Supersingular Elliptic Curves and Moonshine

Supersingular Elliptic Curves and Moonshine††This paper is a contribution to
the Special Issue on Moonshine and String Theory. The full collection is
available at https://www.emis.de/journals/SIGMA/moonshine.html

Victor Manuel ARICHETA †‡

V.M. Aricheta

† Department of Mathematics, Emory University, Atlanta, GA 30322, USA
<EMAIL_ADDRESS>‡ Institute of Mathematics, University of the Philippines,
‡ Diliman 1101, Quezon City, Philippines<EMAIL_ADDRESS>
Received September 30, 2018, in final form January 19, 2019; Published online
January 29, 2019

We generalize a theorem of Ogg on supersingular $j$-invariants to
supersingular elliptic curves with level. Ogg observed that the level one case
yields a characterization of the primes dividing the order of the monster. We
show that the corresponding analyses for higher levels give analogous
characterizations of the primes dividing the orders of other sporadic simple
groups (e.g., baby monster, Fischer’s largest group). This situates Ogg’s
theorem in a broader setting. More generally, we characterize, in terms of
supersingular elliptic curves with level, the primes arising as orders of
Fricke elements in centralizer subgroups of the monster. We also present a
connection between supersingular elliptic curves and umbral moonshine.
Finally, we present a procedure for explicitly computing invariants of
supersingular elliptic curves with level structure.

moonshine; modular curves; supersingular elliptic curves; supersingular
polynomials

14H52; 11F06; 11F11; 11F22; 11F37; 20D08

##  1 Introduction and results

_Moonshine_ refers to unexpected connections between disparate areas of
mathematics (e.g., modular objects, sporadic groups) and physics (e.g.,
2-dimensional conformal field theories). This paper deals with a theorem of
Ogg and his subsequent observation, which may be regarded as the earliest
occurrence of moonshine. In this paper we show that these constitute the first
case of a broader phenomenon.

In his 1975 inaugural lecture at the Collège de France, Tits mentioned that
the (then conjectural) _monster_ group $\mathbb{M}$, if it exists, is a simple
sporadic group of order

$\displaystyle|\mathbb{M}|=2^{46}\cdot 3^{20}\cdot 5^{9}\cdot 7^{6}\cdot
11^{2}\cdot 13^{3}\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot
47\cdot 59\cdot 71.$

Ogg, who was in the audience, recognized that the primes in this factorization
are precisely the ones he had recently obtained geometrically from his work on
supersingular $j$-invariants [24]. In his work he showed that the following
statements O1 and O2 are equivalent.

  1. (O1)

All the supersingular $j$-invariants in characteristic $p$ are in the prime
field $\mathbb{F}_{p}$.

  2. (O2)

The genus of the modular curve $X_{0}^{+}(p)$ is zero.

Here $X_{0}^{+}(p)$ is the quotient of $X_{0}(p)$ by its Fricke involution
$w_{p}$. (See Section 2.1 for definitions.) We refer to this equivalence in
this paper as _Ogg’s theorem_. _Ogg’s observation_ is that the primes
satisfying either O1 or O2 are the ones dividing the order of the
monster.111For a short note on Ogg’s observation, see [12].

A few years later, in 1979, Conway and Norton – encouraged by observations of
McKay and Thompson [31] – published their _monstrous moonshine_ conjecture
[10]. They postulated the existence of an infinite-dimensional representation
of the monster, with a $\mathbb{Z}$-grading, whose graded dimension function
is the modular $j$-invariant. Moreover, they gave precise predictions for the
other graded trace functions. Similar to the $j$-function, the other graded
trace functions were expected to be _principal moduli_ for genus zero
quotients of the upper half-plane. A candidate for such a representation of
the monster, called the _moonshine module_ , was constructed by Frenkel,
Lepowsky and Meurman in 1984 [16]. Borcherds showed in 1992 that the graded
trace functions of the moonshine module are indeed the principal moduli
predicted by Conway and Norton, thereby proving the monstrous moonshine
conjecture [1].

Ogg’s observation is partly explained by monstrous moonshine. For each prime
divisor $p$ of the order of the monster, there is an element of the monster
whose graded trace function is the principal modulus for $X_{0}^{+}(p)$. The
genus of $X_{0}^{+}(p)$ is then forced to be zero. In other words, the primes
dividing the order of the monster are necessarily included in Ogg’s list of
primes, i.e, the set of primes satisfying either of the equivalent statements
O1 and O2. This however does not explain why Ogg’s list does not contain more
primes. In this paper we show that both Ogg’s theorem and Ogg’s observation
generalize naturally. In so doing we provide further evidence that Ogg’s
observation is more than just a coincidence.

It is useful, for purposes of generalization, to state Ogg’s theorem in terms
of modular curves. The modular curve $X_{0}(1)$, whose non-cuspidal points
parametrize isomorphism classes of elliptic curves, has good reduction modulo
any prime $p$ (cf. Section 2.1). The supersingular points of $X_{0}(1)$ modulo
$p$ correspond to supersingular elliptic curves in characteristic $p$, and a
supersingular point is defined over the prime field $\mathbb{F}_{p}$ if and
only if the $j$-invariant of the corresponding elliptic curve is in
$\mathbb{F}_{p}$. Thus, we can restate Ogg’s theorem as follows: the
supersingular points of $X_{0}(1)$ modulo $p$ are all defined over
$\mathbb{F}_{p}$ if and only if the genus of $X_{0}^{+}(p)$ is zero.

The main idea of this paper is to consider elliptic curves with level
structure by replacing $X_{0}(1)$ by more general modular curves $X$. We give
a characterization for the primes $p$ that have the _rationality property for
$X$_, by which we mean that all the supersingular points of $X$ modulo $p$ are
defined over $\mathbb{F}_{p}$. This analysis yields several consequences.
First, as mentioned earlier, we find that Ogg’s theorem generalizes naturally
(Theorem 1.1). Second, we discover that Ogg’s observation also generalizes
naturally (Theorem 1.3). Third, we find that the primes that have the
rationality property detect the existence of _mock modular forms_ with nonzero
_shadows_ in _umbral moonshine_ (Theorem 1.5). (See Sections 2.2 and 2.3 for
definitions.)

In this paper, we consider the quotient of $X_{0}(N)$ by Atkin–Lehner
involutions $w_{e},w_{f},\ldots$ which we denote by $X_{0}(N){+}e,f,\ldots$.
The non-cuspidal points of these modular curves represent isomorphism classes
of elliptic curves with level structure. By a theorem of Igusa, these modular
curves have good reductions modulo primes $p$ not dividing $N$. Let
$X=X_{0}(N){+}e,f,\ldots$. Denote by: $Q_{p}(X)$ the number of supersingular
points of $X$ modulo $p$ – or equivalently the number of isomorphism classes
of supersingular elliptic curves with level structure in characteristic $p$ –
that are not defined over $\mathbb{F}_{p}$; $\operatorname{\textsl{genus}}(X)$
the genus of $X$; and $X^{p}$ the modular curve $X_{0}(Np){+}p,e,f,\ldots$
obtained by taking the quotient of $X_{0}(Np)$ by Atkin–Lehner involutions
$w_{p},w_{e},w_{f},\ldots$ where $\\{p,e,f,\ldots\\}$ is understood to be the
set $\\{e,f,\ldots,p,pe,pf,\ldots\\}$. Our first theorem gives a
characterization for the primes that have the rationality property for these
modular curves.

######  Theorem 1.1 (Theorem 3.1).

Let $N$ be a positive integer, let $e,f,\ldots$ be exact divisors of $N$, and
let $X=X_{0}(N){+}e,f,\ldots$. If $p$ is a prime not dividing $N$, then

$\displaystyle\dfrac{1}{2}Q_{p}(X)=\operatorname{\textsl{genus}}(X^{p})-\operatorname{\textsl{genus}}(X).$

Consequently, $p$ has the rationality property for $X$ if and only if the
modular curves $X$ and $X^{p}$ have the same genus.

Theorem 1.1 naturally generalizes Ogg’s theorem. Indeed, by letting
$X=X_{0}(1)$, so that $\operatorname{\textsl{genus}}(X)=0$ and
$X^{p}=X_{0}^{+}(p)$, this theorem says that $p$ has the rationality property
for $X_{0}(1)$ if and only if the genus of $X_{0}^{+}(p)$ is zero, which is
precisely Ogg’s theorem.

We now discuss a generalization of Ogg’s observation. For this, we consider
modular curves $X$ of the form $X_{0}(N){+}e,f,\ldots$ with genus equal to
zero. All such modular curves except three (i.e., $X_{0}(25)$,
$X_{0}(49){+}49$, $X_{0}(50){+}50$) arise in monstrous moonshine. By Theorem
1.1, identifying the primes that have the rationality property for $X$ reduces
to determining the complete list of primes $p$ such that $X^{p}$ has genus
zero. We obtain the complete list of primes that have the rationality property
for all such $X$, and we collect this information in a table in Appendix A.
Note from this table that there are no primes that have the rationality
property for the three non-monstrous modular curves.

Table 1: Primes that have the rationality property for low levels. $X$ | primes $p$ that have the rationality property for $X$
---|---
$X_{0}^{+}(2)$ | 3, 5, 7, 11, 13, 17, 19, 23, 31, 47 1pt1pt
$X_{0}(2)$ | 3, 5, 7, 11, 23
$X_{0}^{+}(3)$ | 2, 5, 7, 11, 13, 17, 23, 29 1pt1pt
$X_{0}(3)$ | 2, 5, 11

In Table 1, we list the primes that have the rationality property for some
modular curves. Notice that the primes on the first row – the primes that have
the rationality property for $X_{0}^{+}(2)$ – are exactly the odd primes
dividing the order of the _baby monster_ sporadic group. Similarly, the primes
in the third row are the primes, not equal to $3$, that divide the order of
the largest _Fischer group_ , another sporadic group. Thus, the primes that
have the rationality property for modular curves of higher level characterize
the primes dividing the orders of other sporadic groups; these are natural
generalizations of Ogg’s observation.

######  Remark 1.2.

These generalizations of Ogg’s observation to the baby monster group and the
largest Fischer group have been found, recently and independently, by Nakaya
using analytic methods (e.g., class number estimates) [23]. He conjectured
generalizations of Ogg’s observation to the Harada–Norton and Held sporadic
groups. Theorem 1.1 (cf. entries for $X_{0}(5){+}$ and $X_{0}(7){+}$ in
Appendix A) confirms this conjecture.

In fact, we generalize Ogg’s observation to all _monstrous modular curves_ –
i.e. modular curves arising in monstrous moonshine – of the form
$X_{0}(N){+}e,f,\ldots$ as follows. Under the correspondence given by
monstrous moonshine, if $X\neq X_{0}(27){+}27$, then there is a unique
conjugacy class of the monster whose graded trace function is the principal
modulus for $X$, and we denote this conjugacy class by $C(X)$. If
$X=X_{0}(27){+}27$, then there are exactly two conjugacy classes of the
monster whose graded trace functions are both equal to the principal modulus
for $X$. We let $C(X_{0}(27){+}27)$ be the smaller of these conjugacy classes;
this class is labelled 27A in the ATLAS [9]. Finally, by a Fricke element of
the monster of prime order $p$, we mean an element of the monster whose graded
trace function is the principal modulus for $X_{0}^{+}(p)$. A Fricke element
of order $p$ is the same as any representative of the conjugacy class labelled
$pA$ in the ATLAS.

######  Theorem 1.3 (Theorem 3.3).

Let $X$ be a monstrous modular curve $X$ of the form $X_{0}(N){+}e,$
$f,\ldots$. If $p$ is a prime that does not divide $N$, then $p$ has the
rationality property for $X$ if and only if the centralizer of $g\in C(X)$ in
the monster contains a Fricke element of order $p$.

Ogg’s observation is again recovered from Theorem 1.3 by setting $X=X_{0}(1)$,
and thus we find that Ogg’s observation, just like Ogg’s theorem, is the first
case of a general phenomenon. Theorem 1.3 suggests that Ogg’s observation is
not a statement about the primes dividing the order of the monster per se, but
a statement about the Fricke elements of the monster. This hints that a better
understanding of the Fricke elements is necessary for a full understanding of
Ogg’s observation.

######  Remark 1.4.

For the monstrous modular curves of the form $X=X_{0}(n|h){+}e,f,\ldots$ one
can analogously define $X^{p}$ to be the modular curve
$X_{0}(np|h){+}p,e,f,\ldots$ where as before $\\{p,e,f,\ldots\\}$ is the set
$\\{e,f,\ldots,p,pe,pf,\ldots\\}$. In view of the characterization given by
Theorem 1.1, we may say that a prime $p$ has the rationality property for a
monstrous modular curve $X$ if $X$ and $X^{p}$ have the same genus. With this
generalized definition, Theorem 1.3 holds for any monstrous modular curve $X$.

Theorem 1.3 gives a connection between supersingular elliptic curves with
level structure and monstrous moonshine. We point out another connection, an
unexpected one, between supersingular elliptic curves and umbral moonshine.
(See Section 2.3 for definitions.) Each of the 23 cases of moonshine in umbral
moonshine has an associated genus zero modular curve called its _lambency_. In
this paper, we refer to these 23 modular curves as _umbral modular curves_.
For each umbral modular curve $X$, there is an _umbral group_ $G^{X}$ that
serves as an analogue of the monster group, i.e., there exists a graded
representation of $G^{X}$ such that the graded trace function $H_{g}^{X}$ of
$g\in G^{X}$ is a distinguished vector-valued mock modular form of weight 1/2.
There is also a naturally defined quotient of $G^{X}$ denoted $\bar{G}^{X}$,
and we denote by $n_{g}$ the order of the image of $g\in G^{X}$ in
$\bar{G}^{X}$. This integer $n_{g}$ is the level of $H_{g}^{X}$.

######  Theorem 1.5 (Theorem 3.4).

Let $X$ be an umbral modular curve, and let $p$ be a prime not dividing the
level of $X$. Then the prime $p$ has the rationality property for $X$ if and
only if there exists an element $g\in G^{X}$ such that $n_{g}=p$ and the
shadow of $H^{X}_{g}$ is nonzero.

A mock modular form is a classical modular form if and only if its shadow is
zero. Therefore, Theorem 1.5 says that the primes that have the rationality
property for the umbral modular curves are exactly the primes that occur as
levels of strictly mock modular graded trace functions.

Theorem 1.5 gives a connection between supersingular elliptic curves and
umbral moonshine, while Theorem 1.3 provides another connection between
supersingular elliptic curves and monstrous moonshine. We may then regard
supersingular elliptic curves as a link between monstrous and umbral
moonshine.

Lastly, we consider the modular curves of the form $X_{0}(N)$ of genus zero,
and present another way of checking whether a prime has the rationality
property for $X_{0}(N)$. This alternative method explicitly computes
_supersingular polynomials for $X_{0}(N)$_, which we will define shortly. The
point is that a prime $p$ has the rationality property for $X_{0}(N)$ if and
only if the $p$th supersingular polynomial for $X_{0}(N)$ splits completely
into linear factors over $\mathbb{F}_{p}$. Note that these supersingular
polynomials have already appeared in the literature, for example, in relation
to the Kaneko–Zagier differential equations for low level Fricke groups [28],
and in connection to Atkin orthogonal polynomials [27, 32]. However, the
methods for computing these supersingular polynomials have been written down
only for low levels. We give here a way for calculating these polynomials for
all $X_{0}(N)$ of genus zero.

Let $p$ be a prime not dividing $N$, and let $T_{N}$ be the principal modulus
for $X_{0}(N)$ given in Appendix B. Let $\text{SS}(N)$ be the set of
supersingular points of $X_{0}(N)$ modulo $p$. The $p$th supersingular
polynomial for $X_{0}(N)$ is the polynomial

$\displaystyle\text{ss}_{p}^{(N)}(x):=\prod_{E\in\text{SS}(N)}(x-T_{N}(E)).$

That the rationality property can be determined by looking at the splitting
property of this polynomial follows from the definition, because a point $x$
in $X_{0}(N)$ modulo $p$ is defined over $\mathbb{F}_{p}$ if and only if
$T_{N}(x)\in\mathbb{F}_{p}$.

In Section 4.1, we describe how a certain polynomial
$f_{p}^{(N)}\in\mathbb{Q}[x]$ arises from a modular form $f$ of weight $p-1$
and level 1. If $f$ is chosen to be the weight $p-1$ Eisenstein series (among
others), then this polynomial turns out to encode almost all the
$T_{N}$-values of supersingular points on $X_{0}(N)$ modulo $p$. The few
supersingular points not covered by this polynomial is encoded in another
polynomial $g_{p}^{(N)}$ which we give in Appendix D. (See also Section 4.2.)

######  Theorem 1.6.

Let $p\geq 5$ be a prime, and let $f$ be any of $E_{p-1}$, $G_{p-1}$, or
$H_{p-1}$ as defined in Section 4. Then

$\displaystyle\text{ss}_{p}^{(N)}\equiv\pm f_{p}^{(N)}g_{p}^{(N)}\pmod{p}.$

It is straightforward to write an algorithm in a computer algebra system
(e.g., Sage [26], PARI/GP [30]) that takes a modular form $f$ of weight $p-1$
and level one as an input and produces the polynomial $f^{(N)}_{p}$ as an
output. Therefore, Theorem 1.6 provides a simple way of explicitly computing
supersingular polynomials for $X_{0}(N)$. The first few supersingular
polynomials for $X_{0}(2)$ are given in Table 2. Note that these polynomials
split completely into linear factors over $\mathbb{F}_{p}$ for $p=5,7,11,23$.
Therefore these primes have the rationality property for $X_{0}(2)$, a fact
that agrees with the information in Table 1. Moreover, the number of quadratic
factors in $\text{ss}_{p}^{(2)}$ coincides with the genus of
$X_{0}(2)^{p}=X_{0}(2p){+}p$ (cf. Table 3 in Section 3.1) which is consistent
with Theorem 1.1.

Table 2: Supersingular polynomials for $X_{0}(2)$. $p$ |  $\text{ss}^{(2)}_{p}(x)$3pt
---|---
5 |  $(x+1)$1pt1pt
7 |  $(x+1)\cdot(x+6)$1pt1pt
11 |  $(x+3)\cdot(x+5)\cdot(x+9)$ 1pt1pt
13 |  $(x+1)\cdot\big{(}x^{2}+8x+1\big{)}$2pt1pt
17 |  $(x+1)\cdot(x+16)\cdot\big{(}x^{2}+13x+16\big{)}$ 2pt1pt
19 |  $(x+1)\cdot(x+7)\cdot(x+11)\cdot\big{(}x^{2}+9x+11\big{)}$2pt1pt
23 |  $(x+3)\cdot(x+5)\cdot(x+15)\cdot(x+16)\cdot(x+17)\cdot(x+18)$1pt1pt
29 |  $(x+16)\cdot(x+23)\cdot(x+24)\cdot\big{(}x^{2}+24x+16\big{)}\cdot\big{(}x^{2}+25x+23\big{)}$ 2pt1pt

The rest of the paper is organized as follows. The necessary background on
modular curves, mock modular forms and moonshine is given in Section 2.
Theorems 1.1, 1.3 and 1.5 are proven in Section 3. Finally, the proof of
Theorem 1.6 is given in Section 4.

##  2 Moduli spaces, mock modular forms, moonshine

###  2.1 Moduli spaces

In this subsection, we define the modular curves $X_{0}(N)$ and the
Atkin–Lehner involutions on them. We also describe the Deligne–Rapoport model
for $X_{0}(pN)$ modulo a prime $p$ not dividing $N$, closely following Ogg’s
description in [25].

For every positive integer $N$, the congruence subgroup $\Gamma_{0}(N)$ acts
on the complex upper half-plane $\mathbb{H}$ by linear fractional
transformations. The orbit space $Y_{0}(N):=\Gamma_{0}(N)\backslash\mathbb{H}$
is naturally a Riemann surface, and it admits a moduli interpretation: the
points of $Y_{0}(N)$ parametrize isomorphism classes of cyclic isogenies, of
degree $N$, of complex elliptic curves. This Riemann surface can be
compactified by adjoining the orbits of $\Gamma_{0}(N)$ on
$\mathbb{Q}\cup\\{\infty\\}$, called the cusps of $\Gamma_{0}(N)$, and we
denote this compactification by $X_{0}(N)$.

Let $a,b,c,d\in\mathbb{Z}$ and let $e$ be an exact divisor of $N$, by which we
mean that $e|N$ and $(e,N/e)=1$. If the matrix
$w_{e}:=(\begin{smallmatrix}ae&b\\\ cN&de\end{smallmatrix})$ has determinant
$e$, then $\frac{1}{\sqrt{e}}w_{e}$ lies in the normalizer of $\Gamma_{0}(N)$
in $\text{SL}_{2}(\mathbb{R})$. This element of the normalizer induces an
involution on $X_{0}(N)$, called an Atkin–Lehner involution of $X_{0}(N)$,
which is also denoted by $w_{e}$. (The Atkin–Lehner involution $w_{N}$ of
$X_{0}(N)$ is also known as the Fricke involution.) In order to describe the
action of $w_{e}$ using the moduli interpretation of $X_{0}(N)$, separate an
$N$-isogeny into sub-isogenies of degrees $e$ and $N/e$. The involution
$w_{e}$ acts as the transpose on the $e$-part of the isogeny and leaves the
$N/e$-part fixed.

By a theorem of Igusa, for any prime $p$ not dividing $N$, the modular curve
$X_{0}(N)$ has good reduction modulo $p$. On the other hand, the reduction of
$X_{0}(pN)$ modulo $p$ is a singular curve obtained from glueing two copies of
$X_{0}(N)$ modulo $p$ at the supersingular points. More precisely, the non-
cuspidal points of $X_{0}(pN)$ modulo $p$ parametrize $pN$-isogenies of
elliptic cuves over $\mathbb{F}_{p}$. We separate a $pN$-isogeny into its
$N$-part and its $p$-part. There are as many $N$-isogenies in characteristic 0
as in characteristic $p$, but there are only two $p$-isogenies in
characteristic $p$, namely the Frobenius and its transpose. The first copy of
$X_{0}(N)$ modulo $p$ parametrizes those isogenies whose $p$-part is the
Frobenius; the second copy, those whose $p$-part is the transpose of the
Frobenius. Their intersection consists of isogenies whose $p$-part may be
thought of as either the Frobenius or its transpose – the supersingular
points.

###  2.2 Mock modular forms

In this subsection, we recall the definition of a mock modular form and its
shadow. These objects, which have their origins in the last letter of
Ramanujan to Hardy, are now found in several areas of contemporary mathematics
including moonshine. We refer to [2] for more details about mock modular forms
and their applications.

Let $\Gamma$ be a discrete subgroup of $\text{SL}_{2}(\mathbb{R})$ and let $k$
be a half-integer. We say that a holomorphic function $f$ on $\mathbb{H}$ is a
_mock modular form of weight $k$ for $\Gamma$_ if it has at most exponential
growth as $\tau$ approaches any cusp of $\Gamma$, and if there exists a
modular form $S(f)$ of weight $2-k$ on $\Gamma$ such that the sum $f+S(f)^{*}$
transforms like a holomorphic modular form of weight $k$ on $\Gamma$. Here,
the function $S(f)^{*}$ is a solution to the differential equation:

$\displaystyle(4\pi y)^{k}\dfrac{\partial
S(f)^{*}(\tau)}{\partial\overline{\tau}}=-2\pi{\rm
i}\overline{S(f)(\tau)},\qquad\tau=x+{\rm i}y.$

The modular form $S(f)$ is called the _shadow of $f$_, is uniquely determined
by $f$, and is equal to zero if and only if $f$ is a usual modular form.

###  2.3 Umbral moonshine

Over the last 40 years, several cases of moonshine have been observed and
proven. Most notable for the amount of research that their discovery inspired
are the original monstrous moonshine, discussed in Section 1, and the more
recent Mathieu moonshine. The latter is now known to belong to a family of
moonshine collectively called umbral moonshine, which is the topic of this
subsection.

In 2010 Eguchi, Ooguri and Tachikawa observed a numerical coincidence
reminiscent of the McKay–Thompson observation [14]. The decomposition of the
elliptic genus of a K3 surface into irreducible characters of the
$\mathcal{N}=4$ superconformal algebra gives rise to a $q$-series

$\displaystyle
H(\tau)=2q^{-1/8}\big{(}{-}1+45q+231q^{2}+770q^{3}+2277q^{4}+\cdots\big{)}.$

It was noted that $H(\tau)$ is a mock modular form and its first few
coefficients are dimensions of irreducible representations of the largest
Mathieu group $M_{24}$. Mathieu moonshine, formulated in a series of papers
[3, 13, 18, 19] and proven by Gannon [20], is the statement that there exists
an infinite-dimensional graded representation of $M_{24}$ with the following
property: the graded trace functions are certain distinguished mock modular
forms of weight $1/2$. These graded trace functions are Rademacher sums which
are natural weight 1/2 analogues of the principal modulus property [4].
Mathieu moonshine thus expanded the class of automorphic objects considered in
moonshine to include mock modular forms and other weights. In a series of
papers Cheng, Duncan and Harvey identified Mathieu moonshine as one of a
family of correspondences between finite groups and mock modular forms [6, 7,
8]. They referred to this conjectured family of correspondences as
umbral222The word umbral was chosen to highlight the existence of _shadows_ in
this moonshine. moonshine.

Briefly, umbral moonshine is a collection of 23 cases of moonshine relating
groups arising from lattices to (vector-valued) mock modular forms. The
lattices in umbral moonshine are the _Niemeier lattices_ , which are the even
unimodular self-dual lattices with roots (i.e. lattice vectors of length 2),
and they are determined by their _Niemeier root systems_. (For more
information about root systems and their Dynkin diagrams, see [21].) These
root systems have rank 24 and their simple components are root systems of ADE
type of the same Coxeter number. A Niemeier root system is said to be of _$A$
-type_ if it has a simple component of type $A$, it is said to be of _$D$
-type_ if it has a simple component of type $D$ but no type $A$ component, and
it is said to be of _$E$ -type_ if it only has type $E$ components.

Given a Niemeier root system, there is a group of genus zero naturally
attached to it called its _lambency_. The lambencies are defined as follows:
The Coxeter number of a Niemeier lattice is the common Coxeter number of its
simple components. The Coxeter numbers of the $A$-type Niemeier lattices are
the same integers $N$ for which the genus of $\Gamma_{0}(N)$ is zero, and the
lambency of such a Niemeier lattice of Coxeter number $N$ is defined to be
$\Gamma_{0}(N)$. Similarly, the Coxeter numbers of the $D$-type Niemeier
lattices are the same integers $2N$ for which the genus of
$\Gamma_{0}(2N){+}N$ is zero, and the lambency of such a Niemeier lattice of
Coxeter number $2N$ is defined to be $\Gamma_{0}(2N){+}N$. As discussed in
[7], the genus zero groups naturally attached to the Niemeier root systems
$E_{6}^{4}$ and $E_{8}^{3}$ of $E$-type are $\Gamma_{0}(12){+}4$ and
$\Gamma_{0}(30){+}6,10,15$ respectively, and these corresponding groups are
defined to be their lambencies.

Let $X$ be a Niemeier root system and let $L^{X}$ be the associated Niemeier
lattice. The reflections through the roots of $L^{X}$ generate a normal
subgroup of the full automorphism group of $L^{X}$ known as the _Weyl group_
of $X$. The _umbral group_ $G^{X}$, which plays the same role as the monster
in monstrous moonshine, is defined to be the quotient of the group of
automorphisms of $L^{X}$ by the Weyl group of $X$. Umbral moonshine associates
to each element $g\in G^{X}$ a distinguished (vector-valued) mock modular form
$H^{X}_{g}$. Duncan, Griffin and Ono showed the existence of an infinite-
dimensional graded representation of $G^{X}$ with graded trace functions equal
to $H^{X}_{g}$ [11]. There is also a naturally defined quotient of $G^{X}$
denoted $\bar{G}^{X}$, and we denote by $n_{g}$ the order of the image of
$g\in G^{X}$ in $\bar{G}^{X}$. This integer $n_{g}$ is the level of
$H_{g}^{X}$.

##  3 Supersingular elliptic curves and moonshine

###  3.1 Higher level Ogg’s theorem

In this subsection we prove Theorem 1.1, a generalization of Ogg’s theorem,
and use this to obtain the primes that have the rationality property for the
curves $X_{0}(N){+}e,f,\ldots$ of genus zero.

Let $X=X_{0}(N){+}e,f,\ldots$ and let $p$ be a prime not dividing $N$. Recall
from Section 1 the following notations: $Q_{p}(X)$ is the number of
supersingular points of $X$ modulo $p$ that are not defined over
$\mathbb{F}_{p}$; $\operatorname{\textsl{genus}}(X)$ is the genus of $X$; and
$X^{p}$ is the modular curve $X_{0}(Np){+}p,e,f,\ldots$ obtained by taking the
quotient of $X_{0}(Np)$ by Atkin–Lehner involutions $w_{p}$, $w_{e}$, $w_{f}$,
$\ldots$ where $\\{p,e,f,\ldots\\}$ is understood to be the set
$\\{e,f,\ldots,p,pe,pf,\ldots\\}$.

######  Theorem 3.1.

Let $N$ be a positive integer, let $e,f,\ldots$ be exact divisors of $N$, and
let $X=X_{0}(N){+}e,f,\ldots$. If $p$ is a prime not dividing $N$, then

$\displaystyle\dfrac{1}{2}Q_{p}(X)=\operatorname{\textsl{genus}}(X^{p})-\operatorname{\textsl{genus}}(X).$

Consequently, $p$ has the rationality property for $X$ if and only if the
modular curves $X$ and $X^{p}$ have the same genus.

###### Proof.

The proof follows that of Ogg [25]. The ingredient that we need for his proof
to go through is a model for the (singular) curve $X_{0}(Np){+}e,f,\ldots$
modulo $p$. We recall the model for this curve here which is explained in [17,
Section 5]. If $e,f,\ldots$ are exact divisors of $N$, so that each of them is
coprime with $p$, and if

$\displaystyle X:=X_{0}(N){+}e,f,\ldots\mod p,\qquad
X(p):=X_{0}(Np){+}e,f,\ldots\mod p,$

then $X(p)$ consists of two copies of $X$ that intersect at the supersingular
points of $X$ modulo $p$.

From here, the proof of Ogg goes through: If $X_{1}$ and $X_{2}$ are the
components of $X(p)$, then $w_{p}$ defines an isomorphism of $X_{1}$ onto
$X_{2}$ that acts as the Frobenius on $X_{1}\cap X_{2}$ – the supersingular
points of $X$. Therefore the model of $X(p)/(w_{p})=X^{p}$ modulo $p$ is given
by one copy of $X$ which intersects itself at each point corresponding to a
pair of conjugate supersingular points of $X$, i.e., the supersingular points
of $X$ not defined over $\mathbb{F}_{p}$. From this model of $X^{p}$, we
obtain
$\operatorname{\textsl{genus}}(X^{p})=\operatorname{\textsl{genus}}(X)+\frac{1}{2}Q_{p}(X)$,
which is the first part of Theorem 1.1. The second part of the theorem readily
follows since by definition $p$ has the rationality property for $X$ if and
only if $Q_{p}(X)=0$. ∎

We apply Theorem 1.1 to the case when the genus of $X=X_{0}(N){+}e,f,\ldots$
is zero. Considering such cases leads to a generalization of Ogg’s observation
which is the subject of the next subsection. According to Theorem 1.1, given
such an $X$ the prime $p$ has the rationality property for $X$ if and only if
the genus of $X^{p}$ is zero. For example, Table 3 gives the genus of $X^{p}$
for $X$ of level 2 and for the first few primes. From this we see that the
primes 3, 5, 7, 11, 13, 17, 19, 23, 31, 47 have the rationality property for
$X_{0}^{+}(2)$ and the primes 3, 5, 7, 11, 23 have the rationality property
for $X_{0}(2)$. Moreover we know from [15] that this list is complete since 94
is the largest level of the form $2p$ for which there exists a genus zero
quotient of $X_{0}(2p)$ by Atkin–Lehner involutions. Similarly, we enumerate
all the primes $p$ for which $X^{p}$ has genus zero for all
$X=X_{0}(N){+}e,f,\ldots$ of genus zero, and we compile the results in
Appendix A.

######  Remark 3.2.

We identified the primes that have the rationality property for modular curves
$X_{0}(N){+}e,f,\ldots$ of genus zero. We mention here that Ogg already
obtained the primes that have the rationality property for modular curves
$X_{0}(N)$ regardless of the genus [25]. He did not however consider their
quotients by Atkin–Lehner involutions. His result is as follows: $2$ is the
only prime that has the rationality property for $X_{0}(11)$ and $X_{0}(17)$;
also, if $N\neq 11,17$ and if the genus of $X_{0}(N)$ is positive, then there
are no primes that have the rationality property for $X_{0}(N)$. We point out
here that there is a typo in his list for $X_{0}(4)$; the prime $5$ should not
be in the list.

Table 3: Genus of $X^{p}$. $p$ | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\text{gen}\big{(}\big{(}X_{0}^{+}(2)\big{)}^{p}\big{)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 1pt1pt
$\text{gen}\big{(}\big{(}X_{0}(2)\big{)}^{p}\big{)}$ | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 2 | 1 | 4 | 3 | 4 | 1 1pt1pt

###  3.2 Higher level Ogg’s observation

Suppose that $X=X_{0}(N){+}e,f,\ldots$ is a monstrous modular curve (cf.
Section 1). In this subsection we provide a characterization, which
generalizes Ogg’s observation, of the primes that have the rationality
property for $X$.

Recall that in Section 1, we defined $C(X)$ to be the unique conjugacy class
of the monster associated via monstrous moonshine to $X$ if $X\neq
X_{0}(27){+}27$, and we defined $C(X_{0}(27){+}27)$ to be the conjugacy class
of the monster labelled 27A in the ATLAS [9]. We also defined a Fricke element
of prime order $p$ to be an element of the monster whose graded trace function
is the principal modulus for $X^{+}_{0}(p)$.

######  Theorem 3.3.

Let $X$ be a monstrous modular curve $X$ of the form $X_{0}(N){+}e,f,\ldots$.
If $p$ is a prime that does not divide $N$, then $p$ has the rationality
property for $X$ if and only if the centralizer of $g\in C(X)$ in the monster
contains a Fricke element of order $p$.

###### Proof.

Given a conjugacy class $C$ of a finite group $G$, denote by $C^{n}$ the
conjugacy class of $G$ containing the $n$th powers of elements of $C$. Suppose
$C_{1}$ and $C_{2}$ are conjugacy classes of orders $n_{1}$ and $n_{2}$ such
that $(n_{1},n_{2})=1$. Then the classes $C_{1}$ and $C_{2}$ have
representatives that commute if and only if there exists a conjugacy class $C$
of order $n_{1}n_{2}$ such that $C^{n_{2}}=C_{1}$ and $C^{n_{1}}=C_{2}$.

Now, the centralizer of $g\in C(X)$ in the monster contains a Fricke element
of order $p$ if and only if the conjugacy classes $C(X)$ and $pA$ have
representatives that commute. From the previous paragraph, this occurs if and
only if there exists a conjugacy class of order $pN$ whose $N$th power is $pA$
and whose $p$th power is $C(X)$. One can manually check from the power maps of
the conjugacy classes of the monster, using GAP [29] for instance, that this
latter condition occurs if and only if $p$ has the rationality property for
$X$. ∎

We illustrate by way of an example how this proof works. We consider the case
when $N=2$. This implies that $X=X_{0}^{+}(2)$ or $X_{0}(2)$, and the
conjugacy class $C(X)$ is $2A$ or $2B$ respectively. In Table 4, we label the
columns by the primes $p$ dividing the order of the monster – these are the
primes that can occur as prime orders of Fricke elements – and we label the
rows by $C(X)$. The entries in row $C(X)$ and column $p$ are the conjugacy
classes of the monster of order $2p$ whose $p$th power is $C(X)$. For each
cell, we can check which of those conjugacy classes have 2nd power equal to
$pA$, and write these in boldface. Then from the proof, there is a conjugacy
class in row $C(X)$ and column $p$ in boldface if and only if the centralizer
of $g\in C(X)$ contains a Fricke element of order $p$.

Table 4:

| 3 | 5 | 7 | 11 | 13 | 17 | 19
---|---|---|---|---|---|---|---
$2A$ |  $6A$ $6D$ |  $10A$ $10C$ | $14A$ | $22A$ | $26A$ | $34A$ | $38A$
$2B$ |  $6B$ $6C$ $6E$ $6F$ |  $10B$ $10D$ $10E$ |  $14B$ $14C$ | $22B$ | $26B$ |  |

| 23 | 29 | 31 | 41 | 47 | 59 | 71
---|---|---|---|---|---|---|---
$2A$ | $46CD$ |  | $62AB$ |  | $94AB$ |  |
$2B$ | $46AB$ |  |  |  |  |  |

From this table, we find that the odd primes arising as order of Fricke
elements in the centralizer of $g\in C(X_{0}^{+}(2))$ in the monster are 3, 5,
7, 11, 13, 17, 19, 23, 31, 47. These are the same primes that have the
rationality property for $X_{0}^{+}(2)$. Similarly, the Fricke elements of odd
prime orders in the centralizer of $g\in C(X_{0}(2))$ in the monster have
orders 3, 5, 7, 11, 23, and these coincide with the primes that have the
rationality property for $X_{0}(2)$.

###  3.3 Observation related to umbral moonshine

In this subsection, we consider the modular curves that occur (as lambencies)
in the theory of umbral moonshine. Given such an umbral modular curve $X$, we
present another characterization of the primes that have the rationality
property for $X$. This will be in terms of the modularity properties of
certain graded trace functions.

Recall from Section 2.3 that associated to each of the 23 cases of umbral
moonshine are: a modular curve $X$ of genus zero called its lambency; an
umbral group $G^{X}$; and a set of graded trace functions $H^{X}_{g}$ for each
$g\in G^{X}$. The functions $H^{X}_{g}$ are vector-valued mock modular forms
of weight 1/2 and level $n_{g}$, and a formula for the shadow of $H^{X}_{g}$
may be given in terms of naturally defined characters of $G^{X}$ called
_twisted Euler characters_ (cf. Section 5.1 of [7]). The formulas show that
the shadow of $H_{g}^{X}$ is zero if and only if the value of all the twisted
Euler characters at $g$ is 0.

Consider the case of umbral moonshine of lambency $X_{0}(2)$ (i.e., Mathieu
moonshine). This has umbral group equal to the largest Mathieu group $M_{24}$.
The following table gives the conjugacy classes $[g]$ of $M_{24}$ with $n_{g}$
an odd prime, and the values of the twisted Euler character
$\overline{\chi}_{g}^{A}$ at these conjugacy classes.

$[g]$ | $3A$ | $3B$ | $5A$ | $7AB$ | $11A$ | $23AB$
---|---|---|---|---|---|---
$n_{g}$ | 3 | 3 | 5 | 7 | 11 | 23
$\overline{\chi}_{g}^{A}$ | 6 | 0 | 4 | 3 | 2 | 1

From this table, we see that there exists an element $g$ of $G^{X}$, where
$n_{g}$ is an odd prime, for which the shadow $H_{g}^{X}$ is _nonzero_ if and
only if $p=3,5,7,11,23$. These primes are precisely the ones that have the
rationality property for $X_{0}(2)$. In this example, one could argue that
these primes are also simply the primes appearing as $n_{g}$, but as the next
example shows, in some cases there are primes that appear as $n_{g}$ but not
as the level of a mock modular form with non-vanishing shadow.

Consider the umbral moonshine case of lambency $X=X_{0}(5)$. The following
table gives the conjugacy classes $[g]$ of $G^{X}$ whose order $n_{g}$ is a
prime $p\neq 5$, and the values of the twisted Euler characters
$\overline{\chi}_{g}^{A}$ and $\chi_{g}^{A}$ at these conjugacy classes.

$[g]$ | $2B$ | $2C$ | $3A$ | $6A$
---|---|---|---|---
$n_{g}$ | 2 | 2 | 3 | 3
$\overline{\chi}_{g}^{A}$ | 2 | 2 | 0 | 0
$\chi_{g}^{A}$ | -2 | 2 | 0 | 0

In this case of umbral moonshine, the graded trace function $H_{g}^{X}$ for
any order 3 element $g$ of $G^{X}$ has shadow equal to zero, i.e., $H_{g}^{X}$
is a classical modular form. There is an element $g$, of prime order $p\neq
5$, of $G^{X}$ for which the shadow of $H_{g}^{X}$ is nonzero if and only if
$p=2$. The prime 2 happens to be the only prime that has the rationality
property for $X_{0}(5)$.

In fact, this pattern persists and we have the following theorem.

######  Theorem 3.4.

Let $X$ be an umbral modular curve, and let $p$ be a prime not dividing the
level of $X$. Then the prime $p$ has the rationality property for $X$ if and
only if there exists an element $g\in G^{X}$ such that $n_{g}=p$ and the
shadow of $H^{X}_{g}$ is nonzero.

###### Proof.

From the tables of values of the twisted Euler characters in [7], one can
enumerate the primes $p$ not dividing $N$ with the following properties: (1)
there is a $g\in G^{X}$ with $n_{g}=p$; and (2) there is a twisted Euler
character that does not vanish at $g$, or equivalently, the shadow of
$H_{g}^{X}$ is nonzero. By inspection, the primes that satisfy these
properties are the same primes that have the rationality property for $X$. ∎

######  Remark 3.5.

We considered only the primes that do not divide the level in the formulation
of our notion of rationality. It would be interesting to extend this notion so
as to include all primes. Can this be done in such a way that Theorems 1.3 and
1.5 also generalize?

######  Remark 3.6.

As explained in a recent work of Cheng and Duncan, naturally attached to any
lambency $X$ – i.e., a genus zero quotient of $X_{0}(N)$ by a set of
Atkin–Lehner involutions that does not include the Fricke involution – is an
“optimal mock Jacobi form” $\phi^{X}$ of level 1 with integer coefficients
[5]. These optimal mock Jacobi forms allow the recovery of the graded trace
functions in umbral moonshine; if $X$ is an umbral lambency, then the
components of $H^{X}_{e}$ are the coefficients in the theta decomposition of
$\phi^{X}$, and the components of the other graded trace functions $H_{g}^{X}$
may be obtained, via certain multiplicative relations (cf. Tables 8 and 9 of
[7]), from $\phi^{X^{\prime}}$ where $X^{\prime}$ is of lower lambency. There
are extra lambencies in [5] that do not occur in umbral moonshine. A natural
guess is that there is a generalization of umbral moonshine that incorporates
these more general lambencies, and Theorem 1.5 could serve as a consistency
check for this generalization.

##  4 Generalized supersingular polynomials

In this section we prove Theorem 1.6 mentioned in the introduction. For an
even positive integer $k$, we denote by: $E_{k}$ the normalized Eisenstein
series of weight $k$; $G_{k}$ the coefficient of $X^{k}$ in
$\big{(}1-3E_{4}(\tau)X^{4}+2E_{6}(\tau)X^{6}\big{)}^{-1/2}$; and $H_{k}$ the
coefficient of $X^{k}$ in
$\big{(}1-3E_{4}(\tau)X^{4}+2E_{6}(\tau)X^{6}\big{)}^{k/2}$.

###  4.1 The polynomial $\bm{f_{p}^{(N)}}$

Suppose that the genus of $X_{0}(N)$ is zero, and let $T_{N}$ be the principal
modulus for $X_{0}(N)$. There is a normalized modular form $\Delta_{N}\in
M_{12}(\Gamma_{0}(N))$, whose formula is given in Appendix B, that vanishes
only at the infinite cusp of $\Gamma_{0}(N)$ and nowhere else. Since $\Delta$
is a modular form of weight 12 and level 1 that vanishes only at the infinite
cusp and nowhere else, the modular form $\Delta_{N}$ may be considered as
higher level analogues of $\Delta$, and hence the choice for its notation.

Let $p$ be prime. Note that we can uniquely write $p-1$ in the form

$\displaystyle p-1=12m+4\delta+6\epsilon,\qquad\text{where}\quad
m\geq\mathbb{Z}_{\geq 0},\quad\delta,\epsilon\in\\{0,1\\}.$

Using the classical valence formula, if $f\in M_{p-1}(\Gamma_{0}(1))$, then
$f/\big{(}E_{4}^{\delta}E_{6}^{\epsilon}\big{)}\in M_{12m}(\Gamma_{0}(1))$. We
get a modular function on $\Gamma_{0}(N)$ by dividing
$f/\big{(}E_{4}^{\delta}E_{6}^{\epsilon}\big{)}$ by $\Delta_{N}^{m}$.
Moreover, since $\Delta_{N}$ vanishes only at $\infty$, the poles of
$f/\big{(}E_{4}^{\delta}E_{6}^{\epsilon}\Delta_{N}^{m}\big{)}$ are supported
at $\infty$. Thus there exists a polynomial $f_{p}^{(N)}\in\mathbb{C}[x]$ such
that

$\displaystyle\dfrac{f}{E_{4}^{\delta}E_{6}^{\epsilon}\Delta_{N}^{m}}=f_{p}^{(N)}(T_{N}).$

Note that if $f$ has integral Fourier coefficients, which is true for the
modular forms $E_{p-1}$, $G_{p-1}$, $H_{p-1}$, then $f_{p}^{(N)}$ has rational
coefficients.

###  4.2 The polynomial $\bm{g_{p}^{(N)}}$

Later in proving Theorem 1.6, we will be using the following result – which is
the $N=1$ case of Theorem 1.6 – due to Deuring, Hasse, Deligne, Kaneko and
Zagier [22].

######  Proposition 4.1.

Let $p\geq 5$ be a prime and let $f$ be any of $E_{p-1}$, $G_{p-1}$ or
$H_{p-1}$. Let $f_{p}^{(1)}$ be the polynomial defined in Section 4.1. Then

$\displaystyle ss_{p}^{(1)}(x)\equiv\pm
f_{p}^{(1)}(x)x^{\delta}(x-1728)^{\epsilon}\pmod{p}.$

We denote by $g_{p}^{(N)}$ the higher level analogues of the factor
$x^{\delta}$ and $(x-1728)^{\epsilon}$ found in this proposition which we
obtain as follows. In this proposition, the factor $x$ corresponds to the
(isomorphism class of the) elliptic curve $y^{2}=x^{3}+1$ with $j$-invariant
equal to 0. The exponent $\delta$ is equal to 1 when $p\equiv 2\pmod{3}$. The
proposition says that these are precisely the primes $p$ for which the
elliptic curve $y^{2}=x^{3}+1$ is supersingular in characteristic $p$.
Similarly, the factor $x-1728$ corresponds to the (class of) curve
$y^{2}=x^{3}+x$ with $j$-invariant 1728, and the proposition says that this
elliptic curve is supersingular when $\epsilon=1$ or when $p\equiv 3\pmod{4}$.

These distinguished isomorphism classes of elliptic curves with $j$-invariants
0 and 1728 – or equivalently points on the moduli space $X_{0}(1)$ with
$j$-values 0 or 1728 – break up into several isomorphism classes when we
consider them as elliptic curves with level structure $N$, or as points on the
moduli space $X_{0}(N)$. The $T_{N}$-values of these points constitute the
roots of the polynomial analogue of $x^{\delta}(x-1728)^{\epsilon}$ that we
seek.

To obtain the $T_{N}$-values given a $j$-value, we need a _modular relation_
between $j$ and $T_{N}$, by which we mean a relation
$j(\tau)=r_{N}(T_{N}(\tau))$ for some rational function
$r_{N}\in\mathbb{Q}(x)$. In Appendix E, we present the complete list of
modular relations for $j$ and $T_{N}$ for the $N$’s such that $X_{0}(N)$ has
genus zero. One can verify these identities by checking that the Fourier
coefficients for the left and the right hand side coincide up to the Sturm
bound.

To find the $T_{N}$-values of the points with $j$-invariants 0 and 1728, we
need only to solve the equations $0=r_{N}(T_{N})$ and $1728=r_{N}(T_{N})$. For
example, for $N=2$, we have $r_{2}(T)=(T+256)^{3}/T^{2}$ and so:

$\displaystyle r_{2}(T)=0\ \Rightarrow\ T+256=0,\qquad r_{2}(T)=1728\
\Rightarrow\ (T-512)(T+64)=0.$

Therefore the level 2 analogue of $x^{\delta}(x-1728)^{\epsilon}$ is the
polynomial $(x+256)^{\delta}(x-512)^{\epsilon}(x+64)^{\epsilon}$. One can do
the same for the other levels to obtain all the polynomials $g_{p}^{(N)}$.

###  4.3 Proof of Theorem 1.6

Note that the supersingular points of $X_{0}(N)$ modulo $p$ are the points
lying above the supersingular points of $X_{0}(1)$ modulo $p$. Therefore,
Proposition 4.1 tells us that the roots of the equation
$f_{p}^{(1)}(j)j^{\delta}(j-1728)^{\epsilon}=0$ are the $j$-values of the
supersingular points of $X_{0}(N)$ modulo $p$. From the definition of
$f^{(N)}_{p}$, we have the relation

$f^{(N)}_{p}(T_{N})\Delta^{m}_{N}=f^{(1)}_{p}(j)\Delta^{m}.$ (4.1)

Therefore $f_{p}^{(1)}(j)=0$ if and only if $f_{p}^{(N)}(T_{N})=0$. Also by
definition of $g_{p}^{(N)}$, the equation $j^{\delta}(j-1728)^{\epsilon}=0$ if
and only if $g_{p}^{(N)}(T_{N})=0$. Therefore, the roots of
$g_{p}^{(N)}(T_{N})f_{p}^{(N)}(T_{N})=0$ are the $T_{N}$-values of the
supersingular points of $X_{0}(N)$ modulo $p$. Finally, the coefficient of
$g_{p}^{(N)}(T_{N})f_{p}^{(N)}(T_{N})$ is $\pm 1$ because: $g_{p}^{(N)}$ is
monic; and by (1) the leading term of $f_{p}^{(N)}$ is the same as the leading
term of $f_{p}^{(1)}$, which is $\pm 1$ by Proposition 4.1. $\blacksquare$

##  Appendix A Primes that have the rationality property

In the following table, we use the notation $N{+}e,f,\ldots$ for the modular
curve $X_{0}(N){+}e,f,\ldots$. Moreover, we use the notation $N{+}$ when all
the exact divisors of $N$ are included, and the notation $N{-}$ when no exact
divisors are included. The following table lists the primes that have the
rationality property for genus zero modular curves of the form
$N{+}e,f,\ldots$. There are no primes that have the rationality property for
the genus zero modular curves of the form $N{+}e,f,\ldots$ not listed in this
table.

$X$ |  $\begin{array}[]{c}\text{primes that have}\\\\[-2.0pt] \text{the rat.\ prop.\ for $X$}\end{array}$
---|---
$2{+}$ | 3, 5, 7, 11, 13, 17, 19, 23, 31, 47
$2-$ | 3, 5, 7, 11, 23
$3{+}$ | 2, 5, 7, 11, 13, 17, 23, 29
$3-$ | 2, 5, 11
$4{+}$ | 3, 5, 7, 11, 23
$4{-}$ | 3, 7
$5{+}$ | 2, 3, 7, 11, 19
$5{-}$ | 2
$6{+}$ | 5, 7, 11, 13
$6{+}6$ | 5, 11
$6{+}3$ | 5
$7{+}$ | 2, 3, 5, 17
$7{-}$ | 3
$8{+}$ | 3, 7
$9{+}$ | 2, 5
$9{-}$ | 2
$10{+}$ | 3, 7, 11
$10{+}5$ | 3
$11{+}$ | 2, 3, 5
$12{+}$ | 5
$X$ |  $\begin{array}[]{c}\text{primes that have}\\\\[-2.0pt] \text{the rat.\ prop.\ for $X$}\end{array}$
---|---
$13{+}$ | 2, 3
$14{+}$ | 3, 5
$14{+}14$ | 3
$15{+}$ | 2, 7
$15{+}15$ | 2
$17{+}$ | 2, 3, 7
$19{+}$ | 2, 5
$20{+}$ | 3
$21{+}$ | 2, 5
$22{+}$ | 3, 5
$23{+}$ | 2, 3
$25{+}$ | 2
$26{+}$ | 3
$27{+}$ | 2
$29{+}$ | 3
$31{+}$ | 2
$33{+}$ | 2
$35{+}$ | 2, 3
$47{+}$ | 2
$55{+}$ | 2

##  Appendix B Principal moduli for $\bm{X_{0}(N)}$

Let $\eta(\tau)$ be the Dedekind eta function. In this table, we employ the
following notation for an eta-product: $n_{1}^{d_{1}}\cdots
n_{l}^{d_{l}}:=\eta(n_{1}\tau)^{d_{1}}\cdots\eta(n_{l}\tau)^{d_{l}}$. The
following table shows the principal moduli for the modular curves $X_{0}(N)$.

$N$ | $T_{N}$
---|---
2 | $1^{24}/2^{24}$
3 | $1^{12}/3^{12}$
4 | $1^{8}/4^{8}$
5 | $1^{6}/5^{6}$
6 | $2^{8}3^{4}/1^{4}6^{8}$
7 | $1^{4}/7^{4}$
8 | $1^{4}4^{2}/2^{2}8^{4}$
$N$ | $T_{N}$
---|---
9 | $1^{3}/9^{3}$
10 | $2^{4}5^{2}/1^{2}10^{4}$
12 | $3^{3}4^{1}/1^{1}12^{3}$
13 | $1^{2}/13^{2}$
16 | $1^{2}8^{1}/2^{1}16^{2}$
18 | $2^{2}9^{1}/1^{1}18^{2}$
25 | $1/25$

##  Appendix C Higher level analogues of $\bm{\Delta}$

We again use the following notation for an eta-product: $n_{1}^{d_{1}}\cdots
n_{l}^{d_{l}}:=\eta(n_{1}\tau)^{d_{1}}\cdots\eta(n_{l}\tau)^{d_{l}}$. The
following table lists the modular (non-cuspidal) forms $\Delta^{(N)}\in
M_{12}(\Gamma_{0}(N))$ that vanish only at the cusp $\infty$ of
$\Gamma_{0}(N)$ and nowhere else.

$N$ | $\Delta^{(N)}$
---|---
2 | $2^{48}/1^{24}$
3 | $3^{36}/1^{12}$
4 | $4^{48}/2^{24}$
5 | $5^{30}/1^{6}$
6 | $1^{12}6^{72}/2^{24}3^{36}$
7 | $7^{28}/1^{4}$
8 | $8^{48}/4^{24}$
$N$ | $\Delta^{(N)}$
---|---
9 | $9^{36}/3^{12}$
10 | $1^{6}10^{60}/2^{12}5^{30}$
12 | $2^{12}12^{72}/4^{24}6^{36}$
13 | $13^{26}/1^{2}$
16 | $16^{48}/8^{24}$
18 | $3^{12}18^{72}/6^{24}9^{36}$
25 | $25^{30}/5^{6}$

##  Appendix D The polynomials $\bm{g_{p}^{(N)}}$

In the following table, we list the polynomials $g_{p}^{(N)}$ whose roots are
the $T_{N}$-invariants of the characteristic $p$ supersingular elliptic curves
with level $N$ structure, such that the $j$-invariant is 0 or 1728.

$N$ |  $g_{p}^{(N)}$ where $p-1=12m+4\delta+6\epsilon$ 3pt1pt
---|---
2 | $(x+256)^{\delta}(x-512)^{\epsilon}(x+64)^{\epsilon}$
3 | $(x+27)^{\delta}(x+243)^{\delta}(x^{2}-486x-19683)^{\epsilon}$
4 | $(x^{2}+256x+4096)^{\delta}(x+32)^{\epsilon}(x^{2}-512x-8192)^{\epsilon}$
5 | $(x^{2}+250x+3125)^{\delta}(x^{2}-500x-15625)^{\epsilon}(x^{2}+22x+125)^{\epsilon}$
6 | $(x+3)^{\delta}(x^{3}+225x^{2}-405x+243)^{\delta}(x^{2}+18x-27)^{\epsilon}(x^{4}-540x^{3}+270x^{2}-972x+729)^{\epsilon}$
7 | $(x^{2}+13x+49)^{\delta}(x^{2}+245x+2401)^{\delta}(x^{4}-490x^{3}-21609x^{2}-235298x-823543)^{\epsilon}$
8 | $(x^{4}+256x^{3}+5120x^{2}+32768x+65536)^{\delta}(x^{2}+32x+128)^{\epsilon}(x^{4}-512x^{3}-10240x^{2}-65536x-131072)^{\epsilon}$
9 | $(x+9)^{\delta}(x^{3}+243x^{2}+2187x+6561)^{\delta}(x^{6}-486x^{5}-24057x^{4}-367416x^{3}-2657205x^{2}-9565938x-14348907)^{\epsilon}$
10 |  $(x^{6}+230x^{5}+275x^{4}-1500x^{3}+4375x^{2}-6250x+3125)^{\delta}(x^{2}+2x+5)^{\epsilon}(x^{2}+20x-25)^{\epsilon}$ $(x^{4}-540x^{3}+1350x^{2}-1500x+625)^{\epsilon}(x^{2}-2x+5)^{\epsilon}$
12 |  $(x^{2}+4x-8)^{\delta}(x^{6}+228x^{5}-408x^{4}-128x^{3}-192x^{2}+768x-512)^{\delta}$ $(x^{4}+20x^{3}-48x^{2}+32x-32)^{\epsilon}(x^{8}-536x^{7}-272x^{6}+3328x^{5}+6400x^{4}-20480x^{3}+4096x^{2}+16384x-8192)^{\epsilon}$
13 |  $(x^{2}+5x+13)^{\delta}(x^{4}+247x^{3}+3380x^{2}+15379x+28561)^{\delta}$ $(x^{6}-494x^{5}-20618x^{4}-237276x^{3}-1313806x^{2}-3712930x-4826809)^{\epsilon}(x^{2}+6x+13)^{\epsilon}$
16 |  $(x^{8}+256x^{7}+5632x^{6}+53248x^{5}+282624x^{4}+917504x^{3}+1835008x^{2}+2097152x+1048576)^{\delta}(x^{4}+32x^{3}+192x^{2}+512x+512)^{\epsilon}$ $(x^{8}-512x^{7}-11264x^{6}-106496x^{5}-565248x^{4}-1835008x^{3}-3670016x^{2}-4194304x-2097152)^{\epsilon}$
18 |  $(x^{3}+3x^{2}-9x+9)^{\delta}(x^{9}+225x^{8}-1080x^{7}+3348x^{6}-8262x^{5}+16038x^{4}-23328x^{3}+26244x^{2}-19683x+6561)^{\delta}$ $(x^{6}+18x^{5}-81x^{4}+216x^{3}-405x^{2}+486x-243)^{\epsilon}$ $(x^{12}-540x^{11}+1890x^{10}-4212x^{9}+13527x^{8}-48600x^{7}+129276x^{6}-262440x^{5}$ $+413343x^{4}-498636x^{3}+433026x^{2}-236196x+59049)^{\epsilon}$
25 |  $(x^{10}+250x^{9}+4375x^{8}+35000x^{7}+178125x^{6}+631250x^{5}+1640625x^{4}+3125000x^{3}+4296875x^{2}+3906250x+1953125)^{\delta}$ $(x^{4}+10x^{3}+45x^{2}+100x+125)^{\epsilon}(x^{2}+2x+5)^{\epsilon}$ $(x^{10}-500x^{9}-18125x^{8}-163750x^{7}-871875x^{6}-3137500x^{5}-8203125x^{4}-15625000x^{3}-21484375x^{2}-19531250x-9765625)^{\epsilon}$

##  Appendix E Modular relations

The following table gives $j$ as rational functions of $T_{N}$.

$N$ | modular relations between $j$ and $T=T_{N}$
---|---
2 | $j=\dfrac{(T+256)^{3}}{T^{2}}$
3 | $j=\dfrac{(T+27)(T+243)^{3}}{T^{3}}$
4 | $j=\dfrac{(T^{2}+256T+4096)^{3}}{(T+16)T^{4}}$
5 | $j=\dfrac{(T^{2}+250T+3125)^{3}}{T^{5}}$
6 | $j=\dfrac{(T+3)^{3}(T^{3}+225T^{2}-405T+243)^{3}}{(T-1)^{2}(T-9)^{6}T^{3}}$
7 | $j=\dfrac{(T^{2}+13T+49)(T^{2}+245T+2401)^{3}}{T^{7}}$
8 | $j=\dfrac{(T^{4}+256T^{3}+5120T^{2}+32768T+65536)^{3}}{(T+4)(T+8)^{2}T^{8}}$
9 | $j=\dfrac{(T+9)^{3}(T^{3}+243T^{2}+2187T+6561)^{3}}{(T^{2}+9T+27)T^{9}}$
10 | $j=\dfrac{(T^{6}+230T^{5}+275T^{4}-1500T^{3}+4375T^{2}-6250T+3125)^{3}}{(T-1)^{2}(T-5)^{10}T^{5}}$
12 | $j=\dfrac{(T^{2}+4T-8)^{3}(T^{6}+228T^{5}-408T^{4}-128T^{3}-192T^{2}+768T-512)^{3}}{(T-2)(T-1)^{3}(T+2)^{3}(T-4)^{12}T^{4}}$
13 | $\dfrac{(T^{2}+5T+13)(T^{4}+247T^{3}+3380T^{2}+15379T+28561)^{3}}{T^{13}}$
16 | $j=\dfrac{(T^{8}+256T^{7}+5632T^{6}+53248T^{5}+282624T^{4}+917504T^{3}+1835008T^{2}+2097152T+1048576)^{3}}{(T+2)(T+4)^{4}(T^{2}+4T+8)T^{16}}$
18 | $j=\dfrac{(T^{3}+3T^{2}-9T+9)^{3}(T^{9}+225T^{8}-1080T^{7}+3348T^{6}-8262T^{5}+16038T^{4}-23328T^{3}+26244T^{2}-19683T+6561)^{3}}{(T-1)^{2}T^{9}(T-3)^{18}(T^{2}-3T+3)(T^{2}+3)^{2}}$
25 | $j=\dfrac{(T^{10}+250T^{9}+4375T^{8}+35000T^{7}+178125T^{6}+631250T^{5}+1640625T^{4}+3125000T^{3}+4296875T^{2}+3906250T+1953125)^{3}}{(T^{4}+5T^{3}+15T^{2}+25T+25)T^{25}}$

### Acknowledgments

We are immensely indebted to John Duncan for his constant advice and
encouragement throughout the project, without which this work would not have
been possible. We are also grateful to Luca Candelori, Scott Carnahan, Simon
Norton, Preston Wake, Robert Wilson and David Zureick-Brown for taking time to
answer our questions, and to Ken Ono for posing the original problem which
motivated this research. We thank the referees for the helpful suggestions and
comments. We wrote part of this paper during our stay at the Erwin Schrödinger
International Institute for Mathematics and Physics (ESI) from 10 to 14
September 2018; we are thankful to ESI for their support.

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