2. Characterisation by splitting conditions

In each of the three cases discussed in the introduction, we can characterise 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 $m\in{\mathbf{Z}}_{\ge1}$ 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 $\ell\ne p$ 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 $p=\mathrm{char} ({\mathfrak{p}})$ , 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 $\ell=\mathrm{char}\ {{\mathfrak{p}}}$ .

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 $m\in{\mathbf{Z}}_{\ge1}$ , 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 [ Reference Lenstra10 ] that the set $S_{K, x}$ has a natural density that is given by the inclusion-exclusion sum

(2·1) \begin{equation}\delta_{K,x}=\sum_{m=1}^{\infty} \frac{\mu(m)}{[K_m\,:\,K]}\end{equation}

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

(2·2) \begin{equation}\delta_{K,x}=\sum_{m|N} \frac{\mu(m)}{[K_m\,:\,K]}\cdot \prod_{\ell\nmid N \text{ prime}} \left(1-\frac{1}{\ell(\ell-1)}\right).\end{equation}

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 $\mathrm{Gal}(K_\ell/K)\cong \mathrm{Aff}_1({\mathbf{F}}_\ell)={\mathbf{F}}_\ell\rtimes {\mathbf{F}}_\ell^*$ 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 $m\in{\mathbf{Z}}_{\ge1}$ , 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 [ Reference Campagna and Stevenhagen2 , 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 $\ell$ .

As in the multiplicative case I, cyclicity of the p-primary part of the groups $E(k_{\mathfrak{p}})$ is automatic for $p=\mathrm{char} ({\mathfrak{p}})$ . Also here, total splitting of ${\mathfrak{p}}$ in non-trivial extensions $K\subset K_p$ for $p=\mathrm{char}\ {{\mathfrak{p}}}$ 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 $\ell=\mathrm{char}\ {{\mathfrak{p}}}$ . 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 [ Reference Campagna and Stevenhagen2 , Section 2] by an inclusion-exclusion sum that we already know from (2·1):

(2·3) \begin{equation}\delta_{E/K}=\sum_{m=1}^{\infty} \frac{\mu(m)}{[K_m\; :\; K]}.\end{equation}

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 factorisation of $\delta_{E/K}$ that is typographically identical to (2·1), provided that N is divisible by all primes from an appropriately defined finite set of critical primes [ Reference Campagna and Stevenhagen2 , theorems 1.1 and 1.2]. If E has CM by an order $\mathcal{O}\not\subset K$ , there is a hybrid formula [ Reference Campagna and Stevenhagen2 , theorem 1.4] with different contributions from ordinary and supersingular primes.

A ‘factorisation formula’ for $\delta_{K, x}$ and $\delta_{E/K}$ as in (2·2) shows that the vanishing of these densities is always caused by an obstruction at some finite level N. For such N, no element in $\mathrm{Gal}(K_N/K)$ restricts for all $\ell|N$ to a non-trivial element of $\mathrm{Gal}(K_\ell/K)$ . As a consequence, there are no non-critical primes in $S_{K,x}$ or $S_{E/K}$ : the Frobenius elements of such primes in $\mathrm{Gal}(K_N/K)$ 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 [ Reference Campagna and Stevenhagen2 , 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 $m\in{\mathbf{Z}}_{\ge1}$ , 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 ${\mathbf{Z}}/m{\mathbf{Z}}$ -linear action of the absolute Galois group $G_K$ of K, and this mod-m Galois representation induces an embedding

(2·4) \begin{equation}G_m=\mathrm{Gal}(K_m/K)\hookrightarrow \mathrm{GL}(V_m)\cong \mathrm{GL}_3({\mathbf{Z}}/m{\mathbf{Z}}).\end{equation}

As $G_K$ stabilises the rank 2 subspace $U_m=E[m](\overline K)\subset V_m$ , this is a ‘reducible’ representation. Write $V_m=U_m\oplus ({\mathbf{Z}}/m{\mathbf{Z}})\cdot \bar Q$ with $\bar Q = (Q\bmod \langle P\rangle) \in V_m$ the image of a point $Q\in E(\overline K)$ satisfying $m Q=P$ . We then have a split exact sequence

\[0\to U_m \longrightarrow V_m \longrightarrow {\mathbf{Z}}/m{\mathbf{Z}}\cdot \bar Q \to 0\]

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 a ${\mathbf{Z}}/m{\mathbf{Z}}$ -basis $\{T_1, T_2\}$ of $U_m$ and extending it by some $\bar Q$ as above to a ${\mathbf{Z}}/m{\mathbf{Z}}$ -basis for $V_m$ , the matrix representation of $\sigma\in G_m$ becomes

(2·5) \begin{equation}\sigma=\sigma_{A,b}=\begin{pmatrix}a_{11} & & & a_{12} && & b_1 \\a_{21} && & a_{22} && & b_2 \\0 && & 0 && & 1\end{pmatrix},\end{equation}

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 $A=\begin{pmatrix}a_{11} && & a_{12} \\a_{21} && & a_{22}\end{pmatrix}\in \mathrm{GL}(U_m)$ , and

\[b=b_1T_1+b_2T_2=\sigma Q - Q\]

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 $G_m\subset \mathrm{GL}(V_m)$ that is contained in the 2-dimensional affine group

\[\mathrm{Aff}_2({\mathbf{Z}}/m{\mathbf{Z}})=({\mathbf{Z}}/m{\mathbf{Z}})^2 \rtimes \mathrm{GL}_2({\mathbf{Z}}/m{\mathbf{Z}}).\]

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 Lemmas 2.1 and 2.2 is a little bit more involved. We have to impose a condition on the Frobenius elements $\mathrm{Frob}_{{\mathfrak{p}}, \ell}\in G_\ell$ at all primes $\ell\ne\mathrm{char}\ {{\mathfrak{p}}}$ different from being equal to the identity element $\mathrm{id}_\ell\in G_\ell$ : 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

\begin{equation*}\ell | [E(k_{\mathfrak{p}})\,:\,\langle {\bar P}\rangle ]\quad \Longleftrightarrow \quad \mathrm{rank} (\mathrm{Frob}_{{\mathfrak{p}},\ell} - \mathrm{id}_\ell)\le 1.\end{equation*}

Conversely, if $\ell$ divides $[E(k_{\mathfrak{p}})\,:\,\langle {\bar P}\rangle]$ then either $E(k_{\mathfrak{p}})$ has complete $\ell$ -torsion or $E(k_{\mathfrak{p}})$ has a cyclic non-trivial $\ell$ -part and $\bar P$ is contained in $\ell\cdot E(k_{\mathfrak{p}})$ . In both cases $\mathrm{Frob}_{{\mathfrak{p}},\ell}$ is the identity on a subspace of $V_\ell$ of dimension at least 2.

This time, it is not clear that in the case $\ell={\mathfrak{p}}$ , the condition $\ell \mid [E(k_{\mathfrak{p}}) : \langle \bar{P} \rangle]$ of the Lemma will automatically not be satisfied for almost all primes ${\mathfrak{p}}$ of $K$ . In order to have this, we will restrict our attention to primes ${\mathfrak{p}}$ of K of degree one, i.e., primes ${\mathfrak{p}}$ of prime norm $\mathrm{char}\ {{\mathfrak{p}}}$ .

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 $\mathrm{char}\ {{\mathfrak{p}}}>5$ for which $\bar P\ne \bar O\in E(k_{\mathfrak{p}})$ . Then we have

\[E(k_{\mathfrak{p}})=\langle {\bar P}\rangle\quad \Longleftrightarrow \quad \mathrm{rank} (\mathrm{Frob}_{{\mathfrak{p}},\ell} - \mathrm{id}_\ell)\ge 2\text{ for all primes $\ell$}.\]

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 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 $S_\ell\subset G_\ell=\mathrm{Gal}(K_\ell/K)$ of ‘bad’ elements at the prime $\ell$ as

\[S_\ell=\{\sigma\in G_\ell\,:\,\mathrm{rank}_{{\mathbf{F}}_\ell} (\sigma - \mathrm{id}_\ell)\le 1\}.\]

For arbitrary $m\in{\mathbf{Z}}_{\ge1}$ 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

(2·6) \begin{equation}\delta_{E/K,P}=\sum_{m=1}^{\infty} \frac{\mu(m)s_m}{[K_m\,:\,K]}.\end{equation}

It is the elliptic analogue of the multiplicative primitive root density (2·1). 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 [ Reference Campagna and Stevenhagen2 ]. 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 factorisation

(2·7) \begin{equation}\delta_{E/K,P}=\sum_{m|N} \frac{\mu(m)s_m}{[K_m\,:\,K]}\cdot \prod_{\ell\nmid N \text{ prime}} \left(1-\frac{s_\ell}{[K_\ell\,:\,K]}\right),\end{equation}

as in (2·2), 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

\[R_P=\{Q\in E(\bar K)\,:\, m Q\in \langle P\rangle\text{ for some } m\in {\mathbf{Z}}_{\ge1}\} \cong ({\mathbf{Q}}/{\mathbf{Z}})^2\times {\mathbf{Q}}\]

of $E(\bar K)$ generated by all the roots of P in $E(\bar K)$ . 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

(2·8) \begin{equation}\rho_{E,P}\,:\, G_K \longrightarrow \mathrm{Aut}(V)\cong \mathrm{GL}_3({\widehat {{\mathbf{Z}}}}),\end{equation}

which has (2·4) as its mod-m representation. It factors via $\mathrm{Gal}(K(R_P)/K)$ , with $K(R_P)=\bigcup_m K_m\subset \bar K$ the compositum of all ‘m-division fields’ of P inside $\bar K$ . The group $U=E(\overline K)^\text{tor}\cong ({\mathbf{Q}}/{\mathbf{Z}})^2$ 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 $V_m=U_m\oplus {\mathbf{Z}}/m{\mathbf{Z}}\cdot\bar Q$ , the image of $\rho_{E,P}$ is in $\mathrm{Aff}_2({\widehat {{\mathbf{Z}}}})={\widehat {{\mathbf{Z}}}}^2\rtimes \mathrm{GL}_2({\widehat {{\mathbf{Z}}}})$ . For E without CM over $\bar K$ , one deduces from Serre’s open image theorem that this image is of finite index in $\mathrm{Aff}_2({\widehat {{\mathbf{Z}}}})$ , which yields (2·7) for any N divisible by some finite product $N_{E/K,P}\in{\mathbf{Z}}_{>0}$ of critical primes. As in [ Reference Campagna and Stevenhagen2 ], one deduces that all non-CM-densities $\delta_{E,P}$ are rational multiples of a universal constant. If E has CM over $\bar K$ by an order $\mathcal{O}\subset K$ , one replaces $\mathrm{Aff}_2({\widehat {{\mathbf{Z}}}})$ by $\mathrm{Aff}_1(\mathcal{O})$ , 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 [ Reference Campagna and Stevenhagen2 ].

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 $\delta_{K,x}$ .

The assumption of global primitivity in Theorem 1.1 cannot be weakened to the assumption $K\ne K_\ell$ for all prime numbers $\ell$ . 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 $\varepsilon=({1+\sqrt 5})/{2}$ . The element $\pi=\varepsilon^2-4=({-5+\sqrt 5})/{2}\in K$ 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 $\mu_K=\{\pm1\}$ 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 $[{\mathbf{F}}_p\,:\,\langle\bar x\rangle]$ 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 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 $\mathrm{Frob}_{{\mathfrak{p}}, \ell}\in G_\ell$ 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 $G_\ell=\mathrm{Gal}(K_\ell/K)$ 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

(4·1) \begin{equation}\dim V^{\langle\sigma\rangle}\ge n-1\quad \Longleftrightarrow\quad \dim (\sigma-1)V \le 1.\end{equation}

Proof. The implication $(\textrm{ii})\Rightarrow(\textrm{i})$ is immediate, as the inequality $\dim V_G\ge n-1$ implies that, for all $\sigma\in G$ , we have $\dim (\sigma-1)V \le 1$ and, by (4·1), $\dim V^{\langle\sigma\rangle}\ge n-1$ .

For $(\textrm{i})\Rightarrow (\textrm{ii})$ , 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 (ii), 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 (i) means that every element of G has a matrix representation with respect to a suitable basis that, according to (4·1), can be given in one of the equivalent forms

(4·2)

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 $\mathrm{Frob}_{\mathfrak{p}}\in G$ pointwise fixes a 1-dimensional subspace of V. We find that $E(k_{\mathfrak{p}})$ has order divisible by $\ell$ for almost all primes ${\mathfrak{p}}$ of K 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 $E'=E/\langle T\rangle$ is of the second kind, with the dual isogeny $E'\to E$ being the $\ell$ -isogeny in question. This is a well-known fact that occurs as the very first exercise in [ Reference Serre15 , 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 $G_\ell=\mathrm{Gal}(K_\ell/K)\subset \mathrm{GL}(V_\ell)$ 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 $Q'=aQ+bP\in E(K)$ satisfies $\ell Q'= a\ell Q+b\ell P=amP+b\ell P=P$ , 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 $E\to E'=E/T_\ell$ 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 $\phi\,:\, E'\to E$ dual to $\widehat\phi\,:\, E\to E'=E/T_\ell$ . If $Q\in E(\overline K)$ satisfies $\ell Q=P$ , then $P'=\widehat\phi(Q)$ 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 $\phi(P')=\phi\widehat\phi(Q)=\ell Q=P$ , 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 $\phi\,:\, E'\to E$ defined over K and a point $P'\in E'(K)$ with $\phi(P')=P$ . Pick $Q'\in E'(\overline K)$ with $\ell Q'=P'$ and put $Q=\phi(Q')\in E(\overline K)$ . Writing $\widehat\phi\,:\, E\to E'$ for the dual isogeny, we are in the situation of Example 4.2, and we have

\[\ell Q=\phi(\ell Q')=\phi(P')=P\qquad \text{and} \qquad \widehat\phi Q= \widehat\phi \phi Q'= \ell Q'=P'\in E'(K).\]

As Q is in the fibre $\widehat\phi^{-1}(P')$ , 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 $\ell | [E(k_{\mathfrak{p}})\,:\,\langle {\bar P}\rangle ]$ 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 $E\to E'$ , and the Galois representation $\bar\rho_{E, \ell}$ of $G_K$ on $U_\ell=E[\ell](\bar K)$ is non-trivial, with image contained in a Borel subgroup of $\mathrm{GL}(U_\ell)$ .

In the case $\ell=2$ , a Borel subgroup of $\mathrm{GL}(U_2)\cong \mathrm{GL}_2({\mathbf{F}}_2)$ has order 2, and the existence of a non-trivial locally 2-imprimitive point $P\in E(K)$ implies that we have $\#E(K)[2]=2$ , and that $\bar\rho_{E, 2}$ is a non-trivial quadratic character.

If $T\in E(K)$ is a point of order 2, we can choose a Weierstrass model

(5·1) \begin{equation}E\,:\, y^2=x(x^2+ax+b) \qquad \hbox{with } b, d=a^2-4b\in K^*\end{equation}

for $E/K$ in which we have $T=(0,0)$ . The representation $\bar \rho_{E, 2}$ corresponds to the 2-division field of E over K, which equals $K(\sqrt d)$ , and we have $\#E(\overline K)[2]=2$ if and only $d\in K^*$ is not a square.

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 $E'=E/\langle(0,0)\rangle$ . Choosing $u=x+x_1+a$ and $v=y+y_1$ as generators for K( $E^{\prime}$ ), we obtain a Weierstrass model

(5·2) \begin{equation}E'\,:\, v^2=u(u^2-2au+d) \qquad \hbox{with } d, d'=({-}2a)^2-4d=16b\in K^*\end{equation}

for $E^{\prime}$ that is of the same form (5·1), and an explicit 2-isogeny $\varphi\,:\, E\to E'$ given by

(5·3) \begin{equation}\varphi\,:\, (x,y)\longmapsto (u, v) = (x+x_1+a, y+y_1)=\left( \frac {y^2}{x^2}, \left(1-\frac{b}{x^2}\right)y \right) .\end{equation}

An affine point $(u,v)\in E'(K)$ 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.

As $E^{\prime}$ is again of the form (5·1), one sees that the isogeny $\widehat\varphi\,:\, E'\to E$ dual to $\varphi$ is given by

(5·4) \begin{equation}\widehat\varphi\,:\, (u,v)\longmapsto\left( \frac {v^2}{4u^2}, \left(1-\frac{b}{u^2}\right)\frac{v}{8} \right) .\end{equation}

Proof of Theorem 1.3. Let $E/K$ be an elliptic curve with $\#E(K)[2]=2$ . 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 (5·1), 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 $\varphi\,:\, E_D \to E_D'=E_D/\langle (0,0)\rangle$ as above by replacing (a, b) in (5·1), (5·2) and (5·3) by $(aD, bD^2)$ . Any point $P\in E_D(K)$ satisfying condition C from Theorem 1.2 is in the image $\widehat\varphi(E_D'(K))$ of the isogeny $\widehat\varphi\,:\, E_D'\to E_D$ 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 $P=(1,\pm Y)\in E_D(\bar K)$ 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

(5·5) \begin{equation}D_\lambda=\frac{a-2\lambda}{\lambda^2-b} \qquad \text{with }\lambda\in K\setminus\{a/2,\pm\sqrt b\}\end{equation}

come by construction with a K-rational point

(5·6) \begin{equation}P_\lambda=(1,Y_\lambda)=\left(1,\frac{\lambda^2-a\lambda+b}{\lambda^2-b}\right)\in E_\lambda(K).\end{equation}

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 $E'_\lambda(K)$ , since we chose for the first coordinate of $P_\lambda$ the square value 1. By (5·4), 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 $E'_\lambda$ . 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 $u_1u_2=dD_\lambda^2$ . Writing $D_\lambda$ and $P_\lambda$ as in (5·5) and (5·6), 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

\[K_2 = K\left( \dfrac{1}{2} P_\lambda\right)= K(\sqrt {u_1}, \sqrt{u_2}) = K(\sqrt d, \sqrt{\lambda^2 - b}).\]

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 (5·5) 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 (2·4) and (2·5) of $G_2=\mathrm{Gal}(K_2/K)$ becomes

\[\mathrm{Gal}(K_2/K)=\left\{ \begin{bmatrix}1 & && x &&& y \\0 &&& 1 &&& 0 \\0 &&& 0 &&& 1\end{bmatrix}\,:\, x,y\in{\mathbf{F}}_2 \right\},\]

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 (5·1), 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 $E'\to E$ is generated by an $\ell$ -torsion point $T\in E'(K)$ . In this situation, the Galois representations of $G_K$ on the $\ell$ -torsion subgroups of $E^{\prime}$ and $E=E'/\langle T\rangle$ are, with respect to a suitable basis, of the form

(6·1) \begin{equation}\rho_{E'/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}.\end{equation}

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 $\ell\ne2$ . In this section we focus on the case $\ell=3$ .

Proof. Let $H_3=\mathrm{Gal}(K(E[3])/K)\subset \mathrm{GL}(U_3)\cong \mathrm{GL}_2({\mathbf{F}}_3)$ 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

\[\alpha_1=x_1x_2+x_3x_4, \qquad \alpha_2=x_1x_3+x_2x_4, \qquad \alpha_3=x_1x_4+x_2x_3\]

as its roots. Under the permutation action of $\mathrm{GL}(U_3)/\langle -1\rangle=S_4$ 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

(6·2) \begin{equation}H_3 \to \mathrm{Gal}(\psi_3)\to \mathrm{Gal}(\delta_3)\end{equation}

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

\[\mathrm{GL}(U_3)\to \mathrm{GL}(U_3)/\langle-1\rangle=S_4 \to S_4/V_4=S_3.\]

More precisely, the first surjection in (6·2) is injective as we have $-1\notin H_3$ for $H_3$ as in (6·1), and the second is because we have $\mathrm{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 $\mathrm{Gal}(\psi_3)$ . Viewing $H_3$ as $\mathrm{Gal}(\delta_3)$ , we may finish the proof by quoting a classical formula [ Reference Serre14 , 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 $K(E[3])=\mathrm{Split}_K(\delta_3)=\mathrm{Split}_K(X^3-\Delta_{E})$ .

From Lemma 6.1, we see that for the representations in (6·1), 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 $\Delta=\Delta_{E'}$ and $\Delta_E$ .

We can write the curve $E^{\prime}$ in Deuring normal form [ Reference Husemöller7 , p. 89] as

(6·3) \begin{equation}E'\,:\, y^2+axy+by=x^3 \quad \hbox{with } a,b \in K\quad \hbox{and}\quad \Delta_{E'}=b^3(a^3-27b)\in K^*.\end{equation}

Here $T=(0,0)\in E'(K)$ is the point of order 3, and the quotient curve $E=E'/\langle T\rangle$ has Weierstrass equation

(6·4) \begin{equation}E\,:\, y^2+axy+by=x^3-5abx-(a^3+7b)b,\end{equation}

with the explicit formula for the 3-isogeny $\varphi_3\,:\, E'\to E=E'/\langle T\rangle$ being given by

(6·5) \begin{equation}\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).\end{equation}

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 $(x, y)\mapsto (a^2x, a^3y)$ and simplify (6·3) to

(6·6) \begin{equation}E_b'\,:\, y^2+xy+by=x^3.\end{equation}

For K a number field, we have infinitely many pairwise different 3-isogenous images $(E_b, P_b)=\varphi_3[(E_b', P_b')]$ 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 specialisation $E_{b(t)}$ of $E_b$ is an elliptic curve over K for which

(6·7) \begin{equation} \left(\frac{(t^2+t)(t^2-1)+1}{t^2}, \frac{(1-t^2)\left(t^4+2 t^3+t-1\right)}{t^3}\right)\in E_{b(t)}(K)\end{equation}

is a non-trivial locally 3-imprimitive point.

Proof. For the curve $E'_b$ in (6·6) with $b\in K(X)$ as defined, the point $P'_b=(1, X)$ lies on $E'_b$ by the very choice of b: it satisfies $X^2+X+bX=1$ . Under the 3-isogeny $\varphi_3\,:\, E'_b\to E_b$ from (6·5) to the curve $E_b$ obtained by putting $a=1$ in (6·4), 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 specialisation $X=t\in K^*$ , we obtain a point $P_{b(t)}$ , on the curve $E_{b(t)}$ defined over K that is given by (6·7). We are only interested in those specialisations 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

\[K(X)\left(\dfrac13 P_b\right)=K(\zeta_3, \root 3 \of b, \root 3 \of X)=K(\zeta_3, \root 3 \of X, \root 3 \of {X^2+X-1}).\]

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

\[\mathrm{Gal}\left(K(X)\left(\dfrac13 P_b\right)/K(X)\right)=\left\{ \begin{bmatrix}\omega_3 &&& x &&& y \\0 &&& 1 &&& 0 \\0 &&& 0 &&& 1\end{bmatrix}\,:\, x,y\in{\mathbf{F}}_3 \right\},\]

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 (6·7) is a non-trivial locally 3-imprimitive point.

Remark 6.3. The construction in the proof of Theorem 6.2 excludes all specialisations for which $b=b(t)\in K^*$ is a cube and the elliptic curve in (6·6) 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 $E'\,:\,y^2+cxy+y=x^3$ . As in the previous case, we have $P'=(1,t)\in E'(K)$ for $c=({-}t^2-t+1)/t$ , and the image of $P^{\prime}$ under the map

\[\varphi\,:\, E'\to E= E'/\langle(0,0)\rangle\,:\, y^2+cxy+y=x^3-5cx-(c^3+7)\]

is the point $P=\varphi_3(P')=(({-}t^2 + t + 1)/t, (t^2 - 1)/t^2)\in E(K)$ , for which the 3-division field $K(\frac13P)$ 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 (6·3) 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 $E'\,:\, y^2=x^3+c^2$ , 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

\begin{align*} \varphi_3\,:\, E'_s\,:\, y^2= x^3 + (s^2-1)^2 &\longrightarrow E_s\,:\, y^2= x^3 - 27 (s^2-1)^2 \\ P'_s=(s^2-1, s(s^2-1))&\longmapsto P_s=(s^2+3, s(s^2-9))\end{align*}

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 $\ell\le 7$ . 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

(7·1) \begin{equation}E'\,:\, y^2+(1-c)xy-cy=x^3-cx^2\end{equation}

that parametrises elliptic curves with (0, 0) as a 5-rational point (see Kulesz [ Reference Kulesz8 ]). It has further points (0, c), (c, 0) and $(c,\, c^2)$ of order 5, and its discriminant equals $\Delta_{E'}=c^5 \left(c^2-11 c-1\right)$ . Using Vélu’s formula [ Reference Vélu20 ] or invoking Pari-GP, we compute the Weierstrass equation for the 5-isogenous curve $E=E'/\langle(0,0)\rangle$

\begin{align*}E=E_c\,: \quad y^2+&(1-c)xy-cy \\&= x^3-cx^2 -5 c (c^2+2 c-1)x-c (c^4+ 10 c^3 - 5 c^2+ 15 c -1),\end{align*}

and also the explicit 5-isogeny $\varphi_5\,:\, E'\rightarrow E$ . The discriminants involved are $\Delta_{E'}=c^5 (c^2-11c-1)$ 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 (6·1), and even though the proof of Lemma 6 for $\ell=3$ does not generalise to $\ell=5$ , we found by a direct calculation that the 5-division field is $K(E[5])=K(\zeta_5,\sqrt[5]{c})$ : generated over $K(\zeta_5)$ by the 5-th root of the discriminant.

We can equip $E^{\prime}$ with a K-rational point $P_t'=(t,t)$ by putting $c=t(2-t)$ , and compute its image $P_t=\varphi_5(P')\in E_{t(2-t)}(K)$ as

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

The corresponding 5-division field of $P_t$ is

$$K\left(\dfrac15 P_t\right)=K(\zeta_5, \root 5 \of t, \root 5 \of {t-2}).$$

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 $P'_t$ 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 [ 19 ]. Note that for $c=-15$ we have

\[c(c^2-11c-1)=-5835=-3\cdot5\cdot389.\]

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 $t\in{\mathbf{Z}}_{\ge5}$ 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

$$E'\,:\, y^2+(1-c)xy-by=x^3-bx^2$$

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

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

The curve $E'=E_d'$ parametrises [ Reference Kulesz8 ] elliptic curves with (0, 0) as point of order 7. A Weierstrass equation for the 7-isogenous curve $E=E'/\langle(0,0)\rangle$ is

\begin{align*}E\,:\, & y^2+(1-c)xy-by=x^3-bx^2-5 (2 b^2+b (c^2-3 c-2)+c (c^2+4 c+1))x\\& -b^2 (12 c^2+c+24)-6 b^3+b ({-}c^4+9 c^3+46 c^2+24 c+2)\\& - c (c^4+16 c^3+36 c^2+16 c+1).\end{align*}

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 $P_t'=(d^2t,d^3t)$ by putting $d=d(t)=({t+1)/(t^2-t+1})$ . The image of $P_t'$ under the 7-isogeny $E'\to E$ is

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

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

\[K\left( \dfrac17 P_t\right)=K(E[7])\left(\root 7\of {\frac{t(t+1)}{t-2}}\right)=K\left(\zeta_7, \root 7 \of {\frac{t(t^2-t+1)}{(t+1)}}, \root 7 \of {\frac{t(t+1)}{(t-2)}}\right).\]

The point $P_t$ is a globally 7-primitive but locally 7-imprimitive point when the extension $K(\zeta_7)\subset K(\frac17 P_t)$ 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 $P'_t$ is 7-torsion, and for $t=2$ the curve $E^{\prime}$ is singular. For $t=3$ and $d=4/7$ however we obtain the point $P_3=(286019/490^2, 15951227/490^3)$ on

\[E\,:\, y^2+\dfrac{61}{7^2}xy+\dfrac{48}{7^3}y=x^3+\dfrac{48}{7^3}x^2-\dfrac{774780}{7^7}x-\dfrac{1047829260}{7^{11}},\]

which is the curve 20622.j1 with minimal model

\[E_0\,:\, y^2+xy=x^3-5455771x-5039899603,\]

in the LMFDB-database [ 19 ]. 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 $K(\frac12 P)$ of P is of maximal degree 4 over it. Then $G_2=\mathrm{Gal}(K(\frac12 P)/K)$ is a dihedral group of order 8 for which the matrix representation (2·5) on $V_2=\langle\frac12 P\rangle/\langle P\rangle$ has the form

(8·1) \begin{equation}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 \mathrm{GL}_3({\mathbf{F}}_2).\end{equation}

There is a unique subfield of $L\subset K(\frac12 P)$ 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 (8.1) of $G_2$ as ${\mathbf{F}}_2$ -valued quadratic characters on $G_2$ that generate the character group of the quotient $\mathrm{Gal}(L/K)\cong V_4$ 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 $\mathrm{Frob}_{{\mathfrak{p}},2}\in G_2$ , viewed as a matrix as in (8.1), the endomorphism $(\mathrm{Frob}_{{\mathfrak{p}},2}-\mathrm{id}_2)\,:\, V_2\to V_2$ has ${\mathbf{F}}_2$ -rank at least 2 (Lemma 2.3). We obtain the criterion

(8·2) \begin{equation}2\nmid [E(k_{\mathfrak{p}})\,:\,\langle \bar P\rangle]\quad \Longleftrightarrow \quad a(\mathrm{Frob}_{{\mathfrak{p}},2})=b(\mathrm{Frob}_{{\mathfrak{p}},2})=1\in {\mathbf{F}}_2.\end{equation}

More precisely, $a(\mathrm{Frob}_{{\mathfrak{p}},2})=1$ implies that $E(k_{\mathfrak{p}})$ does not have full 2-torsion, and $b(\mathrm{Frob}_{{\mathfrak{p}},2})=1$ 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 $\varphi_3\,:\,E'\to E$ be the isogeny dual to the quotient map $\phi\,:\,E\to E'=E/T$ . Assume that the point P is in $\varphi_3[E'(K)]$ but not in 3E(K). Then the 3-division field $K(\frac13 P)$ of P has Galois group $G_3=\mathrm{Gal}(K(\frac13 P)/K)$ for which the matrix representation on $V_3=\langle\frac13 P\rangle/\langle P\rangle$ will ‘generically’ be the group

(8·3) \begin{equation}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 \mathrm{GL}_3({\mathbf{F}}_3)\end{equation}

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

(8·4) \begin{equation}g(\mathrm{Frob}_{{\mathfrak{p}},3})=1\in {\mathbf{F}}_3^*\quad \Longrightarrow \quad 3 | [E(k_{\mathfrak{p}})\,:\,\langle \bar P\rangle].\end{equation}

Thus, for primes ${\mathfrak{p}}\nmid 6\Delta_E$ , a necessary condition for $\bar P \in E(k_{\mathfrak{p}})$ to be an elliptic primitive root is that the three quadratic characters a, b and g occurring in (8.2) and (8.4) do not vanish on the Frobenius automorphism of ${\mathfrak{p}}$ in $K_6=K(\frac16P)$ . 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 $K(\frac12 P)$ or $K(\frac13 P)$ : 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

$$\Delta_E=2^4\cdot5^9\cdot11^6,$$

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 $P=({-13475}/{36}, {1249325}/{216})$ 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 ${\mathbf{Q}}(\frac12P)$ of P is the $V_4$ extension

$${\mathbf{Q}}(E[2])={\mathbf{Q}}(\sqrt{5})\subset{\mathbf{Q}}\left(\frac12P\right)={\mathbf{Q}}(\sqrt{5},\sqrt\pi, \sqrt{\bar\pi})$$

generated by the square roots of $\pi=3+2\sqrt5$ and its conjugate. From $\pi\bar\pi=-11$ we see that ${\mathbf{Q}}(\frac12P)$ 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 $T_3=({55}/{3}, {275}\sqrt{165}/{9})$ 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 ${\mathbf{Q}}\subset K_aK_b$ . Over the full 3-division field of E, the 3-division field of P is the cubic extension

$${\mathbf{Q}}(E[3])={\mathbf{Q}}(\sqrt{-3}, \sqrt{-55},\sqrt[3]{2})\subset{\mathbf{Q}}\left(\frac13P\right)={\mathbf{Q}}(E[3], \sqrt[3]{\alpha})$$

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 (8.3). We conclude that P is a locally never-primitive point of $E({\mathbf{Q}})$ as the index of $\langle\bar P\rangle$ in $E({\mathbf{F}}_p)$ is always divisible by 2 or 3.