# A positive proportion of monic odd-degree hyperelliptic curves of genus
$g\geq 4$ have no unexpected quadratic points

Jef Laga and Ashvin A. Swaminathan

###### Abstract

Let $\mathcal{F}_{g}$ be the family of monic odd-degree hyperelliptic curves
of genus $g$ over $\mathbb{Q}$. Poonen and Stoll have shown that for every
$g\geq 3$, a positive proportion of curves in $\mathcal{F}_{g}$ have no
rational points except the point at infinity. In this note, we prove the
analogue for quadratic points: for each $g\geq 4$, a positive proportion of
curves in $\mathcal{F}_{g}$ have no points defined over quadratic extensions
except those that arise by pulling back rational points from $\mathbb{P}^{1}$.

##  1 Introduction

Let $C/\mathbb{Q}$ be a smooth projective curve of genus $g\geq 2$. Faltings
has shown that the set of rational points $C(\mathbb{Q})$ is finite [Fal83,
Satz 7]. It is therefore natural to ask how the quantity $\\#C(\mathbb{Q})$ is
distributed as $C$ varies in a family of curves. Many authors (see, e.g.,
[PV04, Conjecture 2.2], [Poo06], [Gra07, Conjecture 1.3(ii)], [Sto09,
Conjecture 1]) have formulated heuristics or conjectures suggesting that in a
family of curves $C$ of genus $g\geq 2$, the quantity $\\#C(\mathbb{Q})$ is
very often as small as possible: i.e., most members ought to possess no
rational points beyond those that exist generically among all members in the
family. Put rather simply, most curves in a family should be as “pointless” as
can be.

This pithy principle has been proven (or partially proven) for several
families of curves of interest; see, e.g., [Bha13], [BGW17] (cf. [BSS21,
Remark 3.3]), [RT21], [SW18], [Lag22]. Of particular relevance to this note is
the major result of Poonen and Stoll [PS14], which concerns the family of
monic odd-degree hyperelliptic curves

$\displaystyle\mathcal{F}_{g}\vcentcolon=\\{C\colon
y^{2}=f(x)=x^{2g+1}+a_{1}x^{2g}+\dots+a_{2g+1}\mid
a_{i}\in\mathbb{Z},\,\text{disc}(f)\neq 0\\},$ (1)

ordered by the height $0pt(C)=\max|a_{i}|^{1/i}$. Here $C$ denotes the unique
smooth projective curve with affine equation $y^{2}=f(x)$; it has a unique
point at infinity $P_{\infty}$, which is $\mathbb{Q}$-rational. If $g\geq 3$,
Poonen and Stoll showed that a positive proportion of curves
$C\in\mathcal{F}_{g}$ satisfy $C(\mathbb{Q})=\\{P_{\infty}\\}$, and that this
proportion tends to $1$ exponentially fast as $g\rightarrow\infty$.

In this note, we consider quadratic points on curves in the family
$\mathcal{F}_{g}$. Recall that a point $P\in C(\overline{\mathbb{Q}})$ is
quadratic if its field of definition $\mathbb{Q}(P)$ is a quadratic extension
of $\mathbb{Q}$. When $C$ is hyperelliptic, it visibly has infinitely many
quadratic points: since $C$ has a degree-$2$ map $\pi\colon
C\to\mathbb{P}^{1}$ defined over $\mathbb{Q}$, for all but finitely many
points $P\in\mathbb{P}^{1}(\mathbb{Q})$ the preimage $\pi^{-1}(P)$ consists of
a Galois-conjugate pair of quadratic points. Explicitly, if
$P=(x:1)\in\mathbb{P}^{1}(\mathbb{Q})$ then $\pi^{-1}(P)=(x,\pm\sqrt{f(x)})\in
C(\overline{\mathbb{Q}})$. It follows from a more general result of Faltings
[Fal91, Theorem 1 & Example 4.5] that if $C/\mathbb{Q}$ is a hyperelliptic
curve of genus $g\geq 4$, then $C$ has only finitely many quadratic points not
arising as pullbacks from rational points on $\mathbb{P}^{1}$.

###  1.1 Results

If $C\in\mathcal{F}_{g}$, we say a rational point $P$ on $C$ is _expected_ if
$P=P_{\infty}$, and unexpected otherwise. We say a quadratic point $P$ on $C$
is _expected_ if it maps to a rational point on $\mathbb{P}^{1}(\mathbb{Q})$,
and unexpected otherwise. Our main result shows that when $g\geq 4$, a
positive proportion of curves in $\mathcal{F}_{g}$ have no unexpected rational
or quadratic points at all. Let
$\mathcal{F}_{g,X}\vcentcolon=\\{C\in\mathcal{F}_{g}\mid\operatorname{ht}(C)<X\\}$,
and define the lower density

$\displaystyle\delta_{g}\vcentcolon=\liminf_{X\rightarrow\infty}\frac{\\#\\{C\in\mathcal{F}_{g,X}\mid
C\text{ has no unexpected rational points or quadratic
points}\\}}{\\#\mathcal{F}_{g,X}}.$

######  Theorem 1.

Let $g\geq 4$. When ordered by height, a positive proportion of monic odd
hyperelliptic curves $C/\mathbb{Q}$ of genus $g$ possess no unexpected
rational points or quadratic points. In other words, $\delta_{g}>0$.

Theorem 1 is the first unconditional result of its kind to effectivize
Faltings’ theorem on the finiteness of unexpected quadratic points in a
“large” family of curves.

###  1.2 Methods

The proof of Theorem 1 relies on partially generalizing the method of Poonen
and Stoll [PS14] (later dubbed “Selmer group Chabauty” in [Sto17]) to
symmetric squares of curves. Roughly speaking, we proceed in three steps as
follows:

  1. 1.

Given a subvariety $X$ of a $g$-dimensional abelian variety $J/\mathbb{Q}$ and
a prime number $p$, we prove a criterion guaranteeing that
$X(\mathbb{Q}_{p})\cap J(\mathbb{Q})$ is “small”, under suitable assumptions
concerning the disjointness of two sets: (1) the image of
$\operatorname{Sel}_{p}J$ in $J(\mathbb{Q}_{p})/pJ(\mathbb{Q}_{p})$, and (2)
the image of $X(\mathbb{Q}_{p})$ under the re-scale and reduce mod-$p$
logarithm map $\rho\log\colon
X(\mathbb{Q}_{p})\dashrightarrow\mathbb{P}^{g-1}(\mathbb{F}_{p})$; see §2 —
specifically, Proposition 2.

  2. 2.

Given a curve $C\in\mathcal{F}_{g}$, we wish to apply the criterion to $J=$
the Jacobian of $C$, $X=$ the image $\operatorname{Sym}^{2}C\rightarrow J$,
and $p=2$. This leads to a criterion for $C$ to have no unexpected rational or
quadratic points; see §3 — specifically, Corollary 5.

  3. 3.

We show that the latter criterion is satisfied for a positive proportion of
$C\in\mathcal{F}_{g}$ when $g\geq 4$ using two inputs: (1) an equidistribution
theorem concerning the image of $\operatorname{Sel}_{2}J\rightarrow
J(\mathbb{Q}_{2})/2J(\mathbb{Q}_{2})$ as $C$ varies in $\mathcal{F}_{g}$ due
to Bhargava–Gross [BG13, Theorem 12.4], and (2) the construction of an
explicit subfamily $\mathcal{F}_{g}^{(2)}\subset\mathcal{F}_{g}$ having
positive density on which we can control the image
$\rho\log(X(\mathbb{Q}_{2}))$; see §5 — specifically, Proposition 8.

By calculating the density of $\mathcal{F}_{g}^{(2)}$, we obtain explicit
lower bounds on the density $\delta_{g}$, proving that $\delta_{g}\geq
2^{-4g^{2}-6g-7}$; since the curves in $\mathcal{F}_{g}^{(2)}$ all have good
reduction at $2$, the bound must be exponentially small in $g$: indeed, it is
exceedingly rare for a hyperelliptic curve in short Weierstrass form to have
good reduction at $2$.

If one would like to go beyond Theorem 1 and prove — analogously to Poonen and
Stoll — that $\delta_{g}\rightarrow 1$ as $g\rightarrow\infty$, one would need
to control the image of $\rho\log\colon
X(\mathbb{Q}_{2})\dashrightarrow\mathbb{P}^{g-1}(\mathbb{F}_{2})$ for a
density $1$ set of curves $C\in\mathcal{F}_{g}$, with arbitrary reduction type
at $2$. The methods of [PS14, §3.2, §5] do not directly apply, due to the lack
of the Weierstrass preparation theorem for power series over $\mathbb{Q}_{p}$
in more than one variable. It therefore seems that new ideas are needed here.

###  1.3 Previous work

Using just their result that the average size of $\operatorname{Sel}_{2}J$ is
at most $3$ in conjunction with an effective version of Chabauty’s method,
Bhargava and Gross show that a positive proportion of curves in
$\mathcal{F}_{g}$ have at most $3$ rational points. It is natural to ask
whether a similar strategy would work to control points of degree $d$. A
version of Chabauty’s method for symmetric powers of curves was developed by
Siksek [Sik09], and Park [Par16] used to tropical intersection theory to
develop an effective version of Siksek’s method. However, it was later
discovered by Gunther and Morrow [GM19, Remark 5.6] that Park’s work is
missing a transversality hypothesis; while it may be feasible to verify this
hypothesis for individual curves, it is currently unknown how to do so in a
family of curves. Conditional on this transversality hypothesis, Gunther and
Morrow prove that a positive proportion of curves in $\mathcal{F}_{g}$ have at
most 24 unexpected quadratic points for each $g\geq 3$. Our main result,
Theorem 1, unconditionally reduces their bound for quadratic points to zero
when $g\geq 4$.

In a different direction, there are several papers in the literature where
versions of Chabauty’s method are used to obtain uniform bounds on the number
of rational points on curves; here the bounds tend to depend on the genus as
well as on the Mordell–Weil rank of the Jacobian. See, e.g., the works of
Stoll [Sto19] and Katz, Rabinoff, and Zureick-Brown [KRZB16]. Analogous
uniform bounds were obtained for quadratic points by Vemulapalli and Wang
[VW21], but their results depend on the aforementioned work of Park [Par16]
and are thus conditional.

We end by mentioning the work of Caro and Pasten [CP23], which has
consequences for quadratic points on non-hyperelliptic curves whose Jacobian
has Mordell–Weil rank at most one.

###  1.4 Acknowledgements

JL thanks Dick Gross for interesting conversations that motivated us to work
on this problem. We also thank Manjul Bhargava, Jackson Morrow, Bjorn Poonen,
Joe Rabinoff, Michael Stoll, Melanie Matchett Wood, and David Zureick-Brown
for helpful conversations. AS was supported by the National Science
Foundation, under the Graduate Research Fellowship as well as Award No.
2202839.

##  2 A generalization of $p$-Selmer-group Chabauty

In this section, we generalize the method of $p$-Selmer group Chabauty —
originally developed by Poonen and Stoll to control rational points on a curve
embedded inside its Jacobian — to arbitrary subvarieties of abelian varieties.
Specifically, we prove Proposition 2, which gives a criterion for a subvariety
of an abelian variety to contain only very few rational points.

Let $J/\mathbb{Q}$ be a $g$-dimensional abelian variety, $X\subset J$ a closed
subvariety, and $p$ a prime number. Let $\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern 1.5mu\subset
J(\mathbb{Q}_{p})$ be the $p$-adic closure of $J(\mathbb{Q})$ in
$J(\mathbb{Q}_{p})$. Consider the following diagram (modeled on [PS14,
(6.1)]):

${{X(\mathbb{Q})}}$${{X(\mathbb{Q}_{p})}}$${{J(\mathbb{Q})}}$${{\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern
1.5mu}}$${{J(\mathbb{Q}_{p})}}$${{\mathbb{Z}_{p}^{g}}}$${{J(\mathbb{Q})/pJ(\mathbb{Q})}}$${{\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern 1.5mu/p\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern
1.5mu}}$${{J(\mathbb{Q}_{p})/pJ(\mathbb{Q}_{p})}}$${{\mathbb{F}_{p}^{g}}}$${{\mathbb{P}^{g-1}(\mathbb{F}_{p})}}$${{\operatorname{Sel}_{p}J}}$$\scriptstyle{\log}$$\scriptstyle{\rho}$$\scriptstyle{\mu}$$\scriptstyle{\delta}$$\scriptstyle{\log\otimes\mathbb{F}_{p}}$$\scriptstyle{\mathbb{P}}$$\scriptstyle{\eta}$$\scriptstyle{\sigma}$$\scriptstyle{\rho\log}$$\scriptstyle{\mathbb{P}\sigma}$
(2)

In the diagram (2), the unlabeled maps in (2) are the natural ones, and the
dashed arrows refer to partially defined maps. A partially defined map
$f\colon Y\dashrightarrow Z$ is a map defined on a subset $S\subset Y$. The
image of such a map is defined to be the image of the subset $S$ and is
denoted by $f(Y)$; the composition of two partially defined maps is again a
partially defined map, defined on the largest subset where evaluation makes
sense. The labeled maps $\log$, $\mathbb{P}$, $\rho$, $\delta$, $\eta$, and
$\sigma$ are defined as follows:

  * •

Write $T_{0}J=\mathrm{H}^{0}(J,\Omega_{J}^{1})^{\vee}$ for the tangent space
at the identity of $J$, and write $\log\colon J(\mathbb{Q}_{p})\rightarrow
T_{0}J$ for the logarithm map; see [Bou98, III, §7.6] and [Zar96] for its
definition. This is the unique $p$-adic Lie group morphism whose derivative
equals the identity. It is a local diffeomorphism, and its kernel equals the
(finite) torsion subgroup $J(\mathbb{Q}_{p})_{\mathrm{tors}}\subset
J(\mathbb{Q}_{p})$. The image of $\log$ is a $\mathbb{Z}_{p}$-lattice in
$T_{0}J$; we fix a basis of $\mathrm{H}^{0}(J,\Omega_{J}^{1})$ so that
$\log\colon J(\mathbb{Q}_{p})\rightarrow T_{0}J\simeq\mathbb{Q}_{p}^{g}$ has
image exactly $\mathbb{Z}_{p}^{g}$.

  * •

For a field $k$ we let $\mathbb{P}\colon
k^{g}\smallsetminus\\{0\\}\rightarrow\mathbb{P}^{g-1}(k)$ be the usual
homogenization map.

  * •

We write $\rho$ for the reduction map
$\mathbb{P}^{g-1}(\mathbb{Q}_{p})\rightarrow\mathbb{P}^{g-1}(\mathbb{F}_{p})$
or the composite
$\mathbb{Q}_{p}^{g}\smallsetminus\\{0\\}\xrightarrow{\mathbb{P}}\mathbb{P}^{g-1}(\mathbb{Q}_{p})\rightarrow\mathbb{P}^{g-1}(\mathbb{F}_{p})$.

  * •

The $\mathbb{F}_{p}$-vector space $\operatorname{Sel}_{p}J$ is the $p$-Selmer
group of $J$, and it comes equipped with a Kummer map $\delta\colon
J(\mathbb{Q})/pJ(\mathbb{Q})\rightarrow\operatorname{Sel}_{p}J$. If
$x\in\operatorname{Sel}_{p}J\subset\mathrm{H}^{1}(\mathbb{Q},J[p])$, then the
local Galois cohomology class $x_{p}$ at $p$ lies in the image of the
injective map $\delta_{p}\colon
J(\mathbb{Q}_{p})/pJ(\mathbb{Q}_{p})\to\mathrm{H}^{1}(\mathbb{Q}_{p},J[p])$.
The map $\eta$ is given by $x\mapsto\delta_{p}^{-1}(x_{p})$. The map $\sigma$
is the composition of $\eta$ with $\log\otimes\mathbb{F}_{p}$.

All the squares and triangles in this diagram are commutative, whenever
compositions are defined.

Observe that $X(\mathbb{Q})$ lies in the intersection
$X(\mathbb{Q}_{p})\cap\overline{J(\mathbb{Q})}\subset J(\mathbb{Q}_{p})$. In
Chabauty’s method, the objective is to bound $X(\mathbb{Q})$ by controlling
this intersection. The idea of $p$-Selmer group Chabauty is to achieve this
control by pushing this intersection forward all the way to
$\mathbb{P}^{g-1}(\mathbb{F}_{p})$ via the maps in the diagram (2).
Specifically, we obtain the following result:

######  Proposition 2.

Suppose $\sigma$ is injective and
$\mathbb{P}\sigma(\operatorname{Sel}_{p}J)\cap\rho\log(X(\mathbb{Q}_{p}))=\varnothing$.
Then $X(\mathbb{Q}_{p})\cap\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern 1.5mu\subset
J(\mathbb{Q}_{p})[p^{\prime}]$, where $J(\mathbb{Q}_{p})[p^{\prime}]$ denotes
the $($finite$)$ subgroup of prime-to-$p$ torsion elements of
$J(\mathbb{Q}_{p})$.

###### Proof.

The proof is essentially identical to that of [PS14, Proposition 6.2], which
never uses the assumption that $X$ is a curve; we include the details for
completeness. By assumption, $\sigma$ is injective, so $\sigma\delta$ is also
injective. It then follows from (2) that $\mu$ is injective too. Thus, $\mu$
is an isomorphism, and the map $\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern 1.5mu/p\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern
1.5mu\rightarrow\mathbb{F}_{p}^{g}$ is injective.

Suppose that the conclusion fails, and let $P\in X(\mathbb{Q}_{p})\cap\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern 1.5mu$ be an
element not contained in $J(\mathbb{Q}_{p})[p^{\prime}]$. Then $P$ is not
infinitely $p$-divisible in $J(\mathbb{Q}_{p})$, and hence also not infinitely
$p$-divisible in $\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern 1.5mu$. Thus,
there is an element $Q\in\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern 1.5mu$ and an
integer $n\geq 0$ such that $P=p^{n}Q$ and $Q\not\in p\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern 1.5mu$. The image
$\overline{Q}$ of $Q$ in $\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern 1.5mu/p\mkern
1.5mu\overline{\mkern-1.5muJ(\mathbb{Q})\mkern-1.5mu}\mkern 1.5mu$ is nonzero,
so its image under $\log\otimes\mathbb{F}_{p}$ in $\mathbb{F}_{p}^{g}$ is
nonzero. Consequently, $\mathbb{P}\sigma\delta\mu^{-1}(\overline{Q})$ is well-
defined and lies in $\mathbb{P}\sigma(\operatorname{Sel}_{p}J)$. On the other
hand,

$\displaystyle\mathbb{P}\sigma\delta\mu^{-1}(\overline{Q})=\rho\log(Q)=\rho\log(p^{n}Q)=\rho\log(P)\in\rho\log(X(\mathbb{Q}_{p})),$
(3)

contradicting our assumption that
$\mathbb{P}\sigma(\operatorname{Sel}_{p}J)\cap\rho\log(X(\mathbb{Q}_{p}))=\varnothing$.
∎

##  3 A criterion for every rational and quadratic point to be expected

In this section, we specialize the content of §2 to the case of Jacobians of
monic odd-degree hyperelliptic curves over $\mathbb{Q}$, and we obtain a
criterion — namely, Corollary 5 — for every rational point and every quadratic
point on such a curve to be expected.

We start slightly more generally and introduce some notation that we will use
in the remainder of the paper. Let $C$ be a hyperelliptic curve of genus
$g\geq 2$. Denote its Jacobian variety by $J$. There is by definition a
degree-$2$ map $\pi\colon C\to\mathbb{P}^{1}$ defined over $\mathbb{Q}$,
called the hyperelliptic map, which is unique up to automorphism of
$\mathbb{P}^{1}$. We call the pullback along $\pi$ of a $\mathbb{Q}$-point of
$\mathbb{P}^{1}$ a hyperelliptic divisor, and the linear equivalence class of
such a divisor the hyperelliptic divisor class $H$. The associated linear
system is denoted
$|H|\vcentcolon=\mathbb{P}(\mathrm{H}^{0}(C,\mathcal{O}_{C}(H)))\simeq\mathbb{P}^{1}$,
and consists of divisors of the form $\pi^{*}(t)$, where $t$ is a point of
$\mathbb{P}^{1}$. (When $C\in\mathcal{F}_{g}$, a convenient choice for $H$ is
the class $[2P_{\infty}]$.)

The symmetric square $\operatorname{Sym}^{2}C$ is a smooth, projective, and
geometrically integral variety parametrizing effective divisors of degree $2$
on $C$. We may therefore view $|H|$ as a closed subscheme of
$\operatorname{Sym}^{2}C$. Let
$\operatorname{AJ}\colon\operatorname{Sym}^{2}C\rightarrow J$ be the
Abel–Jacobi morphism sending an effective degree-$2$ divisor $D$ to the
divisor class $[D]-H$. Let

$\displaystyle X\vcentcolon=\operatorname{AJ}(\operatorname{Sym}^{2}C)$ (4)

be the scheme-theoretic image of $\operatorname{Sym}^{2}C$ under
$\operatorname{AJ}$. The tangent space of the point $0$ in $X$ has dimension
$g$ [ACGH85, Chapter IV, Proposition (4.2)], so if $g\geq 3$, then $X$ has a
singular point, resolved by the map
$\operatorname{AJ}\colon\operatorname{Sym}^{2}C\rightarrow X$.

######  Lemma 3.

The morphism $\mathrm{AJ}\colon\operatorname{Sym}^{2}C\rightarrow J$ is
birational onto its image $X$. It is an isomorphism above
$X\smallsetminus\\{0\\}$ and the $($scheme-theoretic$)$ fiber above $\\{0\\}$
is the closed subscheme $|H|\subset\operatorname{Sym}^{2}C$.

###### Proof.

The fiber above $[E]\in J$ is the projective space of divisors
$D\in\operatorname{Sym}^{2}C$ in the linear system $|E+H|$ by [ACGH85, Chapter
IV, Lemma (1.1)]. Since $|H|$ is the only positive dimensional degree-$2$
linear system, the lemma follows. ∎

######  Lemma 4.

Suppose that $X(\mathbb{Q})=\\{0\\}$. Then $C(\mathbb{Q})$ consists of at most
one Weierstrass point, and every quadratic point on $C$ maps to
$\mathbb{P}^{1}(\mathbb{Q})$ under $\pi$.

###### Proof.

Using Lemma 3, the condition $X(\mathbb{Q})=\\{0\\}$ implies that
$(\operatorname{Sym}^{2}C)(\mathbb{Q})=|H|(\mathbb{Q})$. In other words, every
$\mathbb{Q}$-rational effective degree-$2$ divisor on $C$ is of the form
$\pi^{*}(t)$ for some $t\in\mathbb{P}^{1}(\mathbb{Q})$. One can check that
this implies the conclusions of the lemma. ∎

Applying Proposition 2 to our choice of $X$ and using Lemma 4, we obtain the
following criterion for the absence of unexpected rational and quadratic
points on curves $C\in\mathcal{F}_{g}$ in terms of the diagram (2):

######  Corollary 5.

Let $C\in\mathcal{F}_{g}$. Suppose that:

  * •

$\sigma\colon\operatorname{Sel}_{p}J\rightarrow\mathbb{F}_{p}^{g}$ is
injective;

  * •

$\mathbb{P}\sigma(\operatorname{Sel}_{p}J)\cap\rho\log(X(\mathbb{Q}_{p}))=\varnothing$;
and

  * •

$X(\mathbb{Q}_{p})\cap J(\mathbb{Q}_{p})[p^{\prime}]=\\{0\\}$.

Then $C$ has no unexpected rational or quadratic points.

##  4 Controlling the image $(\operatorname{Sym}^{2}C)(\mathbb{Q}_{2})$ under
$\rho\log$

The purpose of this section is to construct a subfamily
$\mathcal{F}_{g}^{(2)}\subset\mathcal{F}_{g}$ of positive density for which
(in the notation of §3) $X(\mathbb{Q}_{2})\cap
J(\mathbb{Q}_{2})[2^{\prime}]=\\{0\\}$ and for which we can control the image
$\rho\log(X(\mathbb{Q}_{2}))$.

In [PS14], which concerns rational points on $C$, the corresponding problem is
to control the set $\rho\log(C(\mathbb{Q}_{2}))$. To do this, Poonen and Stoll
consider a particular curve $C_{0}/\mathbb{Q}_{2}$ — namely, the curve with
long Weierstrass equation $C_{0}\colon y^{2}+y=x^{2g+1}+x+1$ — and they prove
for this choice of $C_{0}$ that $\\#\rho\log(C_{0}(\mathbb{Q}_{2}))=1$. They
then separately prove that as $C$ varies in the moduli space of monic odd
hyperelliptic curves of genus $g$ over $\mathbb{Q}_{2}$, the set
$\rho\log(C(\mathbb{Q}_{2}))$ is locally constant, from which it follows that
for every $g\geq 3$ a positive proportion of curves $C\in\mathcal{F}_{g}$ have
$\\#\rho\log(C(\mathbb{Q}_{2}))=1$.

Here, we adopt a slightly different approach: we construct a subfamily
$\mathcal{F}_{g}^{(2)}\subset\mathcal{F}_{g}$, defined by congruence
conditions, such that the image
$\rho\log((\operatorname{Sym}^{2}C)(\mathbb{Q}_{2}))$ can be explicitly
bounded for _all_ curves $C\in\mathcal{F}_{g}^{(2)}$, much as Poonen and Stoll
did for the single curve $C_{0}$. All curves in these subfamilies have good
reduction at $2$, and their Jacobians all have trivial $2$-torsion over both
$\mathbb{Q}_{2}$ and $\mathbb{F}_{2}$. Thus, while $\mathcal{F}_{g}^{(2)}$ has
positive density in $\mathcal{F}_{g}$, this density is exponentially small in
$g$.

The curves in $\mathcal{F}_{g}^{(2)}$ will be those that arise by completing
the square in long Weierstrass equations considered in the following
definition:

######  Definition 6.

We say a monic polynomial $h\in\mathbb{Z}_{2}[x]$ of degree $2g+1$ is _good_
if it satisfies the following property: letting $S$ be the set of exponents of
the mod-$2$ reduction of $h$, and for each $i\in\\{0,1,2\\}$, letting
$S_{i}\subset S$ be the subset of exponents congruent to $i\pmod{3}$, we have
$S\supset\\{0,2g+1\\}$, $\\#S_{0}\equiv 0\pmod{2}$, and $\\#(S_{1}\sqcup
S_{2})\equiv 1\pmod{2}$. For such an $h$, let $C_{h}/\mathbb{Q}_{2}$ be the
hyperelliptic curve with long Weierstrass model

$y^{2}+y=h(x).$ (5)

Let
$P_{g}=\\{h(x)=x^{2g+1}+c_{1}x^{2g}+\cdots+c_{2g+1}\in\mathbb{Z}_{2}[x]\\}=\mathbb{Z}_{2}^{2g+1}$
be the parameter space of monic polynomials of degree $2g+1$ over
$\mathbb{Z}_{2}$, and let $U_{g}\subset P_{g}$ be the open subset of good
polynomials.

Let $C=C_{h}$, where $h\in U_{g}$ is good. The model (5) has good reduction,
hence defines a smooth proper model $\mathcal{C}$ of $C$ over
$\mathbb{Z}_{2}$. Let
$\mathcal{J}=\operatorname{Pic}^{0}_{\mathcal{C}/\mathbb{Z}_{2}}$ be the Néron
model of the Jacobian $J$ of $C$. We write
$P_{\infty}\in\mathcal{C}(\mathbb{Z}_{2})$ for the unique point at infinity.

The next lemma records the useful properties satisfied by good curves:

######  Lemma 7.

Let $C/\mathbb{Q}_{2}$ be a hyperelliptic curve of the form $C_{h}$ for some
good $h\in\mathbb{Z}_{2}[x]$. Then:

  1. $(a)$

$J(\mathbb{Q}_{2})[2]=\\{0\\}=\mathcal{J}(\mathbb{F}_{2})[2]$.

  2. $(b)$

$\mathcal{C}(\mathbb{F}_{2})=\\{P_{\infty}\\}$ and
$\mathcal{C}(\mathbb{F}_{4})=\\{P_{\infty},(0,\alpha),(0,\alpha+1),(1,\alpha),(1,\alpha+1)\\}$,
where $\alpha\in\mathbb{F}_{4}\smallsetminus\\{1\\}$ denotes a nontrivial
third root of unity.

  3. $(c)$

$X(\mathbb{Q}_{2})\cap J(\mathbb{Q}_{2})[2^{\prime}]=\\{0\\}$, where $X$ is
defined by (4) and $J(\mathbb{Q}_{2})[2^{\prime}]$ is the prime-to-$2$ torsion
subgroup of $J(\mathbb{Q}_{2})$.

###### Proof.

Completing the square on the left-hand side of (5), we obtain a short
Weierstrass equation for $C$ of the form $y^{2}=f(x)\vcentcolon=h(x)+1/4$,
thus realizing $C$ has a monic odd-degree hyperelliptic curve.

To verify $(a)$, note that the Newton polygon of $f$ is the line segment from
$(0,-2)$ to $(2g+1,0)$, which contains no lattice points, implying that $f$ is
irreducible over $\mathbb{Q}_{2}$. Consequently, we have
$J(\mathbb{Q}_{2})[2]=0$. Since the hyperelliptic involution on
$\mathcal{C}_{\mathbb{F}_{2}}$ is given by $(x,y)\mapsto(x,y+1)$, its only
fixed point is $P_{\infty}\in\mathcal{C}(\mathbb{F}_{2})$, implying that
$\mathcal{J}(\overline{\mathbb{F}}_{2})[2]=0$, hence
$\mathcal{J}(\mathbb{F}_{2})[2]=0$. The verification of $(b)$ is a
computation. To verify $(c)$, note that the restriction of the reduction map
$\operatorname{red}\colon
J(\mathbb{Q}_{2})[2^{\prime}]\rightarrow\mathcal{J}(\mathbb{F}_{2})$ is
injective, so it suffices to prove that
$\operatorname{red}(X(\mathbb{Q}_{2})\cap
J(\mathbb{Q}_{2})[2^{\prime}])=\\{0\\}$. Every element in
$\operatorname{red}(X(\mathbb{Q}_{2})\cap J(\mathbb{Q}_{2})[2^{\prime}])$ is
represented by a divisor of the form $P_{1}+P_{2}-2P_{\infty}$, where either
$P_{1},P_{2}\in\mathcal{C}(\mathbb{F}_{2})$, or $P_{1},P_{2}$ form a pair of
Galois conjugate points in $\mathcal{C}(\mathbb{F}_{4})$. In both cases, the
divisor class $[P_{1}+P_{2}-2P_{\infty}]$ is trivial by $(b)$. ∎

Lemma 7 allows us to choose a basis of differentials uniformly for all $C_{h}$
such that the logarithm map is surjective. If $h\in U_{g}$ and $C=C_{h}$ is
the hyperelliptic curve with equation (5), the change-of-variables
$s\vcentcolon=1/x$ and $t\vcentcolon=y/x^{g+1}$ applied to (5) yields the
equation

$t^{2}+s^{g+1}t=s^{2g+2}h(1/s).$ (6)

Define

$\displaystyle\omega_{1}$
$\displaystyle\vcentcolon=\frac{dt}{\frac{\partial}{\partial
s}\left(t^{2}+s^{g+1}t-s^{2g+2}h(1/s)\right)},$ (7)

and $\omega_{j}\vcentcolon=s^{j-1}\omega_{1}$ for each $j\in\\{2,\dots,g\\}$.
By the general theory of hyperelliptic curves, $\omega_{1},\dots,\omega_{g}$
is a $\mathbb{Z}_{2}$-basis of
$\mathrm{H}^{0}(\mathcal{C},\Omega^{1}_{\mathcal{C}/\mathbb{Z}_{2}})$. By
Lemma 7 in conjunction with [PS14, Lemma 10.1], the logarithm map $\log\colon
J(\mathbb{Q}_{2})\rightarrow\mathrm{H}^{0}(J,\Omega_{J}^{1})^{\vee}\simeq\mathrm{H}^{0}(C,\Omega^{1}_{C})^{\vee}\simeq\mathbb{Q}_{2}^{g}$
corresponding to this choice of basis satisfies
$\log(J(\mathbb{Q}_{2}))=(2\mathbb{Z}_{2})^{g}$. Using this identification, we
may consider the subset
$\rho\log(X(\mathbb{Q}_{2}))\subset\mathbb{P}^{g-1}(\mathbb{F}_{2})$ like we
did in §3.

######  Proposition 8.

As $h$ varies in $U_{g}$, the subset
$\rho\log(X(\mathbb{Q}_{2}))\subset\mathbb{P}^{g-1}(\mathbb{F}_{2})$ is
locally constant and satisfies

$\displaystyle\\#\rho\log(X(\mathbb{Q}_{2}))\leq 5.$ (8)

###### Proof.

Let $h\in U_{g}$ be good and $C=C_{h}$. Since
$\rho\log(X(\mathbb{Q}_{2}))=\rho\log(\mathrm{AJ}((\operatorname{Sym}^{2}C)(\mathbb{Q}_{2})))$,
it suffices to determine the latter set. Given a finite extension
$K/\mathbb{Q}_{2}$ with residue field $k$ and an element $x\in\mathcal{C}(k)$,
let $D_{x}(K)$ be the fiber of the reduction map
$C(K)\rightarrow\mathcal{C}(k)$. Similarly if
$x\in(\operatorname{Sym}^{2}\mathcal{C})(k)$, let $D_{x}(K)$ be the fiber of
the reduction map
$(\operatorname{Sym}^{2}C)(K)\rightarrow(\operatorname{Sym}^{2}\mathcal{C})(k)$
above $x$. Lemma 7$(b)$ shows that
$(\operatorname{Sym}^{2}\mathcal{C})(\mathbb{F}_{2})$ consists of $3$
elements, represented by the degree-$2$ divisors
$2P_{\infty},(0,\alpha)+(0,\alpha+1)$ and $(1,\alpha)+(1,\alpha+1)$. We bound
$\rho\log(X(\mathbb{Q}_{2}))$ by computing $\rho\log$ explicitly on each
residue polydisk $D_{x}(\mathbb{Q}_{2})$ for
$x\in(\operatorname{Sym}^{2}\mathcal{C})(\mathbb{F}_{2})$.

_Residue polydisk centered at $2P_{\infty}$._ Take a point $P+Q\in
D_{2P_{\infty}}(\mathbb{Q}_{2})$. There are two possibilities: either $P,\,Q$
are both defined over $\mathbb{Q}_{2}$, or $P,\,Q$ are conjugates over a
quadratic extension of $\mathbb{Q}_{2}$. Let $K$ denote the field of
definition of $P$, and let $\pi$ be a uniformizer for $K$. We use the model
(6) of $\mathcal{C}$. In that model, the point
$P_{\infty}\in\mathcal{C}(\mathbb{F}_{2})$ has coordinates $(s,t)=(0,0)$, and
the hyperelliptic involution $\iota\colon\mathcal{C}\rightarrow\mathcal{C}$
has the form $\iota(s,t)=(s,s^{g+1}-t)$. The coordinate
$t\vcentcolon=y/x^{g+1}$ is a uniformizer at $P_{\infty}$ and remains so when
reduced mod $\pi$, so the assignment $R\mapsto t(R)$ induces a bijection
$D_{P_{\infty}}(K)\simeq\pi\mathcal{O}_{K}$.

We now compute the restriction of the logarithm map to $D_{P_{\infty}}(K)$
explicitly as a $g$-tuple of power series in $t$. Using (6), we can expand $s$
as a power series in $t$ centered at $P_{\infty}$; this expansion has
$\mathbb{Z}_{2}$-coefficients and is of the form $s=t^{2}+2ct^{3}+\cdots$ for
some $c\in\mathbb{Z}_{2}$. Substituting this expansion of $s$ into (7), we
find that $\omega_{1}=(1+0t+0t^{2}+2c^{\prime}t^{3}+\cdots)dt$ for some
$c^{\prime}\in\mathbb{Z}_{2}$, and hence that
$\omega_{j}=(t^{2j-2}+2c(j-1)t^{2j-1}+\cdots)dt$ for each
$j\in\\{1,\dots,g\\}$. By the theory of Coleman integration [Col85, §II], if
$R\in D_{P_{\infty}}(K)$ has coordinate $t\in\pi\mathcal{O}_{K}$, then
$\log(R-P_{\infty})=\big{(}\int_{0}^{t}\omega_{1},\dots,\int_{0}^{t}\omega_{g}\big{)}$.

Denote the coordinates of the points $P,Q,\iota(Q)\in D_{P_{\infty}}(K)$ by
$t_{1},t_{2},t_{3}\in\pi\mathcal{O}_{K}$ respectively. Since
$[Q+\iota(Q)-2P_{\infty}]=0$ in $J(\mathbb{Q}_{2})$, we have
$\log(P+Q-2P_{\infty})=\log(P-P_{\infty})-\log(\iota(Q)-P_{\infty})$, hence

$\log(P+Q-2P_{\infty})=\left((t_{1}-t_{3})+\cdots,\frac{t_{1}^{3}-t_{3}^{3}}{3}+\cdots,\dots,\frac{t_{1}^{2g-1}-t_{3}^{2g-1}}{2g-1}+\cdots\right).$
(9)

Dividing each component on the right-hand side of (9) by $t_{1}-t_{3}$, we
deduce that $\rho\log(P+Q-2P_{\infty})$ is given by $\rho(1+\cdots,\dots)$;
here, the ellipses are used to abbreviate terms of the form

$\frac{c^{\prime\prime}}{i+1}\cdot\big{(}t_{1}^{i}+t_{1}^{i-1}t_{3}+\cdots+t_{1}t_{3}^{i-1}+t_{3}^{i-1}\big{)},$
(10)

where $c^{\prime\prime}\in\mathbb{Z}_{2}$ for all $i\geq 1$ and
$c^{\prime\prime}\in 2\mathbb{Z}_{2}$ when $i\in\\{1,3\\}$. For
$t_{1},\,t_{3}\in\pi\mathcal{O}_{K}$, the valuation of the expression (10) is
always positive. We conclude that the map $\rho\log$ is constant on
$D_{2P_{\infty}}(\mathbb{Q}_{2})$, with value $(1:0:\cdots:0)$.

_Residue polydisk above $(0,\alpha)+(0,\alpha+1)$._ In this case, we use the
model (5). Let $K\vcentcolon=\mathbb{Q}_{4}$ be the unramified quadratic
extension of $\mathbb{Q}_{2}$. Any point in
$D_{(0,\alpha)+(0,\alpha+1)}(\mathbb{Q}_{2})$ is of the form $P+\overline{P}$,
where $P\in C(K)$ reduces to $(0,\alpha)$ and $\overline{P}$ is the
$\operatorname{Gal}(K/\mathbb{Q}_{2})$-conjugate of $P$. Let
$\gamma\in\mathcal{O}_{K}$ be an element with $\gamma^{2}+\gamma=h(0)$; such
an element exists by Hensel’s lemma and reduces to $\alpha\in\mathbb{F}_{4}$.
Let $P_{0}\vcentcolon=(0,\gamma)\in C(K)$. Then the
$\operatorname{Gal}(K/\mathbb{Q}_{2})$-conjugate
$\overline{P}_{0}=(0,-1-\gamma)\in C(K)$ is in fact the image of $P_{0}$ under
the hyperelliptic involution $\iota$. The coordinate $x$ is a uniformizer at
both $P_{0}$ and $\overline{P}_{0}$, giving identifications
$D_{(0,\alpha)}(K)\simeq 2\mathcal{O}_{K}$ and $D_{(0,\alpha+1)}(K)\simeq
2\mathcal{O}_{K}$.

By the general theory of hyperelliptic curves, the elements

$\eta_{1}\vcentcolon=\frac{dx}{2y+1}=\frac{dy}{-h^{\prime}(x)},$ (11)

and $\eta_{j}\vcentcolon=x^{j-1}\eta_{1}$ for $j\in\\{2,\dots,g\\}$ form a
$\mathbb{Z}_{2}$-basis of
$\mathrm{H}^{0}(\mathcal{C},\Omega^{1}_{\mathcal{C}/\mathbb{Z}_{2}}\big{)}$.
Using (5), we can expand $y$ as a power series in $x$ on $D_{(0,\alpha)}(K)$
centered at $P_{0}$; this expansion has $\mathcal{O}_{K}$-coefficients and
constant term $\gamma$. Substituting this expansion of $y$ into (11), we find
a power series expansion of $\eta_{1}/dx$ with $\mathcal{O}_{K}$-coefficients
and constant coefficient $(2\gamma+1)\in\mathcal{O}_{K}^{\times}$. Now if
$R\in D_{(0,\alpha)}(K)$ corresponds to $x\in 2\mathcal{O}_{K}$, then
$\phi(x)\vcentcolon=\log(P-P_{\infty})=\log(P_{0}-P_{\infty})+\big{(}\int_{0}^{x}\eta_{1},\dots,\int_{0}^{x}\eta_{g}\big{)}$.

Let the points $P\in D_{(0,\alpha)}(K)$ and $\overline{P}\in
D_{(0,\alpha+1)}(K)$ have coordinates $x_{1},x_{2}\in 2\mathcal{O}_{K}$
respectively. Then $x_{2}$ is the
$\operatorname{Gal}(K/\mathbb{Q}_{2})$-conjugate of $x_{1}$, and the
hyperelliptic conjugate $\iota(\overline{P})\in D_{(0,\alpha)}(K)$ has
coordinate $x_{2}$. Since the hyperelliptic involution acts as $-1$ on the
Jacobian, we have
$\log(P+\overline{P}-2P_{\infty})=\log(P-P_{\infty})-\log(\iota(\overline{P})-P_{\infty})=\phi(x_{1})-\phi(x_{2})$,
hence

$\log(P+\overline{P}-2P_{\infty})=\left(\int_{0}^{x_{1}}\eta_{1}-\int_{0}^{x_{2}}\eta_{1},\dots,\int_{0}^{x_{1}}\eta_{g}-\int_{0}^{x_{2}}\eta_{g}\right).$
(12)

Each difference of integrals in the $g$-tuple on the right-hand side of (12)
is a sum of terms of the form

$\frac{c}{i+1}\cdot(x_{1}^{i+1}-x_{2}^{i+1})$ (13)

where $c\in\mathbb{Z}_{2}$. For each $j\in\\{1,\dots,g\\}$, the smallest value
of $i$ occurring in the $j^{\mathrm{th}}$ component is exactly $j-1$, in which
case $c=2\gamma+1$ lies in $\mathbb{Z}_{2}^{\times}$. Dividing each component
on the right-hand side of (12) by $(2\gamma+1)\cdot(x_{1}-x_{2})$, we deduce
that
$\rho\log(P+\overline{P}-2P_{\infty})=\rho(1+\cdots,(x_{1}+x_{2})/2+\cdots,\cdots)$,
where all the power series in the latter $g$-tuple are sums of terms of the
form

$\frac{c}{i+1}\cdot(x_{1}^{i}+x_{1}^{i-1}x_{2}+\cdots+x_{1}x_{2}^{i-1}+x_{2}^{i}),$
(14)

and where once again, $c\in\mathbb{Z}_{2}$ and for each $j\in\\{1,\dots,g\\}$,
the smallest value of $i$ occurring in the $j^{\mathrm{th}}$ component is
$j-1$. For $x_{1},\,x_{2}\in 2\mathcal{O}_{K}$, the valuation of the
expression (14) is positive so long as $i\geq 2$. Therefore
$\rho\log(P+\overline{P}-2P_{\infty})=\rho(1,(x_{1}+x_{2})/2,0,\cdots,0)$. We
conclude that, in the basis $\eta_{1},\dots,\eta_{g}$, we have
$\rho\log(D_{(0,\alpha)+(0,\alpha+1)}(\mathbb{Q}_{2}))=\\{(1:0:0:\cdots:0),(1:1:0:\cdots:0)\\}$.

_Residue polydisk above $(1,\alpha)+(1,\alpha+1)$._ Under the transformation
$x\mapsto x-1$, the conditions on $h$ in the proposition are preserved. Thus,
the analysis for this case is identical to that of the previous case.

Putting the three cases together, we obtain
$\\#\rho\log(X(\mathbb{Q}_{2}))\leq 1+2+2=5$. To show that
$\rho\log(X(\mathbb{Q}_{2}))\subset\mathbb{P}^{g-1}(\mathbb{F}_{2})$ is
locally constant as $h$ varies in $U_{g}$, it suffices to show this on each
residue polydisk separately. This is obvious for
$D_{2P_{\infty}}(\mathbb{Q}_{2})$. For the residue disk
$D_{(0,\alpha)+(0,\alpha+1)}(\mathbb{Q}_{2})$, there exists a matrix
$A_{h}\in\operatorname{GL}_{g}(\mathbb{Z}_{2})$ that transforms the basis
$\eta_{1},\dots,\eta_{g}$ of
$\mathrm{H}^{0}(\mathcal{C},\Omega^{1}_{\mathcal{C}/\mathbb{Z}_{2}})$ into
$\omega_{1},\dots,\omega_{g}$. Since the association $h\mapsto A_{h}$ is
continuous and $\rho\log(D_{(0,\alpha)+(0,\alpha+1)}(\mathbb{Q}_{2}))$ is
constant with respect to the basis $\eta_{1},\dots,\eta_{g}$,
$\rho\log(D_{(0,\alpha)+(0,\alpha+1)}(\mathbb{Q}_{2}))$ is locally constant
with respect to the basis $\omega_{1},\dots,\omega_{g}$ as $h\in U_{g}$
varies. The argument for $D_{(1,\alpha)+(1,\alpha+1)}(\mathbb{Q}_{2})$ is
analogous. ∎

Let $\mathcal{F}_{g}^{(2)}\subset\mathcal{F}_{g}$ be the subset of polynomials
$f$ of the form $4^{2g+1}(h(x/4)+1/4)$ for some $h\in U_{g}\cap\mathbb{Z}[x]$.
Completing the square shows that monic odd hyperelliptic curves with equation
$y^{2}=f(x)$ with $f\in\mathcal{F}_{g}^{(2)}$ are isomorphic to hyperelliptic
curves with of the form (5) for some good $h\in U_{g}$.

######  Lemma 9.

The subfamily $\mathcal{F}_{g}^{(2)}\subset\mathcal{F}_{g}$ is defined by
congruence conditions modulo $2^{4g+2}$ and has density $2^{-4g^{2}-6g-5}$.

###### Proof.

Equip the parameter space $P_{g}=\mathbb{Z}_{2}^{2g+1}$ of all monic
polynomials in $\mathbb{Z}_{2}[x]$ of degree $2g+1$ with the probability Haar
measure. A combinatorial calculation shows that the open subset of good
polynomials $U_{g}\subset P_{g}$ has measure $1/8$. On the other hand, the
$\mathbb{Z}_{2}$-linear map $h\mapsto 4^{2g+1}h(x/4)$ on $P_{g}$ has
determinant $4=4^{2g^{2}+3g+1}$, so the subset of $f\in P_{g}$ of curves of
the form $4^{2g+1}h(x/4)+4^{2g}$ for some $h\in P_{g}$ has measure
$4^{-2g^{2}-3g-1}$. We conclude that the measure of the subset
$\\{4^{2g+1}h(x/4)+4^{2g}\mid h\in U_{g}\\}\subset P_{g}$ is $(1/8)\times
4^{-2g^{2}-3g-1}=2^{-4g^{2}-6g-5}$, so the density of $\mathcal{F}_{g}^{(2)}$
is also $2^{-4g^{2}-6g-5}$. ∎

##  5 Proof of Theorem 1

The proof of Theorem 1 is completed by the next proposition. Just as in
[PS14], a crucial ingredient in the proof is an equidistribution result of
Bhargava–Gross [BG13, Theorem 12.4]. If $S\subset T\subset\mathcal{F}_{g}$ are
subsets, the relative density of $S$ in $T$ is by definition the quantity
$\lim_{X\rightarrow\infty}\\#(S\cap\mathcal{F}_{g,X})/\\#(T\cap\mathcal{F}_{g,X})$,
whenever it exists. Define the relative lower and upper density of $S$ in $T$
by replacing the limit by $\liminf$ and $\limsup$, respectively.

######  Proposition 10.

The relative lower density of the set

$\displaystyle\\{C\in\mathcal{F}_{g}^{(2)}\mid C\text{ has no unexpected
rational or quadratic points}\\}$ (15)

in $\mathcal{F}_{g}^{(2)}$ is at least $1-6\times 2^{1-g}$. Consequently, if
$g\geq 4$, a positive proportion of curves in $\mathcal{F}_{g}$ have no
unexpected rational or quadratic points.

###### Proof.

This is a standard adaptation of [PS14, Proposition 8.13] to our setting.
Proposition 8 shows that $\mathcal{F}_{g}^{(2)}$ is partitioned into finitely
many subsets $\mathcal{G}_{1},\dots,\mathcal{G}_{m}$, each defined by
congruence conditions modulo a power of $2$, such that the subset
$\rho\log(X(\mathbb{Q}_{2}))\subset\mathbb{P}^{g-1}(\mathbb{F}_{2})$ is
constant when $h$ varies in $\mathcal{G}_{i}$. Fix such an
$i\in\\{1,\dots,m\\}$. If $C\in\mathcal{G}_{i}$, then $J(\mathbb{Q}_{2})[2]=0$
by Lemma 7$(a)$, so the logarithm map defined using the basis (7) induces an
isomorphism $J(\mathbb{Q}_{2})/2J(\mathbb{Q}_{2})\simeq\mathbb{F}_{2}^{g}$.
Write $\sigma\colon\operatorname{Sel}_{2}J\rightarrow\mathbb{F}_{2}^{g}$ for
the corresponding composition $\operatorname{Sel}_{2}J\rightarrow
J(\mathbb{Q}_{2})/2J(\mathbb{Q}_{2})\rightarrow\mathbb{F}_{2}^{g}$ considered
in §3. Bhargava–Gross have shown [BG13, Theorem 12.4] that nontrivial elements
of $\operatorname{Sel}_{2}J$ equidistribute under $\sigma$: when $C$ varies in
$\mathcal{G}_{i}$ and $I\subset\mathbb{F}_{2}^{g}$ is a subset, they show that
the average size of
$\\{s\in\operatorname{Sel}_{2}J\setminus\\{0\\}\colon\sigma(s)\in I\\}$ is at
most $\\#I\times 2^{1-g}$. It follows that the relative upper density of
curves $C\in\mathcal{G}_{i}$ such that
$\\{s\in\operatorname{Sel}_{2}J\setminus\\{0\\}\colon\sigma(s)\in I\\}$ is
nonempty is at most $\\#I\times 2^{1-g}$. Applying this to
$I=\mathbb{P}^{-1}(\rho\log(X(\mathbb{Q}_{2}))\cup\\{0\\})$ and using
Proposition 8, we see that the relative lower density of curves
$C\in\mathcal{G}_{i}$ such that the first two conditions of Corollary 5 are
satisfied is at least $1-\\#I\times 2^{1-g}\geq 1-6\times 2^{1-g}$. The third
condition of Corollary 5 is always satisfied for $C\in\mathcal{F}_{g}^{(2)}$
by Lemma 7$(c)$. We conclude that the intersection of the set (15) with
$\mathcal{G}_{i}$ has relative lower density at least $1-6\times 2^{1-g}$ in
$\mathcal{G}_{i}$. Summing over $i$, the set (15) itself has relative lower
density at least $1-6\times 2^{1-g}$ in $\mathcal{F}_{g}^{(2)}$.

Thus, the lower density of curves in $\mathcal{F}_{g}$ that have no unexpected
rational or quadratic points is at least $(1-6\times
2^{1-g})\times(\text{density of }\mathcal{F}_{g}^{(2)})=(1-6\times
2^{1-g})\times 2^{-4g^{2}-6g-5}$ by Lemma 9. This is positive when $g\geq 4$.
∎

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# Monitoring multidimensional phenomena with a multicriteria composite
performance interval approach

Ana Garcia-Bernabeu
Department of Economy and Social Science
Universitat Politècncia de València, Campus d’Alcoi
Spain
<EMAIL_ADDRESS>
&Adolfo Hilario-Caballero
Institute of Control Systems and Industrial Computing (ai2)
Universitat Politècnica de Valencia
Spain
<EMAIL_ADDRESS>
###### Abstract

In the last two decades, composite indicators’ construction to measure and
compare multidimensional phenomena in a broad spectrum of domains has
increased considerably. Different methodological approaches are used to
summarize huge data sets of information in a single figure. This paper
proposes a new approach that consists of computing a multicriteria composite
performance interval based on different aggregation rules. The suggested
approach provides an additional layer of information as the performance
interval displays a lower bound from a non-compensability perspective, and an
upper bound allowing for full-compensability. The outstanding features of this
proposal are: (i) a distance-based multicriteria technique is taken as the
baseline to construct the multicriteria performance interval (ii) the
aggregation of distances/separation measures is made using particular cases of
Minkowski’s $L_{p}$ metrics; (iii) the span of the multicriteria performance
interval can be considered as a sign of the dimensions or indicators balance.

_K_ eywords Composite indicators $\cdot$ Multicriteria decision making $\cdot$
Compensability $\cdot$ Perfomance Interval

##  1 Introduction

Composite indicators provide a one-dimensional metric to assess, monitor and
predict the performance of complex phenomena approached from a
multidimensional perspective such as human development, sustainability,
innovation, or well being. According to [1], a composite index involves the
combination of a set of indicators that represents the different dimensions of
the phenomenon to be measured. Despite the fact that there is no single
commonly accepted definition of a composite indicator, there is a common
pattern in which the starting point is a complex phenomenon that includes
different components that are to be compiled into a single indicator [2].
Although the proliferation of composite indicators has grown exponentially
over the last decade, there is no agreed consensus on an international
standard for their construction.

The first attempt to establish a common guideline to construct a composite
indicator targeted at policy-makers, academics, the media and other interested
parties was the “Handbook of constructing composite indicators" [3]. In this
manual, a ten-step process from the development of a theoretical framework to
the presentation and dissemination of a composite indicator is proposed to
enhance the transparency and the soundness of the selected methodology. Main
concerns around composite indicators are related with the lack of quality
control and training for users, which can lead to send misleading and non-
robust messages. Due to their relevance in policy-making decisions, the
construction of composite indicators seems to be an important research issue
from both theoretical and operational points of view [4].

One of the most frequent criticism about construction of a composite indicator
appears in the aggregation stage because this step defines the tool to add the
criteria. The debate about the compensability, that appear when a deficit in
one dimension can be offset by a surplus in another, supports the development
of two groups: aggregators versus non-aggregators. Basically, an aggregation
approach can be compensatory or non-compensatory depending on whether it
permits compensability or not among indicators or dimensions [5]. While the
compensatory techniques deal with the imbalances of indicators and employ
linear functions, unbalance-adjusted functions are used in non-compensatory
approaches.

Over the last few years, in multidimensional frameworks, when highly different
dimensions should be aggregated, Multi-Criteria Decision Making (MCDM)
methodologies have been claimed as highly suitable alternatives for
constructing composite indicators [6]. Mainly, these methodologies have been
widely used in selecting the weights and in the aggregation stage. As to the
multicriteria methods proposed to derive weights, we observe a high number of
contributions applying data based methods such as Data Envelopment Analysis
(DEA, [7]) as well as participatory based methods such as the Analytical
Hierarchy Process (AHP, [8]). Regarding the aggregation stage, a significant
group of MCDM methodologies adopt distance-based methods to construct
composite indicators.

This work aims to design a Multicriteria Composite Performance Interval (MCPI)
instead of a single composite indicator looking at one of their technical
weaknesses: the aggregation rule used for their construction. In our approach,
we opted for the TOPSIS multicriteria tool, which is based on additive
aggregation functions representing the distances to ideal and anti-ideal
values. The proposal’s novelty is to provide two indicators depending on the
choice of the type of aggregation for the separation measures. The proposed
methodology’s added value implies extending the information provided by the
classical relative clossenest to the ideal solution score. Thus, the MCPI is
composed of a lower and upper bound. The lower bound corresponds to non-
compensability perspective, whereas the upper bound involves a full
compensatory approach. As an example of implementation and to demonstrate the
advantages of using the proposed MCPI, it has been applied to measure the
circular economy performance of EU countries on the basis of the structure of
indicators provided by the EU monitoring framework. The proposed Circular
Economy MCPI is developed to paint a comprehensive picture of the circular
economy performance in 28 European countries and provide warning signals to
policymakers on the areas where the dimensions need further improvements.

The paper is organized as follows. Section 2 reviews the compensability debate
in constructing composite indicators and how some MCDM techniques have been
applied to overcome this problem. In Section 3, the MCPI methodology is
developed. In Section 4, an application of the proposed method to construct a
Circular Economy MCPI is presented. Section 5 provides the conclusions as well
as future lines of research.

##  2 The compensability debate and MCDM approaches

The selection of the aggregation procedure is one of the most discussed issues
when constructing a composite indicator. Based on the literature review of
composite indicators, the most widely applied aggregation procedures are
linear aggregation rules implying complete substitutability among the various
components considered. However, their applicability depends on several strong
theoretical and operational assumptions. The first one is the assumption of
preferential independence, which in practice becomes very difficult to
accomplish. The second one is related to the meaning of weights, which are
viewed as substitution rates instead of important coefficients. Despite these
drawbacks, linear aggregation rules are a very intuitive and easy-to-use
option.

A deep understanding of different aggregation rules in the framework of
constructing composite indicators is provided in [4] and [9]. The authors
revised the debate on the use of aggregation rules by describing their
relative pros and cons. Finally, they concluded that the use of non-
linear/non-compensatory aggregation rules is advisable for reasons of
theoretical consistency when weights are interpreted as importance
coefficients or when the assumption of preferential independence does not
hold. Under some conditions, these authors pointed out the benefits of using
multicriteria aggregation procedures. While the compensatory techniques deal
with the imbalances of indicators and employ linear functions, unbalance-
adjusted functions are used in non-compensatory approaches.

According to [5], in non-compensatory approaches the dimensions to be measured
must be balanced and require the use of non-linear aggregation functions such
as the geometric mean or the minimum to penalize unbalance . The Mazziota
Pareto Index (MPI) introduced in 2007 [10] is one of the first proposals to
address the compensability issue. Later, the same authors develop a newer
variant of the previous method for spatio temporal comparisons known as
Adjusted Mazziota Pareto Index (AMPI). In an attempt to overcome the problem
of calculating a single figure, these authors have recently proposed a
performance interval depending on the level of compensability of individual
indicators [11]. It should be noted that the previous work has inspired our
proposal. However, while it relies on the power mean of order $r$ to deal with
the compensability issue, we have introduced the use of performance intervals
in the field of multicriteria decision making (MCDM) methodologies.

Also in the literature of composite indicators, several authors claim that
MCDM techniques are well suited for aggregating single indicators in a
composite one in multidimensional frameworks (see [1, 3]). An overview of MCDM
methodologies used to construct composite indicators was presented in [6]. On
looking at the analysis made in the previous research, it can be remarkable
the use of distance-based methodologies. Within this category, the reference
point method [12] was adapted by [13] to develop synthetic sustainability
indicators dealing with different compensation degrees between the criteria
and it has recently applied to construct composite indicators in a wide
variety of domains ([14],[15]). These methods allow the decision-maker to
define reference levels and reduce all indicators to a common scale using the
so-called achievement scalarizing functions.

Other outranking multicriteria methodology where a full compensability
prevails is the “Technique for Ranking Preferences by Similarity to Ideal
Solutions" TOPSIS initially introduced by [16]. This technique uses a
compensatory aggregation rule of geometric distances to ideal and anti-ideal
values. Some examples applying TOPSIS to construct composite indicators can be
found in [17],[18], [19], [20].

On the other hand, outranking related approaches such as ELECTRE (Elimination
and Choice Expressing Reality, [21]) or PROMETHEE (Preference Ranking
Organization Method for Enrichment Evaluation, [22]) can avoid compensation
thanks to the presence of veto threshold and ordinal comparison among
alternatives. Some examples of application can be found in [23] to construct a
Land-Use Policy Efficiency Index, or [24] to assess regional tourism
competitiveness.

In synthesis, despite the studies mentioned above, the debate on the
compensability issue is continuing. Some recent studies emphasised the need to
overcome linear aggregation rules in MCDM techniques by incorporating non-
compensatory aggregation functions. However, no previous work has dealt with
integrating both perspectives by providing a multicriteria composite
performance interval for different compensability degrees. Thus, by looking at
the upper bound of the MCPI proposed in this paper, we can get a big picture
of the phenomenon under the compensability logic. In contrast, by looking at
the lower bound, we can track the alternatives’ specific weakness to be
measured as it provides the worst performance by dimension or by indicator.

##  3 Methodology: Constructing the multicriteria composite performance
interval (MCPI)

In this paper, the TOPSIS multicriteria technique is proposed as a starting
point to tackle the construction of a the MCPI. The main feature of the
proposed methodology is the choice of different compensability levels in the
distance aggregation functions. In this way, the alternatives are classified
according to an index that considers the “shortest distance" from the ideal
solution and the “farthest distance" from the “ideal-negative" solution.

In general, TOPSIS [25] is implemented by the following stepwise procedure.
After defining the decision matrix, including alternatives and criteria, the
following step consists of normalizing the data. This is followed by computing
the weighted normalized decision matrix. Next, the positive and negative ideal
solutions are identified to derive the separation measures. Thus, the
aggregation is made for each alternative’s separation measures. Finally, the
procedure ends by computing the relative closeness coefficient. The set of
alternatives can be ranked according to the descending order of the closeness
coefficient. In what follows, we extend the traditional formulation of this
methodology, which in the earliest stages coincides with the steps for
building a composite indicator to derive the proposed MCPI.

###  3.1 Theoretical framework and data issues

In our approach we consider a given theoretical framework from which the
multidimensional phenomenon to be measured can be described in terms of
alternatives, dimensions and indicators. From it, the data are organized in a
decision matrix $X=x_{ij}$ ($i=1,\ldots,n$; $j=1,\ldots m$) where $n$ denotes
the number of alternatives, and $m$ the number of indicators.

A previous data analysis should be taken to study the overall structure of the
data set. At this stage the imputation of missing data and the application of
multivariate analysis techniques are of great importance.

To allow comparability, the normalization of single indicators is a prior an
necessary step as they are often expressed in different units of measurement.
Starting from the initial data decision matrix, the normalized matrix
$N=n_{ij}$ is constructed, where the normalized value of each indicator is
obtained by applying the following transformation:

$n_{ij}=\frac{x_{ij}}{\sqrt{\sum\limits_{i=1}^{n}(x_{ij})^{2}}}\quad(i=1,\ldots,n;\;j=1,\ldots,m)$
(1)

###  3.2 Weighting and distances to ideal and anti-ideal values

Once the normalized matrix is obtained, the next step is the choice of weights
and the aggregation method. The most common weighting and aggregation
techniques rely on the simple arithmetic mean, which involves assigning equal
weights to all indicators. There exist a rich menu of alternative weighting
methods. Some of them are based on statistics such as Principal Component
Analysis (PCA), Factor Analysis (FA). In other cases, subjective opinions from
experts, citizens or politicians are taking into account to derive the weights
through participatory methodologies such as the Benefit of the Doubt (BOD) or
the Analytical Hierarchy Process (AHP) [26].

As a starting point in our proposal, we have opted for the assignment of equal
weights to build the weighted normalized matrix $V=v_{ij}$ as follows:

$v_{ij}=\omega_{j}\,n_{ij}$ (2)

where $\omega^{T}=\omega_{j}=[\omega_{1},\omega_{2},\ldots,\omega_{m}]$, with
$\sum\omega_{j}=1$.

The positive and negative ideal solutions or the ideal and anti-ideal values,
respectively are:

$\displaystyle V^{+}=v_{j}^{+}$
$\displaystyle=\left\\{\begin{array}[]{l}\displaystyle\max_{i}{v_{ij}\quad\forall
j\in J_{p}}\\\\[8.61108pt] \displaystyle\min_{i}{v_{ij}\quad\forall j\in
J_{n}}\end{array}\right.$ $\displaystyle V^{-}=v_{j}^{-}$
$\displaystyle=\left\\{\begin{array}[]{l}\displaystyle\min_{i}{v_{ij}\quad\forall
j\in J_{p}}\\\\[8.61108pt] \displaystyle\max_{i}{v_{ij}\quad\forall j\in
J_{n}}\end{array}\right.$

where $J_{p}$ is associated to benefit criteria and $J_{n}$ is associated to
cost criteria. Therefore, the criteria separation measure to the ideal is
computed for each criteria:

Notice that originally TOPSIS refer to the Euclidean distance to obtain the
separation measures to the ideal and anti-ideal values. In fact, the Euclidean
distance is a special case of Minkowski’s $L_{p}$ metric in an $n$-dimensional
space (see [27]).

$L_{p}(x,y)=\left[\sum_{j=1}^{m}\lvert
x_{j}-y_{j}\rvert^{p}\right]^{\textstyle\frac{1}{p}}$ (3)

where $p\geq 1$. For $p=1$, we have the Manhattan distance involving and
additive function and full compensability. When $p=2$ the Euclidean distance
is obtained, and finally $p=\infty$ refers to the Tchebycheff distance which
implies a non-compensatory approach.

###  3.3 Defining the Multicriteria Composite Performance Interval

At this stage, and to discuss how compensability among indicators can affect
the overall ranking of alternatives, we propose to construct an interval
instead of computing a single composite indicator based on the closeness to
ideal solution index. The MCPI generates a lower and upper bound for the
composite index.

  * •

Lower bound: The strong closeness to the ideal solution $C^{S}_{i}$. It
involves a non-compensability choice in the aggregation stage of the
separation measures. In this case an unbalance among indicators will have a
negative effect on the value of the composite index. When considering the
separation measure to the ideal values, under strong perspective a penalty
should be given to those criteria involving a maximum distance to the ideal
and a minimum distance to the anti-ideal. In this case, we use the
Tchebycheff’s distances assuming a value of $p=+\infty$ to aggregate the
positive distances and a value of $p=-\infty$ for negative distances
respectively.

$\displaystyle C^{S+}_{ik}$
$\displaystyle=L_{+\infty}(v_{ij},v^{+}_{j})=\max_{j\in J_{k}}\,\lvert
v_{ij}-v^{+}_{j}\rvert$ (4) $\displaystyle C^{S-}_{ik}$
$\displaystyle=L_{-\infty}(v_{ij},v^{+}_{j})=\min_{j\in J_{k}}\,\lvert
v_{ij}-v^{-}_{j}\rvert$

where $J_{k}$ denotes the set of criteria belonging to dimension $k$, and
$l_{k}$ is the number of criteria included.

Finally, we derive for each alternative the strong closeness to the ideal
solution $C^{S*}_{ik}$ as follows:

$C^{S}_{ik}=\frac{C^{S-}_{ik}}{C^{S+}_{ik}+C^{S-}_{ik}}$ (5)

where $0\leq C^{S}_{ik}\leq 1$

  * •

Upper bound: The weak closeness to the ideal solution $C^{W}_{i}$. It
corresponds to the upper bound of the multicriteria performance interval
involving a full compensability in the aggregation stage of the separation
measure. In this case an unbalance among indicators has any effect on the
value of the composite index. In the $L_{p}$ metric, now we take the Manhattan
distance with $p=1$, which corresponds to the arithmetic mean of the
separation measures.

$\displaystyle C^{W+}_{ik}$
$\displaystyle=L_{1}(v_{ij},v^{+}_{j})=\frac{1}{l_{k}}\sum_{j\in J_{k}}\lvert
v_{ij}-v^{+}_{j}\rvert$ (6) $\displaystyle C^{W-}_{ik}$
$\displaystyle=L_{1}(v_{ij},v^{-}_{j})=\frac{1}{l_{k}}\sum_{j\in J_{k}}\lvert
v_{ij}-v^{-}_{j}\rvert$

Analogously, we derive the weak closeness to the ideal solution $C^{W*}_{ik}$
as follows:

$C^{W}_{ik}=\frac{C^{W-}_{ik}}{C^{W+}_{ik}+C^{W-}_{ik}}$ (7)

Thus, the MCPI for each dimension, $\delta_{ik}$, takes the following form:

$\delta_{ik}=\left[C^{S}_{ik},C^{W}_{ik}\right]$ (8)

Once, the aggregation rule is made for the first level, namely, from
indicators to dimensions, the modeller has several options to derive the
overall MCPI when the dimensions should be aggregated. In our case, we propose
a lower bound which takes the minimum value of the dimensions, and a upper
bound corresponding to the mean of the weak closeness to the ideal solution.

$\gamma_{i}=\left[\min_{k}{C^{W}_{ik}},\frac{1}{l}\sum_{k=1}^{l}{C^{W}_{ik}}\right]$
(9)

The question now is, how the user of the MCPI should rank the alternatives?
The answer is not trivial and depends on the subject under study and the
modeller’s objectives. A quite sound option is to rank the alternative by the
value of the upper limit, but looking at the span of the MCPI, which can be
considered as a sign of the unbalanced. For example, looking at the
$\delta_{ik}$, the greater the span of the interval, the greater the imbalance
of the single indicators, whereas the unbalance between dimensions comes from
the length of the $\gamma_{i}$.

Therefore, policy-makers setting targets to improve a given multidimensional
phenomenon could prioritise which alternative and in which dimensions to act
on by looking at the span of the MCPI. Based on the balance property of the
MCPI, the alternatives are assigned a Balance Rating ($\beta$–Rating)
according to the following scale.

Table 1: MCPI interval balance rating Category span |  $\beta$–Rating icon
---|---
High balance MCPI | $\star\star\star$
Medium balance MCPI | $\star\star$
Low balance MCPI | $\star$

  * •

Alternatives with a MCPI span ranging from $0\,\%$ to $33.33\,\%$ of maximum
span can receive a high $\beta$–Rating icon (3 stars).

  * •

Alternatives with a MCPI span ranging from $33.33\,\%$ to $66.67\,\%$ of
maximum span can receive a medium $\beta$–Rating icon (2 stars).

  * •

Alternatives with a MCPI span ranging above the $66.67\,\%$ of maximum span
can receive a low $\beta$–Rating icon (1 star).

##  4 Case study: the construction of the Circular Economy- MCPI

This section applies the MCPI approach following a set of Circular Economic
indicators with data extracted from the EU Circular Economy monitoring
framework [28]. Since 2018, and in line with the highlighted relevance of
sustainability concerns in Europe, a monitoring framework to assess Circular
Economic issues’ performance was presented by the European Commission. This is
an instrument for monitoring key trends in the transition towards a more
circular economy model in Europe, which makes it possible to assess whether
the measures put in place and the involvement of all stakeholders have been
sufficiently effective and identify best practices in the Member States. An
initial proposal to provide a composite indicator for EU member states by
aggregating a different set of indicators was presented in [29].

###  4.1 Circular Economy monitoring framework, indicators and data

The circularity assessment of EU countries is accomplished according to the
Circular Economy monitoring framework, which departs from ten key indicators
including other sub-indicators and grouped in the following four broad
dimensions: production and consumption, waste management, secondary raw
materials, and competitiveness and innovation. The indicators’ data come from
Eurostat, the Joint Research Centre, and the European Patent Office. To
evaluate EU countries’ progress towards circular economy, the information is
disseminated throughout tables and graphs for cross-national comparison in the
following web page https://ec.europsa.eu/eurostat/web/circular-
economy/indicators/monitoring-framework.

From this monitoring framework, we select ten sub-indicators grouped into the
four dimensions and aspects of the circular economy. A descriptive analysis of
the selected indicators is provided in Table 2. In our application, a total of
280 observations are available for the 28 EU countries and ten indicators. The
indicators with the highest averages relate to the amount of waste per capita
measured in kilograms, waste per unit of GDP and trade in secondary raw
materials measured in tonnes. It can be seen that the data, when considering
all 28 countries, show little homogeneity, as indicated by the kurtosis
coefficient, especially in the mean of the amount of waste per unit of GDP,
the number of patents and the contribution of recycled raw materials to the
demand for raw materials. This result is indicative of the heterogeneity of
the data set.

Table 2: Selected indicators of Circular Economy and data issues. Source
European Commission Monitoring Framework (2018)

|  Dimension/ Indicator |  Data Source | Mean | st | Max | Min | Kurtosis
---|---|---|---|---|---|---|---
|  Production and consumption (PC) |  |  |  |  |  |
1a |  Generation of municipal waste per capita (Kg per capita) |  European Statistical System | 497.64 | 128.24 | 814.00 | 272.00 | 0.89
1b |  Generation of waste excluding major mineral wastes per GDP unit (Kg per thousand euro, chain linked volumes (2010)) |  European Statistical System | 105.21 | 133.90 | 646.00 | 27.00 | 11.61
1c |  Generation of waste excluding major mineral wastes per domestic material consumption (percentage) |  European Statistical System | 12.51 | 7.47 | 30.50 | 4.80 | 0.77
|  Waste Management (WM) |  |  |  |  |  |
2a |  Recycling rate of municipal waste (percentage) |  European Statistical System | 38.69 | 14.81 | 67.10 | 10.00 | -0.57
2b |  Recycling rate of all waste excluding major mineral waste (percentage) |  European Statistical System | 49.43 | 18.12 | 80.00 | 10.00 | -0.06
|  Secondary Raw Materials (SRM) |  |  |  |  |  |
3a |  Contribution of recycled material to raw materials demand |  European Statistical System | 9.26 | 6.75 | 29.00 | 1.50 | 1.43
3b |  Trade in recyclable raw materials (tonnes) |  European Statistical System | 3437406.64 | 4323147.25 | 14100540.00 | 59472.00 | 0.92
|  Competitiveness and Innovation (CI) |  |  |  |  |  |
4a |  Persons employed (percentage of total employment) |  European Statistical System | 1.82 | 0.46 | 2.82 | 1.10 | 0.28
4b |  Value added at factor cost (percentage of GDP at current prices) |  European Statistical System | 0.97 | 0.24 | 1.30 | 0.36 | 1.41
4c |  Number of patents related to recycling and secondary raw materials |  European Patent Office | 0.59 | 0.64 | 2.58 | 0.00 | 2.40

###  4.2 The Circular Economy – MCPI by dimensions

Table 3 shows the results of the MCPI by country and for the four dimensions
of production and consumption $\delta_{PC}$, waste management $\delta_{WN}$,
secondary raw materials $\delta_{SRM}$, and competitiveness and innovation
$\delta_{CI}$. When the information is presented by dimension, the particular
performance of each country in each of the areas could be analysed. By
applying a compensatory approach to the indicators in each area, the value of
the upper limit of the interval is obtained. Under a non-compensatory
perspective, the lower limit indicates the value of the worst performing
indicator in each dimension. On the other hand, the length of the MCPI
provides information on the balance or imbalance of the indicators. For
example, if we look at the case of Austria, we can see how the information
displayed in the waste management dimension $\left[80.0,81.1\right]$ presents
a high value at both ends of the interval indicating that all indicators of
this dimension are balanced. However, we can observe that for the dimensions
of production and consumption $\left[43.4,83.2\right]$ and innovation and
competitiveness $\left[8.1,30.7\right]$ the length of the MPCI is considerable
and therefore indicative that some indicators in these dimensions perform too
poorly. From a policy perspective, attention should be paid to those
dimensions either where the value is too low or where there is an imbalance.

Table 3: The MCPI of Circular Economy by dimension in Europe (year 2018) | $\delta_{\mathrm{PC}}\;(\%)$ | $\delta_{\mathrm{WM}}\;(\%)$ | $\delta_{\mathrm{SRM}}\;(\%)$ | $\delta_{\mathrm{CI}}\;(\%)$
---|---|---|---|---
Austria | [ | 43.4, | 83.2 ] | [ | 80.0, | 81.8 ] | [ | 23.4, | 29.5 ] | [ | 8.1, | 30.7 ]
Belgium | [ | 4.7, | 63.4 ] | [ | 77.8, | 87.3 ] | [ | 72.6, | 73.8 ] | [ | 0.0, | 37.5 ]
Bulgaria | [ | 23.0, | 44.2 ] | [ | 24.3, | 31.1 ] | [ | 3.5, | 4.1 ] | [ | 0.0, | 24.6 ]
Croatia | [ | 70.5, | 86.3 ] | [ | 26.8, | 43.1 ] | [ | 3.0, | 7.5 ] | [ | 15.5, | 40.5 ]
Cyprus | [ | 32.7, | 87.7 ] | [ | 10.7, | 20.2 ] | [ | 0.0, | 2.1 ] | [ | 0.0, | 22.8 ]
Czechia | [ | 77.5, | 90.2 ] | [ | 42.9, | 56.9 ] | [ | 6.2, | 14.7 ] | [ | 9.3, | 37.1 ]
Denmark | [ | 0.0, | 80.0 ] | [ | 69.9, | 71.3 ] | [ | 9.6, | 16.7 ] | [ | 7.1, | 35.6 ]
Estonia | [ | 0.0, | 12.2 ] | [ | 0.0, | 16.1 ] | [ | 4.0, | 23.7 ] | [ | 28.3, | 61.8 ]
Finland | [ | 48.5, | 85.0 ] | [ | 38.6, | 47.7 ] | [ | 2.3, | 8.9 ] | [ | 24.7, | 61.7 ]
France | [ | 48.6, | 81.6 ] | [ | 61.1, | 62.1 ] | [ | 36.6, | 50.8 ] | [ | 10.8, | 31.4 ]
Germany | [ | 38.4, | 79.5 ] | [ | 61.4, | 81.1 ] | [ | 38.2, | 70.1 ] | [ | 9.2, | 36.7 ]
Greece | [ | 49.5, | 78.3 ] | [ | 17.7, | 23.7 ] | [ | 6.4, | 8.4 ] | [ | 0.0, | 6.8 ]
Hungary | [ | 70.3, | 88.9 ] | [ | 46.6, | 47.6 ] | [ | 5.1, | 12.3 ] | [ | 13.9, | 30.0 ]
Ireland | [ | 39.9, | 88.1 ] | [ | 44.3, | 46.4 ] | [ | 0.3, | 1.7 ] | [ | 9.3, | 30.9 ]
Italy | [ | 30.0, | 70.3 ] | [ | 69.7, | 76.2 ] | [ | 61.9, | 63.2 ] | [ | 8.9, | 30.7 ]
Latvia | [ | 75.1, | 93.1 ] | [ | 0.0, | 13.6 ] | [ | 4.3, | 8.1 ] | [ | 19.8, | 46.1 ]
Lithuania | [ | 59.5, | 83.8 ] | [ | 74.4, | 78.6 ] | [ | 7.8, | 8.9 ] | [ | 0.0, | 34.0 ]
Luxembourg | [ | 2.0, | 78.6 ] | [ | 68.3, | 72.6 ] | [ | 17.7, | 25.5 ] | [ | 100.0, | 100.0 ]
Malta | [ | 27.9, | 82.6 ] | [ | 0.0, | 23.1 ] | [ | 0.8, | 12.0 ] | [ | 0.0, | 0.0 ]
Netherlands | [ | 10.1, | 65.1 ] | [ | 80.4, | 84.4 ] | [ | 88.1, | 93.9 ] | [ | 2.4, | 32.9 ]
Poland | [ | 53.7, | 79.4 ] | [ | 41.7, | 53.5 ] | [ | 17.2, | 23.3 ] | [ | 26.6, | 56.0 ]
Portugal | [ | 56.6, | 85.8 ] | [ | 33.5, | 46.5 ] | [ | 2.2, | 8.5 ] | [ | 0.0, | 17.0 ]
Romania | [ | 64.0, | 90.9 ] | [ | 1.9, | 15.0 ] | [ | 0.0, | 4.0 ] | [ | 5.8, | 17.3 ]
Slovakia | [ | 62.1, | 81.6 ] | [ | 46.1, | 47.3 ] | [ | 3.5, | 8.0 ] | [ | 0.8, | 16.8 ]
Slovenia | [ | 60.5, | 83.6 ] | [ | 85.6, | 92.7 ] | [ | 7.9, | 19.1 ] | [ | 0.0, | 29.9 ]
Spain | [ | 42.6, | 76.9 ] | [ | 43.4, | 47.4 ] | [ | 29.4, | 40.1 ] | [ | 18.1, | 39.8 ]
Sweden | [ | 70.1, | 89.1 ] | [ | 55.7, | 59.3 ] | [ | 16.2, | 17.5 ] | [ | 9.6, | 28.0 ]
United Kingdom | [ | 31.1, | 72.7 ] | [ | 59.7, | 64.1 ] | [ | 53.8, | 73.5 ] | [ | 8.6, | 29.9 ]

###  4.3 The overall Circular Economy – MCPI

To assess the overall performance in terms of EU Member states’ circular
economy, we construct the Circular Economy – MCPI. Table 4 sorts the EU
countries by the upper bound of the $\gamma_{i}$–MCPI . Moreover, we add a
column to highlight the span of the MCPI. In the last column, we highlight
those countries with more balance scores by the $\beta$–rating scale.
Luxembourg, Netherlands and Germany head up the ranking with the highest
scores of the upper bound of the Circular Economy – MCPI. In contrast, Greece,
Estonia and Bulgaria occupy the lowest positions in the ranking. Notice that,
for Luxembourg, the best-positioned country, the span of the MCPI is the
longest, which is indicative of a significant imbalance in one of the
dimensions and for this reason it receive a $\beta$–Rating of 1 star
($\star$). Then, we need to look at the information provided by dimensions in
Table 3. It can be seen that, a great imbalance in the production and
consumption dimension as highlighted in $\delta_{PC}=\left[2.0,78.6\right]$,
which is compensated by the high performance of the competitiveness and
innovation dimension $\delta_{CI}=\left[100.0,100.0\right]$.

Table 4: Multicriteria composite performance interval of Circular Economy in Europe (year 2018) | Rank | $\gamma_{i}$ | Span |  $\beta$–rating
---|---|---|---|---
Luxembourg | 1 | [ | 25.5, | 69.2 ] | 43.7 | $\star$
Netherlands | 2 | [ | 32.9, | 69.1 ] | 36.2 | $\star$
Germany | 3 | [ | 36.7, | 66.8 ] | 30.2 | $\star\star$
Belgium | 4 | [ | 37.5, | 65.5 ] | 28.0 | $\star\star$
Italy | 5 | [ | 30.7, | 60.1 ] | 29.4 | $\star\star$
United Kingdom | 6 | [ | 29.9, | 60.0 ] | 30.2 | $\star\star$
France | 7 | [ | 31.4, | 56.5 ] | 25.1 | $\star\star$
Slovenia | 8 | [ | 19.1, | 56.3 ] | 37.3 | $\star$
Austria | 9 | [ | 29.5, | 56.3 ] | 26.8 | $\star\star$
Poland | 10 | [ | 23.3, | 53.1 ] | 29.7 | $\star\star$
Lithuania | 11 | [ | 8.9, | 51.3 ] | 42.4 | $\star$
Spain | 12 | [ | 39.8, | 51.0 ] | 11.2 | $\star\star\star$
Denmark | 13 | [ | 16.7, | 50.9 ] | 34.2 | $\star$
Finland | 14 | [ | 8.9, | 50.8 ] | 41.9 | $\star$
Czechia | 15 | [ | 14.7, | 49.7 ] | 35.1 | $\star$
Sweden | 16 | [ | 17.5, | 48.5 ] | 30.9 | $\star\star$
Hungary | 17 | [ | 12.3, | 44.7 ] | 32.4 | $\star\star$
Croatia | 18 | [ | 7.5, | 44.4 ] | 36.8 | $\star$
Ireland | 19 | [ | 1.7, | 41.8 ] | 40.1 | $\star$
Latvia | 20 | [ | 8.1, | 40.2 ] | 32.2 | $\star\star$
Portugal | 21 | [ | 8.5, | 39.4 ] | 31.0 | $\star\star$
Slovakia | 22 | [ | 8.0, | 38.4 ] | 30.4 | $\star\star$
Cyprus | 23 | [ | 2.1, | 33.2 ] | 31.1 | $\star\star$
Romania | 24 | [ | 4.0, | 31.8 ] | 27.8 | $\star\star$
Malta | 25 | [ | 0.0, | 29.4 ] | 29.4 | $\star\star$
Greece | 26 | [ | 6.8, | 29.3 ] | 22.5 | $\star\star$
Estonia | 27 | [ | 12.2, | 28.5 ] | 16.2 | $\star\star\star$
Bulgaria | 28 | [ | 4.1, | 26.0 ] | 21.9 | $\star\star\star$

Once we have obtained an overall assessment of countries’ circular economy
performance, we have noticed that the fourth dimension (CI) has a significant
contribution to the final ranking. This means that a country like Luxembourg
can head the ranking with the best performance in this dimension. For this
reason, we have profiled circular economy performance by eliminating this
dimension, namely, taking into account the dimensions that originally
characterise the circular economy (production and consumption, waste
management and secondary raw materials). Table 5 display the MCPI but
excluding the competitiveness and innovation dimension. It also provides the
rank position, and its variation for the overall ranking. We observe how
Luxembourg now is downgraded eight places and, finally, how Netherlands,
Germany and Belgium top the ranking. Moreover, we can also see that they are
more balanced as shown by the $\beta$–rating.

Figure 1 graphically shows the compared MCPI scores when including all four
dimensions and without the competitiveness and innovation dimension. As can be
seen, in the later case, the heading countries presents a shorter MCPI than
when including all dimensions.

It should be noted that the Circular Economy MCPI within dimensions and for
the overall performance provides a much richer information than a single
composite index allowing the modeller to interpret the meaning of the
composite measures and carry out corrective measures where necessary.

Table 5: Multicriteria composite performance interval of Circular Economy in Europe excluding CI dimension (year 2018) | Rank | $\Delta\mathrm{Rank}$ | $\gamma_{i}$ | Span |  $\beta$–rating
---|---|---|---|---|---
Netherlands | 1 | $+1$ | [ | 65.1, | 81.1 ] | 16.1 | $\star\star\star$
Germany | 2 | $+1$ | [ | 70.1, | 76.9 ] | 6.8 | $\star\star\star$
Belgium | 3 | $+1$ | [ | 63.4, | 74.8 ] | 11.5 | $\star\star\star$
United Kingdom | 4 | $+2$ | [ | 64.1, | 70.1 ] | 6.0 | $\star\star\star$
Italy | 5 | $+0$ | [ | 63.2, | 69.9 ] | 6.7 | $\star\star\star$
Slovenia | 6 | $+2$ | [ | 19.1, | 65.1 ] | 46.1 | $\star$
Austria | 7 | $+2$ | [ | 29.5, | 64.8 ] | 35.3 | $\star$
France | 8 | $-1$ | [ | 50.8, | 64.8 ] | 14.1 | $\star\star\star$
Luxembourg | 9 | $-8$ | [ | 25.5, | 58.9 ] | 33.4 | $\star\star$
Lithuania | 10 | $+1$ | [ | 8.9, | 57.1 ] | 48.2 | $\star$
Denmark | 11 | $+2$ | [ | 16.7, | 56.0 ] | 39.3 | $\star$
Sweden | 12 | $+4$ | [ | 17.5, | 55.3 ] | 37.8 | $\star$
Spain | 13 | $-1$ | [ | 40.1, | 54.8 ] | 14.7 | $\star\star\star$
Czechia | 14 | $+1$ | [ | 14.7, | 53.9 ] | 39.3 | $\star$
Poland | 15 | $-5$ | [ | 23.3, | 52.1 ] | 28.7 | $\star\star$
Hungary | 16 | $+1$ | [ | 12.3, | 49.6 ] | 37.3 | $\star$
Finland | 17 | $-3$ | [ | 8.9, | 47.2 ] | 38.3 | $\star$
Portugal | 18 | $+3$ | [ | 8.5, | 46.9 ] | 38.4 | $\star$
Croatia | 19 | $-1$ | [ | 7.5, | 45.6 ] | 38.1 | $\star$
Slovakia | 20 | $+2$ | [ | 8.0, | 45.6 ] | 37.6 | $\star$
Ireland | 21 | $-2$ | [ | 1.7, | 45.4 ] | 43.7 | $\star$
Malta | 22 | $+3$ | [ | 12.0, | 39.3 ] | 27.2 | $\star\star$
Latvia | 23 | $-3$ | [ | 8.1, | 38.2 ] | 30.2 | $\star\star$
Greece | 24 | $+2$ | [ | 8.4, | 36.8 ] | 28.4 | $\star\star$
Cyprus | 25 | $-2$ | [ | 2.1, | 36.6 ] | 34.5 | $\star$
Romania | 26 | $-2$ | [ | 4.0, | 36.6 ] | 32.6 | $\star\star$
Bulgaria | 27 | $+1$ | [ | 4.1, | 26.5 ] | 22.4 | $\star\star$
Estonia | 28 | $-1$ | [ | 12.2, | 17.3 ] | 5.1 | $\star\star\star$

(a) All four dimensions

(b) Without CI dimension

Figure 1: Overall Circular Economy Multicriteria Composite Performance
Interval (MCPI)

##  5 Conclusion

It is increasingly common to assess a multidimensional phenomenon through a
composite indicator, and indeed the number of composite indicators in use
internationally has been growing steadily. Surprisingly, there is no
international agreement on their construction to date, and a different
methodological approach accompanies each phenomenon. Appealing to simplicity
in their construction, a single number is expected to synthesise very complex
phenomena without emphasising their limitations.

An important issue when constructing composite indicators comes from the
choice of the aggregation methodology which involves the discussion about if
compensability should or not be allowed. In this paper, we have proposed an
alternative way to assess a multidimensional issue’s performance by building a
composite performance interval instead of getting only a real number. The MCPI
allows for overcoming the criticism of composite indicators that they show a
“big-picture” by providing a range of values. Making use of distance-based
multicriteria techniques and different version of the Minkowski’s $L_{p}$
metrics we have built two composite indicators which are presented as a lower
and upper bound of the MCPI. In this proposal, we have used the TOPSIS method
primarily because of its ease of implementation in a wide variety of
situations, with no restrictions on the number of alternatives or criteria.
The lower bound corresponds to a non-compensability aggregation rule and
provides the worst performance of the group of indicators or dimensions that
has been aggregated. The upper bound is constructed allowing for full
compensation using a linear rule based o weighted or additive aggregation.
Finally, based on the balance property of the MCPI, we propose a
$\beta$–Rating to identify those alternative in which there exist great MCPI
span and, therefore, need to be analysed in greater depth to detect the
dimension where action is needed with higher priority.

As an example of application we compute the MCPI to assess the Circular
Economy performance of European Member states by using the structure of
indicators and dimensions provided by the European Commission Circular Economy
monitoring framework. The overall Circular Economy MCPI has been calculated
and also at the dimension level, thus providing a valuable tool to identify
areas where countries need to concentrate their efforts to boost their
circular economy performance. If we consider the four dimensions initially
proposed in the circular economy theoretical framework, the ranking is headed
by Luxembourg, although the dimensions are quite unbalanced. In the other
hand, when the MCPI is constructed excluding the competitiveness and
innovation dimension, the results seem more coherent, leading the ranking with
the countries of Netherlands, Germany and Belgium with a better
$\beta$–Rating.

Moving forwards, we see a promising line of future research in testing the
applicability of the proposed MCPI by adopting other multicriteria techniques
such as the reference point-based method or ELECTREE. Besides, in the future,
we plan to investigate if and how the MCPI could be applied to other domains
in which composite indicators have been previously used, from sustainability
to quality of life assessment, to cite but a few relevant applications.

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