Counting elliptic curves with an isogeny of degree three

We count by height the number of elliptic curves over the rationals that possess an isogeny of degree three.


Introduction
Torsion subgroups of elliptic curves have long been an object of fascination for mathematicians. By work of Duke [1], elliptic curves over Q with nontrivial torsion are comparatively rare. Recently, Harron-Snowden [3] have refined this result by counting elliptic curves over Q with prescribed torsion, as follows. Every elliptic curve E over Q is defined uniquely up to isomorphism by an equation of the form with A, B ∈ Z such that 4A 3 + 27B 2 = 0 and there is no prime such that 4 | A and 6 | B. We define the height of such E by (1.2) ht(E) := max(|4A 3 |, |27B 2 |).
For G a possible torsion subgroup (allowed by Mazur's theorem [5]), Harron-Snowden [3, Theorem 1.5] prove that #{E : ht(E) ≤ X and E(Q) tors G} X 1/d(G) for d(G) ∈ Q explicitly given, and f (X) g(X) means that there exist a 1 , a 2 ∈ R >0 such that a 1 g(X) ≤ f (X) ≤ a 2 g(X) for X large. In the case G Z/2Z, i.e., the case of 2-torsion, they show the count is cX 1 In this article, we count elliptic curves with a nontrivial cyclic isogeny defined over Q. An elliptic curve has a 2-isogeny if and only if it has a 2-torsion point, so the above result of Duke, Grant, and Harron-Snowden handles this case. The next interesting case concerns isogenies of degree 3.

Moreover,
where c 0 is an explicitly given integral (4.8) and the constant c 2 is effectively computable.
We obtain the same asymptotic in Theorem 1.3 if we instead count elliptic curves equipped with a 3-isogeny (that is, counting with multiplicity): see Proposition 2.9. Surprisingly, the main term of order X 1/2 counts just those elliptic curves with A = 0 (having j-invariant 0 and complex multiplication by the quadratic order of discriminant −3). Theorem 1.3 matches computations performed out to X = 10 25 ; see section 6.
The difficulty in computing the constant c 2 in the above theorem arises in applying a knotty batch of local conditions; our computations suggest that c 2 ≈ 0.16. If we count without these conditions, for the coefficient of the X 1/3 term we find the explicit constant c 6 = 1.1204 . . . , given in (5.4); it is already quite complicated. Theorem 1.3 may be interpreted in alternative geometric language as follows. Let X 0 (3) be the modular curve parametrizing (generalized) elliptic curves equipped with an isogeny of degree 3. Then N 3 (X) counts rational points of bounded height on X 0 (3) with respect to the height arising from the pullback of the natural height on the j-line X (1). From this vantage point, the main term corresponds to a single elliptic point of order 3 on X 0 (3)! The modular curves X 0 (N ) are not fine moduli spaces (owing to quadratic twists), so our proof of Theorem 1.3 is quite different from the method used by Harron-Snowden: in particular, a logarithmic term presents itself for the first time. We hope that our method and the lowerorder terms in our result will be useful in understanding counts of rational points on stacky curves more generally.
Contents. The paper is organized as follows. We begin in section 2 with a setup and exhibiting the main term. Then in section 3 as a warmup we prove the right order of magnitude for the secondary term. In section 4, we refine this approach to prove an asymptotic for the secondary term, and then we exhibit a tertiary term in section 5. We conclude in section 6 with our computations.

Setup
In this section, we set up the problem in a manner suitable for direct investigation. We continue the notation from the introduction.
Let E denote the set of elliptic curves E over Q in the form (1.1) (minimal, with nonzero discriminant). For X ∈ R ≥1 , let be the set of elliptic curves E over Q with height at most X. We are interested in asymptotics for the functions (2.2) N 3 (X) :=#{E ∈ E ≤X : E has a 3-isogeny defined over Q}, In defining N 3 (X), we note that we may always post-compose (or pre-compose) a 3-isogeny by the automorphism −1, giving a different isogeny (negating the ycoordinate) but with the same kernel. To avoid this overcounting, we count unsigned isogenies (counting an isogeny and its negative just once).
To that end, let E = E A,B ∈ E, with A, B ∈ Z. The 3-division polynomial of E [7, Exercise 3.7] is equal to the roots of ψ(x) are the x-coordinates of nontrivial 3-torsion points on E.
Lemma 2.4. The elliptic curve E has a 3-isogeny defined over Q if and only if ψ(x) has a root a ∈ Q.
Proof. By the rational root test, a 0 = 3a ∈ Z, and so , the special case where A = 0 gives ψ 0,B (x) = 3x(x 3 + 4B), and so a = 0 is automatically a root. We count these easily. Lemma 2.7. Let N 3 (X) A=0 and N 3 (X) A=0 be defined as in (2.2) but restricted to E ∈ E ≤X with A = 0. Then Proof. In light of the above, we have N 3 (X) A=0 = #{B ∈ Z : |27B 2 | ≤ X and 6 B for any prime }.
If such an elliptic curve had another unsigned 3-isogeny over Q (i.e., a 3-isogeny other than ±φ), it would correspond to a root of ψ(x)/x = x 3 + 4B, in which case −4B is a cube; the count of such is O(X 1/6 ).
That is to say, we define We have excluded from N (X) the count for A = 0 from the function N (X); we have handled this in Lemma 2.7. To conclude this section, we summarize and compare N 3 (X) and N 3 (X). Proposition 2.9. We have Proof. For the first equality, the difference N 3 (X) − N 3 (X) counts elliptic curves with more than one unsigned 3-isogeny. Let E be an elliptic curve with 3-isogenies ϕ i : E → E i such that ϕ 1 = ±ϕ 2 and let ker ϕ i = P i for i = 1, 2. Then P 1 , P 2 = E [3], and so the image of Gal Q acting on E [3] is a subgroup of the group of diagonal matrices in GL 2 (F 3 ). This property is preserved by any twist of E, so such elliptic curves are characterized by the form of their j-invariant, explicitly [8, Table 1, Computing an elliptic surface for this j-invariant, we conclude that every such E is of the form Then by Harron-Snowden [3, Proposition 4.1] (with (r, s) = (4, 6) so m = 1 and n = 2), the number of such elliptic curves is bounded above (and below) by a constant times X 1/6 log X, as claimed. The second equality is immediate from Lemmas 2.4, 2.5, and 2.7.
In light of the above, our main result will follow from an asymptotic for the easier function N (X) defined in (2.8), and so we proceed to study this function.

Order of magnitude
In this section, we introduce new variables u, v, w that will be useful in the sequel and provide an argument that shows the right order of magnitude. This argument explains the provenance of the logarithmic term in a natural way and motivates our approach. We recall (2.8), the definition of N (X).
Before proving Theorem 3.1, we begin with a few observations and lemmas. If A, B, a ∈ Z with A = 0, and ψ A,B (a) = 0, then a = 0 and Proof. The verification is straightforward.
The inequality for B and (3.2) imply that The inequality (3.6) fails for |α| large-in fact, we have |α| < 11/8-which proves the first part of (3.5). To get the second part, note that the first part and condition (N2) imply that |Aa| X 1/2 . And since (3.2) implies that Proof of Theorem 3.1. We first prove the upper bound. Every nonzero a ∈ Z can be written uniquely as a = uv 2 , where u ∈ Z is squarefree and v ∈ Z >0 . Replacing a = uv 2 , we see that a | A 2 if and only if uv | A. Therefore A = uvw with w ∈ Z arbitrary. The inequalities in (3.5) imply that there exist c 3 , c 4 > 0 such that Thus, For X ≥ 2, we have For the lower bound, we let u, v, w range over positive, odd, squarefree numbers with 3 | w and let a = uv 2 and A = uvw as in the previous paragraph. These ensure that conditions (B1)-(B3) hold, so by Lemma 3.3 we have B ∈ Z. Conditions (N1) and (N4) are also satisfied, and condition (N3) is negligible. To ensure (N2), we choose (3.9) v ≤ X 1/24 , uv 2 < 1 2 X 1/6 , w < uv 3 .
since 0 < w/uv 3 ≤ 1 and the polynomial 1 + 2t − 1 3 t 2 on [0, 1] is positive and takes the maximum value 8 3 . Thus, all conditions are satisfied. We now count the choices for u, v, w with the above conditions: we have The inner sum on u is X 1/3 /v, so that N (X) X 1/3 log X, which completes the proof of the lower bound.

An asymptotic
In this section, we prove an asymptotic for N (X). We recall some notation introduced in the proof of Theorem 3.1. Let (A, B, a) ∈ Z 3 satisfy (N1), so a = 0 and B is determined by A, a as in Lemma 3.3. Write with u ∈ Z squarefree, v ∈ Z >0 , and w ∈ Z =0 . Then We rewrite condition (N4) and the conditions in Lemma 3.3 in terms of the quantities u, v, w as follows.
The function h(β) is plotted in Figure 1. The relevance of these quantities (as well as their weighting) is made plain by the following lemma. Proof. Since A = uvw = βu 2 v 4 = α 2 βX 1/3 , the first inequality in (N2) is equivalent to In addition, we have so that the second inequality in (N2) is equivalent to The result then follows from (4.10) and (4.11).
We then have the following first version of our main result.  We suppress the reminder that u is taken to be squarefree. The number of triples with w = 0 is negligible, so we ignore this condition. Let X > 0. For (u, v, w) counted by N 0 (X), we organize by the value of β = w/uv 3 ∈ Q. Taking β in an interval I that does not contain a transition point in its interior, the integers u, v are constrained by |a| = |u|v 2 < |α|X 1/6 < h(β)X 1/6 (with h(β) minimal on I, taking left or right endpoint) by Lemma 4.9. Given u, v, we have w = βuv 3 ∈ uv 3 I, giving approximately uv 3 |I| possible values of w. Repeating this argument with Riemann sum estimates, we obtain (4.14) (For a more refined approach with an error term, see (5.7) below.) We now evaluate this integral. Recall that |u|≤t |u| ∼ 6 π 2 t 2 .
Inputting this into (4.14) and letting X → ∞, we obtain Finally, we impose the local constraints (W1)-(W7). The first 6 of these are clear. To impose (W7) note that The sum converges rapidly; in fact, for Z > 1, Further, the proportion of triples u, v, w with d | v and d 3 | w for some d > Z tends to 0 as Z → ∞. So, imposing (W7) introduces the factor 27/(25ζ(4)) as in Lemma 4.4. We conclude that as X → ∞, as claimed.

Secondary term
In this section, we work on a secondary term for N (X) (giving a tertiary term for N 3 (X)).
We start by explaining how this works for the function N 0 (X) defined in (4.13), namely, the triples (u, v, w) ∈ Z 3 such that u is squarefree, v > 0, and |α| ≤ h(β) where α, β are defined by (4.5). We discuss the modifications to this approach for N (X) below.
We begin by working out an analog of Euler's constant for the squarefree harmonic series.
and γ is Euler's constant.
Proof. The integer variables u, v, d in this proof are positive. We have The O-terms add up to O(x −1/2 ). Since the result follows.
where c 0 is defined in (4.8). More precisely, we have where γ 0 is defined in (5.2) and γ is Euler's constant.
Proof. We return to the derivation of the integral expression (4.14) and consider the contribution of a single term a = uv 2 . With α = a/X 1/6 , the contribution of a to the integral is Note that h is continuous. Let h 1 := h| (−∞,−1/3] and h 2 := h| [−1/3,∞) . Then h 1 is strictly increasing and h 2 is strictly decreasing. Letting j 1 , j 2 be the inverses of h 1 , h 2 , respectively, we have for any t ∈ (0, h(−1/3)] that Plugging (5.6) into the integral (5.5), we obtain j 1 (|α|) − j 2 (|α|). For a choice of a = uv 2 , we count the number of nonzero integers w with w/(|u|v 3 ) ∈ [j 2 (|α|), j 1 (|α|)]: this is equal to So, the error when considering the integral in (4.14) is O(X 1/6 ), i.e., We next consider the evaluation of the integrand Using that 0<v≤t v 3 = 1 4 t 4 + O(t 3 ), for a given value of u with |u| ≤ H 2 , Summing this over squarefree numbers u with |u| ≤ H 2 and using Lemma 5.1, we get (5.9) Next we consider S 2 . For a given value of v ≤ H, we have (5.10) using that the number of squarefree numbers up to a bound x is (6/π 2 )x + O(x 1/2 ) and partial summation. Summing for v ≤ H we get (5.11) Finally, for S 3 we have (5.12) ). Since S = S 1 + S 2 − S 3 , combining (5.9), (5.11), and (5.12) we obtain The expression (5.13) is then to be integrated over all β to obtain N 0 (X) as in (5.7). In this integral we may suppose that |β| X 1/4 , since h(β) |β| −2/3 , and we may suppose that h(β)X 1/6 ≥ 1. Thus, integrating the first error term gives O(X 1/4 (log X) 2 ), and integrating the second gives O(X 7/24 ). We conclude that (5.14) We compute numerically that We now prove our main result.
Proof of Theorem 1.3. The asymptotic for N (X) was proven in Theorem 4.12, and a secondary term with power-saving error term for N 0 (X) was proven in Theorem 5.3. To finish, we claim that the local conditions (W1)-(W7) that move us from N 0 (X) to N (X) can be applied in the course of the argument for Theorem 5.3 to obtain an (effectively computable) constant.
Let i, j, k, d ∈ Z >0 satisfy: i | 6, d squarefree and coprime to 6, j | 12, and k | 6 4 . Let N i,j,k,d (X) denote the number of triples u, v, w counted by N 0 (X) with gcd(u, 6) = i, jd | v, and kd 3 | w. Then with i, j, k running over triples consistent with conditions (W1)-(W6), a signed sum of the counts N i,j,k,d (X) gives N (X). For example, take the case of uvw coprime to 6, which satisfies (W1)-(W6). The contribution of these triples to N (X) is We have similar expressions for other portions of the u, v, w-domain of triples.
We now estimate N i,j,k,d and control the contribution to N (X) from large d. For the latter, since |vw| ≤ A X 1/3 , we have d X 1/12 , so we may suppose that d is so bounded. Getting a good estimate for N i,j,k,d follows in exactly the same way as with N 0 . In particular, we have the analog of (5.7): where it is understood that u is squarefree and v > 0. The sum here is estimated in the same way by first considering the contribution when |u| ≤ H 2 , where H = h(β) 1/4 X 1/24 , then the contribution when v ≤ H, and finally the contribution when both |u| ≤ H 2 and v ≤ H. To accomplish this, we use the following estimates: We also need the sum of 1/|u|, accomplished in Lemma 5.17. Putting these ingredients together, we get that (5.20 where c i,j,k , c i,j,k = O(1) uniformly, and summing these contributions gives the result.

Computations
We conclude with some computations that give numerical verification of our asymptotic expression.
We computed the functions N 0 (X) and N (X) as follows. First, we restrict to u > 0 (still squarefree), since this gives exactly half the count. Second, we loop over u up to 11 8 X 1/6 (valid as in the proof of Lemma 3.4) and keep only squarefree u. Then we loop over v from 0 up to 11 8 X 1/6 /u . This gives us the value of a = uv 2 . Then plugging into h gives (6.1) β max ≤ max X 1/3 4 1/3 a 2 , 3 + 12 + 4 √ 3 Then we loop over w from −β max uv 3 to β max uv 3 , ignoring w = 0, and we take A = uvw. We then check that |4A 3 | ≤ X, and letting B = 1 12 we check that |27B 2 | ≤ X and, if so, add to the count for N 0 (X). For N (X), we further check the local conditions (B1)-(B3) and (N4) (or, equivalently, (W1)-(W7)).
In this manner, we thereby compute the data in Table 6.2 for X = 10 m with m ≤ 25. We compute an approximate value for the constant c 2 ≈ 0.16 as indicated in the fourth column.