Groups of generalized $G$-type and applications to torsion subgroups of rational elliptic curves over infinite extensions of $\mathbb{Q}$

Recently there has been much interest in studying the torsion subgroups of elliptic curves base-extended to infinite extensions of $\mathbb{Q}$. In this paper, given a finite group $G$, we study what happens with the torsion of an elliptic curve $E$ over $\mathbb{Q}$ when changing base to the compositum of all number fields with Galois group $G$. We do this by studying a group theoretic condition called generalized $G$-type, which is a necessary condition for a number field with Galois group $H$ to be contained in that compositum. In general, group theory allows one to reduce the original problem to the question of finding rational points on finitely many modular curves. To illustrate this method we completely determine which torsion structures occur for elliptic curves defined over $\mathbb{Q}$ and base-changed to the compositum of all fields whose Galois group is $A_4$.

much work has been put into classifying the torsion structures of elliptic curves defined over cubic extensions of Q with results recently announced by Etropolski, Morrow, and Zureick-Brown, and independently by the second-named author. The case where [K : Q] = 4 is still completely wide open.
Another approach is to start with an elliptic curve defined over Q and consider what torsion subgroups occur when it is base-extended to a field K/Q. The cases when E is defined over Q and K is a number field of degree d with d = 2, 3, 4, 5, 7 or d is not divisible by 2, 3, 5, 7 have been completely settled in [GJT14,Naj16,Cho16,GJ17,GJN16]. There have been a number of papers that consider the question of what torsion structures occur over a fixed infinite extension of Q. For example, in [LL85] and [Fuj04,Fuj05] all the possible torsion structures that occur when E/Q is base-extended to the compositum of all quadratic extensions of Q are classified. More recently, in [DLRNS18] all of the torsion structures that occur when E/Q is base-extended to the compositum of all degree 3 extensions of Q are classified. It is worth noting at this point that when working over an infinite extension of Q, we no longer have the Mordell-Weil Theorem to ensure that the torsion subgroup remains finite. Because of this we have to make sure that we choose an infinite extension of Q carefully.
In this paper, we provide a general framework for studying the torsion subgroups of rational elliptic curves over certain infinite extensions of Q. Before proceeding we remind the reader of the following definition.
Definition 1.2. For n ∈ N and groups G 1 , . . . , G n , let π i : G 1 × · · · × G n → G i be the standard projection maps. A subgroup K of G 1 ×· · ·×G n is a subdirect product of G 1 , . . . , G n if π i (K) = G i .
We start the study of this question in Section 2 by first introducing the notions of weak and strong generalized G-type. Armed with these two new concepts, we find computable necessary and sufficient conditions to determine if a given finite group H is of generalized G-type. These necessary and sufficient conditions drastically generalize the conditions used in [DLRNS18,Lemma 3.2] and [Dan18,Lemma 3.2] where the cases of G = S 3 and D 4 are studied.
Once these necessary and sufficient conditions are established, Question 1.6 reduces to classifying rational points on modular curves. In fact, since there is a computable uniform bound on #E(Q(G ∞ )) tors depending only on Q(G ∞ ) (and so really depending only on G), Question 1.6 is reduced to finding the rational points on a finite and computable list of modular curves depending on G. To explicitly illustrate our method we fully answer Question 1.6 in the case when G = A 4 by proving the following theorem. which occur for 4, 2, 2, and 1 Q-isomorphism classes respectively.
The proof of Theorem 1.7 gives a concrete demonstration of the general strategy for determining the torsion subgroups for curves which have been base-extended to Q(G ∞ ) for more general G. Thus, the results of this paper provide a framework for determining the torsion subgroups for elliptic curves E/Q base-extended to a large family of infinite extensions of Q. We note here that we do not have an algorithm to classify these groups up to isomorphism since there is not an algorithm to find all the rational points on a general modular curve, although in practice this is possible.
To finish off the classification over Q(A ∞ 4 ), in Table 1 we give elliptic curves of minimal conductor with each of the 26 possible torsion structures when base-extended to Q(A ∞ 4 ). Further, in Table 5 we find a complete list of (possibly constant) rational maps that classify the j-invariants of elliptic curves which contain a subgroup isomorphic to one of the possible 26 torsion structures. More information about Table 5 can be found in Section 6.
Once we have settled the classification of torsion over Q(A ∞ 4 ), it is not hard to determine which torsion structures arise over the compositum of all A 4 -extensions of Q, which we denote by Q A 4 . As we explain in Section 7, since we have a proper containment Q A 4 ⊆ Q(A ∞ 4 ), this comes down to determining which torsion subgroups from Theorem 1.7 still arise over the smaller field Q A 4 . The fact that Q A 4 contains no subextensions which are quadratic over Q plays a major role. We obtain the following theorem.
We conclude in Section 7.4 with an observation about torsion over cyclic cubic fields. In particular, letting Q(C ∞ 3 ) denote the compositum of all cyclic cubic extensions of Q, we obtain for free a classification of all torsion subgroups arising as E(Q(C ∞ 3 )) for an elliptic curve E/Q; see Corollary 7.13.

Groups and fields of generalized G-type
Starting in Section 2.1, we will make a general study of groups and fields of generalized G-type by introducing the concepts of weak and strong G-type. With these we find necessary and sufficient conditions to show a group is of generalized G-type. After doing this general study, in Section 2.3 we will examine the particular case when G = A 4 in order to gain more specific information to prove Theorem 1.7.
2.1. Strong and weak G-types.
Definition 2.1. Let G be a group. Then a group H is of weak G-type if there is an integer n such that H is isomorphic to a subquotient of G n . Further, H is of strong G-type if there is an integer N such that H is isomorphic to a quotient of a subdirect product of G N .
The notion of generalized G-type is dependent on a choice of an embedding of G in S d , while in contrast the notions of strong and weak G-type depend only on the group G itself as an abstract group. Immediately from these definitions we get the following lemma.
Lemma 2.2. Let G, H and I be groups with G ⊆ S d a transitive subgroup.
(1) One has the implications: For the rest of this section we will denote the free group with k generators by F k . Also, if H is a group and h 1 , . . . , h k ∈ H are elements, then the morphism F k → H which sends the i-th generator of F k to h i is denoted by eval h 1 ,...,h k .
Definition 2.4. Let G be a finite group and (H, h 1 , . . . , h k ) be a group of weak (resp. generalized or strong) G-type with k distinguished elements. Then (H, h 1 , . . . , h k ) is called a universal group of weak (resp. generalized or strong) G-type if, for every other group of weak (resp. generalized or strong) G-type with k distinguished elements (I, i 1 , . . . , i k ), there exists a unique φ : Notice that if G is finite then it is possible to compute G univ k explicitly as a subgroup of G Hom(F k ,G) . In practice however one does not need a copy of G for every element of Hom(F k , G), since the kernel of a morphism does not change if we compose it with an automorphism of its range. More generally, if f, g ∈ Hom(F k , G) are such that ker(g) ⊆ ker(f ), then the image of G univ k maps isomorphically onto the projection where the copy of G corresponding to f is omitted.
(1) Let G, H be finite groups and suppose that H is k-generated. Then H is of weak G-type if and only if H is isomorphic to a quotient of G univ k .
(2) If G can be generated by k elements then the notion of weak G-type and weak G univ k -type agree.
Remark 2.7. Since G univ k is computable and it is possible to enumerate all normal subgroups of G univ k , this also gives a (possibly very slow) algorithm to check whether a k-generated group H is of weak G-type. One just needs to check whether there is a normal subgroup N of G univ k such that H ≃ G univ k /N . How practical this algorithm is depends on how easily one can compute G univ k in practice, or whether one can give a nice theoretical description of G univ k for the given group G. For example, if it is easy to write down a finite set of generators g 1 , . . . , g n for ker(π G,k ), then determining whether H is of generalized G-type just comes down to checking whether eval h 1 ,...,h k (g i ) = 0 for all 1 ≤ i ≤ n.
Remark 2.8. Let G be a group, then Lemma 2.5 shows that a universal k-generated group of weak G-type exists, but its true power lies in that it gives a concrete description of G univ k . The equivalent of Lemma 2.5 obtained by replacing strong G-type with weak G-type and replacing Hom(F k , G) by the surjective homomorphisms does not hold. However the universal strong G-type group still exists. Indeed this is a quite formal consequence of the fact that the notion of strong G-type is closed under quotients and subdirect products, so that one can describe the universal strong G-type group as the image of G univ k (or F k ) under the product of all surjective maps to strong G-type groups. What goes wrong with the naive generalization of Lemma 2.5 can be seen by computing that Z/2Z is the universal strong G-type group with 1 distinguished element where G is Z/2Z × Z/2Z or SL 2 (F p ) with p ≥ 5.
(1) An element x ∈ F k in the kernel of π G,k is called a weak G-type relation of rank k, or alternatively, it is said that G satisfies the relation x. In this case I G,k := ker(π G,k ) is called the group of G-type relations, so that G univ k = F k /I G,k . (2) If k 1 , . . . , k n are integers, x 1 , . . . , x n are in F k 1 , . . . F kn respectively and (G, g 1 , . . . , g k ) is a group with k distinguished elements such that G is of x 1 , . . . , x n -type, then the tuple (G, g 1 , . . . , g k ) is called a universal group with k-generators satisfying relations x 1 , . . . , x n if for every other group (H, h 1 , . . . , h k ) of x 1 , . . . , x n -type with k distinct elements there is a unique homomorphism φ : . . , k ′ n ′ is another set of integers and x ′ 1 , . . . , x ′ n ′ ∈ F k 1 , . . . F kn is another set of relations, then x 1 , . . . , x n are said to generate the relations • If G is a group of exponent n then x n 1 ∈ F 1 is a weak G-type relation of rank 1.
] is a weak G-type relation.
• If G is an abelian group of exponent n then the notions of strong, weak and generalized G-type agree, and a group H is of weak G-type if and only if it satisfies the relations x n 1 and [x 1 , x 2 ], i.e. x n 1 and [x 1 , x 2 ] generate the weak G-type relations. The above language can be used to restate the classification of groups of generalized D 4 -type given in [Dan18]. There the first named author shows that a group G is of generalized D 4 type if and only if it has exponent 4 and is of nilpotency class at most two. Since all subgroups of D 4 are of generalized D 4 -type one has that the notions of generalized In fact, this is the universal property of the semidirect product.
To simplify the notion that is to come we give the following definition.
Definition 2.12. Let G be a finite group and p a prime. Then we let λ p (G) be the 1st term in the lower central p-series of G; that is, Let p and q be distinct primes, let ζ p be a p-th root of unity, let ρ p,q : Z/pZ → F q (ζ p ) * denote the action of Z/pZ on F q (ζ p ) defined by ρ p,q (i) = ζ i p , and let D p,q := F q (ζ p ) ⋊ ρ Z/pZ denote the corresponding semidirect product of Z/pZ and F q (ζ p ) as groups. The following is a generalization of Proposition 2.19 and also of [DLRNS18, Lemma 3.2].
Proposition 2.13. Let p, q be distinct primes and H a finite group. Then the following are equivalent: (1) H is isomorphic to a subgroup of D k p,q for some integer k.
View V := (Z/qZ) n as a vector space over F q so that ρ can be seen as an F q -linear representation. Since the order of (Z/pZ) m is coprime to q, V decomposes into a direct sum of irreducible representations of (Z/pZ) m . Let V triv ≃ (Z/qZ) s be the part on which (Z/pZ) m acts trivially, write V = V triv ⊕ W , and let G := W ⋊ (Z/pZ) m . Then H decomposes as a direct product H = V triv × G, and since V triv ≃ (Z/qZ) s ⊆ D s p,q it suffices to show that G is isomorphic to a subgroup of D r p,q for some r. Now write W = r i=1 W i with W i irreducible representations of (Z/pZ) m . Then every W i is nontrivial by definition of W . All nontrivial irreducible F q -linear representations of (Z/pZ) m are of the form ρ p,q • π for some π : (Z/pZ) m → Z/pZ; let π i : (Z/pZ) m → Z/pZ be the map such that W i is isomorphic to the representation ρ p,q • π i and let f i : W i → F q (ζ p ) be an F q -linear map witnessing this isomorphism. From the morphisms f i and π i one gets a morphism φ i : G → D p,q by the universal property of the semidirect product.
.., f r ) is an isomorphism one sees that ker φ has empty intersection with W and in particular this means that φ together with the quotient map q W gives an injection (φ, q W ) : G ֒→ D r p,q × G/W and the desired implication finally follows since G/W = (Z/pZ) m ֒→ D m p,q . From the above proof it also follows that as soon as H is non-commutative then H has a surjective map to D p,q , so that in this case D p,q is of strong H-type, and in particular we have the following corollary.
Corollary 2.14. Let H be a non commutative group such that λ q (λ p (H)) = {e}. Then the notions of weak H-type and weak D p,q -type are equivalent.
Remark 2.15. This simple criterion for checking whether a given group is of weak D p,q -type is one of the primary advantages of this theory. For instance, since A 4 ≃ D 3,2 and for A 4 ⊆ S 4 the notions of weak and generalized G-type agree, one immediately has Proposition 2.19.
Lemma 2.16. Let p, q be distinct primes and H a group of weak D p,q -type. Write H ≃ (Z/qZ) n ⋊ ρ (Z/pZ) m for some integers n, m and let ρ : (Z/pZ) m → GL n (Z/qZ) be a representation. Then the following are equivalent: (1) H is of strong D p,q -type.
(2) The trivial representation does not occur as a subrepresentation of ρ.
(3) The trivial representation does not occur as quotient representation of ρ.
(4) H does not have a quotient isomorphic to Z/qZ.
Note that pH ≃ (Z/qZ) n and H/pH ≃ (Z/pZ) m . Furthermore, since pH is abelian, ρ only depends on a choice of basis for pH and H/pH and not on the choice of the section H/pH → H that was used to write H as a semidirect product. In particular items (2) and (3) do not depend on the way in which H was written as a semidirect product.
Proof. The equivalence of (4) and (5) is trivial. The equivalence of (2) and (3) follows because q is coprime to #(Z/pZ) m and hence V can be written as a direct sum of irreducible representations.
The equivalence of (3) and (4) follows directly from the universal property. Indeed, let f 1 : (Z/pZ) m → Z/qZ be a morphism. Then f 1 = 0, so conjugation by f 1 (x) on Z/qZ is the trivial action for all x ∈ (Z/pZ) m . In particular f 1 can be extended to a surjective map f : (Z/qZ) n ⋊ (Z/pZ) m → Z/qZ if and only if the trivial representation is a quotient of V .
The implication (2) ⇒ (1) is similar to the proof of (4) ⇒ (1) in Proposition 2.13 and the same notation will be used. Since V triv = 0 by assumption one has that the G and H in that proof are the same. Now since the π i and the f i are surjective, one sees that the φ i : G → D p,q are surjective. In particular, G is a subdirect product of r i=1 D p,q × G/W . Thus H = G from Lemma 2.2 (6) because D r p,q and G/W = (Z/pZ) m are both of strong D p,q -type. Now we consider the implication (1) ⇒ (2). Since the representation ρ p,q : Z/pZ → F q (ζ p ) * corresponding to D p,q does not contain the trivial representation as a subrepresentation it suffices to show that the property of not having a trivial subrepresentation is maintained under taking quotients and subdirect products.
First we handle quotients. Let H be a group of strong D p,q -type such that the associated representation ρ does not have a trivial subrepresentation and H → H ′ is a surjective map of groups. If H does not have a quotient isomorphic to Z/qZ then neither does H ′ so that now the fact that the representation ρ ′ associated to H ′ does not have a trivial subrepresentation follows from the equivalence between (2) and (4).
For the invariance under subdirect products let H 1 , H 2 be groups of strong D p,q -type such that for i = 1, 2 the associated representations ρ i : H i /pH i → Aut Fq (pH i ) do not contain a trivial subrepresentation and let H ⊂ H 1 × H 2 be a subdirect product. Define H triv := (pH) H/pH to be the part of pH on which H/pH acts trivially.
The extra fact that a strong D p,q -type group has no quotient isomorphic to Z/qZ shows that in the proof of the equivalence of (1) and (2) in Proposition 2.13 one can replace subgroup by subdirect product.
2.3. The case G = A 4 . Using the general theory that we developed in the previous subsections, we now turn our attention to studying the infinite extension Q(A ∞ 4 )/Q. We begin by examining the relationship between Q(A ∞ 4 ) and its more natural counterpart, the compositum of all A 4 extensions of Q, which we will denote by Q A 4 . It is natural to ask whether Q(A ∞ 4 ) is equal to Q A 4 . Indeed, when A 4 is replaced by D 4 , the analogous statement turns out to be true [Dan18, Theorem 1.11]. However, in our present setting the situation is as follows.
First, recall that A 4 has no normal subgroup of index 2. The same remains true for groups of strong A 4 -type by Lemma 2.16 and the fact that A 4 ≃ D 3,2 . In particular, we have the following.
On the other hand, the following characterization of groups of generalized A 4 -type follows immediately from Proposition 2.13; see also Remark 2.15.
As an immediate consequence, we have the following corollary.
Corollary 2.20. If G is of generalized A 4 -type then the exponent of G divides 6.
We also have the following useful lemma.
With this result in hand, the relationship between Q(A ∞ 4 ) and Q A 4 becomes clearer.
Since the only transitive proper normal subgroup of A 4 is isomorphic to the Klein four-group V 4 , the field Q(A ∞ 4 ) can be viewed as the compositum of all A 4 and V 4 extensions of Q. With this we immediately get that . To see the strict inclusion, notice that if Q A 4 were to contain any V 4 extensions of Q, then Q A 4 would contain a quadratic extension of Q, but this can't be the case from Corollary 2.18.
In fact, we have the following description of Q(A ∞ 4 ). Corollary 2.23. Let Q(2 ∞ ) be the compositum of all quadratic extensions of Q and let Q A 4 be as in Proposition 2.22. Then Q(A ∞ 4 ) is the compositum of Q A 4 and Q(2 ∞ ). For n ≥ 1, for the rest of the paper we will write ζ n for an n-th root of unity.

Growth of the torsion subgroup of elliptic curves by base extensions
Here we collect some results from other papers which will be useful in the subsequent sections. We direct the interested reader to the corresponding papers for their proofs. We begin with a result on the relationship between torsion subgroups and roots of unity.
Throughout the rest of the paper we will make extensive use of the classification of isogenies of elliptic curves defined over Q. Given such a curve, we will say that it possesses an n-isogeny if it admits a degree n isogeny with cyclic kernel.    . Let E/Q be an elliptic curve that admits a rational n-isogeny φ, and let R ∈ E[n] be a point of order n in the kernel of φ, and write Q(R) = Q(x(R), y(R)) for the field of definition of R. The field extension Q(R)/Q is Galois with Galois group isomorphic to a subgroup of (Z/nZ) × . In particular, if n is prime then Gal(Q(R)/Q) is cyclic and its order divides n − 1.
Finally, we observe that torsion structures for E(Q(A ∞ 4 )) are almost entirely determined by the j-invariant j(E).
and T max is the smallest group with this property.
. By Lemmas 3.3 and 3.5, we see that Q(R)/Q must be a cyclic extension of degree 6 or 12. Furthermore, since R is defined over Q(A ∞ 4 ), Corollary 2.20 implies that this cyclic extension must have degree 6. Thus, identifying E[13] with column vectors, the image ofρ E,13 is conjugate to a subgroup of GL 2 (Z/13Z) contained inside the matrices of the form a 2 * 0 * . The elliptic curves defined over Q with this property have been completely classified in [Zyw15], and they correspond to the curves with j-invariant of the form given in (1). Further, since there are no elliptic curves defined over Q with a cyclic 169-isogeny, it is not possible for the 13-primary component of E(Q(A ∞ 4 )) to be any larger. 4.1.2. The case when p = 5. (2) j(E) = (t 4 − 12t 3 + 14t 2 + 12t + 1) 3 t 5 (t 2 − 11t − 1) .
Proof. As in the previous proposition, since Q(ζ 5 ) ⊆ Q(A ∞ 4 ) by Lemma 2.24, it follows from Proposition 3.1 that E(Q(A ∞ 4 )) [5] is cyclic of order 5 j for some j, and by Lemma 3.3 and Theorem 3.2 we must have j ≤ 2.
Suppose j = 1, so E(Q(A ∞ 4 ))(5) = Z/5Z, and let R be a point of order 5. Then by Lemma 3.3, E admits a rational 5-isogeny, and by Lemma 3.5 its field of definition Q(R) is an extension of Q of degree at most 4. It follows from Corollary 2.20 that R is defined over a quadratic extension of Q, or in other words, there is a quadratic twist of E that has a rational point of order 5. Such curves are parametrized by the genus 0 modular curve X 1 (5), and the j-map for this curve is given in [Zyw15] as (2).
Now suppose E(Q(A ∞ 4 )) contains a point R of order 25; then E admits a rational 25-isogeny, and by Lemma 3.5 the field of definition Q(R) is Galois with Galois group isomorphic to a subgroup of (Z/25Z) × , which has order 20. But since Q(R) ⊆ Q(A ∞ 4 ), by Corollary 2.20 we must have that Q(R)/Q is a field extension of degree dividing 6. These facts together imply R is defined over a quadratic extension of Q. If E has a point of order 25 defined over a quadratic extension of Q, then there is a quadratic twist of E with a rational point of order 25, but this cannot happen by Theorem 1.1.
If j = k, then E cannot admit a rational 7-isogeny; to see this, assume a rational 7-isogeny does exist, and let R be a non-trivial point in its kernel. Then by Lemma 3. . Thus k = 0 and j = 1 if and only if E admits a rational 7-isogeny, and such elliptic curves correspond to points on the modular curve X 0 (7), whose j-map is given in [LR13, Table 3] as (3). Now suppose j = k = 1. Then Q(E[7]) ⊆ Q(A ∞ 4 ), so Gal(Q(E[7])/Q) has exponent dividing 6 by Corollary 2.20. This implies that for every prime p = 7 of good reduction for E, the elliptic curve E p /F p obtained by reducing E modulo p has its 7-torsion defined over an F p -extension of degree dividing 6, and in particular, admits an F p -rational 7-isogeny. Thus E/Q admits a rational 7-isogeny locally everywhere but not globally, and as proved in [Sut12], this implies j(E) = 2268945/128. It is also proved in loc. cit. that, conversely, for every elliptic curve E/Q with this j-invariant the group Gal(Q(E[7])/Q) is isomorphic to a subgroup of GL 2 (F 7 ) with surjective determinant map whose image in PGL 2 (F 7 ) is isomorphic to S 3 ; up to conjugacy there are exactly two such groups (labeled 7NS.2.1 and 7NS.3.1 in [Sut16]). However, one checks using Magma that neither of these groups is of generalized A 4 -type, and so the case j = k = 1 is impossible.   ; then the subgroup G = Imρ E,9 ⊆ GL 2 (Z/9Z) must satisfy the following conditions: (i) G has a surjective determinant map and an element with trace 0 and determinant −1, and (ii) G contains a normal subgroup N that acts trivially on a Z/9Z-submodule of Z/9Z ⊕ Z/9Z isomorphic to Z/9Z ⊕ Z/9Z for which G/N is of generalized A 4 -type. Enumerating these groups, one finds that, up to conjugation, there is exactly one maximal subgroup H ⊆ GL 2 (Z/9Z) with this property, and it has the following generating set: Thus H is contained in the split Cartan subgroup, and so E has two distinct rational 9-isogenies. We thus have elliptic curves E 1 and E 2 , both defined over Q, and the isogeny graph of E contains a subgraph of the form E 1 9 ←→ E 9 ←→ E 2 where the number over the arrow indicates the degree of the isogeny. But this implies the existence of a rational 81-isogeny between E 1 and E 2 , and this is impossible by Theorem 3.2.
Proof. By Lemma 4.5, if E(Q(A ∞ 4 ))(3) contains a subgroup isomorphic to Z/3Z ⊕ Z/3Z, then E must admit two distinct rational 3-isogenies. If E(Q(A ∞ 4 ))(3) contained a subgroup isomorphic to Z/3Z ⊕ Z/27Z, then by Lemmas 3.3 and 4.5 we see that E would admit a 3-and a 9-isogeny where the kernel of the 3-isogeny is not contained in the kernel of the 9-isogeny (by abuse of terminology we call such isogenies independent). So in fact E would be isogenous to a curve which admits a rational 27-isogeny. Since X 0 (27) is a rank 0 genus 1 curve, there are only finitely many Q-isomorphism classes of elliptic curves that are 3-isogenous to an elliptic curve with a 27-isogeny. Checking the field of definition of these points of order 27, we see that none of them have a point of order 27 defined over Q(A ∞ 4 ). Thus, in light of Lemma 4.6, E(Q(A ∞ 4 ))(3) is indeed a subgroup of Z/3Z ⊕ Z/9Z. The next lemma shows that this bound is sharp. See also Proposition 4.12.  Table 2. Parameterizations of the possible nontrivial 2-primary components

Modular Curve
; then the subgroup G = Imρ E,9 ⊆ GL 2 (Z/9Z) must satisfy the following conditions: (i) G has a surjective determinant map and an element with trace 0 and determinant −1, and (ii) G contains a normal subgroup N that acts trivially on a Z/9Z-submodule of Z/9Z ⊕ Z/9Z isomorphic to Z/3Z ⊕ Z/9Z for which G/N is of generalized A 4 -type.
Enumerating these groups, one finds that, up to conjugation, there is exactly one maximal subgroup H ⊆ GL 2 (Z/9Z) with this property, and it has the following generating set: Thus E admits independent 3-and 9-isogenies, so in fact it is isogenous to a curve E ′ which admits a 27-isogeny. From [LR13, Table 3] we see j(E ′ ) = −12288000, and a simple computation shows that j(E) = 0. Magma confirms that if E is the elliptic curve with Cremona label 27a1, then [2] may be as big as Z/8Z ⊕ Z/128Z. To determine the torsion structures that actually appear, we make use of the results of Rouse and Zureick-Brown [RZB15]. Using Magma, we are able to search through their data and classify all 2-torsion structures that can occur over Q(A ∞ 4 ), yielding the following proposition.
Proposition 4.10. Let E/Q be an elliptic curve that 2 divides #E(Q(A ∞ 4 )) tors . Then E(Q(A ∞ 4 ))(2) is isomorphic to one of the following 8 groups: Corollary 4.11. Let E/Q be an elliptic curve. Using the notation in [RZB15], for each T listed in Proposition 4.10 the group E(Q(A ∞ 4 )) contains a subgroup isomorphic to T if and only if E corresponds to a rational point on the modular curve given in Table 2.
X 6 X 2 Figure 1. Covering relationships between the curves in Corollary 4.11 Table 3. Torsion structures of CM curves with j = 0, 1728 4.3. The case when E has complex multiplication. If E/Q is an elliptic curve with complex multiplication by an imaginary quadratic field of discriminant D, then D belongs to the set {−3, −4, −7, −8, −11, −19, −43, −67, −163}; representative curves for each of these discriminants, along with their j-invariants, are given in [Sil94, Appendix A, § 3]. By Proposition 3.7, if E/Q has j = 0, 1728, then the isomorphism type of E(Q(A ∞ 4 )) tors depends only on j. Thus, apart from j = 0, 1728, the torsion structures associated to curves with complex multiplication can be computed directly in Magma, and they are given (organized by Cremona label) in the Table 3. We deal with the CM curves separately since it is possible that the images of the corresponding Galois representations are properly contained in the groups above and thus might have larger torsion subgroups than those listed above. ) implies j(E) = 0, which is a contradiction. Thus, for a curve with j(E) = 0, we need only consider the primes p = 2 or 3. Now, every elliptic curve E/Q with j(E) = 0 is isomorphic over Q to a curve of the form E s : y 2 = x 3 + s for some s ∈ Z \ {0} that is 6th-power free. Our analysis will proceed as in the proof of [Dan18, Lemma 5.5]. The 3-division polynomial of E s is given by f 3 (x) = 3x(x 3 + 4s), so generically such a curve has a point of order 3 defined over a quadratic field and hence over Q(A ∞ 4 ). It follows that such a curve always has a rational 3-isogeny. Now, if the factor x 3 +4s is irreducible, then the full 3-torsion will be defined over a field contained in an S 3 extension, which is not of generalized A 4 -type. On the other hand, if 4s = t 3 for some t ∈ Q, then we have E[3] ⊆ E(Q(A ∞ 4 )) since this would mean its full 3-torsion is defined over a 2-elementary extension of Q. In this case E has two separate 3-isogenies, and so Imρ E,3 is conjugate to a subgroup of the split Cartan subgroup of GL 2 (Z/3Z). In fact, if 4s is a cube, the factorization of the 9-division polynomial of E s shows that E s has a 3-isogeny and a 9-isogeny that are independent of each other. Notice also that if t, r ∈ Z \ {0}, then the curves E 2t 3 and E 2r 3 are isomorphic over . Therefore, all of these curves have the same torsion subgroup over Q(A ∞ 4 ) as E 2 , and using Magma we find that E 2 (Q(A ∞ 4 )) tors = Z/3Z ⊕ Z/9Z. As noted before, the only other isogeny type that E s can have is a 2-isogeny, which occurs precisely when s = t 3 for some t ∈ Z and thus E s has a point of order 2 defined over Q. As before, where the second isomorphism was computed using Magma.
Putting this all together, we obtain the following result.   )). Searching GL 2 (Z/4Z) in Magma, we see that the only way that this is possible would be for the point of order 4, call it P , to be in the kernel of a 4-isogeny, with 2P ∈ E s (Q) and x(P ) ∈ Q. Since we are assuming ±s is not a square, we know that the only point of order 2 on E defined over Q is the point (0, 0). Letting P = (α, β) and using the duplication formula for E s we have that Clearly the only way that α can be in Q is if s is a square, contradicting our assumption. Therefore, E s cannot have a point of order 4 defined over Q(A ∞ 4 ) in this case. This gives the following lemma. Lemma 4.14. If ±s is not a square in Q, then E s (Q(A ∞ 4 ))(2) ≃ Z/2Z ⊕ Z/2Z. The last thing that we have to show is that if ±s is a rational square, then E(Q(A ∞ 4 )) does not contain a point of order 8. We start by searching Magma to discover that for E s (Q(A ∞ 4 )) to have a subgroup isomorphic to Z/4Z × Z/8Z, E s would have to possess either an 8-isogeny or independent 2-and 4-isogenies. A quick check shows that j(E s ) = 1728 is not in the image of the map j : X 0 (8)(Q) → Q, and thus E s cannot have an 8-isogeny. Further, in order for E s to have independent 2-and 4-isogenies it must be that E[2] ⊆ E(Q), which implies that s = −t 2 for some t ∈ Z. In this case we get that the 4-division polynomial is exactly In order for E s to have a 4-isogeny, this polynomial would need to have a root in Q. Simple inspection shows that this could only happen if t = 0 or if 2 were a square in Q, neither of which is the case. Thus we have shown the following proposition.
Now, the fact that E necessarily admits a rational 2-isogeny allows us to rule out many other ptorsion possibilities. For instance, if 13 | #E(Q(A ∞ 4 )) tors , then E must admit a rational 13-isogeny, and so E must admit a rational 26-isogeny, but this is impossible by Theorem 3.2. Since E[13] cannot be contained in E(Q(A ∞ 4 )), we see that 13 ∤ #E(Q(A ∞ 4 )) tors . Similarly, if E were to admit a rational 7-isogeny (resp. 5-isogeny), then E would have to admit a rational 14-isogeny (resp. 10-isogeny). By investigating [LR13 , Tables 3 and 4], we see that this cannot occur for a curve with j(E) = 1728. Since E[5] ⊆ E(Q(A ∞ 4 )) by Lemma 2.24 and E[7] ⊆ E(Q(A ∞ 4 )) by Proposition 4.4, we conclude that E does not have any 5-or 7-torsion defined over Q(A ∞ 4 ). It remains to consider p = 3. By [LR13, Table 3], a curve with j(E) = 1728 cannot have a 6-isogeny, so if 3 | #E(Q(A ∞ 4 )) tors then by Lemma 3.3 we must have E[3] ⊆ E(Q(A ∞ 4 )). On the other hand, the 3-division polynomial of E s is 3x 4 + 6sx 2 − s 2 . This polynomial has discriminant −2 12 · 3 3 · s 6 , and it is irreducible. To see that it is irreducible one can argue as follows. First of all it has no roots since E s has no 3-isogenies, so the only possibility is that it factors as a product of two quadratic terms. Write s = 3s ′ with s ′ ∈ Q so that it suffices to prove the irreducibility of the monic polynomial x 4 + 6s ′ x 2 − 3s ′2 instead. We want to show that Table 4. Torsion structures of CM curves j = 0 or 1728 has no solutions with a 1 , a 2 , b 1 , b 2 ∈ Q. This is indeed the case since the cubic term implies a 1 = −b 1 , after which the linear term implies either a 1 = 0 or a 2 = b 2 .
1) If a 1 = 0, then x 4 + 6s ′ x 2 − 3s ′2 = (x 2 + a 2 )(x 2 + b 2 ), in particular this means that −a 2 is a root of y 2 + 6s ′ y − 3s ′2 which is irreducible as soon as s ′ = 0 so this cannot happen. 2) If a 2 = b 2 , then looking at the constant term gives a 2 2 = −3s ′2 which has no solutions as soon as s ′ = 0. So the polynomial is indeed irreducible. By a standard result in Galois theory (c.f. [Con13]), the Galois group of this polynomial has exponent dividing 6 if and only if its discriminant is a square in Q, hence we may conclude by Corollary 2.20 that E(Q(A ∞ 4 )) does not have any 3-torsion. Our discussion in this section has proved the following results.  Table 4.

Determining the possible torsion structures
With Proposition 2.19 in hand, we must now determine which subgroups of T max actually occur as torsion subgroups of elliptic curves over Q(A ∞ 4 ). The main task is to determine which combinations of the possible p-primary components are realized by elliptic curves E/Q upon base extension to Q(A ∞ 4 ). Thanks to Section 4.3, we need only consider curves without complex multiplication. Before continuing, let us observe that the results of Section 4 combine to give us the following useful corollaries.
Corollary 5.1. If E/Q is an elliptic curve without CM and p is an odd prime such that E(Q(A ∞ 4 ))(p) is nontrivial, then p ∈ {3, 5, 7, 13} and E has a rational p-isogeny.  )) tors then E must admit a 13-isogeny. The same corollary implies that if any other p-primary component is nontrivial for p an odd prime, then E must admit a rational 13p-isogeny. Now Theorem 3.2 implies that this is impossible.
The result now follows from Proposition 4.2.
5.2. The case when 7 divides #E(Q(A ∞ 4 )) tors . Let us immediately observe that if E(Q(A ∞ 4 )) is nontrivial, then by the same argument as in Proposition 5.3, the 13-and 5-primary components of E(Q(A ∞ 4 )) must both be trivial, but there do exist elliptic curves E/Q with 14-and 21-isogenies, so we must consider the possibility that E has a point of order 14 or 21 defined over Q(A ∞ 4 ); note that by Theorem 3.2, it cannot possess both. As in Section 5.1, we must also consider the possibility that it has square discriminant. Let us first consider the possibility of E possessing a 21-isogeny. Table 3] we see that, up toQ-isomorphism, there are only four curves which possess a 21-isogeny, and they are represented by the curves with Cremona labels 162b1, 162c1, 162b3, and 162c3. Checking each of these with Magma, we obtain the claimed result.
Thus ∆(E) is a nonzero rational square if and only if t is a nonzero rational square, and the result now follows from Corollary 5.2.
Remark 5.6. Using the LMFDB database [LMF], we find that among the curves guaranteed to exist by Lemma 5.5, the curve with smallest conductor is the one with Cremona label 1922e2.
Finally, we must consider the case that E possesses a 14-isogeny.  )).
Example 7.1. Let E be the elliptic curve with Cremona reference 44a1 and let E ′ be the elliptic curve with Cremona reference 176c1. In this case j(E) = j(E ′ ) and The curves E and E ′ are quadratic twists of each other and both have a 3-isogeny. In the case of E, the kernel of its 3isogeny is defined over Q while in the case of E ′ the kernel of its 3-isogeny is defined over Q(i).
While this also happens when considering base-extension to Q(A ∞ 4 ) when j(E) = 0 or 1728, it can happen in many more instances when considering base extension to Q A 4 , and because of this, replicating Table 5 in this context is not possible.
We also note that, while it is true that E(Q A 4 ) tors ⊆ E(Q(A ∞ 4 )) tors for every elliptic curve E/Q (since Q A 4 ⊆ Q(A ∞ 4 )), it is not the case that every group which arises as the torsion subgroup of an elliptic curve base-extended to Q A 4 also arises as the torsion subgroup of an elliptic curve base-extended to Q(A ∞ 4 ), as shown in the following example. Example 7.2. Let E be the elliptic curve with Cremona reference 46a1. From Table 1, we know that E(Q(A ∞ 4 )) tors ≃ Z/2Z ⊕ Z/2Z and from [LMF] we know that  4 )(2) ≃ Z/2Z ⊕ Z/4Z and so E would possess a 2-isogeny, but this would imply that E(Q) [2] was nontrivial, giving a contradiction.
The above discussion immediately leads to the following two lemmas.
N model for the generic curve with a cyclic group of order N in E(Q A 4 ) tors 3 y 2 + axy + by = x 3 (can take a = 1 if j(E) = 0) − 27(t 4 − t 3 + 5t 2 + t + 1) 3 (t 8 − 5t 7 + 7t 6 − 5t 5 + 5t 3 + 7t 2 + 5t + 1)x +54(t 2 + 1)(t 4 − t 3 + 5t 2 + t + 1) 4 (t 12 − 8t 11 + 25t 10 − 44t 9 + 40t 8 + 18t 7 −40t 6 − 18t 5 + 40t 4 + 44t 3 + 25t 2 + 8t + 1) 7.2. The possible prime-to-2 torsion. From Corollary 7.5 and the Weil pairing, if we let E/Q be an elliptic curve and G be the maximal subgroup of E(Q A 4 ) of odd order, we get that G must be cyclic and isomorphic to Z/N Z for some odd N . Further, a generator of G would have to be defined over an abelian extension and so Corollary 7.4 says the field of definition would have exponent dividing 3. Combining this with the classification of isogenies of elliptic curves over Q we get that N = 1, 3, 5, 7, 9, 13, 15, or 21. We start by noticing that the only way E can have a point of order 3 or 5 defined over Q A 4 is if it already had a point of order 3 or 5 defined over Q. Therefore, since there are no elliptic curves E/Q with a point of order 15 defined over Q, N cannot be 15. Further, if N = 21 then E must have a rational point of order 3 and a 7-isogeny defined over a cubic extension of Q. From [Naj16, Theorem 1], we know that there is exactly one such elliptic curve up to Q-isomorphism and that is the curve with Cremona reference 162b1. Therefore, all that is left to do is classify when the other possible N > 1 occur, and the results are summarized in Table 6. The models in this table for N = 3, 5 can be found in [LR11, Appendix E], while the models for N = 7, 13 can be obtained from [Zyw15], and the model for N = 9 is the model for a curve with a 9-isogeny twisted so that the associated 3-isogeny has a Q-rational kernel. 7.3. Torsion subgroups that occur over Q A 4 . All that is left now is to determine what combinations of 2-powered torsion can occur with the prime-to-2 options. Doing so gives the following theorem. Proof. Notice that the above list is the same as the list of torsion structures that occur already over Q (c.f. Theorem 1.1) with the exception of M = 7, 13, 14, 18, and 21. The fact that 7 and 13 torsion occurs follows immediately from Table 6. The fact that 14 and 18 occur follows from the same table with the additional observation that the curve y 2 = f (x, t) has rational 2 torsion if and only if f (x, t) has a zero, and the curves f (x, t) = 0 are birational to P 1 for N = 7, 9.
So what remains to show is that, except for M = 1 · 7, 1 · 13, 2 · 7, and 2 · 9 as explained above, the combinations of 2-torsion and odd torsion that do not occur over Q also do not occur over Q A 4 . We start by ruling out the case where E has full 2-torsion over Q A 4 and a point of order 5, 7, 9, 13, or 21. In the first four cases, using the information in Table 6 we can compute the discriminant of the generic curve with a point of this order and see that there are no rational numbers that make it a square. If E has a point of order 21 over Q A 4 , then by [Naj16, Theorem 1], E is isomorphic to the curve with Cremona reference 162b1. The curve 162b1 has discriminant −2 3 3 4 , and since the discriminant is invariant modulo squares under Q-isomorphism, this case is excluded by Lemma 7.7.
It only remains to show that there are no curves E/Q with E(Q A 4 ) tors ≃ Z/M Z for M = 26 or 42, but this follows from the fact that there are no elliptic curves with 26 or 42 isogenies.
Example 7.9. Let E be the elliptic curve with Cremona reference 49a4 and let F be the splitting field of f (x) = x 3 − x 2 − 2x + 1. Then F ⊆ Q A 4 since Gal(F/Q) ≃ Z/3Z and E(F ) tors ≃ Z/14Z, and therefore E(Q A 4 ) tors ≃ Z/14Z as well. Similarly if E ′ is the elliptic curve with Cremona reference 49a3, then E(Q A 4 ) tors ≃ Z/14Z as well. These are the only elliptic curves with a point of order 14 over Q A 4 without full 2-torsion defined over Q A 4 .

Torsion over Q(C ∞
3 ). Comparing Theorem 7.8 with [Naj16, Theorem 1], we see that all of the groups that occur when base extending an elliptic curve to Q A 4 also occur when base extending elliptic curves to a single cubic extension.
Inspecting the proof of Theorem 7.8 we see that whenever we have growth in the torsion subgroup when extending to Q A 4 it occurs when extending to at most 2 cyclic cubic extensions of Q. That is to say, for any elliptic curve E/Q, we have E(Q A 4 ) tors = E(Q(C ∞ 3 )) tors . With this observation we get the following corollary. Remark 7.14. It is also interesting to compare this to [DN18,Theorem 4.1], which classifies Krational torsion structures of elliptic curves E/K for K a cyclic cubic field. The present question is similar to but essentially different from the one studied in [Naj16]. The main difference is that the condition that E has to be defined over Q is dropped in [DN18]. Indeed, if E is allowed to be defined over a cyclic cubic field K instead of over Q then there are the extra possibilities of Z/16Z, Z/2Z ⊕ Z/10Z and Z/2Z ⊕ Z/12Z for E(K) tors . The analogous question for K = Q(C ∞ 3 ) is one we do not know how to solve using our techniques, since this question cannot be solved solely by studying the action of Gal(Q/Q) on E(Q) tors . Indeed, when E is defined over Q(C ∞ 3 ) then there is only an action of Gal(Q/Q(C ∞ 3 )).