Classification of torsion of elliptic curves over quartic fields

Maarten Derickx; Filip Najman

doi:10.1515/crelle-2025-0069

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Classification of torsion of elliptic curves over quartic fields

Maarten Derickx und Filip Najman

Veröffentlicht/Copyright: 11. Oktober 2025

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal)

Abstract

Let 𝐸 be an elliptic curve over a quartic field 𝐾. By the Mordell–Weil theorem, E ⁢ ( K ) is a finitely generated group. We determine all the possibilities for the torsion group E ⁢ ( K ) tors , where 𝐾 ranges over all quartic fields 𝐾 and 𝐸 ranges over all elliptic curves over 𝐾. We show that there are no sporadic torsion groups, or in other words, that all torsion groups either do not appear or they appear for infinitely many non-isomorphic elliptic curves 𝐸. Proving this requires showing that numerous modular curves X 1 ⁢ ( m , n ) have no non-cuspidal degree 4 points. We deal with almost all the curves using one of 3 methods: a method for the rank 0 cases requiring no computation; the Hecke sieve, a local method requiring computer-assisted computations; and the global method, an argument for the positive rank cases also requiring no computation. We deal with the handful of remaining cases using ad hoc methods.

Funding source: European Regional Development Fund

Award Identifier / Grant number: PK.1.1.02

Funding source: Hrvatska Zaklada za Znanost

Award Identifier / Grant number: IP-2022-10-5008

Funding statement: The authors were supported by the project “Implementation of cutting-edge research and its application as part of the Scientific Center of Excellence for Quantum and Complex Systems, and Representations of Lie Algebras”, PK.1.1.02, European Union, European Regional Development Fund and by the Croatian Science Foundation under the project no. IP-2022-10-5008.

A Sporadic points on X 1 ⁢ ( n ) and X 0 ⁢ ( n )

A.1 Sporadic points on X 1 ⁢ ( n )

In the introduction, we mentioned that sporadic points of degree 5 ≤ d ≤ 13 have been found on X 1 ⁢ ( n ) in [36]. However, [36] only lists points of low degree, and it is not immediately clear from the data that these points are sporadic. We give a table from which it should be clear that our claim is true, and for which 𝑛 there are X 1 ⁢ ( n ) with degree 𝑑 sporadic points. The data in the table about rk ⁡ J 1 ⁢ ( n ) is obtained from [7, Theorem 3.1], the data about gon Q ⁡ X 1 ⁢ ( n ) from [11] and the low degree points themselves are collected from [36]. Recall that, for a point 𝑥 of degree 𝑛 on a curve 𝑋 to be sporadic, it is sufficient that either n < 1 2 ⁢ gon Q ⁡ X (see [14]), or n < gon Q ⁡ X and rk ⁡ J ⁢ ( X ) ⁢ ( Q ) = 0 (see [10, Proposition 2.3]).

Table 5

𝑑	𝑛	rk ⁡ J 1 ⁢ ( n )	gon Q ⁡ X 1 ⁢ ( n )
5	28	0	6
6	37	¿ 0	18
7	33	0	10
8	33	0	10
9	31	0	12
10	29	0	11
11	35	0	12
12	39	0	14
13	39	0	14

There are a number of points of degree d = 14 , 15 , … , which are likely to be sporadic; however, lower bounds on the gonalities of X 1 ⁢ ( n ) for n ≥ 41 that are good enough to prove this have not been determined, and this is the obstacle of going any further than d = 13 at the moment.

A.2 Sporadic points on X 0 ⁢ ( n )

The evidence for the abundance of sporadic points on X 0 ⁢ ( n ) of all degrees is even stronger than for X 1 ⁢ ( n ) , which leads us to conjecture the following.

Conjecture A.1

For every positive integer 𝑑, there exists an 𝑛 such that there exists a sporadic point of degree 𝑑 on X 0 ⁢ ( n ) .

We now give some evidence for the conjecture.

Proposition A.2

If d ≤ 2166 or there exists an imaginary quadratic number field with class number 𝑑, then there exists a degree 𝑑 sporadic point on some X 0 ⁢ ( n ) .

Proof

Let 𝑑 be an integer, K = Q ⁢ ( − n ) , where 𝑛 is square-free, an imaginary quadratic field whose absolute value of the discriminant is Δ and which has class number h Δ = d . Let E Δ be an elliptic curve with complex multiplication by O K .

From Abramovich’s bound, since [ PSL 2 ( Z ) : Γ 0 ( n ) ] ≥ n + 1 , it follows that

gon Q ⁡ X 0 ⁢ ( n ) ≥ gon C ⁡ X 0 ⁢ ( n ) > 325 ⁢ n 2 15 .

By [27, Corollary 4.2], it follows that E Δ has an 𝑛-isogeny over Q ⁢ ( j ⁢ ( E Δ ) ) , a number field of degree [ Q ( j ( E Δ ) ) : Q ] = h Δ .

By the arguments above and Minkowski’s bound, we have

(A.1) d = h Δ ≤ 2 π ⁢ Δ ≤ 4 ⁢ n π < 1 2 ⋅ 325 ⁢ n 2 15 ≤ 1 2 ⋅ gon Q ⁡ X 0 ⁢ ( n )

as soon as

n > 2 18 325 ⁢ π ,

i.e., when n ≥ 65920 . This (and hence (A.1)) is also satisfied as soon as

n ≥ π ⁢ d 4 > 256 ,

i.e., d > 201 .

For d ≤ 201 , it is enough to explicitly find an imaginary quadratic field Q ⁢ ( − n ) with class number 𝑑 and satisfying

Δ 4 ≥ n > 2 16 ⁢ d 325

to prove the existence of a sporadic point of degree 𝑑 on X 0 ⁢ ( n ) . We verify this for all d ≤ 201 .

Furthermore, we verify the assumption that there exists a quadratic imaginary field with class number 𝑑 for all d ≤ 2166 by searching through the LMFDB. ∎

It is widely believed that the assumptions of Proposition A.2 are satisfied, i.e., that every positive integer is a class number. In particular, if one defines F ⁢ ( d ) to be the number of imaginary quadratic fields with class number 𝑑, then it is expected (see [33, (1.4)] and also [16]) that

F ⁢ ( d ) ≍ d log ⁡ d .

Finally, we note that if one assumes the Generalized Riemann Hypothesis, then the assumptions of Proposition A.2 are satisfied for all d ≤ 10 6 (see [16, Section 9]).

B A moduli problem with 2 torsion

For our explicit computations in Proposition 5.11, we need algebraic models for the modular curves X 1 ⁢ ( 2 , 2 ⁢ n ) . In [10, §3.1], there is a description of how to compute a model of X 1 ⁢ ( 2 , 2 ⁢ n ) . The idea is to start with a suitable family 𝐸 of elliptic curves over A 2 possessing a point 𝑃 of order 2 and a point 𝑄 of infinite order and writing down equations for the closed subscheme Y ⊆ A 2 over which 𝑄 has order exactly 2 ⁢ n . The elliptic curve E Y over 𝑌 will then have points P Y and Q Y of order 2 and 2 ⁢ n respectively. If P Y and n ⁢ Q Y generate E Y ⁢ [ 2 ] in all fibers, then by the universal property of Y 1 ⁢ ( 2 , 2 ⁢ n ) , one gets a map Y → Y 1 ⁢ ( 2 , 2 ⁢ n ) . If this map Y → Y 1 ⁢ ( 2 , 2 ⁢ n ) is birational, one can construct X 1 ⁢ ( 2 , 2 ⁢ n ) as the desingularization of a projective closure of 𝑌. In [10, §3.1], the family 𝐸 of elliptic curves used is the one from [17]. However, neither from the arguments in [17] nor those in [10, §3.1] is it a priori clear whether Y → Y 1 ⁢ ( 2 , 2 ⁢ n ) is birational. In order to remedy this, we write down a universal family of elliptic curves with points 𝑃 of order 2 and 𝑄 of order greater than 2 (possibly infinite) over some open subset Y 1 ( 2 , > 2 ) ⊂ A 2 in Theorem B.1. Carrying out the same construction with this universal family solves the issue, since the universal property now also guarantees the existence of a map Y 1 ( 2 , 2 n ) → Y 1 ( 2 , > 2 ) whose image lands in 𝑌. Luckily, it turns out that this universal family is birational to the family from [17] as discussed in Remark B.2, implying that the equations obtained [10, §3.1] are correct.

Additionally, the proof Theorem B.1 is also used behind the scenes in Proposition 5.11 to compute the Hecke operators directly by using their definition in terms of the moduli interpretation, which is not clear to us how to do using the family in [17].

Let R : = Spec Z [ 1 / 2 ] and consider the category Ell R whose objects are pairs ( E , T ) , where 𝑇 is an 𝑅 scheme and 𝐸 is an elliptic curve over 𝑇; see [23, Section 4.13]. A morphism in Ell R between f 1 : E 1 → T 1 and f 2 : E 2 → T 2 is a Cartesian square

A moduli problem on Ell R is a contravariant functor P : Ell R op → Sets . We define

[ Γ 1 ( 2 , > 2 ) ] : Ell / R → Sets

to be the moduli problem which sends ( E , T ) to the set of pairs P , Q ∈ E ⁢ ( T ) such that 𝑃 is of exact order 2 in all fibers and 𝑄 is of order greater than 2 in all fibers.

Define

Δ ⁢ ( b , c ) := 16 ⁢ b 2 ⁢ ( b + c − 1 ) 2 ⁢ ( 4 ⁢ b 2 + 4 ⁢ b ⁢ c − 4 ⁢ b + c 2 ) ∈ Z ⁢ [ b , c ] , Y 1 ( 2 , > 2 ) := Spec ⁡ Z ⁢ [ 1 2 , b , c , 1 Δ ⁢ ( b , c ) ] , E 1 ( 2 , > 2 ) : y 2 = x 3 + c ⁢ x 2 + ( 1 − b − c ) ⁢ b ⁢ x , P 1 ( 2 , > 2 ) := ( 0 , 0 ) , Q 1 ( 2 , > 2 ) := ( b , b ) .

Theorem B.1

The curve E 1 ( 2 , > 2 ) is an elliptic curve over Y 1 ( 2 , > 2 ) . The points P 1 ( 2 , > 2 ) and Q 1 ( 2 , > 2 ) on E 1 ( 2 , > 2 ) have order 2 and greater than 2, respectively, in all fibers. Furthermore, the triple ( E 1 ( 2 , > 2 ) , P 1 ( 2 , > 2 ) , Q 1 ( 2 , > 2 ) ) over Y 1 ( 2 , > 2 ) is universal in the sense that it represents the moduli problem [ Γ 1 ( 2 , > 2 ) ] .

Proof

We start with E / S written in Weierstrass form

y 2 + a 1 ⁢ x ⁢ y + a 3 ⁢ y = x 3 + a 2 ⁢ x 2 + a 4 ⁢ x + a 6 ,

together with two points P , Q ∈ E ⁢ ( S ) of order 2 in all fibers and order greater than 2 in all fibers, respectively. This equation can be mapped via an isomorphism 𝑓 to another equation in Weierstrass form, where the isomorphism 𝑓 is represented by a quadruple ( u , r , s , t ) such that 𝑓 maps ( x , y ) to

( ( x − r ) ⁢ u 2 , ( y − s ⁢ ( x − r ) − t ) ⁢ u 3 ) ;

see [32, Proposition VII.1.3]. The effect of this isomorphism on the coefficients a i is

(B.1) a 1 ′ = ( a 1 + 2 ⁢ s ) ⁢ u , a 2 ′ = ( a 2 − a 1 ⁢ s + 3 ⁢ r − s 2 ) ⁢ u 2 , a 3 ′ = ( a 3 + a 1 ⁢ r + 2 ⁢ t ) ⁢ u 3 , a 4 ′ = ( a 4 + 2 ⁢ a 2 ⁢ r − a 1 ⁢ ( r ⁢ s + t ) − a 3 ⁢ s + 3 ⁢ r 2 − 2 ⁢ s ⁢ t ) ⁢ u 4 , a 6 ′ = ( a 6 − a 1 ⁢ r ⁢ t + a 2 ⁢ r 2 − a 3 ⁢ t + a 4 ⁢ r + r 3 − t 2 ) ⁢ u 6 .

We move the point 𝑃 to ( 0 , 0 ) by selecting an appropriate 𝑟 (to move x ⁢ ( P ) to 0) and 𝑡 (to move y ⁢ ( P ) to 0, after 𝑥 has been moved to 0). Now our equation is of the form

y 2 + a 1 ′ ⁢ x ⁢ y = x 2 + a 2 ′ ⁢ x 2 + a 4 ′ ⁢ x .

Notice that a 6 ′ = 0 since ( 0 , 0 ) is a point on the curve and a 3 ′ = 0 since ( 0 , 0 ) is of order 2. We can move a 1 ′ to 0 by selecting an appropriate 𝑠 in the quadruple corresponding to the isomorphism by (B.1). Now the point 𝑄 is of the form ( v , w ) for some v , w ∈ O S ⁢ ( S ) since it is of order greater than 2 in all fibers. We can scale the coordinates of 𝑄 by ( v , w ) ↦ ( v ⁢ u 2 , w ⁢ u 3 ) . Hence we can choose u = w / v , and letting b : = v 3 / w 2 , we get that 𝑄 is now of the form ( b , b ) . Note here that w , v , u ∈ O S ⁢ ( S ) × , since otherwise 𝑄 would be a point of order 2 in the fiber where one of w , v , u vanishes. Let

y 2 = x 3 + a 2 ′′ ⁢ x 2 + a 4 ′′ ⁢ x

be the equation we get after this transformation. Let c = a 2 ′′ . Since Q = ( b , b ) lies on this equation, we get that

b 2 = b 3 + c ⁢ b 2 + b ⁢ a 4 ′′ , so a 4 ′′ = b 2 − b 3 − c ⁢ b 2 b = ( 1 − b − c ) ⁢ b ,

as claimed. ∎

Remark B.2

Jain gives in [17] a (two-variable) family of elliptic curves that is birational to Y 1 ( 2 , > 2 ) . His family is given in terms of variables 𝑡 and 𝑞, and to get from Y 1 ( 2 , > 2 ) (given in terms of 𝑏 and 𝑐) to his family and vice versa, one uses the following transformations:

t := ( − 2 ⁢ b − c + 1 ) / ( c − 1 ) , q := b + 1 / 2 ⁢ c + 1 / 2 − 2 ⁢ b / ( 2 ⁢ b + c − 1 ) , b := ( t + 1 ) ⁢ ( q ⁢ t + 1 ) / t 2 , c := ( t 2 − 2 ⁢ q ⁢ t − 2 ) / t 2 .

Acknowledgements

We would like to thank Andrew Sutherland for computing the explicit equations for X 1 ⁢ ( m , n ) that we used, and Barinder S. Banwait, Abbey Bourdon and Michael Stoll for helpful comments on an earlier version of the paper. We are grateful to the anonymous referees for their very helpful comments.

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Received: 2025-02-18

Revised: 2025-09-03

Published Online: 2025-10-11

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https://doi.org/10.1515/crelle-2025-0069