Locally imprimitive points on elliptic curves

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

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 $K={\mathbf{Q}}$.

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.

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.

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

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 $\ell=\operatorname{char}{{\mathfrak{p}}}$.

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 $\ell$.

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):

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.

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

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

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

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

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

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

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.

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

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

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

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

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

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

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].

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.