# Locally imprimitive points on elliptic curves

Nathan Jones  ,  Francesco Pappalardi  and  Peter Stevenhagen  Department of
Mathematics, Statistics and Computer Science, University of Illinois at
Chicago, 851 S Morgan St, 322 SEO, Chicago, IL 60607, USA  Dipartimento di
Matematica, Università Roma Tre, Largo S. L. Murialdo 1, I–00146 Rome, Italy
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The
Netherlands<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Under GRH, any element in the multiplicative group of a number field $K$ that
is globally primitive (i.e., not a perfect power in $K^{*}$) is a primitive
root modulo a set of primes of $K$ of positive density.

For elliptic curves $E/K$ that are known to have infinitely many primes
${\mathfrak{p}}$ of cyclic reduction, possibly under GRH, a globally primitive
point $P\in E(K)$ may fail to generate any of the point groups
$E(k_{\mathfrak{p}})$. We describe this phenomenon in terms of an associated
Galois representation
$\rho_{E/K,P}:G_{K}\to\operatorname{GL}_{3}({\widehat{{\mathbf{Z}}}})$, and
use it to construct non-trivial examples of global points on elliptic curves
that are locally imprimitive.

###### Key words and phrases:

Elliptic curves, primitive points, Galois representation

###### 2010 Mathematics Subject Classification:

Primary 11G05; Secondary 11F80

##  1\. Introduction

Under the Generalized Riemann Hypothesis (GRH), every non-zero rational number
that is not $-1$ or a square is a primitive root modulo infinitely many primes
$p$. This was proved in 1967 by Hooley [Hooley], forty years after Artin had
stated it as a conjecture. For general number fields $K$, there are
counterexamples to the direct analogue of this statement, i.e., number fields
$K$ with non-torsion elements $x\in K^{*}$ that are not $\ell$-th powers for
any prime $\ell$ for which $K$ contains a primitive $\ell$-th root of unity,
but that are nevertheless a primitive root in only finitely many residue class
fields $k_{\mathfrak{p}}$. The direct analogue of Artin’s conjecture does
however hold for $x\in K^{*}$ that are _globally primitive_ , i.e., not in
${K^{*}}^{\ell}$ for any prime $\ell$.

######  Theorem 1.1.

Let $K$ be a number field and $x\in K^{*}$ globally primitive, and assume GRH.
Then $x$ is a primitive root modulo ${\mathfrak{p}}$ for a set primes
${\mathfrak{p}}$ of $K$ of positive density.

Informally stated: _globally primitive_ elements in $K^{*}$ are _locally
primitive_ in $k_{\mathfrak{p}}^{*}$ at infinitely many places
${\mathfrak{p}}$, with an element being ‘primitive’ in $K^{*}$ or
$k_{\mathfrak{p}}^{*}$ meaning that it generates a subgroup that is _not_
contained in a strictly larger cyclic subgroup of $K^{*}$ or
$k_{\mathfrak{p}}^{*}$. Section 3 provides a proof of Theorem 1.1, and
counterexamples to stronger statements.

Now replace the multiplicative group $K^{*}$ by the point group $E(K)$ of an
elliptic curve $E/K$, and the unit group $k_{\mathfrak{p}}^{*}$ of the residue
class field at ${\mathfrak{p}}$ by the point group $E(k_{\mathfrak{p}})$ at
the primes of good reduction ${\mathfrak{p}}$ of $E$. Then we can add two
natural questions to Artin’s to obtain the following three problems: for a
number field $K$, determine the infinitude (or natural density) of the set of
primes ${\mathfrak{p}}$ in $K$ for which

  1. I.

(Artin) a given element $x\in K^{*}$ is a primitive root modulo
${\mathfrak{p}}$, i.e., $k_{\mathfrak{p}}^{*}=\langle\mathchoice{x\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm x}}$}}{x\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm x}}$}}{x\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm x}}$}}{x\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm x}}$}}\rangle$;

  2. II.

(Serre [Serre3]) a given elliptic curve $E/K$ has cyclic reduction modulo
${\mathfrak{p}}$;

  3. III.

(Lang-Trotter [Lang-Trotter]) a given point $P\in E(K)$ generates the group
$E(k_{\mathfrak{p}})$ at ${\mathfrak{p}}$.

We will refer to the cases I, II, and III as the _multiplicative primitive
root_ , the _cyclic reduction_ and the _elliptic primitive root_ case, and
denote the corresponding sets of primes by $S_{K,x}$, $S_{E/K}$, and
$S_{E/K,P}$, respectively. By definition, the finitely many primes of bad
reduction for which we have $\operatorname{ord}_{\mathfrak{p}}(x)\neq 0$ (in
case I) or $\operatorname{ord}_{\mathfrak{p}}(\Delta_{E})\neq 0$ (in case II
and III, with $\Delta_{E}$ the discriminant of $E$) are excluded from these
sets. Note that we have an obvious inclusion $S_{E/K,P}\subset S_{E/K}$.

In each of the three cases, we have a group theoretical statement that can be
checked prime-wise at the primes $\ell$ dividing the order of the groups
$k_{\mathfrak{p}}^{*}$ and $E(k_{\mathfrak{p}})$ involved. The statement ‘at
$\ell$’ has a translation in terms of the splitting behavior of
${\mathfrak{p}}$ in a finite Galois extension $K\subset K_{\ell}$ that we
describe in Section 2. Combining the requirements for all $\ell$ leads to
(conjectural) density statements based on the Chebotarev density theorem
[Stev-Lenstra].

Imposing infinitely many splitting conditions, one for each prime $\ell$,
leads to analytic problems with error terms that have been mastered under
assumption of GRH in Cases I [Hooley, Cooke-Weinberger, Lenstra] and II
[Serre3, Gupta-Murty-cycl, Campagna-Stevenhagen], and that remain open in Case
III. In each case, there is a conjectural density $\delta_{K,x}$,
$\delta_{E/K}$, or $\delta_{E/K,P}$ that is an upper bound for the upper
density of the set of primes ${\mathfrak{p}}$. Proving unconditionally that
the set is infinite in case the conjectural density is positive is an open
problem. If it is zero, we can however prove unconditionally that the
corresponding set of primes is finite.

This paper focuses on the vanishing of $\delta_{E/K,P}$ in the elliptic
primitive root case, which is much more subtle than the vanishing in the cases
I and II. We call a global point $P\in E(K)$ _locally imprimitive_ if it is a
generator of the local point group $E(k_{\mathfrak{p}})$ for only finitely
many primes ${\mathfrak{p}}$ of $K$. Our analysis will yield ‘elliptic
counterexamples’ $(E/K,P)$ to Theorem 1.1, i.e., elliptic curves $E/K$ for
which the cyclic reduction density $\delta_{E/K}$ is positive but for which a
globally primitive point $P\in E(K)$ is locally imprimitive.

Just as in the multiplicative primitive root and the cyclic reduction cases I
and II, obstructions to local primitivity of a point $P\in E(K)$ become
visible in an associated Galois representation. In the elliptic primitive root
case, the absolute Galois group $G_{K}=\operatorname{Gal}(\overline{K}/K)$ of
the number field $K$ acts on the subgroup of $E(\overline{K})$ consisting of
the points $Q\in E(\overline{K})$ satisfying $kQ\in\langle P\rangle\subset
E(K)$ for some $k\in{\mathbf{Z}}_{\geq 1}$. This yields a representation
$\rho_{E/K,P}:G_{K}\to\operatorname{GL}_{3}({\widehat{{\mathbf{Z}}}}).$ Just
as in the two other cases [Campagna-Stevenhagen], it suffices to consider the
residual representation

(1) $\mathchoice{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm\rho}}$}}_{E,P}:G_{K}\to\operatorname{GL}_{3}({\widehat{{\mathbf{Z}}}})\to\operatorname{GL}_{3}({\mathbf{Z}}/N{\mathbf{Z}})$

modulo a suitable squarefree integer $N$ that is divisible by all ‘critical
primes’.

Unlike the cases I and II, case III already allows non-trivial obstructions to
local primitivity at prime level $N=\ell$. In the multiplicative case I, the
index $[k_{\mathfrak{p}}^{*}:\langle\overline{x}\rangle]$ can only be
divisible by $\ell$ for almost all ${\mathfrak{p}}$ for the ‘trivial reason’
that $K$ contains an $\ell$-th root of unity and $x$ is an $\ell$-th power in
$K^{*}$. In the cyclic reduction case II, the group $E(k_{\mathfrak{p}})$ can
only have a non-cyclic $\ell$-part for almost all ${\mathfrak{p}}$ for the
‘trivial reason’ that the full $\ell$-torsion of $E/K$ is $K$-rational. In the
elliptic primitive root case III however, there is a third reason why a point
$P\in E(K)$ can be _locally $\ell$-imprimitive_, meaning that $\ell$ divides
the index $[E(k_{\mathfrak{p}}):\langle\overline{P}\rangle]$ for all but
finitely many ${\mathfrak{p}}$. It is a less obvious one, and it was
numerically discovered in 2015 in the case $\ell=2$ by Meleleo [Meleleo], who
restricted himself to the basic case $K={\mathbf{Q}}$.

######  Theorem 1.2.

Let $P\in E(K)$ be a non-torsion point of an elliptic curve $E$ defined over a
number field $K$, and $\ell$ a prime number. Then $P$ is locally
$\ell$-imprimitive if and only if at least at least one of the following
conditions holds

  1. A.

$E(K)$ contains a torsion point of order $\ell$ and $P\in\ell\cdot E(K)$;

  2. B.

$E$ has complete rational $\ell$-torsion over $K$;

  3. C.

there exists an isogeny $\phi:E^{\prime}\to E$ defined over $K$ with kernel
generated by a torsion point of $E^{\prime}(K)$ of order $\ell$ and
$P\in\phi[E^{\prime}(K)]$.

Condition A in Theorem 1.2 is the analogue of the trivial condition from case
I: if $E(K)$ has non-trivial $\ell$-torsion, then almost all
$E(k_{\mathfrak{p}})$ are groups of order divisible by $\ell$, and for these
${\mathfrak{p}}$ a point $P\in\ell\cdot E(K)$ will have its reduction in the
subgroup $\ell E(k_{\mathfrak{p}})\subset E(k_{\mathfrak{p}})$ of index
divisible by $\ell$. Condition B bears no relation to $P$, and is well known
from case II: non-cyclicity of the $\ell$-part of the global torsion subgroup
$E(K)^{\textup{tor}}$ implies non-cyclicity of the $\ell$-part of
$E(k_{\mathfrak{p}})$ at almost all ${\mathfrak{p}}$. At these
${\mathfrak{p}}$, no single point $P$ can generate it.

Condition C has no analogue in the multiplicative primitive root case, and it
is a truly different condition as it includes cases in which we have both
$P\notin\ell\cdot E(K)$ and $E[\ell](K)=0$. If it holds, the dual isogeny
$\widehat{\phi}:E\to E^{\prime}$ maps $P$ into $\ell E^{\prime}(K)$, and the
pair $(E,P)$ is $\ell$-isogenous to the curve-point pair
$(E^{\prime},\widehat{\phi}(P))$ satisfying Condition A. We call a locally
$\ell$-imprimitive non-torsion point $P\in E(K)$ _non-trivial_ if Condition C
is satisfied, but _not_ Condition A or B.

By Theorem 1.2, non-trivial locally 2-imprimitive points $P\in E(K)$ can only
exist for $E/K$ having a single $K$-rational point of order 2, i.e., a
2-torsion subgroup of order $\\#E(K)[2]\neq 1,4$. Examples of such points are
actually surprisingly abundant.

######  Theorem 1.3.

Let $E/K$ be any elliptic curve with $\\#E(K)[2]=2$. Then there are infinitely
many quadratic twists of $E$ over $K$ that have a non-trivial locally
$2$-imprimitive point.

The proof of this Theorem, which we give in Section 5, uses the fact that it
is easy to create non-torsion points on twists of $E$, and exploits the
particularly explicit description of $K$-rational 2-isogenies.

For primes $\ell>2$, it is harder to obtain families of elliptic curves with
points of infinite order that are locally $\ell$-imprimitive in non-trivial
ways. In Section 6 we provide an approach in the case $\ell=3$. It can be
extended to higher values of $\ell$ (Section 7), but the examples rapidly
become unwieldy.

Non-torsion points that are locally imprimitive but not locally
$\ell$-imprimitive for any single prime $\ell$ do exist, but they are not
easily found. They involve restrictions arising from reductions of
$\rho_{E/K,P}$ of composite level caused by non-trivial _entanglement_ between
the fields $K_{\ell}$. In the context of the easier cyclic reduction case II,
this is discussed in [Campagna-Stevenhagen], and we present a first type of
examples for our Lang-Trotter case III in our final Section 8. Such higher
level obstructions will be explored in more detail in a forthcoming paper.

Acknowledgements. _All authors received support from the Max-Planck-Institut
für Mathematik in Bonn while working on this paper. They thank the institute
for its financial support and for its very inspiring atmosphere._

##  2\. Characterization by splitting conditions

In each of the three cases discussed in the introduction, we can characterize
the corresponding sets of primes $S_{K,x}$, $S_{E/K}$, and $S_{E/K,P}$ of $K$
in terms of the splitting behaviour of their elements ${\mathfrak{p}}$ in
suitable extensions $K\subset K_{\ell}$, with $\ell$ ranging over all prime
numbers.

I. Multiplicative primitive root case. Let $K$ be a number field and $x\in
K^{*}$ non-torsion. Define $K_{m}={\mathbf{Q}}(\zeta_{m},\root m \of{x})$ for
$m\in{\mathbf{Z}}_{\geq 1}$ as the ‘$m$-division field of $x$’, i.e., the
splitting field over $K$ of the polynomial $X^{m}-x\in K[X]$. If
${\mathfrak{p}}$ is a prime of $K$ of characteristic $p$ for which $x$ is a
${\mathfrak{p}}$-adic unit, the index
$[k_{\mathfrak{p}}^{*}:\langle\overline{x}\rangle]$ is divisible by a prime
$\ell\neq p$ if and only if ${\mathfrak{p}}$ splits completely in $K\subset
K_{\ell}$.

Note that $[k_{\mathfrak{p}}^{*}:\langle\overline{x}\rangle]$ is never
divisible by $p=\operatorname{char}({\mathfrak{p}})$, even though
${\mathfrak{p}}$ may split completely in $K_{p}$. Example: $x=17$ is a
primitive root modulo the prime ${\mathfrak{p}}_{3}$ of norm 3 in
$K={\mathbf{Q}}(\sqrt{-21})$, but ${\mathfrak{p}}_{3}$ splits completely in
the sextic extension

$K\subset K_{3}=K(\zeta_{3},\root 3 \of{17})=K(\sqrt{7},\root 3 \of{17}).$

This can however only happen for primes ${\mathfrak{p}}|2\Delta_{K}$, with
$\Delta_{K}$ the discriminant of $K$, since $K\subset K_{p}$ is ramified at
all ${\mathfrak{p}}|p$ for $p$ coprime to $2\Delta_{K}$. In other words: for
almost all ${\mathfrak{p}}$, the ‘condition at $\ell$’ in the following Lemma
is automatically satisfied at $\ell=\operatorname{char}{{\mathfrak{p}}}$.

######  Lemma 2.1.

For ${\mathfrak{p}}$ a prime of $K$ outside the support of $x$, we have
$k_{\mathfrak{p}}^{*}=\langle\overline{x}\rangle$ if and only if
${\mathfrak{p}}$ does not split completely in $K\subset K_{\ell}$ for any
prime $\ell\neq\operatorname{char}{{\mathfrak{p}}}$. ∎

By Lemma 2.1, the set $S_{K,x}$ of primes in $K$ for which $x$ is a primitive
root is up to finitely many primes equal to the set of primes that do not
split completely in $K\subset K_{\ell}$ for any prime $\ell$. For
$m\in{\mathbf{Z}}_{\geq 1}$, the set of primes ${\mathfrak{p}}$ of $K$ that
split completely in $K\subset K_{m}$ has natural density $1/[K_{m}:K]$. Under
GRH, it follows from [Lenstra] that the set $S_{K,x}$ has a natural density
that is given by the inclusion-exclusion sum

(2) $\delta_{K,x}=\sum_{m=1}^{\infty}\frac{\mu(m)}{[K_{m}:K]}$

that converges slowly, but that can be rewritten in ‘factored form’ as

(3) $\delta_{K,x}=\sum_{m|N}\frac{\mu(m)}{[K_{m}:K]}\cdot\prod_{\ell\nmid
N\text{ prime}}(1-\frac{1}{\ell(\ell-1)}).$

Here we can take for $N$ any integer divisible by the primes in some finite
set of _critical primes_. One may take for this set the set of primes that are
either in the support of $x$ or divide $2\Delta_{K}$, together with those
primes $\ell$ for which $x$ is in ${K^{*}}^{\ell}$. The essential feature of
$N$ is that the family $\\{K_{\ell}\\}_{\ell\nmid N}$ of ‘$\ell$-division
fields of $x$ outside $N$’ is a linearly disjoint family over $K$ with each
$K_{\ell}$ having the full Galois group
$\operatorname{Gal}(K_{\ell}/K)\cong\operatorname{Aff}_{1}({\mathbf{F}}_{\ell})={\mathbf{F}}_{\ell}\rtimes{\mathbf{F}}_{\ell}^{*}$
of order $\ell(\ell-1)$, and that the compositum $L$ of the fields in this
family satisfies $L\cap K_{N}=K$.

II. Cyclic reduction case. For an elliptic curve $E/K$, we consider the set
$S_{E/K}$ of primes of cyclic reduction of $E$, i.e., the primes
${\mathfrak{p}}$ of $K$ for which $E$ has good reduction and the reduced
elliptic curve point group $E(k_{\mathfrak{p}})$ is cyclic. The condition that
$E$ have good reduction modulo ${\mathfrak{p}}$ only excludes the finitely
many primes dividing the discriminant $\Delta_{E}$ of $E$.

For $m\in{\mathbf{Z}}_{\geq 1}$, we define
$K_{m}=K(E[m](\overline{\mathbf{Q}}))$ in this case to be the $m$-division
field of $E$ over $K$. The following elementary lemma [Campagna-
Stevenhagen]*Corollary 2.2 is the analogue of Lemma 2.1. It expresses the fact
that a finite abelian group is cyclic if and only if its $\ell$-primary part
is cyclic for all primes $\ell$.

######  Lemma 2.2.

A prime ${\mathfrak{p}}$ of good reduction of $E/K$ is a prime of cyclic
reduction if and only if ${\mathfrak{p}}$ does not split completely in
$K\subset K_{\ell}$ for any prime
$\ell\neq\operatorname{char}{{\mathfrak{p}}}$. ∎

As in the multiplicative Case I, cyclicity of the $p$-primary part of the
groups $E(k_{\mathfrak{p}})$ is automatic for
$p=\operatorname{char}({\mathfrak{p}})$. Also here, total splitting of
${\mathfrak{p}}$ in non-trivial extensions $K\subset K_{p}$ for
$p=\operatorname{char}{{\mathfrak{p}}}$ does occur: it suffices to base change
any elliptic curve $E/K$ with $K\subset K_{p}=K[X]/(f)$ non-trivial by an
extension $K\subset L=K[X]/(g)$ with $f$ and $g$ polynomials of the same
degree that are ${\mathfrak{p}}$-adically close, but with $g$ Eisenstein at a
prime ${\mathfrak{q}}$ that is unramified in $K\subset K_{p}$. For $E/L$, the
non-trivial extension $L\subset L_{p}$ will be totally split at all primes
dividing ${\mathfrak{p}}$. Again, this can happen only at primes
${\mathfrak{p}}|2\Delta_{K}$, as otherwise $K\subset K_{p}$ will be ramified
at all primes ${\mathfrak{p}}$ of characteristic $p$ by the fact that $K_{p}$
contains $\zeta_{p}$. Thus, for almost all ${\mathfrak{p}}$, the ‘condition at
$\ell$’ in Lemma 2.2 is again automatically satisfied at
$\ell=\operatorname{char}{{\mathfrak{p}}}$. The finitely many primes dividing
$2\Delta_{K}$ are clearly irrelevant when dealing with the density of the set
$S_{E/K}$, which, just like in the previous case, coincides up to finitely
many primes with the set of primes ${\mathfrak{p}}$ of $K$ that do not split
completely in $K\subset K_{\ell}$ for any prime $\ell$.

Under GRH, the density of $S_{E/K}$ is again given [Campagna-
Stevenhagen]*Section 2 by an inclusion-exclusion sum that we already know from
(2):

(4) $\delta_{E/K}=\sum_{m=1}^{\infty}\frac{\mu(m)}{[K_{m}:K]}.$

If $E$ is without CM over $\overline{\mathbf{Q}}$, or has CM by an order
$\mathcal{O}\subset K$, there is in each case a factorization of
$\delta_{E/K}$ that is typographically identical to (2), provided that $N$ is
divisible by all primes from an appropriately defined finite set of critical
primes [Campagna-Stevenhagen]*Theorems 1.1 and 1.2. If $E$ has CM by an order
$\mathcal{O}\not\subset K$, there is a hybrid formula [Campagna-
Stevenhagen]*Theorem 1.4 with different contributions from ordinary and
supersingular primes.

A ‘factorization formula’ for $\delta_{K,x}$ and $\delta_{E/K}$ as in (3)
shows that the vanishing of these densities is always caused by an obstruction
at some finite level $N$. For such $N$, no element in
$\operatorname{Gal}(K_{N}/K)$ restricts for all $\ell|N$ to a non-trivial
element of $\operatorname{Gal}(K_{\ell}/K)$. As a consequence, there are no
non-critical primes in $S_{K,x}$ or $S_{E/K}$: the Frobenius elements of such
primes in $\operatorname{Gal}(K_{N}/K)$ cannot exist for group theoretical
reasons.

An obstruction at prime level $N=\ell$, which means an equality $K=K_{\ell}$,
amounts in the cases I and II to a ‘trivial’ reason that we already mentioned
in the context of Conditions A and B in Theorem 1.2. For $K={\mathbf{Q}}$,
vanishing of $\delta_{{\mathbf{Q}},x}$ and $\delta_{E/{\mathbf{Q}}}$ only
occurs if we have $K=K_{\ell}$ for a prime $\ell$, which in this case has to
be $\ell=2$.

Over general number fields, vanishing may be caused by obstructions that occur
_only_ at composite levels. Typical examples can be constructed by base
changing a non-vanishing example $(K,x)$ or $E/K$ to a suitable extension
field $K\subset L$. Example 3.1 accomplishes vanishing of $\delta_{K,x}$ by an
obstruction at level $30=2\cdot 3\cdot 5$ for the field
$K={\mathbf{Q}}(\sqrt{5})$ that does not arise at lower level by cleverly
choosing $x$. In [Campagna-Stevenhagen]*Example 5.4 we find a base change of
an elliptic curve $E/{\mathbf{Q}}$ to a field $K$ of degree 48 with a similar
level 30 obstruction to cyclic reduction.

III. Elliptic primitive root case. In addition to the elliptic curve $E/K$, we
are now given a point $P\in E(K)$ of infinite order. We consider the set
$S_{E/K,P}$ of primes ${\mathfrak{p}}$ of $K$ for which $E$ has cyclic
reduction and the reduction of the point $P$ modulo ${\mathfrak{p}}$ generates
the group $E(k_{\mathfrak{p}})$. Note the obvious inclusion $S_{E/K,P}\subset
S_{E/K}$.

For $m\in{\mathbf{Z}}_{\geq 1}$, we let $K_{m}=K(m^{-1}P)$ be the _$m$
-division field of $P$_, i.e., the extension of $K$ generated by the points of
the subgroup of $E(\overline{K})$ defined as

$\langle m^{-1}P\rangle=\\{Q\in E(\overline{K}):mQ\in\langle P\rangle\\}.$

Note that this extension $K_{m}$ contains the $m$-division field of the
elliptic curve $E$ that we encountered in the cyclic reduction case II. The
$m$-division field $K_{m}=K(m^{-1}P)$ of $P$ is again unramified over $K$ at
primes ${\mathfrak{p}}$ of good reduction coprime to $m$. The proof of this
fact is as for the $m$-division field of $E$: as the $m$-th roots $Q\in
E(\overline{K})$ of $P$ that generate $K_{m}$ over $K$ differ by $m$-torsion
points, their reductions modulo a prime over ${\mathfrak{p}}$ remain
different, so inertia acts trivially on the set of such $Q$.

The quotient group $V_{m}=\langle m^{-1}P\rangle/\langle P\rangle$ is a free
module of rank 3 over ${\mathbf{Z}}/m{\mathbf{Z}}$. It comes with a natural
linear action of the absolute Galois group $G_{K}$ of $K$, and this mod-$m$
Galois representation induces an embedding

(5)
$G_{m}=\operatorname{Gal}(K_{m}/K)\hookrightarrow\operatorname{GL}(V_{m})\cong\operatorname{GL}_{3}({\mathbf{Z}}/m{\mathbf{Z}}).$

As $G_{K}$ stabilizes the rank 2 subspace $U_{m}=E[m](\overline{K})\subset
V_{m}$, this is a ‘reducible’ representation. Write
$V_{m}=U_{m}\oplus({\mathbf{Z}}/m{\mathbf{Z}})\cdot\mathchoice{Q\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm Q}}$}}$ with
$\mathchoice{Q\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm
Q}}$}}{Q\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm Q}}$}}=(Q\bmod\langle
P\rangle)\in V_{m}$ the image of a point $Q\in E(\overline{K})$ satisfying
$mQ=P$. We then have a split exact sequence

$0\to U_{m}\longrightarrow
V_{m}\longrightarrow{\mathbf{Z}}/m{\mathbf{Z}}\cdot\mathchoice{Q\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm Q}}$}}\to 0$

of free ${\mathbf{Z}}/m{\mathbf{Z}}$-modules that is split as a sequence of
$({\mathbf{Z}}/m{\mathbf{Z}})[G_{m}]$-modules if and only if we have $P\in
m\cdot E(K)$. After choosing an ${\mathbf{Z}}/m{\mathbf{Z}}$-basis
$\\{T_{1},T_{2}\\}$ of $U_{m}$ and extending it by some $\mathchoice{Q\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm Q}}$}}$ as above to a
${\mathbf{Z}}/m{\mathbf{Z}}$-basis for $V_{m}$, the matrix representation of
$\sigma\in G_{m}$ becomes

(6) $\sigma=\sigma_{A,b}=\begin{pmatrix}a_{11}&a_{12}&b_{1}\\\
a_{21}&a_{22}&b_{2}\\\ 0&0&1\end{pmatrix},$

in which the linear action of $\sigma$ on $U_{m}$ with respect to some
${\mathbf{Z}}/m{\mathbf{Z}}$-basis $\\{T_{1},T_{2}\\}$ of $U_{m}$ is described
by $A=\begin{pmatrix}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{pmatrix}\in\operatorname{GL}(U_{m})$, and

$b=b_{1}T_{1}+b_{2}T_{2}=\sigma Q-Q$

is the translation action of $\sigma$ on some chosen ‘$m$-th root’ $Q$ of $P$
with respect to that same basis. In other words, $V_{m}$ gives a Galois
representation of $G_{K}$ with image $G_{m}\subset\operatorname{GL}(V_{m})$
that is contained in the 2-dimensional affine group

$\operatorname{Aff}_{2}({\mathbf{Z}}/m{\mathbf{Z}})=({\mathbf{Z}}/m{\mathbf{Z}})^{2}\rtimes\operatorname{GL}_{2}({\mathbf{Z}}/m{\mathbf{Z}}).$

In the important case where $m=\ell$ is prime, we are in the classical
situation of a 3-dimensional Galois representation over the finite field
${\mathbf{F}}_{\ell}$.

The analogue in the elliptic primitive root case of the Lemmas 2.1 and 2.2 is
a little bit more involved. We have to impose a condition on the Frobenius
elements $\operatorname{Frob}_{{\mathfrak{p}},\ell}\in G_{\ell}$ at all primes
$\ell\neq\operatorname{char}{{\mathfrak{p}}}$ different from being _equal_ to
the identity element $\operatorname{id}_{\ell}\in G_{\ell}$: in this case it
only needs to be ‘sufficiently close’ to it.

######  Lemma 2.3.

For $P\in E(K)$ of infinite order and ${\mathfrak{p}}$ a prime of good
reduction of $E$ of characteristic different from $\ell$ we have

$\ell|[E(k_{\mathfrak{p}}):\langle{\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm
P}}$}}}\rangle]\quad\Longleftrightarrow\quad\operatorname{rank}(\operatorname{Frob}_{{\mathfrak{p}},\ell}-\operatorname{id}_{\ell})\leq
1.$

###### Proof.

As all $V_{\ell}$ are 3-dimensional over ${\mathbf{F}}_{\ell}$, the condition
$\operatorname{rank}(\operatorname{Frob}_{{\mathfrak{p}},\ell}-\operatorname{id}_{\ell})\leq
1$ means that $\operatorname{Frob}_{{\mathfrak{p}},\ell}$ is the identity on
an ${\mathbf{F}}_{\ell}$-subspace of $V_{\ell}$ of dimension at least 2. If it
equals $U_{\ell}$, then $E(k_{\mathfrak{p}})$ has complete $\ell$-torsion of
order $\ell^{2}$ and every cyclic subgroup $\langle{\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm P}}$}}}\rangle$ has
index divisible by $\ell$. If not, it intersects $U_{\ell}$ in a 1-dimensional
subspace, so we have a point of order $\ell$ in $E(k_{\mathfrak{p}})$ and a
point $\mathchoice{Q\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm
Q}}$}}{Q\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm Q}}$}}\in
E(k_{\mathfrak{p}})$ satisfying $\ell\mathchoice{Q\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm
Q}}$}}=\mathchoice{P\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm
P}}$}}{P\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm P}}$}}$. This also
implies that $[E(k_{\mathfrak{p}}):\langle{\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm P}}$}}}\rangle]$ is
divisible by $\ell$.

Conversely, if $\ell$ divides $[E(k_{\mathfrak{p}}):\langle{\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm P}}$}}}\rangle]$ then
either $E(k_{\mathfrak{p}})$ has complete $\ell$-torsion or
$E(k_{\mathfrak{p}})$ has a cyclic non-trivial $\ell$-part and
$\mathchoice{P\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm
P}}$}}{P\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm P}}$}}$ is contained in
$\ell\cdot E(k_{\mathfrak{p}})$. In both cases
$\operatorname{Frob}_{{\mathfrak{p}},\ell}$ is the identity on a subspace of
$V_{\ell}$ of dimension at least 2. ∎

######  Corollary 2.4.

Let $P\in E(K)$ be of infinite order and ${\mathfrak{p}}$ a prime of good
reduction of $E$ of prime norm $\operatorname{char}{{\mathfrak{p}}}>5$ for
which $\mathchoice{P\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm
P}}$}}{P\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm
P}}$}}\neq\mathchoice{O\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm
O}}$}}{O\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm O}}$}}{O\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm O}}$}}{O\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm O}}$}}\in
E(k_{\mathfrak{p}})$. Then we have

$E(k_{\mathfrak{p}})=\langle{\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm
P}}$}}}\rangle\quad\Longleftrightarrow\quad\operatorname{rank}(\operatorname{Frob}_{{\mathfrak{p}},\ell}-\operatorname{id}_{\ell})\geq
2\text{ for all primes $\ell$}.$

###### Proof.

By Lemma 2.3, the condition on the right side says that
$p=\operatorname{char}{{\mathfrak{p}}}$ is the only possible prime divisor of
the index $[E(k_{\mathfrak{p}}):\langle{\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm P}}$}}}\rangle]$. For a
prime ${\mathfrak{p}}$ of degree one, i.e., of prime norm $p$, the index of a
subgroup of $E(k_{\mathfrak{p}})=E({\mathbf{F}}_{p})$ can only be divisible by
$p$ if it is the trivial subgroup, as we have
$\\#E({\mathbf{F}}_{p})<p+1+2\sqrt{p}<2p$ for $p>5$. So we have
$E(k_{\mathfrak{p}})=\langle{\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm P}}$}}}\rangle$ unless
${\mathfrak{p}}$ is a prime for which we have $\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm
P}}$}}=\mathchoice{O\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm
O}}$}}{O\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm O}}$}}{O\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm O}}$}}{O\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm O}}$}}\in
E(k_{\mathfrak{p}})$. As we have $P\neq O\in E(K)$, this happens only for
finitely many ${\mathfrak{p}}$. ∎

In density questions, we can disregard any finite set of primes, and more
generally a set of primes of density zero. The set of primes in a number field
of degree bigger than one is such a zero density set. For this reason, the
density of the set $S_{E/K,P}$ only depends on the primes of degree one
outside any finite set of ‘critical primes’ that it contains. Thus, Corollary
2.4 can play the same role as the Lemmas 2.1 and 2.2.

In order to express the ‘heuristical density’ $\delta_{E/K,P}$ of $S_{E/K,P}$,
we define the subset $S_{\ell}\subset G_{\ell}=\operatorname{Gal}(K_{\ell}/K)$
of ‘bad’ elements at the prime $\ell$ as

$S_{\ell}=\\{\sigma\in
G_{\ell}:\operatorname{rank}_{{\mathbf{F}}_{\ell}}(\sigma-\operatorname{id}_{\ell})\leq
1\\}.$

For arbitrary $m\in{\mathbf{Z}}_{\geq 1}$ and $\ell|m$ prime we let
$\pi_{m,\ell}:G_{m}\to G_{\ell}$ be the natural restriction map, and define
$S_{m}\subset G_{m}$ as

$S_{m}=\bigcup_{\ell|m}\pi_{m,\ell}^{-1}[S_{\ell}].$

With $s_{m}=\\#S_{m}$ denoting the cardinality of $S_{m}$, the _elliptic
primitive root density_ is now given by the inclusion-exclusion sum

(7) $\delta_{E,P}=\sum_{m=1}^{\infty}\frac{\mu(m)s_{m}}{[K_{m}:K]}.$

It is the elliptic analogue of the multiplicative primitive root density (2).
It is an upper density for $S_{E/K,P}$ that has not been proven to be its true
density in cases with $\delta_{E/K,P}>0$, not even under GRH.

We can compute $\delta_{E/K,P}$ using the methods of [Campagna-Stevenhagen].
This is not directly relevant for us, as our focus in this paper is on cases
where $\delta_{E/K,P}$ vanishes in ‘non-trivial’ ways, so we will merely
sketch this here. In order to obtain a factorization

(8)
$\delta_{E/K,P}=\sum_{m|N}\frac{\mu(m)s_{m}}{[K_{m}:K]}\cdot\prod_{\ell\nmid
N\text{ prime}}(1-\frac{s_{\ell}}{[K_{\ell}:K]}).$

as in (3), it suffices to have an ‘open-image theorem’ for the Galois
representation $\rho_{E,P}$ arising from the action of $G_{K}$ on the subgroup

$R_{P}=\\{Q\in E(\mathchoice{K\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm K}}$}}):mQ\in\langle
P\rangle\text{ for some }m\in{\mathbf{Z}}_{\geq
1}\\}\cong({\mathbf{Q}}/{\mathbf{Z}})^{2}\times{\mathbf{Q}}$

of $E(\mathchoice{K\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm
K}}$}}{K\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm K}}$}})$ generated by
all the roots of $P$ in $E(\mathchoice{K\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm K}}$}})$. The Galois
action of $G_{K}$ on the quotient group $V=R_{P}/\langle P\rangle$, which is
free of rank 3 over ${\mathbf{Q}}/{\mathbf{Z}}$, gives rise to a Galois
representation

$\rho_{E,P}:G_{K}\longrightarrow\operatorname{Aut}(V)\cong\operatorname{GL}_{3}({\widehat{{\mathbf{Z}}}}),$

which has (5) as its mod-$m$ representation. It factors via
$\operatorname{Gal}(K(R_{P})/K)$, with
$K(R_{P})=\bigcup_{m}K_{m}\subset\mathchoice{K\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm K}}$}}$ the compositum
of all ‘$m$-division fields’ of $P$ inside $\mathchoice{K\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm K}}$}}$. The group
$U=E(\overline{K})^{\textup{tor}}\cong({\mathbf{Q}}/{\mathbf{Z}})^{2}$ is a
direct summand of $V$, and if we choose a ${\mathbf{Q}}/{\mathbf{Z}}$-basis
for $V=U\oplus{\mathbf{Q}}/{\mathbf{Z}}$ as we did for
$V_{m}=U_{m}\oplus{\mathbf{Z}}/m{\mathbf{Z}}\cdot\mathchoice{Q\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm Q}}$}}{Q\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm Q}}$}}$, the image of
$\rho_{E,P}$ is in
$\operatorname{Aff}_{2}({\widehat{{\mathbf{Z}}}})={\widehat{{\mathbf{Z}}}}^{2}\rtimes\operatorname{GL}_{2}({\widehat{{\mathbf{Z}}}})$.
For $E$ without CM over $\mathchoice{K\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm K}}$}}$, one deduces
from Serre’s open image theorem that this image is of finite index in
$\operatorname{Aff}_{2}({\widehat{{\mathbf{Z}}}})$, which yields (8) for any
$N$ divisible by some finite product $N_{E/K,P}\in{\mathbf{Z}}_{>0}$ of
critical primes. As in [Campagna-Stevenhagen], one deduces that all non-CM-
densities $\delta_{E,P}$ are rational multiples of a universal constant. If
$E$ has CM over $\mathchoice{K\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm K}}$}}$ by an order
$\mathcal{O}\subset K$, one replaces
$\operatorname{Aff}_{2}({\widehat{{\mathbf{Z}}}})$ by
$\operatorname{Aff}_{1}(\mathcal{O})$, and in the case of CM by an order
$\mathcal{O}\not\subset K$, one separates the contribution of ordinary and
supersingular primes of $E/K$ as in [Campagna-Stevenhagen].

##  3\. Multiplicative primitivity

Before focusing on the Lang-Trotter case III, we first settle the
multiplicative primitive root case: under GRH, globally primitive elements
$x\in K^{*}$ are locally primitive for a set of primes of positive density
$\delta_{K,x}$.

###### Proof of Theorem 1.1.

Let $x\in K^{*}$ be globally primitive. As we assume GRH, the primitive root
density for $x\in K^{*}$ exists and is equal to $\delta_{K,x}$ defined in (2),
by the results of [Lenstra]. We need to show that $\delta_{K,x}$ does not
vanish. In view of the factorization formula (3), it suffices to show that for
any squarefree integer $N>1$, the fraction $\sum_{m|N}\mu(m)[K_{m}:K]^{-1}$ of
elements in $\operatorname{Gal}(K_{N}/K)$ that have non-trivial restriction to
$K_{\ell}$ for all primes $\ell|N$ does not vanish.

As $x$ is not an $\ell$-th power in $K^{*}$, the polynomial $X^{\ell}-x$ is
irreducible in $K[X]$. It therefore gives rise to an extension $K\subset
K_{\ell}=\operatorname{Split}_{K}(X^{\ell}-x)=K(\zeta_{\ell},\root\ell
\of{x})$ of degree $\ell\cdot c_{\ell}$, with $c_{\ell}$ a divisor of
$\ell-1$. If $\ell$ is the largest prime dividing the squarefree number $N$,
we conclude that $K_{N/\ell}\subset K_{N}$ is Galois of degree divisible by
$\ell$.

Showing that $\operatorname{Gal}(K_{N}/K)$ contains an element of the required
type is now easily done by induction on the number of of primes dividing the
squarefree integer $N>1$. If $N$ is prime, then $\operatorname{Gal}(K_{N}/K)$
contains a non-trivial element of order $N$. If not, we let $\ell$ be the
largest prime dividing $N$ and observe that an automorphism of the required
type in $\operatorname{Gal}(K_{N/\ell}/K)$, which exists by the induction
hypothesis, always possesses an extension to the compositum $K_{N}$ of
$K_{N/\ell}$ and $K_{\ell}$ that is non-trivial on $K_{\ell}$. ∎

The assumption of global primitivity in Theorem 1.1 cannot be weakened to the
assumption $K\neq K_{\ell}$ for all prime number $\ell$. The resulting
stronger statement is correct for $K={\mathbf{Q}}$, but counterexamples to it
exist for general number fields $K$, as the cyclotomic extensions $K\subset
K(\zeta_{\ell})$ for different $\ell$ may all be non-trivial, but ‘entangled’
over $K$. The following counterexample takes $K$ to be quadratic.

######  Example 3.1.

The quadratic field $K={\mathbf{Q}}(\sqrt{5})$ has fundamental unit
$\varepsilon=\frac{1+\sqrt{5}}{2}$. The element
$\pi=\varepsilon^{2}-4=\frac{-5+\sqrt{5}}{2}\in K$ has norm 5 and is a square
modulo 4. The field $K(\sqrt{\pi})$, which is cyclic of degree 4 over
${\mathbf{Q}}$ and unramified outside 5, is therefore equal to $K(\zeta_{5})$.
Take $y=-3\pi\in K$ and choose $x=y^{15}$. We then have

$K_{3}=K(\zeta_{3})=K(\sqrt{-3})\qquad\text{and}\qquad
K_{5}=K(\zeta_{5})=K(\sqrt{\pi}),$

so $K_{2}=K(\sqrt{x})=K(\sqrt{y})=K(\sqrt{-3\pi})$ and $K_{3}$ and $K_{5}$ are
three different quadratic extensions of $K$ contained in the biquadratic
extension $K\subset K_{6}=K_{10}=K_{15}=K_{30}$. We have $\mu_{K}=\\{\pm 1\\}$
and, even though $x$ is not a square in $K^{*}$, there is exactly one prime of
$K$ modulo which $x$ is a primitive root: $(2)$. For the primes
${\mathfrak{p}}=(3)$ and ${\mathfrak{p}}=(\sqrt{5})$ the element $x$ is not in
$k_{\mathfrak{p}}^{*}$, and for all primes of characteristic $p>5$ the index
$[{\mathbf{F}}_{p}:\langle\mathchoice{x\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm x}}$}}{x\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm x}}$}}{x\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm x}}$}}{x\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm x}}$}}\rangle]$ is
divisible by at least one of 2, 3 or 5. Indeed, no prime can be inert in all
three quadratic subfields of an extension with group
${\mathbf{Z}}/2{\mathbf{Z}}\times{\mathbf{Z}}/2{\mathbf{Z}}$.

The simple observation that no prime ${\mathfrak{p}}$ of a number field $K$
can be inert in all three quadratic subextensions of an extension $K\subset L$
with group ${\mathbf{Z}}/2{\mathbf{Z}}\times{\mathbf{Z}}/2{\mathbf{Z}}$
underlies many ‘entanglement obstructions’, including the one in our final
Section 8.

##  4\. Proof of Theorem 1.2

In the Lang-Trotter situation, Lemma 2.3 shows that a point $P\in E(K)$ will
generate a subgroup of the local point group $E(k_{\mathfrak{p}})$ of index
divisible by $\ell$ when $\operatorname{Frob}_{{\mathfrak{p}},\ell}\in
G_{\ell}$ pointwise fixes a 2-dimensional subspace of the 3-dimensional
${\mathbf{F}}_{\ell}$-vector space $V_{\ell}$. Vanishing of the density
$\delta_{E/K,P}$ can therefore occur ‘because of $K_{\ell}$’ in cases where
$G_{\ell}=\operatorname{Gal}(K_{\ell}/K)$ is non-trivial, but only contains
elements that pointwise fix a 2-dimensional subspace of $V_{\ell}$.

Our proof of Theorem 1.2 is based on a general lemma that we phrase and prove
in the generality that was suggested to us by Hendrik Lenstra. It describes
the linear group actions on vector spaces of finite dimension over arbitrary
fields that have ‘many fixpoints’ in the sense of the 3-dimensional example
$V_{\ell}$ that we have at hand.

Let $V$ be any vector space on which a group $G$ acts linearly, and denote by

$V^{G}=\\{v\in V:\sigma v=v\text{ for all }\sigma\in
G\\}\qquad\text{and}\qquad V_{G}=V/(\textstyle\sum_{\sigma\in G}(\sigma-1)V)$

the maximal subspace and quotient space of $V$ on which $G$ acts trivially.
For every $\sigma\in G$, we have an exact sequence of vector spaces

$0\longrightarrow V^{\langle\sigma\rangle}\longrightarrow
V\mathop{\longrightarrow}\limits^{\sigma-1}V\longrightarrow V/(\sigma-1)V\to
0$

showing that for $V$ of finite dimension $n$, we have

(9) $\dim V^{\langle\sigma\rangle}\geq
n-1\quad\Longleftrightarrow\quad\dim(\sigma-1)V\leq 1.$

######  Lemma 4.1.

Let $G$ be a group acting linearly on a vector space $V$ of dimension
$n\in{\mathbf{Z}}_{\geq 0}$. Then the following are equivalent:

  1. (1)

$\dim V^{\langle\sigma\rangle}\geq n-1$ for all $\sigma\in G$;

  2. (2)

$\dim V^{G}\geq n-1$ or $\dim V_{G}\geq n-1$.

###### Proof.

The implication $(2)\Rightarrow(1)$ is immediate, as the inequality $\dim
V_{G}\geq n-1$ implies that, for all $\sigma\in G$, we have
$\dim(\sigma-1)V\leq 1$ and, by (9), $\dim V^{\langle\sigma\rangle}\geq n-1$.

For $(1)\Rightarrow(2)$, we can assume there exists $\sigma\in G$ acting non-
trivially on $V$, and define subgroups $A_{\sigma},B_{\sigma}\subset G$ by

$A_{\sigma}=\\{\tau\in G\colon V^{\langle\tau\rangle}\supset
V^{\langle\sigma\rangle}\\}\qquad\text{and}\qquad B_{\sigma}=\\{\tau\in
G\colon(\tau-1)V\subset(\sigma-1)V\\}.$

The equality $A_{\sigma}=G$ implies that $V^{G}=V^{\langle\sigma\rangle}$ has
dimension $n-1$, and the equality $B_{\sigma}=G$ implies that
$V_{G}=V/(\sigma-1)V$ has dimension $n-1$. In order to show that we have one
of these equalities, and therefore (2), we argue by contradiction. Assume
$A_{\sigma}$ and $B_{\sigma}$ are _strict_ subgroups of $G$, and pick $\tau\in
G$ outside $A_{\sigma}\cup B_{\sigma}$. Then there exist $s\in
V^{\langle\sigma\rangle}\setminus V^{\langle\tau\rangle}$ and $t\in
V^{\langle\tau\rangle}\setminus V^{\langle\sigma\rangle}$, and $(\sigma-1)V$
and $(\tau-1)V$ are _different_ 1-dimensional subspaces of $V$ spanned by
$(\sigma-1)t$ and $(\tau-1)s$, respectively.

The subspace $(\tau\sigma-1)V$ is 1-dimensional and spanned by
$(\tau\sigma-1)s=(\tau-1)s$, so it equals $(\tau-1)V$. It contains
$(\tau\sigma-1)t=\tau(\sigma-1)t$, but since $\tau$ acts on
$(\sigma-1)t\notin(\tau-1)V$ by translation along a vector in $(\tau-1)V$, we
have $\tau(\sigma-1)t\notin(\tau-1)V$. Contradiction. ∎

For those who like to think of Lemma 4.1 in terms of matrices, Condition (1)
means that every element of $G$ has a matrix representation with respect to a
suitable basis that, according to (9), can be given in one of the equivalent
forms

(10) $\left(\begin{array}[]{cc}\framebox[42.67912pt][c]{
\mbox{$\begin{array}[]{c}\\\ I_{n-1}\\\ \\\
\end{array}$}}&\begin{array}[]{c}*\\\ \vdots\\\ \end{array}\\\
\begin{array}[]{ccc}0&\cdots&0\end{array}&*\end{array}\right)\qquad\text{or}\qquad\left(\begin{array}[]{cc}*&\begin{array}[]{ccc}*&\cdots&*\end{array}\\\
\begin{array}[]{c}0\\\ \vdots\\\
0\end{array}&\framebox[42.67912pt][c]{\mbox{$\begin{array}[]{c}\\\ I_{n-1}\\\
\\\ \end{array}$}}\end{array}\right).$

The first form shows $n-1$ linearly independent vectors in
$V^{\langle\sigma\rangle}$, the second starts from a vector spanning
$(\sigma-1)V$. The lemma then states that under this condition, a _single_
basis for $V$ can be chosen such that either all elements of $G$ have a matrix
representation of the first form, or they all have one of the second form.

######  Example 4.2.

For an elliptic curve $E/K$, we can apply Lemma 4.1 to the action of the
Galois group $G$ of the $\ell$-division field of $E$ over $K$ on the
2-dimensional ${\mathbf{F}}_{\ell}$-vector space $V=E[\ell](\overline{K})$ of
$\ell$-torsion points of $E$. In this case the point group
$E(k_{\mathfrak{p}})$ at a prime ${\mathfrak{p}}\nmid\ell$ of good reduction
is of order divisible by $\ell$ if and only if
$\operatorname{Frob}_{\mathfrak{p}}\in G$ pointwise fixes a 1-dimensional
subspace of $V$. We find that almost all local point groups
$E(k_{\mathfrak{p}})$ are of order divisible by $\ell$ if and only if the
Galois representation $\rho_{E/K,\ell}$ of $G_{K}$ on the group of
$\ell$-torsion points of $E$ can be given in matrix form as

$\rho_{E/K,\ell}\sim\begin{pmatrix}1&*\\\
0&*\end{pmatrix}\qquad\text{or}\qquad\rho_{E/K,\ell}\sim\begin{pmatrix}*&*\\\
0&1\end{pmatrix}.$

In words: either $E(K)$ contains an $\ell$-torsion point, or it is
$\ell$-isogenous over $K$ to an elliptic curve with a $K$-rational
$\ell$-torsion point. Moreover, for $E/K$ of the first kind, with a point
$T\in E(K)$ of order $\ell$, the quotient curve $E^{\prime}=E/\langle
T\rangle$ is of the second kind, with the dual isogeny $E^{\prime}\to E$ being
the $\ell$-isogeny in question. This is a well-known fact that occurs as the
very first exercise in [Serre2]*p. I-2.

###### Proof of Theorem 1.2.

Let $E/K$ be an elliptic curve, and $P\in E(K)$ a non-torsion point that is
locally $\ell$-imprimitive. We define $K_{\ell}=K(\ell^{-1}P)$ as in Section
2, and view
$G_{\ell}=\operatorname{Gal}(K_{\ell}/K)\subset\operatorname{GL}(V_{\ell})$ as
a group of ${\mathbf{F}}_{\ell}$-linear automorphisms of the 3-dimensional
vector space $V_{\ell}=\langle\ell^{-1}P\rangle/\langle P\rangle$. As every
element of $G_{\ell}$ occurs as the Frobenius of infinitely many primes of
good reduction, it follows from Lemma 2.3 that all elements of $G_{\ell}$
leave a 2-dimensional subspace of $V_{\ell}$ pointwise invariant. We can now
apply our Lemma 4.1 for $n=3$ with $G=G_{\ell}$ and $V=V_{\ell}$ to conclude
that at least one of the following occurs: either $G_{\ell}$ acts trivially on
a 2-dimensional subspace of $V_{\ell}$, or $G_{\ell}$ acts trivially on a
2-dimensional quotient space of $V_{\ell}$.

In the first case, if $U_{\ell}=E[\ell](\overline{K})\subset V_{\ell}$ is a
subspace with trivial $G_{\ell}$-action, then $E(K)$ has complete
$\ell$-torsion and Condition B is satisfied. If $G_{\ell}$ acts trivially on a
different 2-dimensional subspace $S_{\ell}\subset V_{\ell}$, than $S_{\ell}$
is spanned by a non-zero vector in $U_{\ell}\cap S_{\ell}$ and the non-zero
image of a point of infinite order $Q\in\langle\ell^{-1}P\rangle$ in the
${\mathbf{F}}_{\ell}$-vector space $V_{\ell}$. In other words: $E(K)$ contains
a torsion point of order $\ell$ and a point $Q$ with $\ell Q=mP$ for some
$m\in{\mathbf{Z}}$ that is not divisible by $\ell$. Writing $am+b\ell=1$ in
${\mathbf{Z}}$, the point $Q^{\prime}=aQ+bP\in E(K)$ satisfies $\ell
Q^{\prime}=a\ell Q+b\ell P=amP+b\ell P=P$, so Condition A is satisfied.

In the second case, where $G_{\ell}$ acts trivially on a 2-dimensional
quotient space $V_{\ell}/T_{\ell}$, it acts on $V_{\ell}$ by translation along
the 1-dimensional subspace $T_{\ell}$. We will assume that $G_{\ell}$ does not
act trivially on the subspace $U_{\ell}$, as this would bring us back in the
first case, with Condition B holding. As
$U_{\ell}=E[\ell](\overline{K})\subset V_{\ell}$ is $G_{\ell}$-stable, we have
strict inclusions $0\subsetneq T_{\ell}\subsetneq U_{\ell}$ of
${\mathbf{F}}_{\ell}[G_{\ell}]$-modules, so $T_{\ell}$ is a $K$-rational
subgroup of $E(\overline{K})$ of order $\ell$. The corresponding isogeny $E\to
E^{\prime}=E/T_{\ell}$ is defined over $K$, and identifies the
${\mathbf{F}}_{\ell}[G_{\ell}]$-module $U_{\ell}/T_{\ell}$, which has trivial
$G_{\ell}$-action, with the subgroup of $E^{\prime}(K)$ of order $\ell$ that
is the kernel of the isogeny $\phi:E^{\prime}\to E$ dual to
$\widehat{\phi}:E\to E^{\prime}=E/T_{\ell}$. If $Q\in E(\overline{K})$
satisfies $\ell Q=P$, then $P^{\prime}=\widehat{\phi}(Q)$ is in
$E^{\prime}(K)$, as it is the image of any point in the Galois orbit
$G_{\ell}\cdot Q\subset Q+T_{\ell}$. Moreover, we have
$\phi(P^{\prime})=\phi\widehat{\phi}(Q)=\ell Q=P$, so Condition C of Theorem
1.2 is satisfied.

Conversely, each of the Conditions A, B, and C guarantees that $P\in E(K)$ is
locally $\ell$-imprimitive. For $A$ and $B$ this is immediate. If Condition C
holds, we have an $\ell$-isogeny $\phi:E^{\prime}\to E$ defined over $K$ and a
point $P^{\prime}\in E^{\prime}(K)$ with $\phi(P^{\prime})=P$. Pick
$Q^{\prime}\in E^{\prime}(\overline{K})$ with $\ell Q^{\prime}=P^{\prime}$ and
put $Q=\phi(Q^{\prime})\in E(\overline{K})$. Writing $\widehat{\phi}:E\to
E^{\prime}$ for the dual isogeny, we are in the situation of Example 4.2, and
we have

$\ell Q=\phi(\ell
Q^{\prime})=\phi(P^{\prime})=P\qquad\text{and}\qquad\widehat{\phi}Q=\widehat{\phi}\phi
Q^{\prime}=\ell Q^{\prime}=P^{\prime}\in E^{\prime}(K).$

As $Q$ is in the fibre $\widehat{\phi}^{-1}(P^{\prime})$, the
$G_{\ell}$-action on $Q\bmod\langle P\rangle\in V_{\ell}$, which is by
translation over $\ell$-torsion points, gives rise to a Galois orbit of length
dividing $\ell$. If the length is 1, then Condition A is satisfied, and $P\in
E(K)$ is locally $\ell$-imprimitive. If the length is $\ell$, then $G_{\ell}$
acts on $V_{\ell}$ by translation along the $K$-rational subgroup
$T_{\ell}=\ker\widehat{\phi}\subset U_{\ell}=E[\ell](\overline{K})$, and the
matrix representation of $G_{\ell}$ on $V_{\ell}$ with respect to the
filtration $T_{\ell}\subset U_{\ell}\subset V_{\ell}$ is

$\rho_{E/K,\ell}\sim\begin{pmatrix}*&*&*\\\ 0&1&0\\\ 0&0&1\end{pmatrix}.$

By Lemma 2.3, we find that we have
$\ell|[E(k_{\mathfrak{p}}):\langle{\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm P}}$}}}\rangle]$ for
every prime ${\mathfrak{p}}$ of good reduction of $E$ that is of
characteristic different from $\ell$, so $P\in E(K)$ is locally
$\ell$-imprimitive. ∎

##  5\. Locally 2-imprimitive points

A _non-trivial_ locally $\ell$-imprimitive point on an elliptic curve $E/K$ is
a non-torsion point $P\in E(K)$ for which Condition C of Theorem 1.2 holds,
but not Conditions A or B. If $P$ is such a point, $E$ admits a $K$-rational
$\ell$-isogeny $E\to E^{\prime}$, and the Galois representation
$\mathchoice{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm\rho}}$}}_{E,\ell}$ of
$G_{K}$ on $U_{\ell}=E[\ell](\mathchoice{K\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm K}}$}})$ is non-trivial,
with image contained in a Borel subgroup of $\operatorname{GL}(U_{\ell})$.

Let $P\in E(K)$ be a non-trivial locally 2-imprimitive point. As a Borel
subgroup of
$\operatorname{GL}(U_{2})\cong\operatorname{GL}_{2}({\mathbf{F}}_{2})$ has
order 2, the representation $\mathchoice{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm\rho}}$}}_{E,2}$ is a
non-trivial quadratic character, and as a Weierstrass model for $E$ we can
take

(11) $E:y^{2}=x(x^{2}+ax+b)\qquad\hbox{with }b,d=a^{2}-4b\in K^{*}.$

Here $(0,0)$ is the $K$-rational point of order 2, and $\mathchoice{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm\rho}}$}}{\rho\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm\rho}}$}}_{E,2}$
corresponds to the $2$-division field of $E$ over $K$, which equals
$K(\sqrt{d})$. Addition by $(0,0)$ induces an involution

$(x,y)\mapsto(x_{1},y_{1})=(b/x,-by/x^{2})$

on the function field $K(E)=K(x,y)$ of $E$, and the invariant field
$K(x+x_{1},y+y_{1})$ is the function field of the 2-isogenous curve
$E^{\prime}=E/\langle(0,0)\rangle$. Choosing $u=x+x_{1}+a$ and $v=y+y_{1}$ as
generators for $K(E^{\prime})$, we obtain a Weierstrass model

(12) $E^{\prime}:v^{2}=u(u^{2}-2au+d)\qquad\hbox{with
}d,d^{\prime}=(-2a)^{2}-4d=16b\in K^{*}$

for $E^{\prime}$ that is of the same form (11), and an explicit 2-isogeny
$\varphi:E\to E^{\prime}$ given by

(13)
$\varphi:(x,y)\longmapsto(u,v)=(x+x_{1}+a,y+y_{1})=\left(\frac{y^{2}}{x^{2}},(1-\frac{b}{x^{2}})y\right).$

An affine point $(u,v)\in E^{\prime}(K)$ different from $(0,0)$ is in the
image of $E(K)$ under this isogeny if and only if $u\in K^{*}$ is a square.
The point $(0,0)$ is in the image if and only if $d$ is a square in $K^{*}$,
which amounts to saying that $E(K)$ has full 2-torsion. This not the case for
our $E$.

As $E^{\prime}$ is again of the form (11), one sees that the isogeny
$\widehat{\varphi}:E^{\prime}\to E$ dual to $\varphi$ is given by

(14)
$\widehat{\varphi}:(u,v)\longmapsto\left(\frac{v^{2}}{4u^{2}},(1-\frac{b}{u^{2}})\frac{v}{8}\right).$

###### Proof of Theorem 1.3.

Let $E/K$ be an elliptic curve with $\\#E(\overline{K})=2$. We take $(0,0)$ as
the point of order 2 in a Weierstrass model of $E$ in order to obtain an
equation $E:y^{2}=x(x^{2}+ax+b)$ as in (11), with $d=a^{2}-4b\in
K^{*}\setminus{K^{*}}^{2}$. Any quadratic twist of $E$ over $K$ is of the form

$E_{D}:y^{2}=x(x^{2}+aDx+bD^{2})$

for some $D\in K^{*}$ that we may still rescale by squares in $K^{*}$, and we
can define the 2-isogeny $\varphi:E_{D}\to
E_{D}^{\prime}=E/\langle(0,0)\rangle$ as above by replacing $(a,b)$ in (11),
(12), and (13) by $(aD,bD^{2})$. Any point $P\in E_{D}(K)$ satisfying
Condition C from Theorem 1.2 is in the image
$\widehat{\varphi}(E_{D}^{\prime}(K))$ of the isogeny
$\widehat{\varphi}:E_{D}^{\prime}\to E_{D}$ dual to $\varphi$. This means that
its $x$-coordinate is a square in $K^{*}$, which we can take to be 1 after
rescaling the model of $E$ over $K$. Thus, the twists that are relevant for us
are those for which the point $P=(1,\pm Y)\in E_{D}(\mathchoice{K\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm K}}$}}{K\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm K}}$}})$ is
$K$-rational.

We want $(D,Y)$ to be a $K$-rational point on the conic $Y^{2}=1+aD+bD^{2}$
different from $(0,\pm 1)$. Such points are obtained as the second point of
intersection of this conic with the line $Y-1=\lambda D$ through $(0,1)$ with
slope $\lambda\in K$. We find that the twists $E_{\lambda}=E_{D_{\lambda}}$ of
$E$ by

(15) $D_{\lambda}=\frac{a-2\lambda}{\lambda^{2}-b}\qquad\text{with }\lambda\in
K\setminus\\{a/2,\pm\sqrt{b}\\}$

come by construction with a $K$-rational point

(16)
$P_{\lambda}=(1,Y_{\lambda})=\left(1,\frac{\lambda^{2}-a\lambda+b}{\lambda^{2}-b}\right)\in
E_{\lambda}(K).$

We can find $K_{2}=K(\frac{1}{2}P_{\lambda})$ by solving
$2Q=\widehat{\varphi}(\varphi Q)=P_{\lambda}$ in $E_{\lambda}(\overline{K})$.
The equation $\widehat{\varphi}(u,v)=P_{\lambda}$ has 2 solutions in
$E^{\prime}_{\lambda}(K)$, since we chose for the first coordinate of
$P_{\lambda}$ the square value 1. By (14), these are the $K$-rational points
$(u,v)$ different from $(0,0)$ that have $v^{2}=4u^{2}$ and satisfy the
equation

$v^{2}=u(u^{2}-2aD_{\lambda}u+dD_{\lambda}^{2})$

defining $E^{\prime}_{\lambda}$. With $d=a^{2}-4b$, the resulting equation

$u^{2}-2aD_{\lambda}u-4u+dD_{\lambda}^{2}=(u-aD_{\lambda}-2)^{2}-4(1+aD_{\lambda}+bD_{\lambda}^{2})=0$

for $u$ yields two solutions $u_{1},u_{2}=aD_{\lambda}+2\pm Y_{\lambda}\in K$
with product $u_{1}u_{2}=dD_{\lambda}^{2}$. Writing $D_{\lambda}$ and
$P_{\lambda}$ as in (15) and (16), we find
$aD_{\lambda}+2-Y_{\lambda}=d/(\lambda^{2}-b)$, so the minimal extension
$K\subset K_{2}$ for which the corresponding points are in $\varphi[E(K_{2})]$
is

$K_{2}=K(\textstyle\frac{1}{2}P_{\lambda})=K(\sqrt{u_{1}},\sqrt{u_{2}})=K(\sqrt{d},\sqrt{\lambda^{2}-b}).$

If we avoid the values $\lambda\in K$ for which $\lambda^{2}-b$ is either a
square or $d$ times a square in $K$–this includes the 3 values excluded in
(15)–then $K_{2}$ is a biquadratic extension of $K$ which, unsurprisingly, has
the 2-division field $K(\sqrt{d})$ of $E/K$ as one of its quadratic
subextensions. For these $\lambda$, our matrix representation from (5) and (6)
of $G_{2}=\operatorname{Gal}(K_{2}/K)$ becomes

$\operatorname{Gal}(K_{2}/K)=\left\\{\begin{bmatrix}1&x&y\\\ 0&1&0\\\
0&0&1\end{bmatrix}:x,y\in{\mathbf{F}}_{2}\right\\},$

which implies that $P_{\lambda}\in E_{\lambda}(K)$ is globally 2-primitive,
but locally $2$-imprimitive.

There is still the possibility that $P_{\lambda}$, though globally
2-primitive, is a torsion point of even order $m>2$. Examples: the point
$(1,-3)$ on $y^{2}=x(x^{2}-7x+3)$ has order 4, and the point $(1,1)$ on
$y^{2}=x(x^{2}+3x-3)$ has order 6. However, for fixed $K$ there are only
finitely many possibilities for $m$ by Merel’s theorem. For every given even
$m$, the point $P_{\lambda}$ is of order $m$ if and only if the $m$-th
division polynomial $\psi_{m}(x)=y^{-1}f(x,E_{\lambda})$ vanishes at $x=1$.
This happens for only finitely many values of $\lambda$, as $f(1,E_{\lambda})$
is a non-constant rational function of $\lambda$ if we fix $a,b\in K$. ∎

Our understanding of local 2-imprimitivity is more or less complete, as every
non-trivial 2-locally imprimitive point on an elliptic curve arises as in the
construction in the proof of Theorem 1.3. Indeed, the hypothesis
$\\#E[2](\overline{K})=2$ implies that $E$ has a model as in (11), and as the
$x$-coordinate of a point $P$ satisfying Condition C of Theorem 1.2 is a
square, the model can be scaled over $K$ to have $P=(1,y)\in E(K)$.

##  6\. Locally 3-imprimitive points

By Theorem 1.2, every pair $(E,P)$ of an elliptic curve $E/K$ with a non-
trivial locally $\ell$-imprimitive point $P\in E(K)$ arises as the
$\ell$-isogenous image of a $K$-rational curve-point-pair
$(E^{\prime},P^{\prime})$ for which the kernel $E^{\prime}\to E$ is generated
by an $\ell$-torsion point $T\in E^{\prime}(K)$. In this situation, the Galois
representations of $G_{K}$ on the $\ell$-torsion subgroups of $E^{\prime}$ and
$E=E^{\prime}/\langle T\rangle$ are, with respect to a suitable basis, of the
form

(17) $\rho_{E^{\prime}/K,\ell}\sim\begin{pmatrix}1&*\\\
0&\omega_{\ell}\end{pmatrix}\qquad\text{and}\qquad\rho_{E/K,\ell}\sim\begin{pmatrix}\omega_{\ell}&*\\\
0&1\end{pmatrix}.$

Here $\omega_{\ell}$ is the cyclotomic character corresponding to the
extension $K\subset K(\zeta_{\ell})$. For $\ell>2$, the $K$-rationality of
$\ell$-torsion points of $E$ is not preserved under twisting of $E$, so there
is no direct analogue of Theorem 1.3 for $\ell\neq 2$. In this section we
focus on the case $\ell=3$.

######  Lemma 6.1.

Let $E/K$ be an elliptic curve of discriminant $\Delta_{E}$ for which the
Galois representation $\rho_{E/K,3}$ on $U_{3}=E[3](\overline{K})$ is of one
of the two forms in (17). Then the $3$-division field of $E$ over $K$ equals
the splitting field of the polynomial $X^{3}-\Delta_{E}$.

###### Proof.

Let
$H_{3}=\operatorname{Gal}(K(E[3])/K)\subset\operatorname{GL}(U_{3})\cong\operatorname{GL}_{2}({\mathbf{F}}_{3})$
be the image of $\rho_{E/K,3}$, and denote by
$\psi_{3}=\prod_{i=1}^{4}(X-x_{i})\in K[X]$ the 3-division polynomial of $E$
over $K$. The quartic polynomial $\psi_{3}$ comes with a Galois resolvent
$\delta_{3}\in K[X]$, a cubic having

$\alpha_{1}=x_{1}x_{2}+x_{3}x_{4},\qquad\alpha_{2}=x_{1}x_{3}+x_{2}x_{4},\qquad\alpha_{3}=x_{1}x_{4}+x_{2}x_{3}$

as its roots. Under the permutation action of
$\operatorname{GL}(U_{3})/\langle-1\rangle=S_{4}$ on the roots of $\psi_{3}$,
the normal subgroup $V_{4}\triangleleft S_{4}$ of order 4 fixes each of these
3 roots $\alpha_{i}$. The two natural surjections of Galois groups

(18) $H_{3}\to\operatorname{Gal}(\psi_{3})\to\operatorname{Gal}(\delta_{3})$

are isomorphisms as they arise as a restriction to suitable subgroups of the
generic group theoretical maps

$\operatorname{GL}(U_{3})\to\operatorname{GL}(U_{3})/\langle-1\rangle=S_{4}\to
S_{4}/V_{4}=S_{3}.$

More precisely, the first surjection in (18) is injective as we have $-1\notin
H_{3}$ for $H_{3}$ as in (17), and the second is because we have
$\operatorname{Gal}(\psi_{3})\cap V_{4}=1$ in $S_{4}$: the $x$-coordinate of
the 3-torsion point spanning the Galois invariant subspace corresponding to
the first column of the matrices is fixed by $\operatorname{Gal}(\psi_{3})$.
Viewing $H_{3}$ as $\operatorname{Gal}(\delta_{3})$, we may finish the proof
by quoting a classical formula [Serre]*p. 305 that expresses the three cube
roots of $\Delta_{E}$ as $b_{4}-3\alpha_{i}$ ($i=1,2,3$), with $b_{i}\in K$ a
coefficient from the Weierstrass model of $E$. This yields
$K(E[3])=\operatorname{Split}_{K}(g)=\operatorname{Split}_{K}(X^{3}-\Delta_{E})$.
∎

From Lemma 6.1, we see that for the representations in (17), the subgroup
$\binom{1\ *}{0\ 1}$ corresponds to the extension $K(\zeta_{3})\subset
K(\zeta_{3},\root 3 \of{\Delta})$ for the discriminant values
$\Delta=\Delta_{E^{\prime}}$ and $\Delta_{E}$.

We can write the curve $E^{\prime}$ in Deuring normal form [Husemoller]*p. 89
as

(19) $E^{\prime}:y^{2}+axy+by=x^{3}\qquad\hbox{with }a,b\in K\hbox{ and
}\Delta_{E^{\prime}}=b^{3}(a^{3}-27b)\in K^{*}.$

Here $T=(0,0)\in E^{\prime}(K)$ is the point of order 3, and the quotient
curve $E=E^{\prime}/\langle T\rangle$ has Weierstrass equation

(20) $E:y^{2}+axy+by=x^{3}-5abx-(a^{3}+7b)b,$

with the explicit formula for the 3-isogeny $\varphi_{3}:E^{\prime}\to
E=E^{\prime}/\langle T\rangle$ being given by

(21)
$\varphi_{3}(x,y)=\left(\frac{x^{3}+abx+b^{2}}{x^{2}},\frac{y(x^{3}-abx-2b^{2})-b(ax+b)^{2}}{x^{3}}\right).$

As $E$ has discriminant $\Delta_{E}=b(a^{3}-27b)^{3}$, the 3-division field
$K(E[3])$ over $K$ equals $K(\zeta_{3})$ if and only if $b\in K^{*}$ is a cube
and different from $(a/3)^{3}$.

If $a$ is non-zero, we can rescale $(x,y)\mapsto(a^{2}x,a^{3}y)$ and simplify
(19) to

(22) $E_{b}^{\prime}:y^{2}+xy+by=x^{3}.$

For $K$ a number field, we have infinitely many pairwise different 3-isogenous
images $(E_{b},P_{b})=\varphi_{3}[(E_{b}^{\prime},P_{b}^{\prime})]$ for which
$P_{b}\in E_{b}(K)$ is non-trivially locally 3-imprimitive.

######  Theorem 6.2.

For $K$ a number field, take $b=b(X)=(1-X-X^{2})/X\in K(X)$ and define the
associated elliptic curve over $K(X)$ as

$E_{b}:y^{2}+xy+by=x^{3}-5bx-(1+7b)b.$

Then for infinitely many $t\in K^{*}$, the specialization $E_{b(t)}$ of
$E_{b}$ is an elliptic curve over $K$ for which

(23)
$\left(\frac{(t^{2}+t)(t^{2}-1)+1}{t^{2}},\frac{(1-t^{2})\left(t^{4}+2t^{3}+t-1\right)}{t^{3}}\right)\in
E_{b(t)}(K)$

is a non-trivial locally 3-imprimitive point.

###### Proof.

For the curve $E^{\prime}_{b}$ in (22) with $b\in K(X)$ as defined, the point
$P^{\prime}_{b}=(1,X)$ lies on $E^{\prime}_{b}$ by the very choice of $b$: it
satisfies $X^{2}+X+bX=1$. Under the 3-isogeny $\varphi_{3}:E^{\prime}_{b}\to
E_{b}$ from (21) to the curve $E_{b}$ obtained by putting $a=1$ in (20), it is
mapped to

$P_{b}=(1+b+b^{2},(1-b-2b^{2})X-b(1+b)^{2})\in E_{b}(K(X)).$

Under the specialization $X=t\in K^{*}$, we obtain a point $P_{b(t)}$, on the
curve $E_{b(t)}$ defined over $K$ that is given by (23). We are only
interested in those specializations for which $E_{b(t)}$ is an elliptic curve.
As these are all $t\in K^{*}$ for which $b(t)\notin\\{0,1/27\\}$, at most 4
‘bad’ values of $t$ are excluded. Also, by the same argument as we gave for
$P_{\lambda}$ in the case $\ell=2$, there are only finitely many $t\in K^{*}$
for which $P_{t}$ is torsion point. These finitely many $t$ we also exclude as
‘bad’ values.

We saw already that $E_{b}$ has 3-division field $K(\zeta_{3},\root 3
\of{b})$, and an explicit computation shows that the 3-division field of the
point $P_{b}\in E_{b}(K(X))$ equals

$K(X)(\textstyle\frac{1}{3}P_{b})=K(\zeta_{3},\root 3 \of{b},\root 3
\of{X})=K(\zeta_{3},\root 3 \of{X},\root 3 \of{X^{2}+X-1}).$

Over $K(\zeta_{3},X)$, the elements $X$ and $X^{2}+X-1$ have ‘independent’
cube roots – it suffices look at their ramification locus. It follows that the
Galois group of the 3-division field of $P_{b}$ over $K(X)$ may be described
as

$\operatorname{Gal}(K(X)(\textstyle\frac{1}{3}P_{b})/K(X))=\left\\{\begin{bmatrix}\omega_{3}&x&y\\\
0&1&0\\\ 0&0&1\end{bmatrix}:x,y\in{\mathbf{F}}_{3}\right\\},$

with $\omega_{3}$ denoting the ${\mathbf{F}}_{3}^{*}$-valued character
corresponding to the (possibly trivial) extension $K(X)\subset
K(X,\zeta_{3})$. By Hilbert irreducibility, it follows that for infinitely
many $t\in K^{*}$ outside the finite set of ‘bad’ values, the 3-division field
of the point $P_{b(t)}\in E_{b(t)}(K)$ has the ‘same’ Galois group over $K$,
making it into a point that is globally 3-primitive, but locally
$3$-imprimitive. As $E_{b(t)}$ does not have complete 3-torsion over $K$, we
conclude that the point $P_{b(t)}$ given in (23) is a non-trivial locally
3-imprimitive point. ∎

######  Remark 6.3.

The construction in the proof of Theorem 6.2 excludes all specializations for
which $b=b(t)\in K^{*}$ is a cube and the elliptic curve in (22) has
3-division field $K(\zeta_{3})$. In this special case, we can also equip
$E=E_{b}$ with a non-trivial locally 3-imprimitive point for infinitely many
$b\in{K^{*}}^{3}$. We first write $b=c^{-3}$ and transform the curve under
$(x,y)\mapsto(c^{-2}x,c^{-3}y)$ into $E^{\prime}:y^{2}+cxy+y=x^{3}$. As in the
previous case, we have $P^{\prime}=(1,t)\in E^{\prime}(K)$ for
$c=(-t^{2}-t+1)/t$, and the image of $P^{\prime}$ under the map

$\varphi:E^{\prime}\to
E=E^{\prime}/\langle(0,0)\rangle:y^{2}+cxy+y=x^{3}-5cx-(c^{3}+7)$

is the point $P=\varphi_{3}(P^{\prime})=((-t^{2}+t+1)/t,(t^{2}-1)/t^{2})\in
E(K)$, for which the 3-division field $K(\frac{1}{3}P)$ is equal to
$K(\zeta_{3},\sqrt[3]{t})$. For almost all $t\notin{K^{*}}^{3}$, this makes
$P$ into a globally 3-primitive point that is locally $3$-imprimitive.

We conclude our discussion for $\ell=3$ with the remaining case in which the
curve $E^{\prime}$ in (19) has $a=0$. In this case $E^{\prime}$ has
$j$-invariant 0, and writing $c=b/2$ we may rescale the equation by $y\mapsto
y-c$ to the more familiar shape $E^{\prime}:y^{2}=x^{3}+c^{2}$, with 3-torsion
point $T=(0,c)$ and CM by ${\mathbf{Z}}[\zeta_{3}]$. We equip $E^{\prime}$
with a $K$-rational point $(c,c\sqrt{c+1})$ by putting $c=s^{2}-1$ with $s\in
K$. This leads to a 1-parameter family of 3-isogenies

$\displaystyle\varphi_{3}:E^{\prime}_{s}:y^{2}=x^{3}+(s^{2}-1)^{2}$
$\displaystyle\longrightarrow E_{s}:y^{2}=x^{3}-27(s^{2}-1)^{2}$
$\displaystyle P^{\prime}_{s}=(s^{2}-1,s(s^{2}-1))$ $\displaystyle\longmapsto
P_{s}=(s^{2}+3,s(s^{2}-9))$

between CM-curves with $j$-invariant 0. In this case the 3-division field of
$E_{s}$ over $K$ is $K(\zeta_{3},\root 3 \of{2(s^{2}-1)})$, and the 3-division
field of $P_{s}$ over $K$ equals

$K(\zeta_{3},\root 3 \of{2(s^{2}-1)},\root 3 \of{4(s+1)}).$

Again, for $s\in K$ outside a thin set, the point $P_{s}\in E_{s}(K)$ is
globally 3-primitive but locally 3-imprimitive.

##  7\. Further examples

Over $K={\mathbf{Q}}$, non-trivial locally $\ell$-imprimitive points can only
occur for primes $\ell\leq 7$. Examples for $\ell=5$ and also $\ell=7$ can be
found by the techniques that we employed for $\ell=3$, but the formulas and
resulting curves rapidly become less suitable for presentation on paper.

###  7.1. Curves with a locally 5-imprimitive point

In this case, we start from Tate’s normal form

(24) $E^{\prime}:y^{2}+(1-c)xy-cy=x^{3}-cx^{2}$

that parametrises elliptic curves with $(0,0)$ as a $5$-rational point (see
Kulesz [Kulesz]). It has further points $(0,c)$, $(c,0)$ and $(c,c^{2})$ of
order 5, and its discriminant equals
$\Delta_{E^{\prime}}=c^{5}\left(c^{2}-11c-1\right)$. Using Vélu’s formula
[Velu] or invoking Pari-GP, we compute the Weierstrass equation for the
5-isogenous curve $E=E^{\prime}/\langle(0,0)\rangle$

$\displaystyle E=E_{c}:\quad y^{2}+$ $\displaystyle(1-c)xy-cy=$ $\displaystyle
x^{3}-cx^{2}-5c(c^{2}+2c-1)x-c(c^{4}+10c^{3}-5c^{2}+15c-1),$

and also the explicit 5-isogeny $\varphi_{5}:E^{\prime}\rightarrow E$. The
discriminants involved are $\Delta_{E^{\prime}}=c^{5}(c^{2}-11c-1)$ and
$\Delta_{E}=c(c^{2}-11c-1)^{5}$, much like we saw for $\ell=3$. The 5-torsion
representations of $E^{\prime}$ and $E$ are as in (17), and even though the
proof of Lemma 6.1 for $\ell=3$ does not generalize to $\ell=5$, we found by a
direct calculation that the 5-division fields are
$K(E^{\prime}[5])=K(\zeta_{5},\sqrt[5]{c^{2}-11c-1)})$ and
$K(E[5])=K(\zeta_{5},\sqrt[5]{c})$: generated over $K(\zeta_{5})$ by the 5-th
root of the discriminant.

We can equip $E^{\prime}$ with a $K$-rational point $P_{t}^{\prime}=(t,t)$ by
putting $c=t(2-t)$, and compute its image $P_{t}=\varphi_{5}(P^{\prime})\in
E_{t(2-t)}(K)$ as

$P_{t}=\big{(}\frac{2t^{4}-8t^{3}+11t^{2}-6t+2}{(t-1)^{2}},-\frac{t^{8}-7t^{7}+19t^{6}-23t^{5}+4t^{4}+23t^{3}-31t^{2}+19t-4}{(t-1)^{3}}\big{)}.$

The corresponding 5-division field of $P_{t}$ is

$\textstyle K(\frac{1}{5}P_{t})=K(\zeta_{5},\root 5 \of{t},\root 5 \of{t-2}).$

If this is an extension of degree 5 of $K(\zeta_{5},\root 5
\of{c})=K(\zeta_{5},\root 5 \of{t(t-2)})$, then $P_{t}$ is a globally
5-primitive but locally 5-imprimitive point in $E_{t(2-t)}(K)$.

######  Example 7.1.

Take $K={\mathbf{Q}}$. For $t=1$ the point $P_{t}$ above is the zero point as
$P^{\prime}_{t}$ is 5-torsion, for $t=2$ we have $c=0$ and $E$ is singular,
while for $t=3$ and 4 we encounter the ‘accidents’ $t=-c$ and $tc=2^{5}$
leading to points $P_{t}\in 5E_{t(2-t)}({\mathbf{Q}})$. For $t=5$ we obtain
the point $P_{5}=(497/16,-73441/64)$ on

$E_{-15}:y^{2}+16xy+15y=x^{3}+15x^{2}+14550x+232860,$

which is the curve 5835.c2 in the LMFDB-database. Note that for $c=-15$ we
have

$c(c^{2}-11c-1)=-5835=-3\cdot 5\cdot 389.$

The locally 5-imprimitive point $P_{5}$ is a generator of
$E_{-15}({\mathbf{Q}})\cong{\mathbf{Z}}$. In fact, $P_{t}$ will be globally
5-primitive but locally 5-imprimitive in $E_{t(2-t)}({\mathbf{Q}})$ for all
$t\in{\mathbf{Z}}_{\geq 5}$ that are not a fifth power or a fifth power plus
2, as for these $t$ the subgroup of ${\mathbf{Q}}^{*}/{{\mathbf{Q}}^{*}}^{5}$
generated by $t$ and $t-2$ has order 25.

###  7.2. Curves with a locally 7-imprimitive point

Again we start from the Tate’s normal equation

$E^{\prime}:y^{2}+(1-c)xy-by=x^{3}-bx^{2}$

but now we do not impose $b=c$ as for $\ell=5$, but instead

$c=d^{2}-d\quad\text{ and }b=d^{3}-d^{2}.$

The curve $E^{\prime}=E_{d}^{\prime}$ parametrizes [Kulesz] elliptic curves
with $(0,0)$ as point of order 7. A Weierstrass equation for the 7-isogenous
curve $E=E^{\prime}/\langle(0,0)\rangle$ is

$E:y^{2}+(1-c)xy-
by=x^{3}-bx^{2}-5\left(2b^{2}+b\left(c^{2}-3c-2\right)+c\left(c^{2}+4c+1\right)\right)x$
$-b^{2}\left(12c^{2}+c+24\right)-6b^{3}+b\left(-c^{4}+9c^{3}+46c^{2}+24c+2\right)-$
$c\left(c^{4}+16c^{3}+36c^{2}+16c+1\right).$

It has discriminant $\Delta_{E}=d(d-1)(d^{3}-8d^{2}+5d+1)^{7}$, and this time
we find its 7-division field to be
$K(E[7])=K(\zeta_{7},\sqrt[7]{d(d-1)^{2}}).$

We equip $E^{\prime}$ with a $K$-rational point
$P_{t}^{\prime}=(d^{2}t,d^{3}t)$ by putting $d=d(t)=({t+1)/(t^{2}-t+1})$. The
image of $P_{t}^{\prime}$ under the 7-isogeny $E^{\prime}\to E$ is

$P_{t}=\left(-\frac{C(t)}{(2t-1)^{2}(t-1)^{2}\left(t^{2}-t+1\right)^{4}},\frac{D(t)}{(t-1)^{3}(2t-1)^{3}\left(t^{2}-t+1\right)^{6}}\right)\in
E(K)$

for certain polynomials $C(t)$ and $D(t)$ in ${\mathbf{Z}}[t]$ of degree 12
and 18. In terms of $t$, the 7-division field is $K(E[7])=K(\zeta_{7},\root 7
\of{t^{2}(t+1)(t-2)^{2}(t^{2}-t+1)^{4}})$, and the 7-division field of $P_{t}$
is

$K(\textstyle\frac{1}{7}P_{t})=K(E[7])\left(\root 7
\of{\frac{t(t+1)}{t-2}}\right)=K\left(\zeta_{7},\root 7
\of{\frac{t(t^{2}-t+1)}{(t+1)}},\root 7 \of{\frac{t(t+1)}{(t-2)}}\right).$

The point $P_{t}$ is a globally 7-primitive but locally 7-imprimitive point
when the extension $K(\zeta_{7})\subset K(\frac{1}{7}P_{t})$ has its generic
degree $7^{2}$.

######  Example 7.2.

Take $K={\mathbf{Q}}$. For $t=1$ the point $P_{t}$ above is the zero point as
$P^{\prime}_{t}$ is 7-torsion, and for $t=2$ the curve $E^{\prime}$ is
singular. For $t=3$ and $d=\frac{4}{7}$ however we obtain the point
$P_{3}=(286019/490^{2},15951227/490^{3})$ on

$\textstyle
E:y^{2}+\frac{61}{7^{2}}xy+\frac{48}{7^{3}}y=x^{3}+\frac{48}{7^{3}}x^{2}-\frac{774780}{7^{7}}x-\frac{1047829260}{7^{11}},$

which is the curve 20622.j1 with minimal model

$E_{0}:y^{2}+xy=x^{3}-5455771x-5039899603,$

in the LMFDB-database. Our locally 7-imprimitive point $P_{3}$ is a generator
of $E({\mathbf{Q}})\cong{\mathbf{Z}}$. On $E_{0}$ the corresponding generator
is $(328219/10^{2},109777927/10^{3})$.

##  8\. A composite level obstruction

So far we have focused on non-trivial obstructions to local primitivity at
prime level $\ell$, as this is a new phenomenon in the elliptic primitive root
case III that does not arise in the multiplicative primitive root case I and
the cyclic reduction case II.

In all three cases, there exist obstructions of different nature at composite
levels that arise from the _entanglement_ between finitely many of the
corresponding division fields $K_{\ell}$. These obstructions do not arise over
$K={\mathbf{Q}}$, and most examples in the cases I and II are created by base
changing to a well-chosen finite extension of the fields of definition
${\mathbf{Q}}(x)$ and ${\mathbf{Q}}(j_{E})$. Again, case III is different
here, as entanglement obstructions already occur over ${\mathbf{Q}}$. In this
Section we construct a level 6 obstruction.

Let $E/K$ be an elliptic curve with $\\#E[2](K)=2$, and $P\in E(K)$ a point of
infinite order. Then the 2 division field $K(E[2](\overline{K}))$ is a
quadratic extension of $K$. Assume that the 2-division field $K(\frac{1}{2}P)$
of $P$ is of maximal degree 4 over it. Then
$G_{2}=\operatorname{Gal}(K(\frac{1}{2}P)/K)$ is a dihedral group of order 8
for which the matrix representation (6) on
$V_{2}=\langle\frac{1}{2}P\rangle/\langle P\rangle$ has the form

(25) $G_{2}=\left\\{\begin{bmatrix}1&a&c\\\ 0&1&b\\\
0&0&1\end{bmatrix}:a,b,c\in{\mathbf{F}}_{2}\right\\}\subset\operatorname{GL}_{3}({\mathbf{F}}_{2}).$

There is a unique subfield of $L\subset K(\frac{1}{2}P)$ with Galois group
over $K$ isomorphic to the Klein 4-group $V_{4}=C_{2}\times C_{2}$, and we can
view $a$ and $b$ in the matrix representation (25) of $G_{2}$ as
${\mathbf{F}}_{2}$-valued quadratic characters on $G_{2}$ that generate the
character group of the quotient $\operatorname{Gal}(L/K)\cong V_{4}$ of
$G_{2}$.

For a prime ${\mathfrak{p}}\nmid 2\Delta_{E}$, the point $P$ generates a
subgroup of odd index in $E(k_{\mathfrak{p}})$ if and only if for its
Frobenius $\operatorname{Frob}_{{\mathfrak{p}},2}\in G_{2}$, viewed as a
matrix as in (25), the endomorphism
$(\operatorname{Frob}_{{\mathfrak{p}},2}-\operatorname{id}_{2}):V_{2}\to
V_{2}$ has ${\mathbf{F}}_{2}$-rank at least 2 (Lemma 2.3). We obtain the
criterion

(26) $2\nmid[E(k_{\mathfrak{p}}):\langle\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm
P}}$}}\rangle]\quad\Longleftrightarrow\quad
a(\operatorname{Frob}_{{\mathfrak{p}},2})=b(\operatorname{Frob}_{{\mathfrak{p}},2})=1\in{\mathbf{F}}_{2}.$

More precisely, $a(\operatorname{Frob}_{{\mathfrak{p}},2})=1$ implies that
$E(k_{\mathfrak{p}})$ does not have full 2-torsion, and
$b(\operatorname{Frob}_{{\mathfrak{p}},2})=1$ implies that $P$ is not only not
in $2E(k_{\mathfrak{p}})$, but also not a 2-isogenous image as in Condition C
of Theorem 1.2.

Suppose further that $E$ has a $K$-rational 3-torsion subgroup $T$, and let
$\varphi_{3}:E^{\prime}\to E$ be the isogeny dual to the quotient map
$\phi:E\to E^{\prime}=E/T$. Assume that the point $P$ is in
$\varphi_{3}[E^{\prime}(K)]$ but not in $3E(K)$. Then the 3-division field
$K(\frac{1}{3}P)$ of $P$ has Galois group
$G_{3}=\operatorname{Gal}(K(\frac{1}{3}P)/K)$ for which the matrix
representation on $V_{3}=\langle\frac{1}{3}P\rangle/\langle P\rangle$ will
‘generically’ be the group

(27) $G_{3}=\left\\{\begin{bmatrix}d&e&f\\\ 0&g&0\\\ 0&0&1\end{bmatrix}:\quad
d,g\in{\mathbf{F}}_{3}^{*},e,f\in{\mathbf{F}}_{3}\right\\}\subset\operatorname{GL}_{3}({\mathbf{F}}_{3})$

of order 36. In this case $d$ and $g$ can be viewed as a quadratic characters
$G_{3}\to{\mathbf{F}}_{3}^{*}$, and another application of Lemma 2.3 shows
that for primes ${\mathfrak{p}}\nmid 3\Delta_{E}$, we have

(28)
$g(\operatorname{Frob}_{{\mathfrak{p}},3})=1\in{\mathbf{F}}_{3}^{*}\quad\Longrightarrow\quad
3|[E(k_{\mathfrak{p}}):\langle\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm P}}$}}\rangle].$

Thus, for primes ${\mathfrak{p}}\nmid 6\Delta_{E}$, a necessary condition for
$\mathchoice{P\hbox to0.0pt{\hss$\overline{\phantom{\displaystyle\rm
P}}$}}{P\hbox to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm P}}$}}\in
E(k_{\mathfrak{p}})$ to be an elliptic primitive root is that the three
quadratic characters $a$, $b$ and $g$ occurring in (26) and (28) do not vanish
on the Frobenius automorphism of ${\mathfrak{p}}$ in $K_{6}=K(\frac{1}{6}P)$.
In other words: the prime ${\mathfrak{p}}$ has to be inert in the quadratic
extensions $K_{a}$, $K_{b}$ and $K_{g}$ of $K$ corresponding to these 3
characters.

Primes ${\mathfrak{p}}$ satisfying the condition above exist if the quadratic
extensions $K_{a}$, $K_{b}$ and $K_{g}$ are linearly disjoint over $K$, but
_not_ if they are the three quadratic subfields of a $V_{4}$-extension
$K\subset K_{a}K_{b}K_{g}$. In the latter case, we have a splitting
obstruction to local primitivity of $P$ in $K_{6}$ that does not exist in one
of the smaller fields $K(\frac{1}{2}P)$ or $K(\frac{1}{3}P)$: it has level 6,
but not 2 or 3, making it an obstruction caused by _entanglement_ of division
fields.

######  Example 8.1.

An example is provided by the elliptic curve $E/{\mathbf{Q}}$ with label
12100.j1 in the LMFDB data base. The curve $E$ has discriminant

$\Delta_{E}=2^{4}\cdot 5^{9}\cdot 11^{6},$

and if we take $(0,0)$ to be its unique ${\mathbf{Q}}$-rational 2-torsion
point it has Weierstrass model

$E:y^{2}=x^{3}+605x^{2}-3025x.$

For this curve we have $E({\mathbf{Q}})=\langle T_{2}\rangle\times\langle
P\rangle\cong{\mathbf{Z}}/2{\mathbf{Z}}\times{\mathbf{Z}}$ with $T_{2}=(0,0)$
of order $2$ and $P=(\frac{-13475}{36},\frac{1249325}{216})$ a generator of
infinite order. We have
$K_{a}={\mathbf{Q}}(E[2])={\mathbf{Q}}(\sqrt{\Delta_{E}})={\mathbf{Q}}(\sqrt{5})$,
and over this field the 2-division field ${\mathbf{Q}}(\frac{1}{2}P)$ of $P$
is the $V_{4}$ extension

${\mathbf{Q}}(E[2])={\mathbf{Q}}(\sqrt{5})\subset{\mathbf{Q}}(\textstyle\frac{1}{2}P)={\mathbf{Q}}(\sqrt{5},\sqrt{\pi},\sqrt{\mathchoice{\pi\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm\pi}}$}}{\pi\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm\pi}}$}}{\pi\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm\pi}}$}}{\pi\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm\pi}}$}}})$

generated by the square roots of $\pi=3+2\sqrt{5}$ and its conjugate. From
$\pi\mathchoice{\pi\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm\pi}}$}}{\pi\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm\pi}}$}}{\pi\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm\pi}}$}}{\pi\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm\pi}}$}}=-11$ we see that
${\mathbf{Q}}(\frac{1}{2}P)$ is cyclic of degree 4 over
${\mathbf{Q}}(\sqrt{-55})$, and that we have $K_{b}={\mathbf{Q}}(\sqrt{-11})$.

As $E$ acquires a 3-torsion point
$T_{3}=(\frac{55}{3},\frac{275}{9}\sqrt{165})$ over the quadratic field
${\mathbf{Q}}(\sqrt{165})={\mathbf{Q}}(\sqrt{-3\cdot-55})$ that generates a
${\mathbf{Q}}$-rational torsion subgroup of order 3, the 3-division field of
$E$ has quadratic subfields $K_{d}={\mathbf{Q}}(\sqrt{165})$ and
$K_{g}={\mathbf{Q}}(\sqrt{-55})$, making $K_{g}$ the third quadratic subfield
in the $V_{4}$-extension ${\mathbf{Q}}\subset K_{a}K_{b}$. Over the full
3-division field of $E$, the 3-division field of $P$ is the cubic extension

${\mathbf{Q}}(E[3])={\mathbf{Q}}(\sqrt{-3},\sqrt{-55},\sqrt[3]{2})\subset{\mathbf{Q}}(\textstyle\frac{1}{3}P)={\mathbf{Q}}(E[3],\sqrt[3]{\alpha})$

generated by a cube root of an element $\alpha=(3+\sqrt{-55})/2\in K_{g}$ of
norm 16, which shows that its Galois group over ${\mathbf{Q}}$ is the group
$G_{3}$ in (27). We conclude that $P$ is a locally never-primitive point of
$E({\mathbf{Q}})$ as the index of $\langle\mathchoice{P\hbox
to0.0pt{\hss$\overline{\phantom{\displaystyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\textstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptstyle\rm P}}$}}{P\hbox
to0.0pt{\hss$\overline{\phantom{\scriptscriptstyle\rm P}}$}}\rangle$ in
$E({\mathbf{F}}_{p})$ is always divisible by 2 or 3.

An upcoming paper will have further details on obstructions to primitivity of
composite level, and on how to find explicit examples.

## References