Cartan images and \(\ell \)-torsion points of elliptic curves with rational j-invariant
- Oron Y. Propp^{1}Email authorView ORCID ID profile
Received: 14 February 2017
Accepted: 27 November 2017
Published: 20 February 2018
Abstract
Let \(\ell \) be an odd prime and d a positive integer. We determine when there exists a degree-d number field K and an elliptic curve E / K with \(j(E)\in \mathbb {Q}\setminus \{0,1728\}\) for which \(E(K)_\mathrm {tors}\) contains a point of order \(\ell \), conditionally on a conjecture of Sutherland. We likewise determine when there exists such a pair (K, E) for which the image of the associated mod-\(\ell \) Galois representation is contained in a Cartan subgroup or its normalizer. We do the same under the stronger assumption that E is defined over \(\mathbb {Q}\).
1 Introduction
The problem of identifying the possible images of these Galois representations is closely related to that of determining the possible torsion subgroups and isogenies admitted by elliptic curves over a particular number field. Mazur famously classified the possible torsion subgroups \(E(\mathbb {Q})_\mathrm {tors}\) in [25] and the possible \(\ell \)-isogenies of an elliptic curve \(E/\mathbb {Q}\) in [26]. Kamienny, Kenku and Momose generalized Mazur’s results on torsion subgroups to quadratic number fields in [22, 23], and though no complete characterization for higher-degree number fields is known, there has been recent progress towards characterizing the cubic case [6, 17, 18, 29, 30, 35], the quartic case [7, 14, 19, 28], and the quintic case [8, 12]. In particular, the set of torsion subgroups that arise for infinitely many \(\overline{\mathbb {Q}}\)-isomorphism classes of elliptic curves defined over number fields of degree d has been determined for \(d=3,4,5,6\) [10, 20, 21]. The growth of torsion subgroups of elliptic curves upon base change was further studied in [13, 16]. Using Galois representations, Lozano-Robledo has explicitly characterized the set of primes \(\ell \) for which an elliptic curve \(E/\mathbb {Q}\) has a point of order \(\ell \) over a number field of degree at most d, assuming a positive answer to Serre’s uniformity problem [24]. The same was done by Najman for cyclic \(\ell \)-isogenies of elliptic curves with rational j-invariant [27].
We aim to give a more precise characterization of the degrees of number fields over which elliptic curves exhibit certain properties deducible from the images of their Galois representations. Recently, Sutherland developed an algorithm to efficiently compute Galois images of elliptic curves, and conjectured the following stronger version of Serre’s uniformity problem in [34, Conj. 1.1] based on data from some 140 million elliptic curves, including all curves of conductor up to 400,000 [9]:
Conjecture 1.1
(Sutherland) Let \(E/\mathbb {Q}\) be an elliptic curve without complex multiplication and let \(\ell \) be a prime. Then the image of \(\rho _{E,\ell }\) is either equal to \({\text {GL}}_2(\mathbb {Z}/\ell \mathbb {Z})\) or conjugate to one of the 63 exceptional groups listed in Table 1 (see also [34, Tables 3, 4]).
Known exceptional images of \(\rho _{E,\ell }\) for non-CM elliptic curves \(E/\mathbb {Q}\) (for \(\ell \le 11\), the list is complete) [34, Tables 3,4]
Combining this conjecture with the precise characterization of Galois images for elliptic curves over \(\mathbb {Q}\) with complex multiplication (CM) given in [36, Theorem 1.14–1.16], we are able to deduce the possible subgroups of \({\text {GL}}_2(\mathbb {Z}/\ell \mathbb {Z})\) arising as images of \(\rho _{E,\ell }\) for base changes of elliptic curves \(E/\mathbb {Q}\) to a number field K, and for elliptic curves E / K with \(j(E)\in \mathbb {Q}\), assuming \(j(E)\ne 0,1728\).^{1} Thus, all results in this paper hold unconditionally for elliptic curves with CM.
Values of \({\mathcal {S}}_M(\ell )\) (see Theorem 1.2)
\(\varvec{\ell }\) | \({\varvec{Z}}\) | \({\varvec{C}}_\textit{s}\) | \({\varvec{C}}_\textit{ns}\) | \({\varvec{N}}_\textit{s}\) | \({\varvec{N}}_\textit{ns}\) |
---|---|---|---|---|---|
3 | 2 | 1 | 2 | 1 | 1 |
5 | 4 | 1 | 2 | 1 | 1 |
7 | 6, 14, 16 | 2, 3, 7 | 2 | 1 | 1 |
13 | 24,28,52 | 2,13 | 2 | 1 | 1 |
17 | 32, 36, 272 | 2, 17 | 2 | 1 | 1 |
37 | 72, 76, 1332 | 2, 37 | 2 | 1 | 1 |
11, 19, 43, 67, 163 | \(2(\ell -1),2\ell ,2(\ell +1)\) | \(2,\ell \) | 2 | 1 | 1 |
\(\in {\mathcal {P}}\) | \(2(\ell -1)\) | 2 | \(\ell -1\) | 1 | \((\ell -1)/2\) |
\(\in {\mathcal {Q}}\) | \(2(\ell +1)\) | \(\ell +1\) | 2 | \((\ell +1)/2\) | 1 |
Other | \(2(\ell -1),2(\ell +1)\) | 2 | 2 | 1 | 1 |
Values of (see Theorem 1.2)
\(\varvec{\ell }\) | \({\varvec{Z}}\) | \({\varvec{C}}_\textit{s}\) | \({\varvec{C}}_\textit{ns}\) | \({\varvec{N}}_\textit{s}\) | \({\varvec{N}}_\textit{ns}\) |
---|---|---|---|---|---|
3, 7, 11, 19, 67, 163 | \(2(\ell -1),2\ell ,2(\ell +1)\) | \(2,\ell \) | 2 | 1 | 1 |
\(\in {\mathcal {P}}\) | \(2(\ell -1)\) | 2 | \(\ell -1\) | 1 | \((\ell -1)/2\) |
\(\in {\mathcal {Q}}\) | \(2(\ell +1)\) | \(\ell +1\) | 2 | \((\ell +1)/2\) | 1 |
Other | \(2(\ell -1),2(\ell +1)\) | 2 | 2 | 1 | 1 |
Values of (see Theorem 1.2)
\(\varvec{\ell }\) | \({\varvec{Z}}\) | \({\varvec{C}}_\textit{s}\) | \({\varvec{C}}_\textit{ns}\) | \({\varvec{N}}_\textit{s}\) | \({\varvec{N}}_\textit{ns}\) |
---|---|---|---|---|---|
3 | 2 | 1 | 2 | 1 | 1 |
5 | 4 | 1 | 2 | 1 | 1 |
7 | 6, 14, 16 | 2, 3, 7 | 2 | 1 | 1 |
11 | 24, 110 | 11, 12 | 2 | 6 | 1 |
13 | 24, 52 | 6, 8, 13 | 12 | 3, 4 | 6, 26 |
17 | 272 | 17 | 272 | 153 | 136 |
37 | 1332 | 37 | 1332 | 703 | 666 |
Other | \(\ell (\ell +1)(\ell -1)\) | \(\ell (\ell +1)\) | \(\ell (\ell -1)\) | \(\ell (\ell +1)/2\) | \(\ell (\ell -1)/2\) |
Our first result determines the degrees of number fields K over which the image of the Galois representation attached to an elliptic curve can belong to each of these subgroups M, or equivalently, the degrees of K-points on the modular curves \(X_M\):
Theorem 1.2
Assume Conjecture 1.1. Let \(\ell \) be an odd prime, and let M be one of the subgroups Z, \(C_\textit{s}\), \(C_\textit{ns}\), \(N_\textit{s}\), or \(N_\textit{ns}\) of \({\text {GL}}_2(\mathbb {Z}/\ell \mathbb {Z})\). There exists a set \({\mathcal {S}}_M(\ell )\), identified explicitly in Table 2, such that the following holds: there exists a degree-d number field K and an elliptic curve \(E/\mathbb {Q}\) with \(j(E)\ne 0,1728\) for which \(\rho _{E,\ell }({\text {Gal}}(\overline{K}/K))\) belongs to M if and only if \(s\mid d\) for some \(s\in {\mathcal {S}}_M(\ell )\). The set \({\mathcal {S}}_M(\ell )\) satisfies the same condition with \(E/\mathbb {Q}\) replaced by an elliptic curve E / K with \(j(E)\in \mathbb {Q}\setminus \{0,1728\}\).
Consequently, there exists a non-cuspidal K-point on the modular curve \(X_M\) that is not in the image of any of the morphisms in (1.2) for some degree-d number field K if and only if \(s\mid d\) for some \(s\in {\mathcal {S}}_M(\ell )\).
We also identify analogous sets and for elliptic curves with and without CM, listed in Tables 3 and 4, respectively. To replace “belonging” in the theorem with the ordinary notion of containment, and remove the corresponding restriction related to the morphisms in (1.2), simply take unions of the \({\mathcal {S}}_M(\ell )\) along the subgroup inclusion lattice in (1.2) (e.g., replace \({\mathcal {S}}_{N_\textit{ns}}(\ell )\) with \({\mathcal {S}}_{N_\textit{ns}}(\ell )\cup {\mathcal {S}}_{C_\textit{ns}}(\ell )\cup {\mathcal {S}}_{Z}(\ell )\)). For interpretations of the latter statement on modular curves in terms of moduli data, see the table on page 2 of [32].
Our next result characterizes the degrees of abelian sub-extensions of \(\ell \)-torsion fields \(\mathbb {Q}(E[\ell ])\):
Theorem 1.3
Assume Conjecture 1.1. Given an odd prime \(\ell \), there exists a set \({\mathcal {K}}(\ell )\), identified explicitly in (3.5), such that the following holds: there exists a degree-d number field K contained in the \(\ell \)-torsion field of an elliptic curve \(E/\mathbb {Q}\) with \(j(E)\ne 0,1728\) for which \(\mathbb {Q}(E[\ell ])/K\) is an abelian extension precisely when \(d\in {\mathcal {K}}(\ell )\). The set \({\mathcal {K}}(\ell )\) satisfies the same condition with \(E/\mathbb {Q}\) replaced by an elliptic curve E / K with \(j(E)\in \mathbb {Q}\setminus \{0,1728\}\).
Finally, we have the following result, which is a refinement of [24, Thm. 1.7] on degrees of number fields over which elliptic curves with rational j-invariant contain torsion points of order \(\ell \):
Theorem 1.4
Assume Conjecture 1.1. Given an odd prime \(\ell \), there exists a set \({\mathcal {T}}(\ell )\), identified explicitly in Table 6, such that the following holds: there exists a degree-d number field K and an elliptic curve E / K with \(j(E)\in \mathbb {Q}\setminus \{0,1728\}\) for which \(E(K)_\mathrm {tors}\) contains a point of order \(\ell \) if and only if \(t\mid d\) for some \(t\in {\mathcal {T}}(\ell )\).
The analogous result for elliptic curves defined over \(\mathbb {Q}\) is proven unconditionally in [16, Cor. 6.1]; our methods can be easily made to give a proof of this case, which we omit, instead reproducing the resultant sets \({\mathcal {T}}_\mathbb {Q}(\ell )\) in Table 7. We again identify analogous sets , , , for elliptic curves with and without CM, listed with \({\mathcal {T}}(\ell )\) and \({\mathcal {T}}_\mathbb {Q}(\ell )\). These results agree with, and in some cases sharpen, recent work on CM elliptic curves [3–5]. For instance, [4, Thm. 1.5] and [5, Thm. 1.2] imply that there exists a number field of odd degree d over which a CM elliptic curve attains an \(\ell \)-torsion point if and only if \(\ell \equiv 3\bmod 4\) and d is divisible by \(h_\ell \cdot \frac{\ell -1}{2}\), where \(h_\ell \) denotes the class number of \(\mathbb {Q}(\sqrt{-\ell })\) (see also [3, Thm. 6.2]). But by Theorem 1.4, for certain primes \(\ell \equiv 3\bmod 4\), all odd degrees d divisible by \(h_\ell \cdot \frac{\ell -1}{2}\) have this property even if we restrict to CM elliptic curves E with \(j(E)\in \mathbb {Q}\setminus \{0,1728\}\); for other primes \(\ell \equiv 3\bmod 4\), no such degrees do. Table 8 illustrates this phenomenon for all such \(\ell \le 83\). The entries in its final column can also be derived from [4, 5]: they are “all” when \(h_\ell =1\) and “none” when \(h_\ell >1\).
Values of \({\mathcal {T}}(\ell )\), , and (see Theorem 1.4)
Values of \({\mathcal {T}}_\mathbb {Q}(\ell )\), , and (see Theorem 1.4)
Primes \(\ell \equiv 3\bmod 4\), along with whether “all” or “none” of the odd numbers d divisible by \(h_\ell \cdot \frac{\ell -1}{2}\) have the property that there exists a degree-d number field over which a CM elliptic curve E with \(j(E)\in \mathbb {Q}\setminus \{0,1728\}\) attains an \(\ell \)-torsion point
2 Background
The following proposition, originally due to Dickson [11], classifies subgroups of \({\text {GL}}_2(\ell )\) in terms of their images in \({\text {PGL}}_2(\ell )\):
Proposition 2.1
- (1)
H is cyclic and a conjugate of G lies in \(C_\textit{s}(\ell )\) or \(C_\textit{ns}(\ell )\);
- (2)
H is dihedral and a conjugate of G lies in \(N_\textit{s}(\ell )\) or \(N_\textit{ns}(\ell )\), but not in \(C_\textit{s}(\ell )\) or \(C_\textit{ns}(\ell )\);
- (3)
H is isomorphic to \(A_4\), \(S_4\), or \(A_5\), and no conjugate of G is contained in the normalizer of any Cartan subgroup.
Proof
See [33, §2]. \(\square \)
An integer \(n\in S\subseteq \mathbb {Z}_{>0}\) is said to be div-minimal in S if it is a minimal element of S in the divisibility lattice of integers. A subgroup \(H\in S\subseteq \{G:G\le {\text {GL}}_2(\ell )\}\) is said to be div-minimal in S if \(\#H\) is div-minimal in \(\{\#G:G\in S\}\). The terms “div-maximal,” “div-minima,” and so forth, are defined as expected. The set S is often omitted if it is clear from context. Note that an element of a set S can be both div-minimal and div-maximal.
3 Proof of Theorems 1.2 and 1.3
The set \(\mathcal {I}({\text {Excep}}(\ell );M(\ell ))\) of indices of subgroups G of some exceptional image \(G_E(\ell )=\rho _{E,\ell }({\text {Gal}}(\overline{\mathbb {Q}}/\mathbb {Q}))\) listed in Table 1 such that G belongs to \(M(\ell )\), for the standard subgroups \(M=Z,C_\textit{s},C_\textit{ns},N_\textit{s},N_\textit{ns},C_{\textit{r}}\)
\(\varvec{\ell }\) | \({\varvec{Z}}(\varvec{\ell })\) | \({\varvec{C}}_\textit{s}(\varvec{\ell })\) | ||
---|---|---|---|---|
3 | \(\Delta (2\mid 12,16)\) | \(\Delta (1\mid 6,8)\) | ||
5 | \(\Delta (4\mid 80,96)\) | \(\Delta (1\mid 40,48)\setminus \{3\}\) | ||
7 | \(\Delta (6,14,16\mid 72,96,252)\) | \(\Delta (2,3,7\mid 36,48,84,126)\) | ||
11 | \(\Delta (24,110\mid 220,240)\) | \(\Delta (11,12\mid 44,110,120)\) | ||
13 | \(\Delta (24,52\mid 288,1872)\) | \(\Delta (6,8,13\mid 96,144,624,936)\) | ||
17 | \(\Delta (272\mid 1088)\) | \(\Delta (17\mid 544)\) | ||
37 | \(\Delta (1332\mid 15984)\) | \(\Delta (37\mid 5328,7992)\) | ||
Other | \(\varnothing \) | \(\varnothing \) |
\(\varvec{\ell }\) | \({\varvec{C}}_\textit{ns}(\varvec{\ell })\) | \({\varvec{N}}_\textit{s}(\varvec{\ell })\) | \({\varvec{N}}_\textit{ns}(\varvec{\ell })\) | \({\varvec{C}}_\mathsf{r}(\varvec{\ell })\) |
---|---|---|---|---|
3 | \(\Delta (2\mid 4)\) | \(\Delta (1\mid 4)\) | \(\Delta (1\mid 6,8)\) | \(\Delta (2\mid 4)\) |
5 | \(\Delta (2\mid 12,32)\) | \(\Delta (1\mid 12)\) | \(\Delta (1\mid 40,48)\setminus \{5\}\) | \(\Delta (4\mid 16)\) |
7 | \(\Delta (2\mid 18,24)\) | \(\Delta (1\mid 18,24)\) | \(\Delta (1\mid 36,48,126)\) | \(\Delta (2\mid 26)\) |
11 | \(\Delta (2\mid 60,80)\) | \(\Delta (6\mid 60)\) | \(\Delta (1\mid 110,120)\setminus \{11,22\}\) | \(\Delta (10\mid 20)\) |
13 | \(\Delta (12\mid 36)\) | \(\Delta (3,4\mid 36)\) | \(\Delta (6,26\mid 144,936)\) | \(\Delta (4\mid 144)\) |
17 | \(\varnothing \) | \(\varnothing \) | \(\Delta (136\mid 544)\) | \(\Delta (16\mid 64)\) |
37 | \(\varnothing \) | \(\varnothing \) | \(\Delta (666\mid 7992)\) | \(\Delta (36\mid 432)\) |
Other | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
The set of indices of subgroups G of \(G_E(\ell )\) such that G belongs to \(M(\ell )\), for each possible non-exceptional image \(G_E(\ell )=\rho _{E,\ell }({\text {Gal}}(\overline{\mathbb {Q}}/\mathbb {Q}))\) and for the standard subgroups \(M=Z,C_\textit{s},C_\textit{ns},N_\textit{s},N_\textit{ns},C_{\textit{r}}\) (the final row should be interpreted in the manner of (3.2))
\({\varvec{G}}_E\) | \({\varvec{M}}\) | \(\mathcal {I}({\varvec{G}}_E(\varvec{\ell });{\varvec{M}}(\varvec{\ell }))\) |
---|---|---|
\({\text {GL}}_2\) | Z | \(\Delta (\ell (\ell +1)(\ell -1)\mid \ell (\ell +1)(\ell -1)^2)\) |
\(C_\textit{s}\) | \(\Delta (\{\ell (\ell +1)\},\{\ell (\ell +1)(\ell -1)^2/q:q\in \Pi (\ell -1)\})\) | |
\(C_\textit{ns}\) | \(\Delta (\{\ell (\ell -1)\},\{\ell (\ell +1)(\ell -1)^2/q:q\in \Pi _\mathrm {odd}(\ell +1)\cup \{2^{\nu _2(\ell -1)+1}\}\})\) | |
\(N_\textit{s}\) | \(\Delta (\{\ell (\ell +1)/2\},\{\ell (\ell +1)(\ell -1)^2/2q:q\in \Pi _\mathrm {odd}(\ell -1)\cup \{4\text { if }4\mid (\ell -1),\text { else }2\}\})\) | |
\(N_\textit{ns}\) | \(\Delta (\ell (\ell -1)/2\mid \ell (\ell +1)(\ell -1)^2/2)\) | |
\(C_{\textit{r}}\) | \(\Delta (\ell ^2-1\mid (\ell +1)(\ell -1)^2)\) | |
\(N_\textit{s}\) | Z | \(\Delta (2(\ell -1)\mid 2(\ell -1)^2)\) |
\(C_\textit{s}\) | \(\Delta (\{2\},\{2(\ell -1)^2/q:q\in \Pi (\ell -1)\})\) | |
\(C_\textit{ns}\) | \(\Delta (\ell -1\mid (\ell -1)^2/2^{\nu _2(\ell -1)})\) | |
\(N_\textit{s}\) | \(\Delta (\{1\},\{(\ell -1)^2/q:q\in \Pi _\mathrm {odd}(\ell -1)\cup \{4\text { if }4\mid (\ell -1),\text { else }2\}\})\) | |
\(N_\textit{ns}\) | \(\Delta ((\ell -1)/2\mid (\ell -1)^2)\) | |
\(C_{\textit{r}}\) | \(\varnothing \) | |
\(N_\textit{ns}\) | Z | \(\Delta (2(\ell +1)\mid 2(\ell ^2-1))\) |
\(C_\textit{s}\) | \(\Delta (\ell +1\mid \ell ^2-1)\) | |
\(C_\textit{ns}\) | \(\Delta (\{2\},\{2(\ell ^2-1)/q:q\in \Pi _\mathrm {odd}(\ell +1)\cup \{2^{\nu _2(\ell -1)+1}\}\})\) | |
\(N_\textit{s}\) | \(\Delta ((\ell +1)/2\mid (\ell ^2-1)/2^{\nu _2(\ell -1)})\) | |
\(N_\textit{ns}\) | \(\Delta (1\mid (\ell ^2-1))\) | |
\(C_{\textit{r}}\) | \(\varnothing \) | |
\(G,H_1,H_2\) | Z | \(\Delta (2\ell \mid 2\ell (\ell -1))\) |
\(C_\textit{s}\) | \(\Delta (\ell \mid \ell (\ell -1))\) | |
\(C_\textit{ns}\) | \(\varnothing \) | |
\(N_\textit{s}\) | \(\varnothing \) | |
\(N_\textit{ns}\) | \(\Delta (\ell \mid \ell (\ell -1))\) | |
\(C_{\textit{r}}\) | \(\Delta (2\mid 2(\ell -1))\) |
Now let K be a degree-d number field, and E / K be an elliptic curve with \(j(E)\in \mathbb {Q}\setminus \{0,1728\}\). Then E is either a base change of an elliptic curve over \(\mathbb {Q}\), or a quadratic twist of such a curve. By [34, Cor. 5.25], the images \(\rho _{E,\ell }({\text {Gal}}(\overline{K}/K))\) which may arise in this case are precisely the groups G arising for base changes of elliptic curves \(E/\mathbb {Q}\), along with their twists: these are the group \(\langle G,-1\rangle \) and its index-2 subgroups that do not contain \(-1\). The following lemma shows that a subgroup of \({\text {GL}}_2(\ell )\) and its twists always belong to the same subgroup, and therefore all results above hold when \(E/\mathbb {Q}\) is replaced by an elliptic curve E / K with \(j(E)\in \mathbb {Q}\setminus \{0,1728\}\).
Lemma 3.1
Let \(G\le {\text {GL}}_2(\ell )\), let H be a twist of G, and let M be Z, \(C_\textit{s}\), \(C_\textit{ns}\), \(N_\textit{s}\), or \(N_\textit{ns}\). Then G belongs to M if and only if H belongs to M.
Proof
Let H be an index-2 subgroup of \(\langle G,-1\rangle \) not containing \(-\,1\), and suppose \(H\le M\). Then H is an index-2 subgroup of \(\langle H,-1\rangle \simeq H\times \{\pm \,1\}\), so since \(\langle H,-1\rangle \subseteq \langle G,-1\rangle \), it follows that \(\langle H,-1\rangle =\langle G,-1\rangle \). But \(-\,1\in M\) (as is true of all standard subgroups of \({\text {GL}}_2(\ell )\)), hence \(G\le \langle H,-1\rangle \le M\). Conversely, if \(G\le M\), then \(H\le \langle G,-1\rangle \le M\) as \(-\,1\in M\), so altogether \(H\le M\) if and only if \(G\le M\). Since the above argument respects conjugacy, the conclusion follows by definition. The same proof applies if \(H=\langle G,-1\rangle \). \(\square \)
Theorem 3.2
Assume Conjecture 1.1. Let \(\ell \) be an odd prime, and let \(M\in \{C_\textit{s},C_\textit{ns},N_\textit{s},N_\textit{ns}\}\). There exists a degree-d number field K and an elliptic curve \(E/\mathbb {Q}\) with K a subfield of \(\mathbb {Q}(E[\ell ])\) such that \(\rho _{E,\ell }({\text {Gal}}(\overline{K}/K)))\) belongs to \(M(\ell )\) if and only if \(d\in {\mathcal {E}}_M(\ell )\). The same is true if E is replaced by an elliptic curve E / K with \(j(E)\in \mathbb {Q}\setminus \{0,1728\}\).
Naturally, to restrict to an elliptic curve with or without CM, we may simply replace \({\mathcal {E}}_M(\ell )\) by or , respectively. To obtain Theorem 1.2, it remains to choose the div-minimal elements of \({\mathcal {E}}_M(\ell )\), by (3.1). Doing so yields the sets \({\mathcal {S}}_M(\ell )\) in Table 2; the sets and are obtained similarly.
Lemma 3.3
The set of indices of subgroups G of \(G_E(\ell )\) such that G belongs to \(C_{\textit{r}}(\ell )\) is the set of positive integers divisible by \(\ell ^2-1\) (resp. 2) and dividing \((\ell +1)(\ell -1)^2\) (resp. \(2(\ell -1)\)) if \(G_E\) is \({\text {GL}}_2\) (resp. G, \(H_1\), or \(H_2\)), and the empty set if \(G_E\) is \(N_\textit{s}\) or \(N_\textit{ns}\).
Proof
For the cases where \(G_E(\ell )\) is \(N_\textit{s}(\ell )\) or \(N_\textit{ns}(\ell )\), it suffices to note that neither of these subgroups contains an element of order \(\ell \). \(\square \)
3.1 Full image
Lemma 3.4
The set of indices of subgroups G of \({\text {GL}}_2(\ell )\) such that G belongs to \(Z(\ell )\) is the set of positive integers divisible by \(\ell (\ell +1)(\ell -1)\) and dividing \(\ell (\ell +1)(\ell -1)^2\).
Proof
Lemma 3.5
Proof
Since \(C_\textit{s}(\ell )\simeq \mathbb {F}_\ell ^\times \times \mathbb {F}_\ell ^\times \), it contains non-scalar subgroups of every order dividing \(\#C_\textit{s}(\ell )=(\ell -1)^2\) except for 1, which corresponds to the trivial subgroup. The div-minimal such orders are therefore the prime divisors of \(\ell -1\), and the proof concludes as in Lemma 3.4. \(\square \)
Lemma 3.6
Proof
Otherwise, if \(\#G\) is even, then G must contain an element \(\gamma ={\mathcal {D}}_\textit{ns}(x,y)\in C_\textit{ns}(\ell )\setminus Z(\ell )\) squaring to an element of \(Z(\ell )\); the only such elements are those for which \(x=0\), which square to an element of \(\epsilon Z(\ell )^2\), i.e., the non-square elements of \(Z(\ell )\). Picking a primitive root \(\alpha \in \mathbb {F}_\ell ^\times \), we see that G is div-minimal when \(\gamma ^2={\mathcal {D}}_\textit{ns}(\alpha ^{(\ell -1)/2^{\nu _2(\ell -1)}},0)\) and \(G=\langle \gamma \rangle \), that is, \(\#G=2^{\nu _2(\ell -1)+1}\).
Lemma 3.7
Proof
In the former case, let \(\kappa :\mathbb {Z}/r\mathbb {Z}\rightarrow \mathbb {Z}/(r/p)\mathbb {Z}\) be the unique quotient map, and let \(K_1:=\mathbb {Z}/m\mathbb {Z}\times _{\mathbb {Z}/(r/p)\mathbb {Z}}\mathbb {Z}/m\mathbb {Z}\) with maps \(\kappa \circ \psi _i:\mathbb {Z}/m\mathbb {Z}\rightarrow \mathbb {Z}/(r/p)\mathbb {Z}\). As before, \(K_1\) is \(\varphi (\tau )\)-invariant, and \(\#K_1=\frac{m^2}{r/p}=p\cdot \#\pi _1(H)\). Moreover, \(\pi _1(H)\le K_1\), since \(\psi _1(x)=\psi _2(y)\) implies \((\kappa \circ \psi _1)(x)=(\kappa \circ \psi _2)(y)\) for \((x,y)\in \pi _1(H)\). It follows that \(K:=K_1\rtimes _\varphi \langle \tau \rangle \) is a subgroup of G containing H as an index-p subgroup, as desired.
Lemma 3.8
Proof
Lemma 3.9
Proof
As in Lemma 3.7, the projection \(\pi _1(H)\) is a subgroup of \(\mathbb {Z}/n\mathbb {Z}\), hence for any \(d\mid [G:H]=n/\#\pi _1(H)\), there exists a unique subgroup \(K_1\le \mathbb {Z}/n\mathbb {Z}\) of order \(d\cdot \#\pi _1(H)\) containing \(\pi _1(H)\). Letting \(\alpha \) be a generator of \(\mathbb {Z}/n\mathbb {Z}\), observe that any automorphism of \(\mathbb {Z}/n\mathbb {Z}\) is of the form \(\alpha \mapsto \alpha ^i\), where \((i,n)=1\), hence any cyclic subgroup of \(\mathbb {Z}/n\mathbb {Z}\) is \(\varphi (a)\)-invariant for each \(a\in A\). Thus, \(K:=K_1\rtimes _\varphi A\) is a subgroup of G containing H as an index-d subgroup, as desired.
For the second assertion, note that any \(d\mid [G:H]\) factors as a product of \(d_1\mid [\mathbb {Z}/n\mathbb {Z}:\pi _1(H)]\) and \(d_2\mid \# A\). Letting \(K_1\) be the unique order \(d_1\cdot \#\pi _1(H)\) subgroup of \(\mathbb {Z}/n\mathbb {Z}\) containing \(\pi _1(H)\) and \(K_2\) be a subgroup of A of order \(d_2\), we see that \(K:=K_1\rtimes _\varphi K_2\) is again a subgroup of G containing H as an index-d subgroup. \(\square \)
Lemma 3.10
Let \(H\le G\) such that \(H\not \le N\), for some index-2 subgroup N of G. Then \([H:H\cap N]=2\).
Proof
Let \(\gamma \in H\cap (G\setminus N)\). Then \(G=N\sqcup \gamma N\), and left multiplication by \(\gamma \) is a bijective map of sets on G, mapping N to \(\gamma N\) and vice versa. Thus, \(\gamma \cdot (H\cap N)\subset H\cap \gamma N\) has cardinality \(\#(H\cap N)\), hence \([H:H\cap N]\ge 2\). Moreover, \(\gamma \cdot (H\cap \gamma N)\subset H\cap N\) has cardinality \(\#(H\cap \gamma N)\), hence \([H:H\cap N]\le 2\), and the result follows. \(\square \)
Lemma 3.11
The set of indices of subgroups G of \({\text {GL}}_2(\ell )\) such that G belongs to \(N_\textit{ns}(\ell )\) is the set of positive integers divisible by \(\ell (\ell -1)/2\) and dividing \(\ell (\ell +1)(\ell -1)^2/2\).
Proof
3.2 Normalizer of a split Cartan
Lemma 3.12
The set of indices of subgroups G of \(N_\textit{s}(\ell )\) such that G belongs to \(Z(\ell )\) is the set of positive integers divisible by \(2(\ell -1)\) and dividing \(2(\ell -1)^2\).
Proof
Lemma 3.13
Proof
The result follows as in Lemma 3.5 after noting that \([N_\textit{s}(\ell ):C_\textit{s}(\ell )]=2\). \(\square \)
Lemma 3.14
The set of indices of subgroups G of \(N_\textit{s}(\ell )\) such that G belongs to \(C_\textit{ns}(\ell )\) is the set of positive integers divisible by \(\ell -1\) and dividing \((\ell -1)^2/2^{\nu _2(\ell -1)}\).
Proof
Lemma 3.15
Proof
Lemma 3.16
The set of indices of subgroups G of \(N_\textit{s}(\ell )\) such that G belongs to \(N_\textit{ns}(\ell )\) is the set of positive integers divisible by \((\ell -1)/2\) and dividing \((\ell -1)^2\).
Proof
3.3 Normalizer of a non-split Cartan
Lemma 3.17
The set of indices of subgroups G of \(N_\textit{ns}(\ell )\) such that G belongs to \(C_\textit{s}(\ell )\) is the set of positive integers divisible by \(2(\ell +1)\) and dividing \(2(\ell ^2-1)\).
Proof
The result follows as in Lemma 3.12 after noting that \(Z(\ell )=\{{\mathcal {D}}_\textit{ns}(x,0):x\in \mathbb {F}_\ell ^\times \}\le N_\textit{ns}(\ell )\). \(\square \)
Lemma 3.18
The set of indices of subgroups G of \(N_\textit{ns}(\ell )\) such that G belongs to \(C_\textit{s}(\ell )\) is the set of positive integers divisible by \(\ell +1\) and dividing \(\ell ^2-1\).
Proof
Lemma 3.19
Lemma 3.20
The set of indices of subgroups G of \(N_\textit{ns}(\ell )\) such that G belongs to \(N_\textit{s}(\ell )\) is the set of positive integers divisible by \((\ell +1)/2\) and dividing \((\ell ^2-1)/2^{\nu _2(\ell -1)}\).
Proof
Lemma 3.21
The set of indices of subgroups G of \(N_\textit{ns}(\ell )\) such that G belongs to \(N_\textit{ns}(\ell )\) is the set of positive integers dividing \(\ell ^2-1\).
3.4 Borel cases
Lemma 3.22
Let \(G\le B(\ell )\), and suppose that G does not contain an element of order \(\ell \). Then G is conjugate to a subgroup of \(C_\textit{s}(\ell )\).
Proof
Lemma 3.23
The set of indices of subgroups G of \(G(\ell )\), \(H_1(\ell )\), or \(H_2(\ell )\) such that G belongs to a standard subgroup \(M(\ell )\) is the set of positive integers divisible by \(2\ell \) (resp. \(\ell \)) and dividing \(2\ell (\ell -1)\) (resp. \(\ell (\ell -1)\)) if M is Z (resp. \(C_\textit{s}\)); the empty set if M is \(C_\textit{ns}\) or \(N_\textit{s}\); and the set of positive integers divisible by \(\ell \) and dividing \(\ell (\ell -1)\) if M is \(N_\textit{ns}\).
Proof
The case where M is \(N_\textit{s}\) is immediate from Lemma 3.22.
4 Proof of Theorem 1.4
div-Minima of the set of indices of subgroups G of some exceptional image \(G_E(\ell )=\rho _{E,\ell }({\text {Gal}}(\overline{\mathbb {Q}}/\mathbb {Q}))\) listed in Table 1 such that a twist of G (or, in the third column, G itself) fixes a nonzero vector of \(\mathbb {F}_\ell ^2\)
\(\varvec{\ell }\) | div-min \(\varvec{\mathcal {I}}({\text {Excep}}(\ell ))\) | div-min \(\varvec{\mathcal {I}}_\mathbb {Q}({\text {Excep}}(\ell ))\) |
---|---|---|
2, 3, 5, 7 | 1 | 1 |
11 | 5 | 5 |
13 | 2, 3 | 3, 4 |
17 | 4 | 8 |
37 | 6 | 12 |
Other | \(\varnothing \) | \(\varnothing \) |
div-Minima of the set of indices of subgroups G of \(G_E(\ell )\) such that a twist of G (or, in the third column, G itself) fixes a nonzero vector of \(\mathbb {F}_\ell ^2\), for each non-exceptional possible image \(G_E(\ell )=\rho _{E,\ell }({\text {Gal}}(\overline{\mathbb {Q}}/\mathbb {Q}))\) (the final row should be interpreted in the manner of (4.1))
\({\varvec{G}}_E\) | div-min \(\varvec{\mathcal {I}}(G_E(\ell ))\) | div-min \(\varvec{\mathcal {I}}_\mathbb {Q}(G_E(\ell ))\) |
---|---|---|
\({\text {GL}}_2\) | \((\ell ^2-1)/2\) | \(\ell ^2-1\) |
\(N_\textit{s}\) | \(\ell -1\) | \(2(\ell -1)\) |
\(N_\textit{ns}\) | \((\ell ^2-1)/2\) | \(\ell ^2-\)1 |
\(G,H_1,H_2\) | \((\ell -1)/2\) | \((\ell -1)/2\) |
If E does not have CM, then \(G_E(\ell )\) is either \({\text {GL}}_2(\ell )\) or one of the 63 exceptional groups listed in Table 1 by our assumption of Conjecture 1.1. The results for each exceptional subgroup were computed [31, 34] using Magma, and the div-minima of the unions of the two sets (4.1) of indices over each \({\text {Excep}}(\ell )\) are the sets \(\mathcal {I}({\text {Excep}}(\ell ))\) and \(\mathcal {I}_\mathbb {Q}({\text {Excep}}(\ell ))\), respectively, listed in Table 11. The set \({\text {div-min}}\mathcal {I}({\text {GL}}_2(\ell ))\) is determined by Lemma 4.1 below, and appears in Table 12.
Now suppose E has CM. As in the proof of Theorems 1.2 and 1.3, \(G_E(\ell )\) is conjugate to \(N_\textit{s}(\ell )\) (resp. \(N_\textit{ns}(\ell )\)) if \(\ell \in {\mathcal {P}}\) (resp. \(\ell \in {\mathcal {Q}}\)); to either \(N_\textit{s}(\ell )\), \(N_\textit{ns}(\ell )\), \(G(\ell )\), \(H_1(\ell )\), or \(H_2(\ell )\) if \(\ell \in \{3,7,11,19,43,67,163\}\), with each case occurring for some such E; and otherwise to either \(N_\textit{s}(\ell )\) or \(N_\textit{ns}(\ell )\), with each case again occurring for some such E [36, Prop. 1.14]. The set \({\text {div-min}}\mathcal {I}(N_\textit{s}(\ell ))\) is determined by Lemma 4.2; the set \({\text {div-min}}\mathcal {I}(N_\textit{ns}(\ell ))\) by Lemma 4.3; and the set \({\text {div-min}}\mathcal {I}(G(\ell ),H_1(\ell ),H_2(\ell ))\) by Lemma 4.4. All appear in Table 12.
We now turn to the task of proving the lemmas used above, which will complete the proof of Theorem 1.4.
Lemma 4.1
The set of indices of subgroups G of \({\text {GL}}_2(\ell )\) such that a twist of G fixes a nonzero element of \(\mathbb {F}_\ell ^2\) has div-minima \(\{(\ell ^2-1)/2\}\).
Proof
Lemma 4.2
The set of indices of subgroups G of \(N_\textit{s}(\ell )\) such that a twist of G fixes a nonzero element of \(\mathbb {F}_\ell ^2\) has div-minima \(\{(\ell -1)\}\).
Proof
The proof is identical to that of Lemma 4.1, with \({\text {GL}}_2(\ell )\) replaced by \(N_\textit{s}(\ell )\) and T replaced by each of the subgroups identified in [24, Lem. 6.6] in turn. \(\square \)
Lemma 4.3
The set of indices of subgroups G of \(N_\textit{ns}(\ell )\) such that a twist of G fixes a nonzero element of \(\mathbb {F}_\ell ^2\) has div-minima \(\{(\ell ^2-1)/2\}\).
Proof
The proof is identical to that of Lemma 4.1, with \({\text {GL}}_2(\ell )\) replaced by \(N_\textit{ns}(\ell )\) and T replaced by each of the subgroups identified in [24, Lem. 7.4] in turn. \(\square \)
Lemma 4.4
Suppose \(\ell \equiv 3\bmod 4\). The set of indices of subgroups G of \(G(\ell )\), \(H_1(\ell )\), or \(H_2(\ell )\) such that a twist of G fixes a nonzero element of \(\mathbb {F}_\ell ^2\) has div-minima \(\{(\ell -1)/2\}\).
Proof
The same reasoning shows that the subgroup is maximal, and it also has order \(2\ell \), hence index \((\ell -1)/2\).
For \(H_2(\ell )\), we obtain the same characteristic polynomials, but in this case elements with \(a=1\) each generate order-2 subgroups fixing , and the elements of \(H_2(\ell )\) with upper left-hand entry 1 form a subgroup of order \(\ell \), since \(\ell \equiv 3\bmod {4}\) by assumption and therefore \(-1\notin (\mathbb {F}_\ell ^\times )^2\). Thus, the two div-minimal indices of such subgroups of \(H_2(\ell )\) are \(\ell -1\) and \(\ell (\ell -1)/2\).
Repeating the proof of Lemma 4.1 with \({\text {GL}}_2(\ell )\) replaced by \(G(\ell )\) and T replaced by each of the subgroups of \(G(\ell )\) identified above in turn shows that \(\mathcal {I}(G(\ell ))=\{(\ell -1)/2\}\). Since \(-1\notin H_1(\ell ),H_2(\ell )\) as \(\ell \equiv 3\bmod {4}\) by assumption, any twist of a subgroup H of \(H_1(\ell )\) or \(H_2(\ell )\) is a subgroup of H, hence \(\mathcal {I}(H_1(\ell ))=\mathcal {I}_\mathbb {Q}(H_1(\ell ))=\{(\ell -1)/2\}\) and \(\mathcal {I}(H_2(\ell ))=\mathcal {I}_\mathbb {Q}(H_2(\ell ))=\{\ell -1,\ell (\ell -1)/2\}\). Thus, among \(G(\ell )\), \(H_1(\ell )\), and \(H_2(\ell )\), the div-minimal index is \((\ell -1)/2\). \(\square \)
For \(\ell =2\), all conjugacy classes of subgroups of \({\text {GL}}_2(\mathbb {Z}/\ell \mathbb {Z})\) occur as the image of \(\rho _{E,\ell }\) for some elliptic curve \(E/\mathbb {Q}\), so we ignore this case henceforth.
Declarations
Acknowledgements
The author wishes to thank Andrew Sutherland for supervising this research and providing many helpful suggestions and insights; Filip Najman and Álvaro Lozano-Robledo for their comments on an early draft of this paper; and the referees for their comments and careful review. This research was generously supported by MIT’s UROP program and the Paul E. Gray (1954) Endowed Fund for UROP.
Authors’ Affiliations
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