathtt{github.com/ahmeshaf/model\\_in\\_coref}$.

  2. 2.

We compare three existing methods for ECR (differing widely in their
computational costs), by adapting them as the underlying ML models governing
the annotations.

  3. 3.

We introduce a novel methodology for assessing the workflow by simulating the
annotations and then evaluating an annotator-centric Recall-Annotation effort
tradeoff.

##  2 Related Work

Previous work for ECR is largely based on modeling the probability of
coreference between mention pairs. These models are built on supervised
classifiers trained using features extracted from the pairs. Most recent work
uses a transformer-based language model (LM) like BERT Devlin et al. (2018);
Liu et al. (2019) to generate joint representations of mention pairs, a method
known as cross-encoding. The cross-encoder is fine-tuned using a coreference
scoring objective (Barhom et al., 2019; Cattan et al., 2020; Meged et al.,
2020; Zeng et al., 2020; Yu et al., 2020; Caciularu et al., 2021). These
methods use scores generated from the scorer to then agglomeratively cluster
coreferent events.

Over the years, a number of metrics have been proposed to evaluate ECR (Vilain
et al., 1995; Bagga and Baldwin, 1998; Luo, 2005; Recasens and Hovy, 2011; Luo
et al., 2014; Pradhan et al., 2014). An ECR system is evaluated using these
metrics to determine how effectively it can find event clusters (recall) and
how cleanly separated the clusters are (precision). From the perspective of
annotation, it may only be necessary to focus on the system’s recall or its
effectiveness in finding ECR links. However, an annotator might still want to
know how much effort is required to identify these links in a corpus to
estimate their budget. In the remainder of the paper, we attempt to answer
this question by first quantifying annotation effort and analyzing its
relation with recall of the system.

We use the Event Coreference Bank Plus (ECB+; Cybulska and Vossen (2014)) and
the Gun Violence Corpus (GVC; Vossen et al. (2018)) for our experiments. The
ECB+ is a common choice for assessing ECR, as well as the experimental setup
of Cybulska and Vossen (2015) and gold topic clustering of documents and gold
mention annotations for both training and testing222The ECB+ test set has
1,780 event mentions with 5K ECR links among 100K pairwise mentions, while the
GVC test set has 1,008 mentions with 2K ECR links in 20K pairs. Full
statistics in Table 1 in Appendix A. On the other hand, the GVC offers a more
challenging set of exclusively event-specific coreference decisions that
require resolving gun violence-related events.

Figure 1: For the target mention ($m_{1}$), the Annotated Event Cluster store
presents three potential coreferent candidates ($m_{2}$, $m_{4}$ and $m_{*}$).
The ranking module (an ECR scorer) then ranks them based on their semantic
similarity to $m_{1}$. The annotator reviews each candidate one-at-a-time and
makes decisions on coreference. $m_{*}$ is skipped after finding $m_{4}$ as
coreferent. The cluster store is then updated based on these decisions.

##  3 Annotation Methodology

We implement an iterative model-in-the-loop methodology333Utilizing the
prodi.gy annotation tool. See Appendix D for annotating ECR links in a corpus
containing annotated event triggers. This approach has two main components -
(1) the storage and retrieval of annotated event clusters, which are then
compared with each new target event, and (2), an ML model that ranks and
prunes the sampled candidate clusters by evaluating their semantic similarity
to the target mention.

As illustrated in Figure 1, our annotation workflow queries the Annotated
Event Store for the target event ($m_{1}$), retrieving three potential
coreferring candidates ($m_{2}$, $m_{*}$, and $m_{4}$). The ranking module
then evaluates these candidates based on their lexical and semantic
similarities to $m_{1}$. The annotator then compares each candidate to the
target and determines if they are coreferent. Upon finding a coreferent
candidate, the target is merged into the coreferent cluster, and any remaining
option(s) ($m_{*}$) are skipped.

###  3.1 Ranking

We investigate three separate methods to drive the ranking of candidates
distinguished by their computational cost. We use these methods to generate
the average pair-wise coreference scores between mentions of the candidate and
target events, then use these scores to rank candidates. We use a single RTX
3090 (24 GB) for running our experiments.

Cross-encoder (CDLM): In this method, we use the fine-tuned cross-encoder ECR
system of Caciularu et al. (2021) to generate pairwise mention scores444This
method is compute-intensive since the transformer’s encoding process scales
quadratically with the number of mentions. Using the trained weights, running
inference on the two test sets for our experiments takes approximately forty
minutes to calculate the similarities of all the mention pairs. The weights
are provided by Caciularu et al. (2021) here.. Their state of the art system
uses a modified Longformer (Beltagy et al., 2020) as the underlying LM to
generate document-level representations of the mention pairs (detailed in
§B.1). More specifically, we generate a unified representation (Eq. 1) of the
mention pair ($m_{i}$, $m_{j}$) by concatenating the pooled output of the
transformer ($\text{E}_{\text{CLS}}$), the outputs of the individual event
triggers ($\text{E}_{m_{i}}$, $\text{E}_{m_{j}}$), and their element-wise
product. Thereafter, pairwise-wise scores are generated for each mention-pair
after passing the above representations through a Multi-Layer Perceptron
($\mathtt{mlp}$) (Eq. 2) that was trained using the gold-standard labels for
supervision.

$\displaystyle\mathtt{LF}(m_{i},m_{j})=\left<\text{E}_{\text{CLS}},\text{E}_{m_{i}},\text{E}_{m_{j}},\text{E}_{m_{i}}\odot\text{E}_{m_{j}}\right>$
(1)
$\displaystyle\texttt{CDLM}(m_{i},m_{j})=\mathtt{mlp}(\mathtt{LF}(m_{i},m_{j}))$
(2)

BERTScore (BERT): Zhang et al. (2019) BERT-Score ($\mathtt{BS}$) is a NLP
metric that measures pairwise text similarity by exploiting pretrained BERT
models. It calculates cosine similarity of token embeddings with inverse
document frequency weights to rate token importance and aggregates them into
precision, recall, and F1 scores. This method emphasizes semantically
significant tokens, resulting in a more accurate similarity score (details in
§B.2).

$\displaystyle\mathtt{S}_{\text{bert}}($ $\displaystyle
m)=~{}\left<t_{m},~{}\text{[SEP]},~{}S_{m}\right>$ (3)
$\displaystyle\begin{split}\texttt{BERT}(&m_{i},m_{j})=\lambda~{}\mathtt{BS}(t_{m_{i}},t_{m_{j}})\\\
&+(1-\lambda)~{}\mathtt{BS}(\mathtt{S}_{\text{bert}}(m_{i}),\mathtt{S}_{\text{bert}}(m_{j}))\end{split}$
(4)

To calculate the BERTScore between the mentions, we first construct a combined
sentence ($\mathtt{S}_{\text{bert}}(m)$; Shi and Lin (2019)) for a mention
($m$) by concatenating the mention text ($t_{m}$) and its corresponding
sentence ($S_{m}$), as depicted in Equation 3. Subsequently, we compute the
$\mathtt{BS}$ for each mention pair using $\mathtt{S}_{\text{bert}}(m)$ and
$t_{m}$ separately, then extract the F1 from each. We then take the weighted
average of the two scores as shown in Equation 4 as our ranking metric. This
process, carried out using the $\mathtt{distilbert-base-uncased}$ Sanh et al.
(2019) model, requires approximately seven seconds to complete on each test
set.

Lemma Similarity (Lemma): The lemma555We use spaCy 3.4 en_core_web_md
lemmatizer similarity method emulates the annotation process carried out by
human annotators when determining coreference based on keyword comparisons
between two mentions. To estimate this similarity, we compute the token
overlap (Jaccard similarity; $\mathtt{JS}$) between the triggers and sentences
containing the respective mentions and take a weighted average of the two
similarities (like Eq 4) as shown in Eq 5666$\lambda$ is a hyper-parameter to
control the weightage of the trigger and sentence similarities in Equations 4
and 5, which we tune using the development set. See Appendix C..

$\displaystyle\begin{split}\texttt{Lemma}(m_{i},&m_{j})=\lambda~{}\mathtt{JS}(t_{m_{i}},t_{m_{j}})\\\
&+(1-\lambda)~{}\mathtt{JS}(S_{m_{i}},S_{m_{j}})\end{split}$ (5)

No Ranking (Random): For our baseline approach, we employ a method that
directly picks the candidate-mention pairs through random sampling and without
ranking, providing a reference point for evaluating the effectiveness of the
above three ranking techniques.

###  3.2 Pruning

To control the comparisons between candidate and target events, we restrict
our selection to the top-$k$ ranked candidates. To refine our analysis, we
employ non-integer $k$ values, allowing for the inclusion of an additional
candidate with a probability equal to the decimal part of $k$. We vary the
values of $k$ from 2 to 20 on increments of 0.5 and then investigate its
relation to recall and effort in §4.

###  3.3 Simulation

To evaluate the ranking methods, we conduct annotation simulations on the
events in the ECB+ and GVC development and test sets. These simulations follow
the same annotation methodology of retrieving and ranking candidate events for
each target but utilize ground-truth for clustering. By executing simulations
on different ranking methods and analyzing their performance, we effectively
isolate and assess each approach.

##  4 Evaluation Methodology

We evaluate the performance of the model-in-the-loop annotation with the
ranking methods through simulation on two aspects: (1) how well it finds the
coreferent links, and (2) how much effort it would take to annotate the links
using the ranking method.

23456789111418$\displaystyle\operatorname{\mathtt{Comparisons}}~{}\times~{}10^{3}~{}(~{}\log~{})$889092949698100$\displaystyle\operatorname{\mathtt{Recall}}\%$ECB+11.522.53$\displaystyle\operatorname{\mathtt{Comparisons}}~{}\times~{}10^{3}$889092949698100$\operatorname{\mathtt{Recall}}$%GVCCDLMBERTLemmaRandom

Figure 2: $\operatorname{\mathtt{Recall}}$ and
$\operatorname{\mathtt{Comparisons}}$ achieved upon varying the $k$ for each
ranking method in the ECR annotation simulation. The three methods result in
significantly fewer comparisons than the no-ranking Random baseline.

###  4.1 Recall-Annotation Effort Tradeoff

$\operatorname{\mathtt{Recall}}$: The recall metric evaluates the percentage
of ECR links that are correctly identified by the suggestion model. It is
calculated as the ratio of the number of times the true coreferent candidate
is among the suggested candidates. The recall error is introduced when the
coreferent candidate is erroneously removed based on the top-$k$ value777Note
that recall is always 100% if no candidates are ever pruned..

$\operatorname{\mathtt{Comparisons}}$: A unit effort represents the comparison
between a candidate and target mentions that an annotator would have to make
in the annotation process. We count the sampled candidates for each target and
stop counting when the coreferent candidate is found. For example, the number
of comparisons for the target $m_{1}$, in Figure 1, is 2 ($m_{2}$ and
$m_{4}$). We count this number for each target event and present the sum as
$\operatorname{\mathtt{Comparisons}}$.

###  4.2 Analysis and Discussion

We present an analysis of the various ranking methods employed in our study,
highlighting the performance and viability of each approach. We employ the
ranking methods on the test sets of ECB+ and GVC. Then, estimate the
$\operatorname{\mathtt{Recall}}$ and $\operatorname{\mathtt{Comparisons}}$
measures for different $k$ values, and collate them into the plots as shown in
Figure 2.

Performance Comparison: The performance improvement of CDLM over BERT and BERT
over Lemma can be quantified by examining the graph for the ECB+ and GVC
datasets. For example, when targeting a 95% recall for the ECB+ corpus, CDLM
provides an almost 100 percent improvement over BERT reducing the number of
comparisons to almost half of the latter. However, both CDLM and BERT
outperform Lemma by a significant margin while being drastically better than
the Random baseline (See Fig. 2). Interestingly, for GVC, the performance gap
between CDLM and BERT is quite close, both needing at least three-fourths as
many comparisons as the Lemma and crucially outperforming the Random baseline.
CDLM’s inconsistent performance on GVC suggests that a corpus-fine-tuned model
such as itself is more effective when applied to a dataset similar to the one
it was trained on.

Efficiency and Generalizability of BERT:

BERT offers a compelling advantage in terms of efficiency, as it can be run on
low-compute settings. Moreover, BERT exhibits greater generalizability out-of-
the-box when comparing its performance on both the ECB+ and GVC datasets. This
makes it an attractive option for ECR annotation task especially when compute
resources are limited or when working with diverse corpora.

##  5 Conclusion

We introduced a model-in-the-loop annotation method for annotating ECR links.
We compared three ranking models through a novel evaluation methodology that
answers key questions regarding the quality of the model in the annotation
loop (namely, recall and effort). Overall, our analysis demonstrates the
viability of the models, with CDLM exhibiting the best performance on the ECB+
dataset, followed by BERT and Lemma. The choice of ranking method depends on
the specific use case, dataset, and resource constraints, but all three
methods offer valuable solutions for different scenarios.

## Limitations

It is important to note that the approaches presented in this paper have
several constraints. Firstly, the methods presented are restricted to English
language only, as Lemma necessitates a lemmatizer and, BERT and CDLM rely on
models trained exclusively on English corpora. Secondly, the utilization of
the CDLM model demands at least a single GPU, posing potential accessibility
issues. Thirdly, ECR annotation is susceptible to errors and severe
disagreements amongst annotators, which could entail multiple iterations
before achieving a gold-standard quality. Lastly, the generated corpora may be
biased to the model used during the annotation process, particularly for
smaller values of $k$.

## Ethics Statement

We use publicly-available datasets, meaning any bias or offensive content in
those datasets risks being reflected in our results. By its nature, the Gun
Violence Corpus contains violent content that may be troubling for some.

## Acknowledgements

We would like to express our sincere gratitude to the anonymous reviewers
whose insightful comments and constructive feedback helped to greatly improve
the quality of this paper. We gratefully acknowledge the support of U.S.
Defense Advanced Research Projects Agency (DARPA) FA8750-18-2-0016-AIDA –
RAMFIS: Representations of vectors and Abstract Meanings For Information
Synthesis. Any opinions, findings, conclusions, or recommendations expressed
in this material are those of the authors and do not necessarily reflect the
views of DARPA or the U.S. government. We would also like to thank ExplosionAI
GmbH for partly funding this work. Finally, we extend our thanks to the
BoulderNLP group and the SIGNAL Lab at Colorado State for their valuable input
and collaboration throughout the development of this work.

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##  Appendix A ECB+ Corpus Event Statistics

Table 1 contains the detailed statistics for both the ECB+ and the GVC
corpora.

| ECB+ | GVC
---|---|---
| Train | Dev | Test | Train | Dev | Test
T/ST | 25 | 8 | 10 | 170 | 37 | 34
D | 594 | 196 | 206 | 358 | 78 | 74
M | 3808 | 1245 | 1780 | 5313 | 977 | 1008
C | 1464 | 409 | 805 | 991 | 228 | 194
S | 1053 | 280 | 623 | 252 | 70 | 43
P | 300K | 100K | 180K | 100K | 20K | 20K
P+ | 15K | 6K | 6.5K | 24K | 3.7K | 4.1K
Table 1:  ECB+ and GVC Corpus statistics for event mentions. T/ST =
topics/sub-topics, D = documents, M = event mentions, C = clusters, S =
singletons. P = unique mention pairs by topic. P+ = mention pairs that are
coreferent.

##  Appendix B Model Details

Figure 3: Illustration of Cross-encoding with CDLM from Caciularu et al.
(2021). Figure 4: Illustration of the Recall Measure of BERTScore from Zhang
et al. (2019).

###  B.1 CDLM

The CDLM model, based on the Longformer architecture, cleverly uses a
combination of global and local attention for event trigger words and the rest
of the document containing those events respectively. More specifically, the
Longformer’s increased input capacity of 4096 tokens is utilized to encode
much longer documents at finetuning that are usually seen in coreference
corpora like the ECB+. As seen in Fig. 3, apart from the document-separator
tokens like <doc-s> and <doc-s/> that help contextualize each document in a
pair, it adds two special tokens (<m> and </m>) to the model vocabulary while
pretraining to achieve a greater level of contextualization of a document pair
while attending to the event triggers globally at finetuning. Apart from the
event-trigger words, the fine-tuned CDLM model also applies the global
attention mechanism on the [CLS] token resulting in a more refined embedding
for that document pair while maintaining linearity in the transformer’s self-
attention.

###  B.2 BERTScore

BERT-Score is an easy-to-use, low-compute scoring metric that can be used to
evaluate NLP tasks that require semantic-similarity matching. This task-
agnostic metric uses a base language model like BERT to generate token
embeddings and leverages the entire sub-word tokenized reference and candidate
sentences ($x$ and $\hat{x}$ in Fig. 4) to calculate the pairwise cosine
similarity between the sentence pair. It uses a combination of a greedy-
matching subroutine to maximize the similarity scores while normalizing the
generated scores based on the IDF (Inverse Document Frequency) of the sub-
tokens thereby resulting in more human-readable scores. The latter weighting
parameter takes care of rare-word occurrences in sentence pairs that are
usually more indicative of how semantically similar such pairs are. In our
experiments, we use the $\mathtt{distilbert-base-uncased}$ model to get the
pairwise coreference scores, consistent with our goal of deploying an
annotation workflow suitable for resource-constrained settings. Such lighter
and ’distilled’ encoders allow us to optimize resources at inference with
minimal loss in performance.

23456789$\displaystyle\operatorname{\mathtt{Comparisons}}~{}\times~{}10^{3}~{}(~{}\log~{})$$\displaystyle{0.75}$$\displaystyle{0.80}$$\displaystyle{0.85}$$\displaystyle{0.90}$$\displaystyle{0.95}$$\displaystyle{1.00}$Lemma
$\operatorname{\mathtt{Recall}}$ECB+$\displaystyle{1400}$$\displaystyle{1600}$$\displaystyle{1800}$$\displaystyle{2000}$$\displaystyle{2200}$$\displaystyle{2400}$$\displaystyle\operatorname{\mathtt{Comparisons}}$$\displaystyle{0.75}$$\displaystyle{0.80}$$\displaystyle{0.85}$$\displaystyle{0.90}$$\displaystyle{0.95}$$\displaystyle{1.00}$Lemma
$\operatorname{\mathtt{Recall}}$GVC23456789$\displaystyle\operatorname{\mathtt{Comparisons}}~{}\times~{}10^{3}~{}(~{}\log~{})$$\displaystyle{0.75}$$\displaystyle{0.80}$$\displaystyle{0.85}$$\displaystyle{0.90}$$\displaystyle{0.95}$$\displaystyle{1.00}$BERT
$\operatorname{\mathtt{Recall}}$$\displaystyle{1400}$$\displaystyle{1600}$$\displaystyle{1800}$$\displaystyle{2000}$$\displaystyle{2200}$$\displaystyle\operatorname{\mathtt{Comparisons}}$$\displaystyle{0.75}$$\displaystyle{0.80}$$\displaystyle{0.85}$$\displaystyle{0.90}$$\displaystyle{0.95}$$\displaystyle{1.00}$BERT
$\operatorname{\mathtt{Recall}}$0.00.30.50.70.81.0

Figure 5: Trigger and Sentence Similarity weight ($\lambda$) Hyper-parameter
tuning on the development sets of ECB+ and GVC. We deduce $\lambda=0.7$ is
optimal for both methods for both datasets.

##  Appendix C $\lambda$ Hyper-parameter Tuning

We employ the evaluation methodology detailed in §4 to determine the optimal
value of $\lambda$ (the weight for trigger similarity and sentence similarity)
for both BERT and Lemma approaches. By conducting incremental annotation
simulations on the development sets of ECB+ and GVC, we assess $\lambda$
values ranging from 0 to 1. The recall-effort curve is plotted for each
$\lambda$ value, as shown in Figure 5, allowing us to identify the one that
consistently achieves the highest recall with the fewest comparisons.
Remarkably, the optimal value for both methods is found to be 0.7, and this
value remains consistent across the two datasets and approaches.

##  Appendix D Annotation Interface using Prodigy

Figure 6: The model-in-the-loop ECR annotation using the Prodigy Annotation
Tool. The target event is on the left and the Candidate cluster is on the
right.

Figure 6 illustrates the interface design of the annotation methodology on the
popular model-in-the-loop annotation tool - Prodigy (prodi.gy). We use this
tool for the simplicity it offers in plugging in the various ranking methods
we explained. The recipe for plugging it in to the tool along with other
experiment code: $\mathtt{github.com/ahmeshaf/model\\_in\\_coref}$.