A criterion to rule out torsion groups for elliptic curves over number fields

We present a criterion for proving that certain groups of the form $\mathbb Z/m\mathbb Z\oplus\mathbb Z/n\mathbb Z$ do not occur as the torsion subgroup of any elliptic curve over suitable (families of) number fields. We apply this criterion to eliminate certain groups as torsion groups of elliptic curves over cubic and quartic fields. We also use this criterion to give the list of all torsion groups of elliptic curves occurring over a specific cubic field and over a specific quartic field.


Introduction
A fundamental result in the theory of elliptic curves, the Mordell-Weil theorem, states that the Abelian group of points of an elliptic curve (or more generally an Abelian variety) E over a number field K is finitely generated. Thus, E(K) is isomorphic to E(K) tors ⊕ Z r , where E(K) tors is the torsion subgroup of E(K) and r ≥ 0 is the rank of E(K).
We will denote by Φ(d), where d is a positive integer, the set of all the possible isomorphism types of E(K) tors , where K runs through all number fields K of degree d and E runs through all elliptic curves over K. Similarly, for a fixed number field K, we will denote by Φ(K) the set of all the possible isomorphism types of E(K) tors where E runs through all elliptic curves over this fixed field K. Obviously, if K is a number field of degree d, then Φ(K) ⊆ Φ(d), and Φ(d) is the union of the Φ(K) with K running over all number fields of degree d. It is interesting to determine the set Φ(d) for fixed integers d, as well as the set Φ(K) ⊆ Φ(d) for fixed number fields K of degree d.
As for results on Φ(K) for specific K, Mazur [23] determined Φ(Q) = Φ(1), the second author determined Φ(Q(ζ 3 )) and Φ(Q(ζ 4 )) [26], and methods of determining Φ(K) for other quadratic fields K were given by Kamienny and the second author [16]. The second author also tried to determine Φ(K) for certain cubic fields K with small discriminant, but managed to obtain only partial results [28].
In this paper we develop a criterion, based on a careful study of the cusps of modular curves X 1 (m, n), which can tell us that certain groups do not occur as torsion groups of elliptic curves over a number field K. This criterion is essentially a generalization of a criterion of Kamienny [13]. Kamienny showed that for certain n, the curve X 1 (n) cannot have non-cuspidal points over an extension of degree d of Q, where d is less than the gonality of X 1 (n), as points of degree d on X 1 (n) would force functions of a smaller degree than the gonality to exist, which is impossible.
We generalize Kamienny's approach both by using the modular curves X 1 (m, n) instead of X 1 (n) and by viewing the number fields K as extensions of a suitable subfield L of Q(ζ m ). This generalization is somewhat technical (for example, it requires a careful consideration of the fields of definitions of cusps), but gives us more flexibility in ruling out torsion groups of the form C m ⊕ C n . Our criterion, on its own or in combination with other techniques, allows us to advance our understanding of the torsion groups of elliptic curves over K, both when K is a fixed number field and when K runs through all number fields of degree d. In particular, we make progress in determining Φ(3) and Φ(4) by ruling out a number of possibilities for torsion groups of elliptic curves over cubic and quartic fields. As for determining Φ(K) for a fixed cubic or quartic field K, a natural choice for a quartic field K, in view of [26], is the 'next' cyclotomic field, Q(ζ 5 ). Since there are no cubic cyclotomic fields, we choose the cyclic cubic field Q(ζ 13 + ζ 5 13 + ζ 8 13 + ζ 12 13 ). In Section 2, we state and prove our main results (Theorem 1 and Corollary 3). In Sections 3 and 4, we use Theorem 1 to determine Φ(Q(ζ 13 + ζ 5 13 + ζ 8 13 + ζ 12 13 )) and Φ(Q(ζ 5 )), achieving to our knowledge the first determination of Φ(K) for a cubic and a quartic field, respectively. In Sections 5 and 6, we use Corollary 3, together with other techniques, to prove that a large number of finite groups do not occur as torsion groups of elliptic curves over cubic and quartic fields, respectively.
The computer calculations were done using Magma [22]. Showing that the rank of Jacobians over Q is 0, unless otherwise mentioned, has been done by showing that the L-function of (the factors of) J is non-zero. By results of Kato [17], the Birch-Swinnerton-Dyer conjecture is true for quotients of modular Jacobians, so this computation unconditionally proves that the rank is 0.

Main results
Notation. If K is a number field, O K denotes its ring of integers. If p is a prime ideal of O K , we write k(p) for the residue field O K /p, and Nm(p) = #k(p). Furthermore, we denote by O K,p the localization of O K at p.
Definition. Let m and n be positive integers with m | n. Let L be a subfield of Q(ζ m ), and let p 0 be a prime of L. Let X be the curve X 1 (m, n) Q(ζm) viewed as a (proper, smooth, but possibly geometrically disconnected) curve over L. We consider triples (X , X ′ , π), where • X is a flat, proper model of X over O L,p0 such that the j-invariant extends to a map j : X → P 1 OL,p 0 ; • X ′ is a flat, proper and regular curve over O L,p0 such that the curve X ′ = X ′ L over L is geometrically connected; • π : X → X ′ is a proper and generically finite map of O L,p0 -schemes. Given such a triple (X , X ′ , π), we write C for the topological inverse image of the section ∞ under the map j : X → P 1 OL,p 0 , and C ′ for the topological image of C under π equipped with the reduced induced subscheme structure. With this notation, we call (X , X ′ , π) a nice (L, p 0 )-quotient of X 1 (m, n) if the following conditions are satisfied: (1) The scheme C ′ is normal and lies in the smooth locus of X ′ over O L,p0 .
(2) The image of the open subscheme X \ C under π equals X ′ \ C ′ .
Let (X , X ′ , π) be a nice (L, p 0 )-quotient of X 1 (m, n). Let X ′ = X ′ L , and let J ′ be the Jacobian of X ′ . Let h be the least common multiple of the ramification indices e(q/p 0 ) where L(C) is the function field of an irreducible component C of C ′ and q is a prime of L(C) over p 0 . Assume that the following conditions are satisfied: i) The gonality of X ′ over L is at least dh + 1.
ii) The group J ′ (L) has rank 0. iii) If p = 2, then the 2-torsion subgroup of J ′ (L) is trivial. iv) For all primes p | p 0 of K, there does not exist an elliptic curve over k(p) with a subgroup isomorphic to A ′ . v) For all primes p | p 0 of K, neither 3 Nm(p) nor 4 Nm(p) is divisible by #A ′ . vi) For all irreducible components C of C ′ , if the function field L(C) has a prime q over p 0 such that [k(q) : k(p 0 )] is in S K,p0 , then q is the unique prime of L(C) over p 0 . Then there does not exist an elliptic curve over K with a subgroup isomorphic to A.
We will prove Theorem 1 below; we begin with an auxiliary result. Lemma 2. Let A, K, L and p 0 be as in Theorem 1. Under the conditions iv) and v) of Theorem 1, any elliptic curve E over K equipped with an embedding ι : A E(K) has multiplicative reduction at all primes of K lying over p 0 .
Proof. Let p be a prime of K over p 0 . ByẼ p we denote the reduction of E modulo p, i.e. the special fibre of the Néron model of E at p. Then we have a reduction map This map is injective on ι(A ′ ) by [18,Appendix] and the definition of A ′ . The groupẼ p (k(p)) therefore contains a subgroup isomorphic to A ′ .
By assumption iv), E does not have good reduction at p. If E had additive reduction, then by the Kodaira-Néron classification [33, Appendix C, §15],Ẽ p (k(p)) would be a product of the additive group of k(p) and a group of order ≤ 4, contradicting assumption v). We conclude that E has multiplicative reduction at p.
Proof of Theorem 1. Let X p0 and X ′ p0 be the special fibres of X and X ′ over p 0 . Let J ′ be the Néron model of J ′ over O L,p0 . It is known [2, §9.5, Theorem 4] that J ′ represents the functor P/E, where P is the open subfunctor of Pic X ′ /OL,p 0 given by line bundles of total degree 0 and E is the schematic closure in P of the unit section in P (L). We have a commutative diagram By assumptions ii) and iii) and [18,Appendix], the bottom horizontal map is injective. Suppose the theorem is false. Let E be an elliptic curve over K equipped with an embedding A E(K). These data determine a point of X(K) whose Zariski closure is a prime divisor D on X . Let D p0 be the schematic intersection of the divisor D with X p0 , and let (π * D) p0 be the schematic intersection of π * D with X ′ p0 . By Lemma 2, E has multiplicative reduction at all primes of K over p 0 , so the support of D p0 is contained in C. Let Z be the support of (π * D) p0 ; then the definition of C ′ implies that Z is contained in C ′ . We can write (π * D) p0 as a linear combination z∈Z n z z, where the n z are positive integers.
Let z be a point of Z. Since (X , X ′ , π) is a nice (L, p 0 )-quotient, there is a unique irreducible component C z of C ′ containing z, and the coordinate ring of C z is the integral closure of O L,p0 in the function field L(C z ) of C z . Hence k(z) can be identified with the residue field k(q z ) of some prime q z of L(C z ) over p 0 ; in particular, [k(q z ) : k(p 0 )] equals [k(z) : k(p 0 )]. On the other hand, k(z) can also be identified with a subfield of the residue field k(p z ) of some prime p z of K over p 0 , so . By assumption vi), q z is the only prime of L(C z ) over p 0 . This implies that the schematic intersection of C z with X ′ p0 equals e z z, where e z = e(q z /p 0 ). We note that e z divides h.
We consider the effective divisor D ′ on X ′ defined by By the above description of the intersections of D and the C z with X ′ p0 , the divisor hπ * D − D ′ on X ′ specializes to the zero divisor on X ′ p0 . This implies that the class of hπ * D − D ′ in P (k(p 0 )) is zero. By the commutativity of the above diagram and the injectivity of the bottom map, the class of hπ * D − D ′ in J ′ (L) is also zero. By assumption i), we conclude that the divisors hπ * D and D ′ are equal. The generic fibre of D is supported outside C; since (X , X ′ , π) is a nice (L, p 0 )-quotient, the generic fibre of π * D is supported outside C ′ . On the other hand, the generic fibre of D ′ is supported on C ′ , a contradiction.
The following corollary of Theorem 1 is useful for eliminating groups from Φ(d).
Corollary 3. Let A be a group of the form C m ⊕ C n with m | n, let d ≥ 1 be an integer, and let L be a subfield of Q(ζ m ). Let p 0 be a prime of L, let p be the residue characteristic of p 0 , and let q = Nm(p 0 ). Let L , and let J ′ be the Jacobian of X ′ . Let h be the least common multiple of the ramification indices e(q/p 0 ) where L(C) is the function field of an irreducible component C of C ′ and q is a prime of L(C) over p 0 . Assume that the following conditions are satisfied: i) The gonality of X ′ over L is at least dh + 1.
ii Proof. Under the conditions of the corollary, the conditions of Theorem 1 are satisfied for every extension K of degree d over L such that L ⊆ Q(ζ m ) ⊆ K.
We end this section with some remarks on checking the conditions of Theorem 1 and Corollary 3. The conditions are straightforward to check in practice, apart from condition ii) if L = Q. An easy way to make sure that condition vi) holds is to choose p 0 totally inert in Q(ζ n ).
An important special case occurs when L equals Q(ζ m ), the prime p 0 does not divide n, the curve X ′ equals X and π is the identity on X. In this case the conditions simplify as follows: (X, X ′ , π) automatically extends to a nice (L, p 0 )quotient, and we have A ′ = A and h = 1. Moreover, the following remarks are useful to check condition vi) in these cases.
Let r be a divisor of n. The cusps of X 1 (m, n) represented by points (a : b) ∈ P 1 (Q), where a, b are coprime integers with gcd(b, n) = r, all have the same field of definition, which we denote by F m,n,r . By generalities on cusps and by the existence of the Weil pairing, we have Q(ζ m ) ⊆ F m,n,r ⊆ Q(ζ n ). Explicitly, the field F m,n,r can be described as follows. We consider the subgroups H 0 m,n,r ⊆ H m,n,r ⊆ G m,n ⊆ (Z/nZ) × defined by (Note that in the latter case m is 1 or 2, so −1 is in G m,n .) Using the canonical identification of Gal(Q(ζ n )/Q) with (Z/nZ) × , the field F m,n,r is then the field of invariants of H m,n,r acting on Q(ζ n ). In the case m = 1, we have Condition vi) can be checked by computing a defining polynomial for F m,n,r over L and factoring it modulo p 0 .
3.1. The cases C 11 , C 14 and C 15 . In these cases, the modular curve X 1 (n) is an elliptic curve, and an easy computation in Magma shows that X 1 (n)(K) = X 1 (n)(Q). It is well known that X 1 (n)(Q) contains only cusps (see for example [23]), hence Y 1 (n)(K) = ∅.

Torsion groups of elliptic curves over Q(ζ 5 )
In this section, we will determine Φ(K), where K is the field Q(ζ 5 ). We will use the (as of yet unpublished) result that the largest prime dividing the order of a point over a quartic field is 17 [6], and the fact that there is no 17-torsion over cyclic quartic extensions of Q [5].
Theorem 5. Let K = Q(ζ 5 ). Then for every elliptic curve E over K, the torsion group E(K) tors is one of the following groups: where n = 1, . . . , 10, 12, 15, 16, There exist infinitely many elliptic curves with each of the torsion groups from the list (3), except for C 15 and C 16 .
Proof. As mentioned at the beginning of this section, we need only consider primes p ≤ 13 as possible divisors of the order of a torsion point.
Before proceeding further, we find all elliptic curves over K containing a point of order 15 and show that the torsion subgroup of these curves is exactly C 15 . Recall that X 1 (15) is isomorphic to the elliptic curve with Cremona label 15a8. We compute that the group of K-points of this elliptic curve is isomorphic to C 16 . Of the 16 points, 8 are cusps, and we compute that the remaining 8 points correspond to elliptic curves over K with torsion subgroup C 15 . In particular, there exist no curves with torsion C 15n for any integer n > 1 and no curves with torsion C 5 ⊕ C 15 .

4.2.
The case C 21 . The curve X 0 (21) is an elliptic curve and we compute that X 0 (21)(K) = X 0 (21)(Q). However, the difference between this case and the previous ones is that Y 0 (21)(K) is not empty and hence one needs also to check that the elliptic curves with 21-isogenies do not have any K-rational points over K. This can be done by using division polynomials; see [28,29] for details.
4.3. The cases C 13 and C 18 . These cases are dealt with exactly as in the proof of Theorem 4.
4.5. The case C 25 . We apply Theorem 1 with L = Q, p 0 = (2) (which is totally inert in K), X ′ = X and π = id. The modular curve X 1 (25) has gonality at least 5 [12, Theorem 2.6], and J 1 (25)(Q) has rank 0 [21] and trivial 2-torsion; this implies conditions i), ii) and iii). Although 25 is in the Hasse interval of F 16 , a search among all elliptic curves over F 16 shows that all such curves E with 25 points satisfy E(F 16 ) ≃ C 5 ⊕ C 5 . This shows that condition iv) holds. Condition v) clearly holds. Finally, condition vi) is satisfied because 2 is totally inert in Q(ζ 25 ).

4.7.
The case C 5 ⊕ C 10 . We apply Theorem 1 with L = Q(ζ 5 ), p 0 one of the primes above 11 (which is totally split in L), X ′ = X 0 (50) L and π the map defined by the inclusion α −1 Γ 1 (5, 10)α ⊂ Γ 0 (50), where α = 5 0 0 1 . The gonality of X ′ is 2, and we compute in Magma that J 0 (50)(K) has rank 0; this implies conditions i) and ii). One easily checks conditions iii) and v). Condition iv) follows from the Hasse bound over k(p 0 ) = F 11 . Finally, condition vi) follows from the fact that all cusps of X ′ are defined over L.
Theorem 5 follows by combining the above cases.

Results for all cubic fields
We now apply the results of Section 2 to prove that certain groups are not in Φ(3). We note that the cases C 40 , C 49 and C 55 (and more) were also proved independently by Wang [34]. Theorem 6. The groups C 2 ⊕ C 20 , C 40 , C 49 and C 55 do not occur as subgroups of elliptic curves over cubic fields.

Results for quartic fields
In this section, we show that certain groups of the form C m ⊕ C n , with m | n and m ≥ 3, are not in Φ (4). Recall that an elliptic curve E with a subgroup isomorphic to C m ⊕ C n has to be defined over a field containing Q(ζ m ).
Theorem 7. The following groups do not occur as subgroups of elliptic curves over quartic fields: Proof. We consider each of the above cases separately.
6.1. The case C 3 ⊕ C 12 . The curve X = X 1 (3,12) has genus 3 and is nonhyperelliptic [8]. We apply Corollary 3 with L = Q(ζ 3 ), p 0 one of the primes of norm 7 in L, X ′ = X and π = id. We compute that the Jacobian of X has rank 0 over Q(ζ 3 ) (see [9,proof of Lemma 4.4] for details). This shows that conditions i), ii) and iii) are satisfied. For all elliptic curves over fields of 49 elements with 36 points, the group of points is isomorphic to C 6 ⊕ C 6 , proving iv). Condition v) clearly holds. Finally, condition vi) holds because p 0 is inert in Q(ζ 12 ).
A number of 2-descent and L-series computations in Magma shows that the rank of all these B i is 0. This shows that condition ii) is satisfied. If q is one of the primes of norm 7 in L, then J(k(q)) has order 3 14 · 7 3 . This implies that the 2-torsion of J(L) is trivial, so condition iii) holds. The Hasse bound implies condition iv).
6.4. The cases C 3 ⊕ C 33 and C 3 ⊕ C 39 . These cases require a slightly different approach, following the lines of [14]. Let K be a quadratic extension of Q(ζ 3 ), let σ be the non-trivial element of Gal(K/Q(ζ 3 )), and let p be a prime of K above 7. We first decribe the case C 3 ⊕ C 33 . We take the hyperelliptic curve X ′ = X 0 (33) of genus 3, with hyperelliptic involution w 11 . Suppose that y is a non-cuspidal point on X 1 (3,33). By Lemma 2, y maps to the cusp at ∞ mod p, and y σ maps to ∞ mod p. The points y and y σ on X 1 (3,33) map to x and x σ on X ′ , which likewise map to ∞ mod p. Consider the map Then f (x) is Q(ζ 3 )-rational, and f (x) mod p is 0. We compute that J ′ (Q(ζ 3 )) is finite. Since reduction modulo p is injective on the torsion, it follows that f (x) = 0 over Q(ζ 3 ), so there is a function g whose divisor is x + x σ − 2∞. Since g has degree 2, it is fixed by the hyperelliptic involution. It follows that ∞ is fixed by the hyperelliptic involution. But w 11 acts freely on the cusps of X ′ , leading to a contradiction.
The curve X ′ is hyperelliptic of genus 2, and the hyperelliptic involution on X ′ is induced by w 3 . We compute that J ′ (Q(ζ 3 )) is finite. Using the same arguments as above, we conclude that w 3 fixes the cusp at ∞ of X ′ , but w 3 acts by switching the two cusps 0 and ∞, which leads to a contradiction.
6.11. The case C 8 ⊕ C 8 . The only field over which there could exist an elliptic curve with full 8-torsion is Q(ζ 8 ) = Q(i, √ 2). To show that such curves do not exist, we will in fact prove the stronger statement that there does not exist an elliptic curve over Q(ζ 8 ) with a subgroup isomorphic to C 4 ⊕ C 8 . To prove this, we note that the modular curve X 1 (4, 8) is isomorphic (over Q(i)) to the elliptic curve with Cremona label 32a2 [27,Lemma 13]. We compute X 1 (4, 8)(Q(ζ 8 )) ≃ C 4 ⊕ C 4 , and all the points are cusps, which proves our claim.
This finishes the proof of Theorem 7.