# On Congruence classes of orders of reductions of elliptic curves

Antigona Pajaziti  Faculty of Engineering and Natural Sciences, Sabancı
University, Tuzla, İstanbul, 34956 Turkey<EMAIL_ADDRESS>and  Mohammad Sadek<EMAIL_ADDRESS>
###### Abstract.

Let $E$ be an elliptic curve defined over ${\mathbb{Q}}$ and
$\widetilde{E}_{p}({\mathbb{F}_{p}})$ denote the reduction of $E$ modulo a
prime $p$ of good reduction for $E$. Given an integer $m\geq 2$ and any $a$
modulo $m$, we consider how often the congruence
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv a\mod m$ holds. We show that the
greatest common divisor of the integers
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ over all rational primes $p$ cannot
exceed $4$. We then exhibit elliptic curves over ${\mathbb{Q}}(t)$ with
trivial torsion for which the orders of reductions of every smooth fiber
modulo primes of positive density at least $1/2$ are divisible by a fixed
small integer. We also show that if the torsion of $E$ grows over a quadratic
field $K$, then one may explicitly compute
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ modulo the torsion subgroup
$|E(K)_{\operatorname{tors}}|$. More precisely, we show that there exists an
integer $N\geq 2$ such that $|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ is
determined modulo $|E(K)_{\operatorname{tors}}|$ according to the arithmetic
progression modulo $N$ in which $p$ lies. It follows that given any $a$ modulo
$|E(K)_{\operatorname{tors}}|$, we can estimate the density of primes $p$ such
that the congruence $|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv
a\mod|E(K)_{\operatorname{tors}}|$ occurs.

††Mathematics Subject Classification: 11G05, 14H52
Keywords: Elliptic curves, growth of torsion, quadratic fields, order of
reduction

##  1\. Introduction

Given an elliptic curve $E$ over a number field $K$, Mordell-Weil Theorem
asserts that the set of $K$-rational points $E(K)$ of $E$ is a finitely
generated abelian group. In particular, $E(K)\cong{\mathbb{Z}}^{r}\times
E(K)_{\operatorname{tors}}$ where $r$ is a nonnegative integer and
$E(K)_{\operatorname{tors}}$ is the torsion subgroup of $E(K)$. It was long
believed that $E(K)_{\operatorname{tors}}$ is not only finite but rather it is
uniformly bounded by an integer that merely depends on the degree of the
number field $K$. Mazur, [25], proved that this is indeed the case when $K$ is
the rational field ${\mathbb{Q}}$ giving an exhaustive list of $15$
possibilities for $E({\mathbb{Q}})_{\operatorname{tors}}$. Kenku, Kamienny and
Momose, [14, 17], established such an exhaustive list for
$E(K)_{\operatorname{tors}}$ when $K$ is a quadratic field. Later on, the
uniform boundedness of $E(K)_{\operatorname{tors}}$ was proved by Merel, [26],
for any number field $K$ of fixed degree. Although the result is exciting, the
uniform bound introduced is far from being sharp leaving the mathematical
community with the challenge of finding a list of all possible finite abelian
groups that may occur as torsion subgroups of elliptic curves over a number
field of a given fixed degree. Recently, such a list was presented for cubic
fields, [6].

Writing ${\mathfrak{p}}$ for a prime ideal in the ring of integers of $K$ with
a corresponding residue field $k_{{\mathfrak{p}}}$ of characteristic $p$, we
set $\widetilde{E}_{\mathfrak{p}}$ to be the reduction of $E$ modulo
${\mathfrak{p}}$. It can be seen that $E(K)_{\operatorname{tors}}$ embeds in
$\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})$ for every prime
${\mathfrak{p}}$ of good reduction of $E$. This implies the divisibility of
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$ by
$|E(K)_{\operatorname{tors}}|$ for every such ${\mathfrak{p}}$. One may pose a
question on the existence of an elliptic curve over $K$ together with an
integer $m\geq 2$ such that
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$ is divisible by $m$ for
all but finitely many primes ${\mathfrak{p}}$, or more generally, for a set of
primes ${\mathfrak{p}}$ of (natural) density $1$. Serre, [36], answers this
question revealing that such elliptic curves are the ones that are
$K$-isogenous to elliptic curves with nontrivial $K$-rational torsion
subgroups where $m$ is the order of the torsion subgroup. In addition, Serre
shows that if $|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv
a\mod m$ for a set of primes ${\mathfrak{p}}$ of density $1$, then $a$ must be
$0$ and $m$ must be the order of the torsion subgroup of a $K$-isogenous
elliptic curve.

Kubert, [19], parametrized elliptic curves defined over the rational field
${\mathbb{Q}}$ possessing nontrivial torsion subgroups. In other words, it was
proved that any such elliptic curve is a rational specialization of an
elliptic surface defined over ${\mathbb{Q}}$. In [33], elliptic curves with
nontrivial torsion subgroups over quadratic fields were parametrized in a
similar fashion. Following Serre’s work, one may seek an elliptic surface over
${\mathbb{Q}}$ whose smooth fibers are $K$-isogenous to elliptic curves with
fixed nontrivial torsion subgroups, yet the elliptic surface itself has
trivial torsion. According to the discussion above, this might be attained by
local considerations, namely, making sure that the order of the reduction of
the fibers modulo any prime of good reduction is divisible by a fixed integer.
We present an explicit example of an elliptic curve over ${\mathbb{Q}}(t)$
with trivial torsion in which every fiber is ${\mathbb{Q}}$-isogenous to an
elliptic curve whose torsion subgroup contains ${\mathbb{Z}}/3{\mathbb{Z}}$.
Moreover, we display an example of an elliptic curve all of whose quadratic
twists satisfy the following property. The order of the reduction of any of
the quadratic twists is divisible by $5$ for a set of primes of density at
least $1/2$.

As observed by Katz, [15], one may compute the greatest common divisor,
$\gcd_{{\mathfrak{p}}\in
S}|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$, of
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$ where ${\mathfrak{p}}$
runs over a set $S$ of primes of good reduction of $E$ of density one under
mild conditions on the ramification indices of the primes. The integer
$\gcd_{{\mathfrak{p}}\in
S}|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$ is controlled by the
size of the torsion subgroups of the elliptic curves lying in the $K$-isogeny
class of $E$. For certain elliptic curves, the set $S$ can be taken to be the
set of all primes in $K$. This is due to the fact that the phenomenon of
everywhere good reduction of elliptic curves occurs over number fields of
degree at least $2$. However, if $K$ is taken to be the rational field, then
there is no elliptic curve defined over ${\mathbb{Q}}$ that attains good
reduction everywhere. One may still be interested in the integer
$\gcd_{p}|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ when $p$ runs over all
rational primes. The earlier discussion together with Hasse bound on
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ and other local considerations will
imply that $\gcd_{p}|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ cannot exceed $5$.
We prove that $\gcd_{p}|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ can never be $5$
by careful analysis of the special fiber of the Néron model of a family of
elliptic curves over ${\mathbb{Q}}_{p}$ with split multiplicative reduction
for certain primes $p$. We moreover show that the upper bound $4$ is realized.
In case $K$ is a number field of degree at least $2$, one can construct
examples for which
$\gcd_{{\mathfrak{p}}}|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$
exceeds $5$.

One can completely describe the congruence classes attainable by
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$ for all primes
${\mathfrak{p}}$ of good reduction of $E$ modulo an integer $m\geq 2$ if $m$
is a divisor of the order of the torsion subgroup of an elliptic curve that is
$K$-isogenous to $E$, namely
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv 0\mod m$ for all
primes ${\mathfrak{p}}$ of good reduction of $E$. In addition, divisors of the
order of torsion subgroups of elliptic curves over $K$ are the only integers
modulo which only one congruence class is attained by
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$ when ${\mathfrak{p}}$
runs over primes of good reduction of $E$. It follows that there are only
finitely many such integers $m\geq 2$ and they are controlled, in general, by
Merel’s Uniformity Theorem that bounds the size of torsion subgroups of
elliptic curves defined over a number field of fixed degree. Given an elliptic
curve $E$ over $K$, if $m\geq 2$ is not a divisor of the torsion subgroup of a
$K$-isogenous elliptic curve to $E$, then we know that there are at least two
congruence classes modulo $m$ realizable by
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$ when ${\mathfrak{p}}$
runs over the primes of good reduction of $E$; and each class occurs for a set
of primes of positive density. This motivates the following questions. If
$m\geq 2$ is not a divisor of the order of the torsion subgroup of a
$K$-isogenous elliptic curve to $E$, is it possible to determine all the
possible congruence classes of
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\mod m$ for all primes
${\mathfrak{p}}$ of good reduction of $E$? In addition, can one compute the
density of primes ${\mathfrak{p}}$ at which each congruence class modulo $m$
occurs?

Now we fix our number field to be the rational field ${\mathbb{Q}}$ and we
write ${\mathbb{F}_{p}}$ for the finite field with $p$ elements where $p$ is a
rational prime. The number of ${\mathbb{F}_{p}}$-rational points
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|$, where $p$ is a prime of good
reduction of $E$, is governed by Hasse-Weil bound, namely,
$\left||\widetilde{E}_{p}({\mathbb{F}_{p}})|-p-1\right|\leq 2\sqrt{p}$.
Deuring, [7], proved if $E$ has complex multiplication, then
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|=p+1$ for a set of primes of density
$1/2$. In other words, he proved that the set of supersingular primes of an
elliptic curve with complex multiplication has density $1/2$. Counting
rational points of reductions of Elliptic curves with complex multiplication
has received much attention, see for example [30, 34, 31, 5]. Due to the
arithmetic properties of complex multiplication, one may find an explicit
description of supersingular primes using congruence classes modulo a fixed
integer. For example, the elliptic curve $E^{D}:y^{2}=x^{3}-6Dx^{2}-3D^{2}x$
has complex multiplication by $\sqrt{-3}$, more precisely
$\operatorname{End}(E^{D})\cong{\mathbb{Z}}[\sqrt{-3}]$. The supersingular
primes of the curve $E^{D}$ are exactly the primes $p\equiv 2\mod 3$; whereas
if $p\equiv 1\mod 3$, then
$|\widetilde{E}^{D}_{p}({\mathbb{F}_{p}})|=p+1-2c\left(\frac{D}{p}\right)$
where $p=c^{2}+3d^{2}$ with $c\equiv\left(\frac{-1}{p}\right)\mod 3$ and
$\left(\frac{\cdot}{p}\right)$ is the Legendre symbol modulo $p$, [32]. In
particular, $|\widetilde{E}^{D}_{p}({\mathbb{F}_{p}})|\equiv 0\mod 3$ if
$p\equiv 2\mod 3$, while $|\widetilde{E}^{D}_{p}({\mathbb{F}_{p}})|\equiv
0\text{ or }1\mod 3$ if $p\equiv 1\mod 3$. In fact, taking into account the
existence of a $2$-torsion point in $E^{D}({\mathbb{Q}})$, one sees that
$|\widetilde{E}^{D}_{p}({\mathbb{F}_{p}})|\equiv 0\mod 6$ if $p\equiv 5\mod 6$
and $|\widetilde{E}^{D}_{p}({\mathbb{F}_{p}})|\equiv 0\text{ or }4\mod 6$ if
$p\equiv 1\mod 6$. It follows that one may describe the congruence classes of
$|\widetilde{E}^{D}_{p}({\mathbb{F}_{p}})|\mod 6$ and the congruence classes
modulo $6$ of the primes $p$ at which they occur, hence one may get estimates
for the densities of these primes $p$ using Dirichlet’s Theorem on primes in
arithmetic progression. It is worth noting that one of the main tools to
obtain the aforementioned expressions for the number of rational points on
reductions of elliptic curves $E:\ y^{2}=f(x)$ with complex multiplication is
computing the character sum $\sum_{x\mod p}\left(\frac{f(x)}{p}\right)$. The
reason is that the trace of Frobenius
$a_{p}(E):=p+1-|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ of the curve $E$ at $p$
can be expressed in terms of the latter character sum.

Given an integer $m\geq 2$ that is not a divisor of the order of the torsion
subgroup of any elliptic curve in the ${\mathbb{Q}}$-isogeny class of $E$, one
knows that $E$ has a torsion point of order $m$ under base change to the
division field ${\mathbb{Q}}(E[m])$, where $E[m]$ is the group of $m$-torsion
points on $E$. By considering the splitting behavior of the primes in the
latter field, one can easily show that the primes $p$ for which
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ is divisible by $m$ is of positive
density. In this work, we consider the case when
$|E(K)_{\operatorname{tors}}|\equiv 0\mod m$ for some quadratic field $K$. In
[11, 12, 27], the possible torsion subgroups of $E$ that appear when the base
is changed from ${\mathbb{Q}}$ to a quadratic field $K$ are listed. Using the
arithmetic of quadratic fields, it can be seen that orders of reductions of
$E$ modulo primes of good reduction are closely linked to the torsion
subgroups of certain quadratic twists of $E$. This enables us to obtain
information about the congruence classes of
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ modulo $|E(K)_{\operatorname{tors}}|$.
In addition, we show the existence of an integer $N$ that depends on $K$ such
that for primes $p$ modulo $N$, one can determine explicitly the possibilities
for $|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ modulo
$|E(K)_{\operatorname{tors}}|$. Thus, given any $a$ modulo
$|E(K)_{\operatorname{tors}}|$, an estimate for the density of primes $p$ such
that $|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv
a\mod|E(K)_{\operatorname{tors}}|$ can be computed. This provides families of
elliptic curves $E$, with no complex multiplication, for which one can find
integers $m$ such that $|\widetilde{E}_{p}({\mathbb{F}_{p}})|\mod m$ are
completely determined and the primes of occurrence $p$ are explicitly known.
Up to the knowledge of the authors, there are only few examples in the
literature of pairs of elliptic curves with no complex multiplication together
with integers $m$ for which the latter information is available. For such an
example see [18, Theorem 2] where the congruence classes
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|\mod 12$ of the curve $E:\
y^{2}=x^{3}-12x-11$ are given. We can recover the occurrences of these classes
by noticing the growth of the torsion subgroup of a ${\mathbb{Q}}$-isogenous
elliptic curve over the quadratic field ${\mathbb{Q}}(\sqrt{5})$. In fact, our
local computations suggest that studying the growth of the torsion of
quadratic twists of elliptic curves over number fields of small degree might
provide us with similar congruence data for elliptic curves whose torsion
grows over these fields.

Elkies, [9], showed that for elliptic curves over ${\mathbb{Q}}$ without
complex multiplication, there are infinitely many supersingular primes, yet
these primes are of density $0$. He extended this result to any elliptic curve
over any real number field in [10]. A prime $p\geq 7$ is called an anomalous
prime for $E$ over ${\mathbb{Q}}$ if
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|=p$. Mazur proved that if $E$ has
nontrivial torsion over ${\mathbb{Q}}$, then the set of anomalous primes
consists either of a single prime, or none, or else is contained in the set
$\\{2,3,5\\}$. Moreover, given any finite set $S$ of primes, there is an
elliptic curve defined over ${\mathbb{Q}}$ whose set of anomalous primes
contains $S$, [24]. Yet, it is not known whether there is an elliptic curve
over ${\mathbb{Q}}$ that possesses an infinite number of anomalous primes. As
an application of our work, for families of elliptic curves $E$ whose torsion
grows over a quadratic field $K$, we provide the possible congruence classes
of supersingular and anomalous primes of these curves modulo integer divisors
of $|E(K)_{\operatorname{tors}}|$.

For an elliptic curve over ${\mathbb{Q}}$ with a nontrivial torsion subgroup
of order divisible by $N\geq 1$, the trace of Frobenius of $E$ at a prime $p$
of good reduction satisfies that $a_{p}(E)\equiv p+1\mod N$. Therefore, unless
$N=p$, one sees that the trace of Frobenius $a_{p}(E)$ can never be $1$ due to
a congruence obstruction modulo $N$. In fact, Weak Lang-Trotter conjecture,
see [20] and [16, Conjecture 1.3], asserts that it is only congruence
obstructions that prevent an integer $a$ from being a trace of Frobenius
$a_{p}(E)$ of $E$ for infinitely many primes $p$ of good reduction of $E$. In
[4], it was shown that if $E$ has a rational $\ell$-isogeny, $\ell\neq 11$,
the number of primes $p$ such that $a_{p}(E)\equiv r\mod\ell$ is finite, for
some $r$ modulo $\ell$, if and only if $E$ has rational $\ell$-torsion over
the cyclotomic field ${\mathbb{Q}}(\zeta_{\ell})$. In our work, for elliptic
curves $E$ whose torsion subgroups grow over a quadratic field $K$, we provide
other integers $N$, namely divisors of $|E(K)_{\operatorname{tors}}|$, such
that congruences modulo $N$ obstruct certain integers from being the trace of
Frobenius $a_{p}(E)$ for infinitely many primes $p$ of good reduction.

### Acknowledgments

We would like to thank Gökhan Soydan, Mohamed Wafik and Tuǧba Yesin for
several comments and suggestions. M. Sadek is supported by The Scientific and
Technological Research Council of Turkey, TÜBİTAK; research grant: ARDEB
1001/122F312 and BAGEP Award of the Science Academy, Turkey.

##  2\. Divisibility of orders modulo almost all primes

Throughout this work, $K$ will denote a number field with ring of integers
$R$. If $\mathfrak{p}$ is a prime ideal in $R$, then we write
$k_{\mathfrak{p}}$ for the residue field $R/\mathfrak{p}$, and we set
$p=\operatorname{char}k_{{\mathfrak{p}}}$. We denote the corresponding
discrete ${\mathfrak{p}}$-valuation by $\nu_{{\mathfrak{p}}}$.

Given an elliptic curve $E$ defined over $K$, we write $S_{E}$ for the set of
primes of bad reduction of $E$, and $\widetilde{E}_{{\mathfrak{p}}}$ for the
reduction of $E$ mod ${\mathfrak{p}}$.

###  2.1. Density $1$

In this subsection, we consider the following question whose answer is known
to the experts. However, we prefer to display it for the self-containment of
the article.

######  Question 2.1.

Let $m\geq 2$ and $\beta$ be an integer with $0\leq\beta\leq m-1$. Can we
classify all elliptic curves $E$ over $K$ such that
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv\beta\mod{m}$ for
primes $\mathfrak{p}$ in $K$ of density $1$?

The following lemma shows that the only possibility for $\beta$ in Question
2.1 is $\beta=0$.

######  Lemma 2.2.

Let $m\geq 2$ be an integer. Let $E$ be an elliptic curve defined over $K$. If
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv\beta\mod{m}$ for
every prime ${\mathfrak{p}}\not\in S_{E}$ and ${\mathfrak{p}}\nmid(m)$, then
$\beta\equiv 0\mod m$.

Proof: The statement
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv\beta\mod{m}$ for
every prime ${\mathfrak{p}}\not\in S_{E}$ is equivalent to
$1+\det(\sigma)-\operatorname{Tr}(\sigma)\equiv\beta\mod{m}$ for all
$\sigma\in G(E,m)$, where
$G(E,m)\subset\operatorname{GL}(2,\mathbb{Z}/m\mathbb{Z})$ is the subgroup
defined by the action of the absolute Galois group
$\operatorname{Gal}(\overline{K}/K)$ on the set of $m$-torsion points $E[m]$
of $E$, [13, Theorem 2.3]. Taking $\sigma$ to be the identity matrix concludes
the proof. $\Box$

The following can be found in [36, IV-6] or [15, Theorem 2].

######  Theorem 2.3.

Let $m\geq 2$ be an integer. Let $E$ be an elliptic curve defined over $K$.
The following statements are equivalent:

  * a)

$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv 0\mod{m}$ for a
set of primes ${\mathfrak{p}}$ of density $1$ in $K$.

  * b)

There exists an elliptic curve $E^{\prime}$ over $K$ such that:

    * i)

$E$ is $K$-isogenous to $E^{\prime}$; and

    * ii)

$|E^{\prime}(K)_{\operatorname{tors}}|\equiv 0\mod{m}$.

Let $\ell$ be a rational prime. Fixing $r\geq 1$, one remarks that if the set
of primes ${\mathfrak{p}}\not\in S_{E}$ for which
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv 0\mod{\ell^{r}}$
is of density $1$, then
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv 0\mod{\ell^{r}}$
for all ${\mathfrak{p}}\not\in S_{E}$ and ${\mathfrak{p}}\nmid(\ell)$, see
[36, IV-6].

###  2.2. Applications

In the current subsection, we list down some of the consequences of Serre’s
work. The following corollary follows from Theorem 2.3.

######  Corollary 2.4.

Let $m\geq 2$ be an integer. Let $E$ be an elliptic curve over a number field
$K$ such that $[K:\mathbb{Q}]=d$. Assume that $E$ satisfies
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv\beta\mod{m}$ for
primes ${\mathfrak{p}}$ of density $1$ in $K$. Then the following statements
hold:

  * i)

$\beta\equiv 0\mod{m}$.

  * ii)

There is a constant $B_{d}$ such that $|m|<B_{d}$.

  * iii)

If $m=\ell^{n}$, $\ell$ is prime and $n\geq 1$, then $m\leq
129(5^{d}-1)(3d)^{6}$.

###### Proof.

(ii) follows from Merel’s uniform boundedness of torsion orders of elliptic
curves over number fields of degree $d$, see [26, Theorem 1], whereas (iii)
follows from the work of Merel-Oseterlé-Parent, [28]. ∎

The bound $B_{d}$ of Corollary 2.4 is computed explicitly for $d=1$ by Mazur
in [25], for $d=2$, in [17], and for $d=3$ in [6]. Setting

$\mathcal{S}_{d}=\\{m:\text{there is an elliptic curve $E$ defined over a
number field }K\text{ with }[K:\mathbb{Q}]=d\text{ and
}|E(K)_{\operatorname{tors}}|\equiv 0\mod m\\},$

one sees that

$\displaystyle\mathcal{S}_{1}$ $\displaystyle=$
$\displaystyle\\{2,3,4,5,6,7,8,9,10,12,14,16\\},$
$\displaystyle\mathcal{S}_{2}$ $\displaystyle=$
$\displaystyle\\{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,20,24\\},$
$\displaystyle\mathcal{S}_{3}$ $\displaystyle=$
$\displaystyle\\{2,3,4,5,6,7,8,9,10,11,12,13,15,18,20,21,24,28\\}.$

######  Corollary 2.5.

Let $m\geq 2$ be an integer. There exists an elliptic curve $E$ defined over a
number field $K$ with $[K:\mathbb{Q}]=d$ such that
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv 0\mod{m}$ for a
set of primes ${\mathfrak{p}}$ of density $1$ in $K$ if and only if
$m\in\mathcal{S}_{d}$.

######  Corollary 2.6.

Let $E$ be an elliptic curve over a number field $K$ and $m>1$ be an integer
such that $E$ is not $K$-isogenous to an elliptic curve $E^{\prime}$ with
$|E^{\prime}(K)_{\operatorname{tors}}|\equiv 0\mod m.$ Then, there are at
least two possible values $\beta_{i}\mod{m}$, $i=1,2$, such that
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv\beta_{i}\mod{m}$
for all $\mathfrak{p}$ in a set $S_{i}$, $i=1,2$, of primes of positive
density.

In particular, if $E$ is not $K$-isogenous to an elliptic curve whose torsion
subgroup has even order, then
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$ is even, respectively
odd, for a set of primes of positive density $\delta_{1}$, respectively
$\delta_{2}$; and $\delta_{1}+\delta_{2}=1$.

###### Proof.

This follows from Theorem 2.3 and Corollary 2.5. ∎

##  3\. Divisibility in families of elliptic curves

Let $E$ be an elliptic curve defined over a number field $K$. In [19], Kubert
gave a parametrization of all elliptic curves over $K={\mathbb{Q}}$ with a
nontrivial torsion subgroup. In particular, it was proved that elliptic curves
with a fixed torsion subgroup over ${\mathbb{Q}}$ constitute an elliptic
surface over ${\mathbb{Q}}$. Similarly, a parametrization of elliptic curves
defined over a quadratic field with a certain torsion subgroup was given in
[33].

In this section, we aim at constructing families of elliptic curves $E_{t}$
over ${\mathbb{Q}}(t)$ together with a set of primes $S$ of positive density
such that for all possible values of $t=t_{0}\in{\mathbb{Z}}$, one has
$|\widetilde{E}_{t=t_{0},p}({\mathbb{F}_{p}})|$ is divisible by a fixed
integer for all $p\in S$. As an example, one can find the following family of
elliptic curves, [18, Theorem 1].

######  Example 3.1.

Let $p>3$ be a prime and $t$ be an integer such that $t(9t+4)\not\equiv 0\mod
p$. Let $E$ be an elliptic curve given by

$E_{t}:y^{2}=x^{3}-(6t+3)x-(3t^{2}+6t+2).$

Then $|\widetilde{E}_{t,p}(\mathbb{F}_{p})|\equiv 0\mod{3}$.

We recall that if $E_{\textbf{a}}$,
$\textbf{a}=(a_{1},a_{2},a_{3},a_{4},a_{6})$, is an elliptic curve defined by
a Weierstrass equation $y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$
with discriminant $\Delta_{\textbf{a}}$ , then the division polynomial
$\psi_{m}$, $m\geq 2$ is odd, is a polynomial in
$\mathbb{Z}[a_{1},a_{2},a_{3},a_{4},a_{6},x]$ of degree $(m^{2}-1)/2$, see
[37, Chapter III] for the definition and explicit description of $\psi_{m}$.
In addition, if $P=(x(P),y(P))\neq O_{E}$ is a point in $E$, then $P$ is a
point in $E[n]$ if and only if $\psi_{n}(x(P))=0$.

We will need the following standard proposition for our construction.

######  Proposition 3.2.

Let $p\neq 2$ be a rational prime. Let $K$ be a number field, with ring of
integers $R$, and $\mathfrak{p}$ be a prime of $K$ with
$\operatorname{char}k_{\mathfrak{p}}\neq 2,3$.

Fix $T\in R$. Let $\textbf{A}=(A_{1},A_{2},A_{3},A_{4},A_{6})$ be a
$k_{\mathfrak{p}}$-solution of the polynomial equation
$G_{T}(a_{1},a_{2},a_{3},a_{4},a_{6})\equiv 0\mod{\mathfrak{p}}$, where

$G_{T}(a_{1},a_{2},a_{3},a_{4},a_{6})=\psi_{p}(a_{1},a_{2},a_{3},a_{4},a_{6},T)\in
k_{\mathfrak{p}}[a_{1},a_{2},a_{3},a_{4},a_{6}].$

If the following two conditions hold

  * i)

$\Delta_{\textbf{A}}\not\equiv 0\mod\mathfrak{p}$, and

  * ii)

$z_{T}^{2}+A_{1}Tz_{T}+A_{3}z_{T}\equiv
T^{3}+A_{2}T^{2}+A_{4}T+A_{6}\mod{\mathfrak{p}}$ for some $z_{T}\in
k_{\mathfrak{p}}$,

then the elliptic curve
$E_{\textbf{A}}:y^{2}+A_{1}xy+A_{3}y=x^{3}+A_{2}x^{2}+A_{4}x+A_{6}$ over
$k_{\mathfrak{p}}$ satisfies $|E_{\textbf{A}}(k_{\mathfrak{p}})|\equiv
0\mod{p}$.

###### Proof.

Condition i) guarantees that $E_{\textbf{A}}$ is an elliptic curve over
$k_{\mathfrak{p}}$, whereas condition ii) implies that
$P_{T}=(T\mod{\mathfrak{p}},z_{T})\in E_{\textbf{A}}(k_{\mathfrak{p}})$.
Finally, the fact that $G_{T}(A_{1},A_{2},A_{3},A_{4},A_{6})\equiv
0\mod\mathfrak{p}$ asserts that $P_{T}$ is a point of order $p$ in
$E_{\mathbf{A}}(k_{\mathfrak{p}})$. ∎

######  Theorem 3.3.

Consider the elliptic curve

$\displaystyle
E_{t}:y^{2}=f_{t}(x):=x^{3}-3(t^{2}+1)x^{2}+3x-1,\qquad\text{where
}\Delta(E_{t})=-432\ t^{4}(9+4t^{2})\neq 0.$

For any $t\in{\mathbb{Q}}\setminus\\{0\\}$, one has
$|\widetilde{E}_{t,p}(\mathbb{F}_{p})|\equiv 0\mod 3$ for all primes $p\neq
2,3$ such that $\nu_{p}(t(9+4t^{2}))=0$, in particular, $E_{t}$ is
${\mathbb{Q}}$-isogenous to an elliptic curve with a ${\mathbb{Q}}$-torsion
point of order $3$.

Moreover, there are infinitely many rational values of $t$ such that
$|\widetilde{E}_{t,p}({\mathbb{F}_{p}})|\equiv 0\mod 6$ for a set $S$ of
primes $p$ of density at least $2/3$; and
$|\widetilde{E}_{t,p}({\mathbb{F}_{p}})|\equiv 0\mod 12$ for at least half of
the primes in $S$.

Proof: The $3$-rd division polynomial $\psi_{3}$ of the elliptic curve $E_{t}$
is given by

$\displaystyle\psi_{3}(x)$ $\displaystyle=$
$\displaystyle-3(-1+x)(1+4t^{2}-3x+4t^{2}x+3x^{2}+4t^{2}x^{2}-x^{3}).$

One notices that $f_{t}(1)=-3t^{2}$. It follow that for any prime $p$ with
$\nu_{p}(t(9+4t^{2}))=0$ such that $p\equiv 1\mod 3$, one has
$f_{t}(1)\equiv(z_{t})^{2}\mod p$ for some $z_{t}\in{\mathbb{F}_{p}}$. In
other words, $(1,z_{t})\in E_{t}({\mathbb{F}_{p}})$. Now, Proposition 3.2
implies that for all primes $p$ of good reduction of $E_{t}$ with $p\equiv
1\mod 3$, one has that $|\widetilde{E}_{t,p}(\mathbb{F}_{p})|\equiv 0\mod 3$.

For primes $p$ such that $p\equiv 2\mod 3$, we consider
$\phi(x):=\psi_{3}(x)/(-3(-1+x))$. The discriminant of $\phi(x)$ is
$-48t^{4}(9+4t^{2})^{2}$. This means that for any value of $t$ such that
$\nu_{p}(\Delta(E_{t}))\neq 0$, the discriminant of $\phi(x)$ is not a square
in ${\mathbb{F}_{p}}$ because $-3$ is not a square in ${\mathbb{F}_{p}}$. It
follows that $\phi(x)$ has exactly one root modulo $p$, say $x_{0}$, see [2,
Proposition 4.8.10]. Now, for any of the values $t$ described above, one may
see that $(x_{0},t(2+x_{0}))\in\widetilde{E}_{t,p}({\mathbb{F}_{p}})$. Again,
Proposition 3.2 implies that for all primes $p$ such that $p\equiv 2\mod 3$
and $\nu_{p}(t(9+4t^{2}))=0$, one has that
$|\widetilde{E}_{t,p}(\mathbb{F}_{p})|\equiv 0\mod 3$.

Since for any value of $t$ such that $p$ is a prime of good reduction for
$E_{t}$, one has $|E_{t}(\mathbb{F}_{p})|\equiv 0\mod 3$, $i=1,2$, it follows
that $E_{t}$ is ${\mathbb{Q}}$-isogenous to an elliptic curve $E^{\prime}_{t}$
with ${\mathbb{Z}}/3{\mathbb{Z}}\subseteq
E^{\prime}_{t}({\mathbb{Q}})_{\operatorname{tors}}$, see Theorem 2.3.

The polynomial $f_{t}$ is irreducible over ${\mathbb{Q}}(t)$. Moreover, it is
easily seen that the discriminant of $f_{t}$ is not a square in
${\mathbb{Q}}(t)$. It follows that the Galois group of $f_{t}$ over
${\mathbb{Q}}(t)$ is the symmetric group $S_{3}$ on a set of three elements.
Hilbert’s irreducibility theorem implies the existence of infinitely many
rational values $t=t_{0}$ for which $f_{t_{0}}$ has Galois group $S_{3}$. In
accordance with Chebotarev’s density theorem, the density of primes $p$ for
which $f_{t_{0}}$ has at least one root modulo $p$ is $2/3$. It follows that
for each such $t=t_{0}$, the density of primes $p$ such that either
$|\widetilde{E}_{t_{0},p}({\mathbb{F}_{p}})|\equiv 0\mod 6$, when $f_{t_{0}}$
has exactly one root modulo $p$ and hence
$\widetilde{E}_{t_{0},p}({\mathbb{F}_{p}})$ has exactly one nontrivial
$2$-torsion point; or $|\widetilde{E}_{t_{0},p}({\mathbb{F}_{p}})|\equiv 0\mod
12$, when $f_{t_{0}}$ splits modulo $p$ and hence
$\widetilde{E}_{t_{0},p}({\mathbb{F}_{p}})$ has full $2$-torsion, is of
density at least $2/3$. $\Box$

We remark that the elliptic curve $E_{t}$ in Theorem 3.3 might be described by
the short Weierstrass equation $y^{2}=x^{3}+(-3888\ t^{4}-7776\ t^{2})\
x+(-93312\ t^{6}-279936\ t^{4}-139968\ t^{2})$. More importantly, $E_{t}$ does
not possess a torsion point of order $3$ over ${\mathbb{Q}}(t)$.

######  Theorem 3.4.

Let $E_{t}$ be the elliptic curve described by

$\displaystyle E_{t}:y^{2}=g_{t}(x):=x^{3}-7t\ x^{2}+96t^{2}\ x+256\
t^{3},\qquad\text{where }\Delta(E_{t})=-121634816\ t^{6}\neq 0.$

For any $t\in{\mathbb{Q}}\setminus{\mathbb{Q}}^{2}$ and any prime $p\neq
2,3,29$ with $\nu_{p}(t)=0$ such that $\left(\frac{t}{p}\right)=1$, one has
$|\widetilde{E}_{t,p}(\mathbb{F}_{p})|\equiv 0\mod 5$.

Moreover, there are infinitely many rational values of $t$ such that
$|\widetilde{E}_{t,p}({\mathbb{F}_{p}})|\equiv 0\mod 10$ for a set $S$ of
primes $p$ of density at least $1/6$; and
$|\widetilde{E}_{t,p}({\mathbb{F}_{p}})|\equiv 0\mod 20$ for a positive
proportion of the primes in $S$.

Proof: The $5$-th division polynomial $\psi_{5}$ of the elliptic curve $E_{t}$
is given by

$\displaystyle\psi_{5}(x)=$ $\displaystyle-$ $\displaystyle x\ (16\
t-x)(343597383680\ t^{10}+42949672960\ t^{9}\ x-5368709120\ t^{8}\ x^{2}$
$\displaystyle+$ $\displaystyle 1929379840\ t^{7}\ x^{3}-56623104\ t^{6}\
x^{4}+33816576\ t^{5}\ x^{5}-835584\ t^{4}\ x^{6}+135936\ t^{3}\ x^{7}$
$\displaystyle+$ $\displaystyle 5776\ t^{2}\ x^{8}-60\ t\ x^{9}+5\ x^{10}).$

Now one obtains $g_{t}(0)=2^{8}\ t^{3}$. Therefore, if
$\left(\frac{t}{p}\right)=1$, then
$(0,2^{4}tm_{t})\in\widetilde{E}_{t,p}({\mathbb{F}_{p}})$ where $t\equiv
m_{t}^{2}\mod p$. We now conclude using Proposition 3.2.

The polynomial $g_{t}$ is irreducible over ${\mathbb{Q}}(t)$ with Galois group
$S_{3}$. Hilbert’s irreducibility theorem yields the existence of infinitely
many rational values $t=t_{0}$ for which $g_{t_{0}}$ has Galois group $S_{3}$.
According to Chebotarev’s density theorem, the density of primes $p$ for which
$g_{t_{0}}$ has at least one root modulo $p$ is $2/3$. Since the set of primes
$p$ for which $\left(\frac{t}{p}\right)=1$ and hence
$|\widetilde{E}_{t,p}(\mathbb{F}_{p})|\equiv 0\mod 5$ is $1/2$, one gets that
the density of primes $p$ such that either
$|\widetilde{E}_{t_{0},p}({\mathbb{F}_{p}})|\equiv 0\mod 10$ or
$|\widetilde{E}_{t_{0},p}({\mathbb{F}_{p}})|\equiv 0\mod 20$ is at least
$1/6$. $\Box$

######  Remark 3.5.

The curve $E_{t}$ in Theorem 3.4 does not possess a rational point of order
$5$ over ${\mathbb{Q}}(t)$. More precisely, the torsion subgroup of $E_{t}$
over ${\mathbb{Q}}(t)$ is trivial. In addition, for every
$t\in{\mathbb{Q}}\setminus\\{0\\}$, the set of primes $p$ for which
$\left(\frac{t}{p}\right)=1$, hence
$|\widetilde{E}_{t,p}(\mathbb{F}_{p})|\equiv 0\mod 5$, is $1/2$.

One also notices that $E_{t}$ is the quadratic twist by $t$ of the elliptic
curve $E:y^{2}=x^{3}-7\ x^{2}+96\ x+256$. Therefore, for all quadratic twists
of $E$ over ${\mathbb{Q}}$, the order of the reduction of the twist is
divisible by $5$ modulo primes of density at least $1/2$.

##  4\. Orders of reductions modulo all primes

Let $E$ be an elliptic curve defined over $K$. Let $S$ be a set of primes in
$K$. We consider the greatest common divisor

$\gcd_{{\mathfrak{p}}\in
S}|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$

of $|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$ where
${\mathfrak{p}}$ runs over $S$. In view of §2, one has the following.

######  Lemma 4.1.

[15, Theorem 2 (bis)] Let $E$ be an elliptic curve over a number field $K$.
Let $S$ be any set of primes of $K$ of density $1$ which consists entirely of
primes ${\mathfrak{p}}$ at which $E$ has good reduction and whose ramification
indices $e_{{\mathfrak{p}}}$ satisfy $e_{{\mathfrak{p}}}<p-1$ (e.g. $S$ is the
set of all odd unramified primes of good reduction for $E$). Then we have

$\gcd_{{\mathfrak{p}}\in
S}|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|=\sup\\{|E^{\prime}(K)_{\operatorname{tors}}|:E^{\prime}\textrm{
is $K$-isogenous to }E\\}.$

In this section, we will evaluate the positive integer
$\gcd_{{\mathfrak{p}}}|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$
where the greatest common divisor is taken over all primes in $K$.

One knows that there is no elliptic curve $E$ defined over the rational field
$\mathbb{Q}$ that has everywhere good reduction, [37, Chapter VIII, Exercise
8.15]. However, there are elliptic curves defined over number fields of degree
$d\geq 2$ with everywhere good reduction. For such elliptic curves, the
integer
$\gcd_{{\mathfrak{p}}}|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$ is
determined by Lemma 4.1.

######  Example 4.2.

Let $E$ be the elliptic curve defined over $\mathbb{Q}(\sqrt{33})$ by

$E:y^{2}=x^{3}+96(-32^{3}-1728)u^{2}x-2(-32^{3}-1728)^{2}u^{3}$

with $u=-462-84\sqrt{33}$. Using MAGMA , [1], one sees that
$E(K)_{\operatorname{tors}}\cong\mathbb{Z}/3\mathbb{Z}$. Moreover, $E$ has
good reduction everywhere. It follows that
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv 0\mod{3}$ for
every prime $\mathfrak{p}$ in ${\mathbb{Q}}(\sqrt{33})$, hence
$\gcd_{{\mathfrak{p}}}|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|=3$.

######  Example 4.3.

Let $E$ be the elliptic curve defined over $\mathbb{Q}(\sqrt{6})$ by

$E:y^{2}+\frac{3(-5-2\sqrt{6})+31}{2}xy+(-5+2\sqrt{6})^{2}y=x^{3}.$

MAGMA , [1], asserts that
$E(K)_{\operatorname{tors}}\cong\mathbb{Z}/6\mathbb{Z}$ and that $E$ has good
reduction everywhere. This implies that
$\gcd_{{\mathfrak{p}}}|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|=6$.

For elliptic curves with at least one prime of bad reduction, one needs the
following proposition to evaluate
$\gcd_{{\mathfrak{p}}}|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$.

######  Proposition 4.4.

Let $m\geq 2$ be an integer. Let $E$ be an elliptic curve defined over a
number field $K$ with $[K:{\mathbb{Q}}]=d$ such that
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv 0\mod{m}$ for all
primes $\mathfrak{p}$ in $K$. Then $m\in\mathcal{S}_{d}$ and
$m|m_{\mathfrak{p}}$ for every prime ${\mathfrak{p}}$ of bad reduction of $E$;
where $m_{{\mathfrak{p}}}$ is defined as follows

$m_{\mathfrak{p}}=\begin{cases}\operatorname{Norm}_{K/{\mathbb{Q}}}({\mathfrak{p}})&\text{
if E has split multiplicative reduction$\mod{\mathfrak{p}}$}\\\
\operatorname{Norm}_{K/{\mathbb{Q}}}({\mathfrak{p}})+2&\text{if E has non-
split multiplicative reduction$\mod{\mathfrak{p}}$}\\\
\operatorname{Norm}_{K/{\mathbb{Q}}}({\mathfrak{p}})+1&\text{if E has additive
reduction$\mod{\mathfrak{p}}$.}\end{cases}$

In particular, if $K={\mathbb{Q}}$, then $m\leq 5$.

###### Proof.

The proof follows directly from Corollary 2.5 and the fact that
$m_{{\mathfrak{p}}}=|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$ when
${\mathfrak{p}}$ is a prime of bad reduction of $E$, [37, Chapter III,
Exercise 3.5].

Finally, over the rational field $\mathbb{Q}$, one sees that
$m_{2}\in\\{2,3,4\\}$. Moreover, Hasse bound implies that
$|\widetilde{E}_{2}(\mathbb{F}_{2})|\leq 5$ if $E$ has good reduction at $2$.
∎

In what follows, we show that there is no elliptic curve $E$ over
${\mathbb{Q}}$ with $\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|=5$. For this
purpose, we need to analyze the possible reduction types of the following
families of elliptic curves.

######  Lemma 4.5.

Let $k\geq 1$ be an integer and $\epsilon=\pm 1$. Consider the following two
families of elliptic curves defined over ${\mathbb{Q}}$

$\displaystyle E^{1}_{k}$ $\displaystyle:$ $\displaystyle y^{2}+(1-\epsilon
5^{k})xy-\epsilon 5^{k}y=x^{3}-\epsilon 5^{k}x^{2},$ $\displaystyle E^{2}_{k}$
$\displaystyle:$ $\displaystyle y^{2}+(\epsilon
5^{k}-1)xy-5^{2k}y=x^{3}-\epsilon 5^{k}x^{2}.$

For $i=1,2$, one has that

  * i)

$E^{i}_{k}$ has split multiplicative reduction with Kodaira symbol ${\rm
I}_{5k}$ modulo $5$.

  * ii)

$E^{i}_{k}$ has no primes of additive reduction type.

Proof: The discriminant of $E_{k}^{1}$ is $\Delta(E_{k}^{1})=\epsilon
5^{5k}\left(5^{2k}-11\epsilon 5^{k}-1\right)$ while the $c_{4}$-invariant is
given by $c_{4}(E^{1}_{k})=1+12\epsilon 5^{k}+14\cdot 5^{2k}-12\epsilon
5^{3k}+5^{4k}.$

Statement i) is straightforward.

ii) A prime $\ell\neq 5$ is a prime of additive reduction of $E_{k}^{1}$ if it
divides both $c_{4}(E^{1}_{k})$ and $\Delta(E^{1}_{k})$. Therefore, $\ell$
must divide $11\epsilon 5^{k}+1$, the remainder of the quotient
$5^{5k}c_{4}(E^{1}_{k})/\Delta(E^{1}_{k})$, hence $\ell$ must be $5$ which is
a contradiction. It follows that $E_{k}^{1}$ cannot have additive reduction.

The proof is similar for the curve $E^{2}_{k}$. $\Box$

We recall the following fact on isogenous elliptic curves. If $E$ and
$E^{\prime}$ are two $K$-isogenous elliptic curves defined over $K$, then $E$
has good, split multiplicative, non-split multiplicative, or additive
reduction at a prime $p$ if and only if $E^{\prime}$ has the same of these
four reduction types at $p$. This is due to the fact that an isogeny sends a
node, respectively cusp, to a node, respectively cusp, moreover, the field of
definition of the tangent lines at a node is preserved by isogeny.

######  Theorem 4.6.

Let $E$ be an elliptic curve defined over ${\mathbb{Q}}$. One has
$\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|\leq 4$.

Proof: In view of Proposition 4.4, one knows that
$\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|\leq 5$. Therefore, one only needs
to eliminate the occurrence of elliptic curves over ${\mathbb{Q}}$ with
$\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|=5$. If such an elliptic curve
exits, then it follows that $E$ is ${\mathbb{Q}}$-isogenous to an elliptic
curve $E^{\prime}$ over ${\mathbb{Q}}$ with
$|E^{\prime}({\mathbb{Q}})_{\operatorname{tors}}|\equiv 0\mod 5$, see
Proposition 4.4. In addition, one must have that $E$ has good reduction at
$2$, since otherwise $|\widetilde{E}_{p}(\mathbb{F}_{2})|\leq 4$.

Since $E^{\prime}$ has a ${\mathbb{Q}}$-rational point of order $5$, this
yields the existence of at least one prime $\ell$ such that $E^{\prime}$ has
split multiplicative reduction modulo $\ell$ of type $\rm{I}_{e}$ with $5\mid
e$, except for the curve $E_{11}$ labeled $11a3$ in Cremona’s table [3], see
for example [22, Proposition 2.7]. In fact, the curve $11a3$ has split
multiplicative reduction of type $\rm{I}_{1}$ at its only bad prime $11$,
hence $\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|=1$ for any elliptic curve
$E$ over ${\mathbb{Q}}$ that is ${\mathbb{Q}}$-isogenous to $E_{11}$.

Now, that $\ell$ is a prime modulo which $E^{\prime}$ has split multiplicative
reduction, one has that
$|\widetilde{E^{\prime}}_{\ell}(\mathbb{F}_{\ell})|=\ell$. Therefore, as
$\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|=5$, the prime $\ell$ must be the
only prime at which $E^{\prime}$ has split multiplicative reduction, and
$\ell=5$. One knows that an elliptic curve with a rational torsion point of
order $5$ can be written as

$E:y^{2}+(1-\lambda)xy-\lambda y=x^{3}-\lambda x^{2}$

for some $\lambda\in{\mathbb{Q}}$, where
$\Delta(E)=\lambda^{5}(\lambda^{2}-11\lambda-1)$ and
$c_{4}(E)=1+12\lambda+14\lambda^{2}-12\lambda^{3}+\lambda^{4}$, [19, Table 3].
One sees that if for a prime $\ell$, $\nu_{\ell}(\lambda)=e>0$, then the
reduction type of $E$ at $\lambda$ is split multiplicative of type $I_{5e}$.
If $\nu_{\ell}(\lambda)=e<0$, then writing $\mu=1/\lambda$, one sees that $E$
can be described by

$y^{2}+(\mu-1)xy-\mu^{2}y=x^{3}-\mu x^{2},$

with discriminant $\mu^{5}(1-11\mu-\mu^{2})$, and the reduction type is again
split multiplicative of type $I_{-5e}$. Since we know that $5$ is the only
prime of split multiplicative reduction, the elliptic curve $E$ may be
described by either of the equations

$\displaystyle E^{1}_{k}$ $\displaystyle:$ $\displaystyle y^{2}+(1-\epsilon
5^{k})xy-\epsilon 5^{k}y=x^{3}-\epsilon 5^{k}x^{2},$ $\displaystyle E^{2}_{k}$
$\displaystyle:$ $\displaystyle y^{2}+(\epsilon
5^{k}-1)xy-5^{2k}y=x^{3}-\epsilon 5^{k}x^{2},\,k\geq 1.$

In case, $\nu_{\ell}(\lambda)=0$, then the Weierstrass equation describing $E$
is either

$\displaystyle y^{2}-y$ $\displaystyle=$ $\displaystyle x^{3}-x^{2},$
$\displaystyle y^{2}+2xy+y$ $\displaystyle=$ $\displaystyle x^{3}+x^{2}.$

Both curves have split multiplicative reduction at the prime $11$.

As for the curves $E^{1}_{k}$, one easily sees that $5^{2k}-11\epsilon
5^{k}-1\neq\pm 1$ for any $k\geq 1$. Therefore, we may assume that there is a
prime $\ell$ such that $\ell\mid(5^{2k}-11\epsilon 5^{k}-1)$. According to
Lemma 4.5 and the fact that $5$ is the only prime of split multiplicative
reduction, we see that $\ell$ must be a prime of non-split multiplicative
reduction. Since $\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|=5$, we must have
that $\ell\equiv 3\mod 5$. The point $P=(0,0)$ is of order $5$. The reduction
of $P$ cannot be singular, since otherwise the reduction type will be split
multiplicative of type ${\rm I}_{5e}$, for some $e\geq 1$, which is a
contradiction.

It follows that the reduction of $P$ is a non-singular point in
$\widetilde{E}_{\ell}(\mathbb{F}_{\ell})$. Since $\ell\neq 5$, the formal
group $\widehat{E}(\ell{\mathbb{Z}}_{\ell})$ has no $5$-torsion, see [37,
Chapter VII, Proposition 3.1], which implies that the reduction of $P$ is not
the identity in the non-singular locus of
$\widetilde{E}_{\ell}(\mathbb{F}_{\ell})$. It follows that $5$ divides the
order of the non-singular locus of $\widetilde{E}_{\ell}(\mathbb{F}_{\ell})$
which is $\ell+1$ in the case of non-split multiplicative reduction. This
contradicts that we must have $\ell\equiv 3\mod 5$. $\Box$

In the following examples we show the existence of elliptic curves over
${\mathbb{Q}}$ for which $\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|=2,3$ or
$4$.

######  Example 4.7.

Consider the elliptic curve $E:y^{2}+xy+y=x^{3}-x^{2}-199x+510$ over
${\mathbb{Q}}$. We have
$E(\mathbb{Q})_{\operatorname{tors}}\cong\mathbb{Z}/4\mathbb{Z}$. The only bad
prime of $E$ is $17$ where $E$ has additive reduction. Therefore,
$|\widetilde{E}_{p}(\mathbb{F}_{p})|$ is even for every prime $p$, and
$\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|=2$.

######  Example 4.8.

Consider the elliptic curve $E:y^{2}=x^{3}+x^{2}-333x-3537$ over
${\mathbb{Q}}$. The torsion subgroup of $E({\mathbb{Q}})$ is
$E(\mathbb{Q})_{\operatorname{tors}}\cong\mathbb{Z}/3\mathbb{Z}$. The bad
primes of $E$ are $2$, $3$ and $5$, where $E$ has additive reduction at the
primes $2$ and $5$, whereas $E$ has split multiplicative reduction at $3$. It
follows that $|\widetilde{E}_{p}(\mathbb{F}_{2})|=3$,
$|\widetilde{E}_{p}(\mathbb{F}_{3})|=3$ and
$|\widetilde{E}_{p}(\mathbb{F}_{5})|=6$. Therefore, Proposition 4.4 implies
that $\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|=3$.

######  Example 4.9.

The torsion subgroup of the elliptic curve $E:y^{2}+xy=x^{3}-x^{2}-1773x-5270$
over ${\mathbb{Q}}$ satisfies
$E(\mathbb{Q})_{\operatorname{tors}}\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$.
The bad primes of $E$ are $3$ and $7$. $E$ had additive reduction at both
primes $3$ and $7$. It follows that
$\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|=4$, see Proposition 4.4.

######  Example 4.10.

Consider the set of all elliptic curves $E$ defined over the rationals
$\mathbb{Q}$ with conductor $N_{E}=p^{2}$, $7\leq p\leq 5000$, such that
$|c_{6}(E)|\leq 25\times 10^{6}$ as listed in [8]. There are exactly two such
elliptic curves with $\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|=2$. These
curves are $y^{2}+xy=x^{3}-x^{2}-37x-78$ and $y^{2}-xy=x^{3}-x^{2}-2x-1$.

######  Corollary 4.11.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with conductor
$N_{E}=3\times 2^{\lambda}$ or $N_{E}=9\times 2^{\lambda}$. One has
$\gcd_{p}|\widetilde{E}_{p}(\mathbb{F}_{p})|=1$.

###### Proof.

In [29], Ogg showed that all such curves have a rational point of order $2$.
The elliptic curves with conductor $N_{E}=3\times 2^{\lambda}$ have additive
reduction at $2$ and multiplicative reduction at $3$, whereas the elliptic
curves with conductor $N_{E}=9\times 2^{\lambda}$ have additive reduction at
both $2$ and $3$, hence the result, see Proposition 4.4. ∎

##  5\. Local results

In this section, we discuss the relation between the size of the reduction of
an elliptic curve $E$ modulo a rational prime $p$ of good reduction of $E$ and
the size of the reduction of $E$ modulo a prime that lies above $p$.

Let $K$ be a number field of degree $d$ with ring of integers
${\mathcal{O}_{K}}$. Let $p$ be a rational prime. If the prime ideal
factorization of $p$ in ${\mathcal{O}_{K}}$ is given by
$p=\prod{\mathfrak{p}}_{i}^{e({\mathfrak{p}}_{i}|p)}$, we call
$e({\mathfrak{p}}_{i}|p)$ the ramification index of ${\mathfrak{p}}_{i}$ over
$p$, and $f({\mathfrak{p}}_{i}|p)=[k_{{\mathfrak{p}}_{i}}:{\mathbb{F}_{p}}]$
the inertial degree of ${\mathfrak{p}}_{i}$ over $p$, where
$\sum_{i}e({\mathfrak{p}}_{i}|p)f({\mathfrak{p}}_{i}|p)=d$. If
$e({\mathfrak{p}}_{i}|p)>1$ for some $i$, then $p$ is said to ramify in $K$,
otherwise $p$ is said to be unramified in $K$. It is known that there are only
finitely many rational primes that ramify in a number field $K$, [23, Chapter
3].

We will write $E^{d}$ for the quadratic twist of $E$ by $d$, where $d$ is an
element in the base field.

######  Proposition 5.1.

Let $E$ be an elliptic curve defined over ${\mathbb{Q}}$. Let ${\mathfrak{p}}$
be a prime in $K$ of good reduction of $E$ over $K$. Assume that
${\mathfrak{p}}\mid p$ where $p$ is unramified in $K$. The following hold.

  * i)

If $f({\mathfrak{p}}|p)=1$, then
$\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})\cong\widetilde{E}_{p}({\mathbb{F}_{p}})$.

  * ii)

If $f({\mathfrak{p}}|p)=2$, then
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|=|\widetilde{E}_{p}(\mathbb{F}_{p})|\cdot|\widetilde{E}_{p}^{d}(\mathbb{F}_{p})|$,
where $d\in{\mathbb{F}_{p}}$ is a nonsquare.

  * iii)

If $f({\mathfrak{p}}|p)=3$, then
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|=|\widetilde{E}_{p}(\mathbb{F}_{p})|\left(p^{2}-p+1+(p+1)|\widetilde{E}_{p}^{d}(\mathbb{F}_{p})|-|\widetilde{E}_{p}(\mathbb{F}_{p})|\cdot|\widetilde{E}_{p}^{d}(\mathbb{F}_{p})|\right)$,
where $d\in{\mathbb{F}_{p}}$ is a nonsquare.

In general, $|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv
0\mod|\widetilde{E}_{p}(\mathbb{F}_{p})|$.

Proof: i) Since $p$ is unramified in $K$ and $f({\mathfrak{p}}|p)=1$, it
follows that $|k_{{\mathfrak{p}}}|=p$, hence
$k_{{\mathfrak{p}}}\cong{\mathbb{F}_{p}}$.

For ii), if $f({\mathfrak{p}}|p)=2$, then
$k_{{\mathfrak{p}}}\cong\mathbb{F}_{p^{2}}$. We recall that
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|=1+p^{2}-a_{\mathfrak{p}}(E)$,
where $a_{\mathfrak{p}}(E)$ is the trace of Frobenius of $E$ at $\mathfrak{p}$
and $N(\mathfrak{p})=p^{2}$. If $E$ is defined over ${\mathbb{Q}}$ has good
reduction at a rational prime $p$, then one may use the following recurrence

$a_{n+2}(E)=a_{1}(E)a_{n+1}(E)-pa_{n}(E),\text{ for all }n\geq 0$

where $a_{0}=2$ and $a_{1}=1+q-|\widetilde{E}(\mathbb{F}_{q})|$ to compute the
trace of Frobenius $a_{n}(E)$ of $E$ over extensions $\mathbb{F}_{q}$,
$q=p^{n}$, see [37, Chapter V, Exercise 5.13]. It follows that

$\displaystyle
a_{\mathfrak{p}}(E)=(p+1-|\widetilde{E}_{p}({\mathbb{F}_{p}})|)^{2}-2p=p^{2}+1-2(p+1)|\widetilde{E}_{p}({\mathbb{F}_{p}})|+|\widetilde{E}_{p}({\mathbb{F}_{p}})|^{2}.$

It follows that

$\displaystyle|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$
$\displaystyle=$
$\displaystyle|\widetilde{E}_{p}({\mathbb{F}_{p}})|\left(2p+2-|\widetilde{E}_{p}({\mathbb{F}_{p}})|\right).$

We recall that for $d\in{\mathbb{F}_{p}}$, if $d$ is not a square, then one
has
$\widetilde{E}_{p}^{d}({\mathbb{F}_{p}})=2p+2-\widetilde{E}_{p}({\mathbb{F}_{p}})$,
see [35, Proposition 3.21]. A similar argument yields iii).

In general, if $E_{n}=|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|$
when $f({\mathfrak{p}}|p)=n$, then substituting in the recurrence above, one
obtains

$E_{n+2}=E_{n+1}+p\cdot E_{n+1}-p\cdot
E_{n}+|\widetilde{E}_{p}({\mathbb{F}_{p}})|(1+p^{n+1}-E_{n+1})).$

Therefore, a simple induction argument implies that
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv
0\mod|\widetilde{E}_{p}({\mathbb{F}_{p}})|$. $\Box$

The followings are direct consequences of Proposition 5.1.

######  Corollary 5.2.

Let $E$ be an elliptic curve defined over ${\mathbb{Q}}$. Let ${\mathfrak{p}}$
be a prime in $K$ of good reduction of $E$ over $K$. Assume that
${\mathfrak{p}}\mid p$ where $p$ is unramified in $K$ and
$f({\mathfrak{p}}|p)=2$. If
$\nu_{2}(|\widetilde{E}_{p}(\mathbb{F}_{p})|)=r\geq 1$, then
$\nu_{2}(|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|)\geq r+1$;
whereas if $\nu_{2}(|\widetilde{E}_{p}(\mathbb{F}_{p})|)=0$, then
$\nu_{2}(|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|)=0$.

######  Corollary 5.3.

Let $E$ be an elliptic curve defined over ${\mathbb{Q}}$. Let $p$ be a prime
of good reduction of $E$. Let $K=\mathbb{Q}(\sqrt{d})$ where $d$ is a square
free integer, and $\mathfrak{p}\mid p$ be a prime in $K$. Then,

$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|=\begin{cases}|\widetilde{E}_{p}(\mathbb{F}_{p})|&\text{
if $\left(\frac{d}{p}\right)=1,\,p\neq 2$ }\\\
|\widetilde{E}_{p}(\mathbb{F}_{p})|\cdot|\widetilde{E}_{p}^{d}(\mathbb{F}_{p})|&\text{
if $\left(\frac{d}{p}\right)=-1,\,p\neq 2$, }\end{cases}$

where $\left(\frac{d}{p}\right)$ is the Legendre symbol of $d$ modulo $p$.

Proof: This follows from Proposition 5.1 by observing that $p\neq 2$ splits in
$K$ if $\left(\frac{d}{p}\right)=1$, whereas $p\neq 2$ is inert in $K$ if
$\left(\frac{d}{p}\right)=-1$. $\Box$

##  6\. Congruence classes of orders of reduction

In this section, we study the link between the growth of the torsion of an
elliptic curve $E$ over ${\mathbb{Q}}$ under a base change over a number field
of small degree and the congruence classes of orders of reduction of $E$
modulo rational primes.

Theorem 2.3 asserts that the statement $E$ is ${\mathbb{Q}}$-isogenous to an
elliptic curve $E^{\prime}$ with
$|E^{\prime}({\mathbb{Q}})_{\operatorname{tors}}|\equiv 0\mod m$ is equivalent
to $|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv 0\mod{m}$, $m\geq 2$, for all
primes $p$ in a set of primes of density $1$. The following result shows that
if $E$ is not ${\mathbb{Q}}$-isogenous to an elliptic curve whose torsion
order is divisible by $m$, then the set of primes $p$ for which
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv 0\mod{m}$ is still of positive
density.

######  Proposition 6.1.

Let $E$ be an elliptic curve over ${\mathbb{Q}}$ and $m>1$ be an integer such
that $E$ is not ${\mathbb{Q}}$-isogenous to an elliptic curve $E^{\prime}$
with $|E^{\prime}({\mathbb{Q}})_{\operatorname{tors}}|\equiv 0\mod m.$ Then
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv 0\mod{m}$ for all primes $p$ in a
set $S$ of primes of positive density $\delta<1$.

Proof: One considers the Galois field $F={\mathbb{Q}}(E[m])$ obtained by
adjoining the $x$\- and $y$\- coordinates of the $m$-torsion points of $E$ to
${\mathbb{Q}}$, where the $x$-coordinates are the roots of the $m$-th division
polynomial $\psi_{m}$ of $E$. According to Chebotarev Density Theorem, the
primes that split completely in $F$ are of density at least
$1/|\operatorname{Gal}(F/{\mathbb{Q}})|$. Since
$|E(F)_{\operatorname{tors}}|\equiv 0\mod m$, Proposition 5.1 i) implies that
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv 0\mod{m}$ for primes of density
at least $1/|\operatorname{Gal}(F/{\mathbb{Q}})|$. $\Box$

We remark that the statement in Proposition 6.1 is valid if ${\mathbb{Q}}$ is
replaced by any number field $K$.

In this section, we will establish the existence of families of elliptic
curves $E$ over ${\mathbb{Q}}$ together with integers $m$ such that one can
explicitly compute the density of primes $p$ for which
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv r\mod m$, $0\leq r\leq m-1$. For
this purpose, we focus on elliptic curves whose torsion subgroups grow over
quadratic field extensions.

In a series of papers [11, 12], the authors answer the following question.
Given an elliptic curve over ${\mathbb{Q}}$ with torsion group
$E({\mathbb{Q}})_{\operatorname{tors}}$, considering a base change of $E$ to a
quadratic number field $K$, what are the possibilities for
$E(K)_{\operatorname{tors}}$? Moreover, they listed the possible number of
quadratic fields over which the torsion of $E$ grows together with the groups
that are realized as torsion groups of $E$ over these fields. In the following
subsection, we write down the results that will be used in this work.

Let $\Phi$ be the set of possible groups that can appear as the torsion
subgroup of an elliptic curve defined over ${\mathbb{Q}}$. If $G\in\Phi$, we
write $\Phi(d,G)$, $d\geq 2$, for the set of possible groups that can appear
as the torsion subgroup over a number field $K$ of degree $d$ of an elliptic
curve $E$ defined over ${\mathbb{Q}}$ such that
$E({\mathbb{Q}})_{\operatorname{tors}}=G$. The following is [12, Theorem 1].

######  Proposition 6.2.

Let $G\in\Phi$. The set $\Phi(2,G)$ is described as follows

$G$ | $\Phi(2,G)$
---|---
$\\{0\\}$ |  $\\{0\\}$, ${\mathbb{Z}}/3{\mathbb{Z}}$, ${\mathbb{Z}}/5{\mathbb{Z}}$, ${\mathbb{Z}}/7{\mathbb{Z}}$, ${\mathbb{Z}}/9{\mathbb{Z}}$
${\mathbb{Z}}/2{\mathbb{Z}}$ |  ${\mathbb{Z}}/2{\mathbb{Z}}$, ${\mathbb{Z}}/4{\mathbb{Z}}$, ${\mathbb{Z}}/6{\mathbb{Z}}$, ${\mathbb{Z}}/8{\mathbb{Z}}$, ${\mathbb{Z}}/10{\mathbb{Z}}$, ${\mathbb{Z}}/12{\mathbb{Z}}$, ${\mathbb{Z}}/16{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/6{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/10{\mathbb{Z}}$
${\mathbb{Z}}/3{\mathbb{Z}}$ |  ${\mathbb{Z}}/3{\mathbb{Z}}$, ${\mathbb{Z}}/15{\mathbb{Z}}$, ${\mathbb{Z}}/3{\mathbb{Z}}\times{\mathbb{Z}}/3{\mathbb{Z}}$
${\mathbb{Z}}/4{\mathbb{Z}}$ |  ${\mathbb{Z}}/4{\mathbb{Z}}$, ${\mathbb{Z}}/8{\mathbb{Z}}$, ${\mathbb{Z}}/12{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/8{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/12{\mathbb{Z}}$, ${\mathbb{Z}}/4{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$
${\mathbb{Z}}/5{\mathbb{Z}}$ |  ${\mathbb{Z}}/5{\mathbb{Z}}$, ${\mathbb{Z}}/15{\mathbb{Z}}$
${\mathbb{Z}}/6{\mathbb{Z}}$ |  ${\mathbb{Z}}/6{\mathbb{Z}}$, ${\mathbb{Z}}/12{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/6{\mathbb{Z}}$, ${\mathbb{Z}}/3{\mathbb{Z}}\times{\mathbb{Z}}/6{\mathbb{Z}}$
${\mathbb{Z}}/7{\mathbb{Z}}$ | ${\mathbb{Z}}/7{\mathbb{Z}}$
${\mathbb{Z}}/8{\mathbb{Z}}$ |  ${\mathbb{Z}}/8{\mathbb{Z}}$, ${\mathbb{Z}}/16{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/8{\mathbb{Z}}$
${\mathbb{Z}}/9{\mathbb{Z}}$ | ${\mathbb{Z}}/9{\mathbb{Z}}$
${\mathbb{Z}}/10{\mathbb{Z}}$ |  ${\mathbb{Z}}/10{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/10{\mathbb{Z}}$
${\mathbb{Z}}/12{\mathbb{Z}}$ |  ${\mathbb{Z}}/12{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/12{\mathbb{Z}}$
${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$ |  ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/6{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/8{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/12{\mathbb{Z}}$
${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$ |  ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/8{\mathbb{Z}}$, ${\mathbb{Z}}/4{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$
${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/6{\mathbb{Z}}$ |  ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/6{\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/12{\mathbb{Z}}$
${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/8{\mathbb{Z}}$ | ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/8{\mathbb{Z}}$

A special case of [21, Lemma 1.1] asserts that if $K={\mathbb{Q}}(\sqrt{d})$,
where $d$ is a square free integer, then there exist homomorphisms

(1) $\displaystyle E({\mathbb{Q}})\oplus E^{d}({\mathbb{Q}})\to E(K),\qquad
E(K)\to E({\mathbb{Q}})\oplus E^{d}({\mathbb{Q}}),$

where the kernels and cokernels of both homomorphisms are annihilated by the
multiplication by-$2$-map, see [27, Lemma 1]. In particular, if $n$ is an odd
integer, then

$E({\mathbb{Q}})[n]\oplus E^{d}({\mathbb{Q}})[n]\cong E(K)[n].$

We are now in a place to discuss families of elliptic curves for which we can
find a positive integer $m\geq 2$ such that the congruence classes of the
orders of its reductions are explicitly determined modulo $m$, hence we can
compute the density at which each class occurs.

######  Theorem 6.3.

Let $K={\mathbb{Q}}(\sqrt{d})$, where $d$ is a square free integer. Let $E$ be
an elliptic curve defined over $\mathbb{Q}$ such that $E$ is
${\mathbb{Q}}$-isogenous to an elliptic curve $E^{\prime}$ with
$E^{\prime}(\mathbb{Q})_{\operatorname{tors}}\subsetneq
E^{\prime}(K)_{\operatorname{tors}}$. Assume moreover that $\ell$ is an odd
integer such that $|E^{\prime}(K)_{\operatorname{tors}}|\equiv 0\mod\ell$ and
$\gcd\left(|E^{\prime\prime}({\mathbb{Q}})_{\operatorname{tors}}|,\ell\right)=1$
for any ${\mathbb{Q}}$-isogenous elliptic curve $E^{\prime\prime}$ to $E$. If
$p\nmid 2d|E^{\prime}(K)_{\operatorname{tors}}|$ is a prime of good reduction
of $E$, then

$|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv\begin{cases}0\mod\ell&\text{ if
}\left(\frac{d}{p}\right)=1\\\ 2p+2\mod\ell&\text{ if
}\left(\frac{d}{p}\right)=-1.\\\ \end{cases}$

In particular, the density of primes $p$ such that
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv 0\mod\ell$ is at least $1/2$ .

###### Proof.

If the prime $p$ splits in $K$, i.e., $\left(\frac{d}{p}\right)=1$, then it
follows by Theorem 5.3 that
$|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\mod\ell$
for every prime ${\mathfrak{p}}$ lying above $p$. Since
$|E^{\prime}(K)_{\operatorname{tors}}|\equiv 0\mod\ell$ where $E^{\prime}$ is
${\mathbb{Q}}$-isogenous to $E$, it follows that
$|\widetilde{E}_{{\mathfrak{p}}}(k_{{\mathfrak{p}}})|\equiv 0\mod\ell$ for
every prime ${\mathfrak{p}}$ of good reduction of $E$ over $K$ that lies above
a rational prime that splits in $K$, see Theorem 2.3.

Now we assume that the prime $p$ is inert in $K$, i.e.,
$\left(\frac{d}{p}\right)=-1$. One knows that
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|=2p+2-|\widetilde{E}_{p}^{d}({\mathbb{F}_{p}})|$.
Since $\ell$ is odd, the remark before the theorem implies that $|E^{\prime
d}({\mathbb{Q}})_{\operatorname{tors}}|\equiv 0\mod\ell$. For these primes
$p$, Theorem 2.3 implies that $|\widetilde{E}_{p}^{d}({\mathbb{F}_{p}})|\equiv
0\mod\ell$. ∎

The hypotheses of Theorem 6.3 cover the elliptic curves in Proposition 6.2
with $E({\mathbb{Q}})_{\operatorname{tors}}=\\{O_{E}\\}$;
${\mathbb{Z}}/2{\mathbb{Z}}$ and ${\mathbb{Z}}/6{\mathbb{Z}},$ or
${\mathbb{Z}}/10{\mathbb{Z}}\subseteq E(K)_{\operatorname{tors}}$;
${\mathbb{Z}}/3{\mathbb{Z}}$ and
$E(K)_{\operatorname{tors}}={\mathbb{Z}}/15{\mathbb{Z}}$;
${\mathbb{Z}}/4{\mathbb{Z}}$ and ${\mathbb{Z}}/12{\mathbb{Z}}\subseteq
E(K)_{\operatorname{tors}}$; ${\mathbb{Z}}/5{\mathbb{Z}}$ and
$E(K)_{\operatorname{tors}}={\mathbb{Z}}/15{\mathbb{Z}}$;
${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$ and
${\mathbb{Z}}/6{\mathbb{Z}}\subset E(K)_{\operatorname{tors}}$.

We recall that for a prime $p\geq 5$, if the trace of Frobenius
$a_{p}(E)=p+1-|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ of $E$ at a prime of good
reduction $p$ is such that $a_{p}(E)=0$, then $p$ is said to be a
supersingular prime for $E$. Theorem 6.3 yields the following result of
possible congruence values of trace of Frobenius of elliptic curves satisfying
the assumptions of Theorem 6.3.

######  Corollary 6.4.

Let $E$, $E^{\prime}$, $K$, $d$ and $\ell$ satisfy the hypotheses of Theorem
6.3. Let $p\nmid 2d|E^{\prime}(K)_{\operatorname{tors}}|$ be a prime of good
reduction of $E$. If $p$ is a supersingular prime for $E$, then
$p\equiv-1\mod\ell$, where $\ell\in\\{3,5,7,9\\}$.

######  Example 6.5.

Let $E$ be an elliptic curve defined by $y^{2}+xy=x^{3}+x^{2}-700x+34000$ over
$\mathbb{Q}$. One may check that
$E(\mathbb{Q})_{\operatorname{tors}}\cong\mathbb{Z}/2\mathbb{Z}$ and
$E(\mathbb{Q}(\sqrt{5}))_{\operatorname{tors}}\cong\mathbb{Z}/10\mathbb{Z}$.
It follows that the torsion subgroup of the quadratic twist $E^{5}$ is
$E^{5}(\mathbb{Q})_{\operatorname{tors}}\cong\mathbb{Z}/10\mathbb{Z}$. Thus,
by Theorem 6.3, one has that

$|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv\begin{cases}0\mod{10}&\text{ if
}p\equiv 1,4\mod 5\\\ 6\mod{10}&\text{ if }p\equiv 2\mod 5\\\ 8\mod{10}&\text{
if }p\equiv 3\mod 5.\end{cases}$

We notice that $|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv 0\mod 10$ for primes
$p$ of density $\frac{1}{2}$, $|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv
6\mod{10}$ for primes of density $\frac{1}{4}$, and
$|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv 8\mod{10}$ for primes of density
$\frac{1}{4}$.

In addition, if $p$ is a supersingular prime for $E$, then $p\equiv
9\mod{10}$. More precisely, one has the following formula for the trace of
Frobenius $a_{p}(E)=p+1-|\widetilde{E}_{p}({\mathbb{F}_{p}})|$, where $p$ is a
prime of good reduction of $E$

$a_{p}(E)\equiv\begin{cases}0\mod{10}&\text{ if }p\equiv 9\mod 10\\\
2\mod{10}&\text{ if }p\equiv 1,7\mod 10\\\ 6\mod{10}&\text{ if }p\equiv 3\mod
10.\end{cases}$

More generally, one has the following generalized version of Theorem 6.3. The
proof is similar to the proof of Theorem 6.3.

######  Theorem 6.6.

Let $K={\mathbb{Q}}(\sqrt{d})$, where $d$ is a square free integer. Let $E$ be
an elliptic curve defined over $\mathbb{Q}$ such that $E$ is
${\mathbb{Q}}$-isogenous to an elliptic curve $E^{\prime}$ with
$E^{\prime}(\mathbb{Q})_{\operatorname{tors}}\subsetneq
E^{\prime}(K)_{\operatorname{tors}}$. Assume moreover that
$|E^{\prime}(K)_{\operatorname{tors}}|\nmid|E^{\prime\prime}({\mathbb{Q}})_{\operatorname{tors}}|$
for any ${\mathbb{Q}}$-isogenous elliptic curve $E^{\prime\prime}$ to $E$.

If $p\nmid 2d|E^{\prime}(K)_{\operatorname{tors}}|$ is a prime of good
reduction of $E$, then

$|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv\begin{cases}0\mod|E^{\prime}(K)_{\operatorname{tors}}|&\text{
if }\left(\frac{d}{p}\right)=1\\\ 2p+2\mod M&\text{ if
}\left(\frac{d}{p}\right)=-1,\\\ \end{cases}$

where the integer $M$ can be taken to be the order of the maximal
${\mathbb{Q}}$-torsion subgroup of a ${\mathbb{Q}}$-isogenous elliptic curve
to $E^{d}$.

In particular, the density of primes $p$ such that
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv
0\mod|E^{\prime}(K)_{\operatorname{tors}}|$ is at least $1/2$ .

One notices that over a number field $L$, the $2$-torsion $E^{d}(L)[2]$ of
$E^{d}(L)$ is the same as the $2$-torsion $E(L)[2]$ of $E(L)$. A quadratic
twist $E^{d}$ is said to have large torsion if
$E^{d}(L)_{\operatorname{tors}}\neq E(L)[2]$. In [27], the maximum number of
quadratic twists of a given elliptic curve over ${\mathbb{Q}}$ with large
torsion was determined. One may use this observation to refine the
possibilities for $|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ when $d$ in Theorem
6.6 is such that $E^{d}({\mathbb{Q}})$ has large torsion.

If $p$ divides $|\widetilde{E}_{p}({\mathbb{F}_{p}})|$, then $p$ is called an
anomalous prime for $E$. By the Hasse inequality, we see that a prime $p\geq
7$ is an anomalous prime for $E$ if and only if
$a_{p}(E)=p+1-|\widetilde{E}_{p}({\mathbb{F}_{p}})|=1$.

If $E({\mathbb{Q}})$ has nontrivial torsion points, then the set of anomalous
primes consists either of a single element, or none, or else is contained in
the set $\\{2,3,5\\}$, see [24]. As a direct result of Theorem 6.3 and
Proposition 6.2, we obtain the following.

######  Corollary 6.7.

Let $K={\mathbb{Q}}(\sqrt{d})$, where $d$ is a square free integer. Let $E$ be
an elliptic curve defined over $\mathbb{Q}$ such that
$E^{\prime}(\mathbb{Q})_{\operatorname{tors}}$ is trivial for any
${\mathbb{Q}}$-isogenous elliptic curve $E^{\prime}$. Assume moreover that
$E^{\prime}(K)_{\operatorname{tors}}$ is nontrivial for a
${\mathbb{Q}}$-isogenous elliptic curve $E^{\prime}$. Let $p>7$ be a prime of
good reduction of $E$ with $p\nmid d|E^{\prime}(K)_{\operatorname{tors}}|$. If
$p$ is an anomalous prime for $E$, then $\left(\frac{d}{p}\right)=-1$. In the
latter case,
$p\equiv-2\mod|E^{\prime}(K)_{\operatorname{tors}}|\in\\{3,5,7,9\\}$.

The following example, conjectured in [38] and proved in [18, Theorem 2],
displays an elliptic curve over ${\mathbb{Q}}$ with exactly two $\beta_{i}$,
$i=1,2$, as described in Corollary 2.6. Moreover, the congruence classes of
orders of reductions of the curve are determined completely modulo the integer
$12$ according to the congruence classes of the primes of good reduction when
considered modulo $20$. We give an explanation for the occurrence of these
congruence classes using the results above.

######  Example 6.8.

Let $E:y^{2}=x^{3}-12x-11$ be an elliptic curve defined over $\mathbb{Q}$. The
elliptic curve $E$ is $\mathbb{Q}$-isogenous to the elliptic curve
$E^{\prime}:y^{2}=x^{3}-372x+2761$ with torsion $\mathbb{Z}/6\mathbb{Z}$.
Therefore, one knows that $|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv 0\mod 6$
for all primes of good reduction, see Theorem 2.3. If one considers
$|\widetilde{E}_{p}(\mathbb{F}_{p})|\mod 12$, then the two possible congruence
classes are $0$ and $6$. Moreover, one knows that the congruence $6\mod 12$
must occur with positive density, see Corollary 2.6.

Now one considers the curves $E$ and $E^{\prime}$ over the quadratic extension
$K={\mathbb{Q}}(\sqrt{5})$. One sees that
$E^{\prime}(K)_{\operatorname{tors}}\cong{\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/6{\mathbb{Z}}$.
According to Theorem 6.6, one has that
$|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv 0\mod 12$ for primes $p\equiv
1,4\mod 5$. For the primes $p\equiv 2,3\mod 5$, one may have either
possibilities for $|\widetilde{E}_{p}(\mathbb{F}_{p})|\mod 12$. For this
reason we investigate the divisibility of
$|\widetilde{E}_{p}(\mathbb{F}_{p})|$ by $4$ in each case. For
$|\widetilde{E}_{p}(\mathbb{F}_{p})|$ to have a point of order $4$, the
$2$-torsion point $(-1,0)$ must be divisible by $2$, i.e., there is a point
$P\in\widetilde{E}_{p}(\mathbb{F}_{p})$ such that $2P=(-1,0)$. Using the
duplication formula on $E$, one knows that the $x$-coordinate of $P$ must be a
root of the polynomial $(x^{2}+2x+10)^{2}$. The latter is equivalent to $-1$
being a square modulo $p$, i.e., $p\equiv 1\mod 4$. Therefore, one has

$|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv\begin{cases}0\mod 12&\text{ if
}p\equiv 1,9,11,13,17,19\mod 20\\\ 6\mod 12&\text{ if }p\equiv 3,7\mod
20.\end{cases}$

In particular, $|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv 0\mod 12$ for primes
of density $3/4$, whereas $|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv 6\mod 12$
for primes of density $1/4$.

######  Example 6.9.

Let $E$ be the elliptic curve defined by
$y^{2}+xy+y=x^{3}-x^{2}+47245x-2990253$ over $\mathbb{Q}$. It is easily seen
that $E(\mathbb{Q})_{\operatorname{tors}}\cong\mathbb{Z}/2\mathbb{Z}$ and
$E(\mathbb{Q}(\sqrt{-15}))_{\operatorname{tors}}\cong\mathbb{Z}/16\mathbb{Z}$.
Also, the torsion subgroup of the quadratic twist $E^{-15}$ is given by
$E^{-15}(\mathbb{Q})_{\operatorname{tors}}\cong\mathbb{Z}/8\mathbb{Z}$.
Moreover, if $p\equiv 7,11,13,14\mod 15$, or equivalently
$\left(\frac{-15}{p}\right)=-1$, then
$|\widetilde{E}_{p}^{-15}(\mathbb{F}_{p})|\equiv 0\mod 8$. It follows that in
the latter case $|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv 0,2,4,6\mod 8$.
However, the elliptic curve $E$ is ${\mathbb{Q}}$-isogenous to the curve
$y^{2}+xy+y=x^{3}-x^{2}-240755x-26606253$ whose torsion subgroup is isomorphic
to ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$. This implies
that $|\widetilde{E}_{p}({\mathbb{F}_{p}})|\equiv 0\mod 4$ for any prime $p$
of good reduction of $E$. Thus, Theorem 6.6 yields that

$|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv\begin{cases}0\mod 16&\text{ if
}p\equiv 1,2,4,8\mod 15\\\ 0,4,8,12\mod 16&\text{ if }p\equiv 7,11,13,14\mod
15.\end{cases}$

######  Example 6.10.

Let $E:y^{2}=x^{3}+20148x+586096$ be an elliptic curve defined over
$\mathbb{Q}$ with
$E(\mathbb{Q})_{\operatorname{tors}}\cong\mathbb{Z}/2\mathbb{Z}$ and
$E(\mathbb{Q}(\sqrt{-6}))_{\operatorname{tors}}\cong\mathbb{Z}/8\mathbb{Z}$.
Also, the torsion subgroup of the quadratic twist of $E^{-6}$ is given by
$E^{-6}(\mathbb{Q})_{\operatorname{tors}}\cong\mathbb{Z}/8\mathbb{Z}$. A prime
$p$ splits in $\mathbb{Q}(\sqrt{-6})$ if $p\equiv 1,5,7,11\mod 24$, otherwise
$p$ is inert. Therefore, Theorem 6.6 gives that

$\\#\widetilde{E}_{p}(\mathbb{F}_{p})\equiv\begin{cases}0\mod 8&\text{ if
}p\equiv 1,5,7,11,19,23\mod 24\\\ 4\mod 8&\text{ if }p\equiv 13,17\mod
24.\end{cases}$

One notices that $E$ is ${\mathbb{Q}}$-isogenous to an elliptic curve whose
torsion subgroup is isomorphic to
${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$.

The following observations can be found in [27, Proposition 1] or [12].

######  Remark 6.11.

  * i)

If $E({\mathbb{Q}})_{\operatorname{tors}}\cong{\mathbb{Z}}/3{\mathbb{Z}}$ is
such that
$E(K)_{\operatorname{tors}}\cong{\mathbb{Z}}/3{\mathbb{Z}}\times{\mathbb{Z}}/3{\mathbb{Z}}$,
then $K={\mathbb{Q}}(\sqrt{-3})$.

  * ii)

The curves 50a3 and 450b4 are the only elliptic curves $E$ with
$E({\mathbb{Q}})_{\operatorname{tors}}\cong{\mathbb{Z}}/3{\mathbb{Z}}$ such
that there exists $d=5$ and $d=-15$, respectively, with
$E^{d}({\mathbb{Q}})_{\operatorname{tors}}\cong{\mathbb{Z}}/5{\mathbb{Z}}$.
Moreover, $E^{d}({\mathbb{Q}})_{\operatorname{tors}}=\\{O_{E}\\}$ for any
other square free integer $d$. The reader may refer to the table in §7 for the
explicit computations of the congruence classes of
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ modulo $15$ for the curve $E$ defined
over ${\mathbb{Q}}$ and labeled as 50a3.

In the following corollary, we explicitly compute the congruence classes of
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|$ for the elliptic curves $E$ in Remark
6.11.

######  Corollary 6.12.

Let $E$ be an elliptic curve defined over ${\mathbb{Q}}$.

  * i)

If $E({\mathbb{Q}})_{\operatorname{tors}}\cong{\mathbb{Z}}/3{\mathbb{Z}}$ is
such that
$E(K)_{\operatorname{tors}}\cong{\mathbb{Z}}/3{\mathbb{Z}}\times{\mathbb{Z}}/3{\mathbb{Z}}$,
then

$\\#\widetilde{E}_{p}(\mathbb{F}_{p})\equiv\begin{cases}0\mod 9&\text{ if
}p\equiv 1\mod 3\\\ 0,3,6\mod 9&\text{ if }p\equiv 2\mod 3.\end{cases}$

  * ii)

If $E({\mathbb{Q}})_{\operatorname{tors}}\cong{\mathbb{Z}}/6{\mathbb{Z}}$ and
${\mathbb{Z}}/6{\mathbb{Z}}\subseteq
E^{d}({\mathbb{Q}})_{\operatorname{tors}}$, then $d=-3$.

  * iii)

If
$E({\mathbb{Q}})_{\operatorname{tors}}\cong{\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$
and
$E^{d}({\mathbb{Q}})_{\operatorname{tors}}\cong{\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$,
then $d=-1$.

Proof: i) This is Remark 6.11 i) and Theorem 6.6.

ii) When $\left(\frac{d}{p}\right)=-1$, Theorem 6.6 implies that
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|=2p+2+s|E^{d}({\mathbb{Q}})_{\operatorname{tors}}|$
for some $s\in{\mathbb{Z}}$. Considering the last equality modulo 6, one has
$p\equiv 2\mod 3$. It follows that $d=-3$.

For iii) Theorem 6.6 implies that
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|=2p+2+s|E^{d}({\mathbb{Q}})_{\operatorname{tors}}|$
for some $s\in{\mathbb{Z}}$ when $\left(\frac{d}{p}\right)=-1$. Now
considering the last equality modulo 8, one has $p\equiv 3\mod 4$. It follows
that $d=-1$. $\Box$

One remarks that the possibilities for the elliptic curves in ii) and iii) in
Corollary 6.12 can be found in [27, Proposition 2, Proposition 3]. It is known
that the torsion subgroups
${\mathbb{Z}}/3{\mathbb{Z}}\times{\mathbb{Z}}/3{\mathbb{Z}}$ and
${\mathbb{Z}}/4{\mathbb{Z}}\times{\mathbb{Z}}/4{\mathbb{Z}}$ occur only over
the quadratic fields ${\mathbb{Q}}(\sqrt{-3})$ and ${\mathbb{Q}}(\sqrt{-1})$,
respectively, see [14, 17]. Our method gives the result without relying on
these facts.

##  7\. Numerical Examples

In the following section, we consider elliptic curves $E$ over ${\mathbb{Q}}$
with torsion subgroup $E({\mathbb{Q}})_{\operatorname{tors}}$ for which there
exists a quadratic field $K={\mathbb{Q}}(\sqrt{D})$, where $D$ is a square
free integer, over which one has
$E({\mathbb{Q}})_{\operatorname{tors}}\subsetneq E(K)_{\operatorname{tors}}$.
We provide an example of such an elliptic curve $E$ with every possibility of
the pair $(E({\mathbb{Q}})_{\operatorname{tors}},E(K)_{\operatorname{tors}})$
as listed in Proposition 6.2 together with the congruence classes of
$|\widetilde{E}_{p}({\mathbb{F}_{p}})|\mod m$,
$m=|E(K)_{\operatorname{tors}}|$, and the primes $p$ at which the latter
classes occur computed in view of Theorem 6.3 and Theorem 6.6. The examples
are listed in a table where

  * i)

the first column contains the label of the elliptic curve $E$ with minimal
conductor satisfying the conditions above on the pair
$(E({\mathbb{Q}})_{\operatorname{tors}},E(K)_{\operatorname{tors}})$, where
the labelling is following Cremona’s tables, [3],

  * ii)

the second column contains the torsion subgroup
$E({\mathbb{Q}})_{\operatorname{tors}}$ of $E$ over $\mathbb{Q}$,

  * iii)

the third column contains $E(K)_{\operatorname{tors}}$,

  * iv)

the forth column contains $D$,

  * v)

the fifth column contains the congruence classes of
$|\widetilde{E}_{p}(\mathbb{F}_{p})|\mod m$, where
$m=|E(K)_{\operatorname{tors}}|$,

  * vi)

the sixth column contains the congruence classes of primes $p$ modulo an
integer $a$ such that $|\widetilde{E}_{p}(\mathbb{F}_{p})|\equiv t\mod m$ if
$p\equiv s\mod a$.

label | $E(\mathbb{Q})_{\operatorname{tors}}$ | $E(K)_{\operatorname{tors}}$ | $D$ | $|\widetilde{E}_{p}(\mathbb{F}_{p})|\mod m$ | $p$
---|---|---|---|---|---
175b2 | $\\{O_{E}\\}$ | $\mathbb{Z}/3\mathbb{Z}$ | $5$ | $0\mod 3$ | $p\equiv 1,4\mod 5$
|  |  | $0,1\mod 3$ | $p\equiv 2,3\mod 5$
75a2 | $\\{O_{E}\\}$ | $\mathbb{Z}/5\mathbb{Z}$ | $5$ | $0\mod 5$ | $p\equiv 1,4\mod 5$
|  |  | $1\mod 5$ | $p\equiv 2\mod 5$
|  |  |  | $3\mod 5$ | $p\equiv 3\mod 5$
208d1 | $\\{O_{E}\\}$ | $\mathbb{Z}/7\mathbb{Z}$ | $-1$ | $0\mod 7$ | $p\equiv 1\mod 4$
|  |  | $0,1,3,4,5,6\mod 7$ | $p\equiv 3\mod 4$
54a2 | $\\{O_{E}\\}$ | $\mathbb{Z}/9\mathbb{Z}$ | $-3$ | $0\mod 9$ | $p\equiv 1\mod 3$
|  |  | $0,3,6\mod 9$ | $p\equiv 2\mod 3$
98a4 | $\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/6\mathbb{Z}$ | $-7$ | $0\mod 6$ | $p\equiv 1,2,4\mod 7$
|  |  | $0,4\mod 6$ | $p\equiv 3,5,6\mod 7$
2880r6 | $\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/8\mathbb{Z}$ | $-6$ | $0\mod 8$ | $p\equiv 1,5,7,11,19,23\mod 24$
|  |  | $4\mod 8$ | $p\equiv 13,17\mod 24$
150b3 | $\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/10\mathbb{Z}$ | $5$ | $0\mod 10$ | $p\equiv 1,4\mod 5$
|  |  | $6\mod 10$ | $p\equiv 2\mod 5$
|  |  |  | $8\mod 10$ | $p\equiv 3\mod 5$
3150bk1 | $\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/16\mathbb{Z}$ | $-15$ | $0\mod 16$ | $p\equiv 1,2,4,8\mod 15$
|  |  | $0,4,8,12\mod 16$ | $p\equiv 7,11,13,14\mod 15$
14a3 | $\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ | $-7$ | $0\mod 4$ | $p\equiv 1,2,4\mod 7$
|  |  | $0,2\mod 4$ | $p\equiv 3,5,6\mod 7$
36a3 | $\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ | $-3$ | $0\mod 12$ | $p\equiv 1\mod 3$
|  |  | $0,6\mod 12$ | $p\equiv 2\mod 3$
450a3 | $\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/10\mathbb{Z}$ | $-15$ | $0\mod 20$ | $p\equiv 1,2,4,8\mod 15$
|  |  | $6,16\mod 20$ | $p\equiv 7\mod 15$
|  |  | $4,14\mod 20$ | $p\equiv 11\mod 15$
|  |  | $8,18\mod 20$ | $p\equiv 13\mod 5$
|  |  | $0,10\mod 20$ | $p\equiv 14\mod 15$
50a3 | $\mathbb{Z}/3\mathbb{Z}$ | $\mathbb{Z}/15\mathbb{Z}$ | $5$ | $0\mod 15$ | $p\equiv 1,4\mod 5$
|  |  | $3\mod 15$ | $p\equiv 3\mod 5$
|  |  | $6\mod 15$ | $p\equiv 2\mod 5$
19a1 | $\mathbb{Z}/3\mathbb{Z}$ | $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$ | $-3$ | $0\mod 9$ | $p\equiv 1\mod 3$
|  |  | $0,3,6\mod 9$ | $p\equiv 2\mod 3$
17a1 | $\mathbb{Z}/4\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ | $-1$ | $0\mod 8$ | $p\equiv 1\mod 4$
|  |  | $0,4\mod 8$ | $p\equiv 3\mod 4$
192c6 | $\mathbb{Z}/4\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}$ | $-2$ | $0\mod 16$ | $p\equiv 1,3\mod 8$
|  |  | $4,12\mod 16$ | $p\equiv 5\mod 8$
|  |  | $0,8\mod 16$ | $p\equiv 7\mod 8$
150c3 | $\mathbb{Z}/4\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z}$ | $-15$ | $0\mod 24$ | $p\equiv 1,2,4,8\mod 15$
|  |  | $0,12\mod 24$ | $p\equiv 11,14\mod 15$
|  |  | $4,16\mod 24$ | $p\equiv 7,13\mod 15$
50b1 | $\mathbb{Z}/5\mathbb{Z}$ | $\mathbb{Z}/15\mathbb{Z}$ | $5$ | $0\mod 15$ | $p\equiv 1,4\mod 5$
|  |  | $0,10\mod 15$ | $p\equiv 2,3\mod 5$
14a4 | $\mathbb{Z}/6\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ | $-7$ | $0\mod 12$ | $p\equiv 1,2,4\mod 7$
|  |  | $0,6\mod 12$ | $p\equiv 3,5,6\mod 7$
14a1 | $\mathbb{Z}/6\mathbb{Z}$ | $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ | $-3$ | $0\mod 18$ | $p\equiv 1\mod 3$
|  |  | $0,6,12\mod 18$ | $p\equiv 2\mod 3$
15a4 | $\mathbb{Z}/8\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}$ | $-1$ | $0\mod 16$ | $p\equiv 1\mod 4$
|  |  | $0,8\mod 16$ | $p\equiv 3\mod 4$
63a2 | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}$ | $-3$ | $0\mod 16$ | $p\equiv 1\mod 3$
|  |  | $0,4,8,12\mod 16$ | $p\equiv 2\mod 3$
960o6 | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z}$ | $6$ | $0\mod 24$ | $p\equiv 1,5,19,23\mod 24$
|  |  | $4,16\mod 24$ | $p\equiv 7,13\mod 24$
|  |  | $0,12\mod 24$ | $p\equiv 11,17\mod 24$
15a3 | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}$ | $5$ | $0\mod 16$ | $p\equiv 1,4\mod 5$
|  |  | $0,8\mod 16$ | $p\equiv 2,3\mod 5$
90c6 | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z}$ | $6$ | $0\mod 24$ | $p\equiv 1,5,19,23\mod 24$
|  |  | $0,12\mod 24$ | $p\equiv 7,11,13,17\mod 24$

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