Counting elliptic curves with an isogeny of degree three

By Maggie Pizzo, Carl Pomerance, John Voight

Abstract

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

1. Introduction

Torsion subgroups of elliptic curves have long been an object of fascination for mathematicians. By work of Duke Reference 1, elliptic curves over with nontrivial torsion are comparatively rare. Recently, Harron–Snowden Reference 3 have refined this result by counting elliptic curves over with prescribed torsion, as follows. Every elliptic curve over is defined uniquely up to isomorphism by an equation of the form

with such that and there is no prime such that and . We define the height of such by

For a possible torsion subgroup (allowed by Mazur’s theorem Reference 5), Harron–Snowden Reference 3, Theorem 1.5 prove that

for explicitly given, and means that there exist such that for large. In the case , i.e., the case of 2-torsion, they show the count is for an explicit constant Reference 3, Theorem 5.5. (For weaker but related results, see also Duke Reference 1, Proof of Theorem 1 and Grant Reference 2, section 2.)

In this article, we count elliptic curves with a nontrivial cyclic isogeny defined over . An elliptic curve has a -isogeny if and only if it has a -torsion point, so the above result of Duke, Grant, and Harron–Snowden handles this case. The next interesting case concerns isogenies of degree .

For , let count the number of elliptic curves over in the form Equation 1.1 with that possess a -isogeny defined over . Our main result is as follows.

Theorem 1.3.

There exist such that for ,

Moreover,

where is an explicitly given integral Equation 4.8 and the constant is effectively computable.

We obtain the same asymptotic in Theorem 1.3 if we instead count elliptic curves equipped with a -isogeny (that is, counting with multiplicity): see Proposition 2.9. Surprisingly, the main term of order counts just those elliptic curves with (having -invariant and complex multiplication by the quadratic order of discriminant ). Theorem 1.3 matches computations performed out to ; see section 6.

The difficulty in computing the constant in the above theorem arises in applying a knotty batch of local conditions; our computations suggest that . If we count without these conditions, for the coefficient of the term we find the explicit constant , given in Equation 5.4; it is already quite complicated.

Theorem 1.3 may be interpreted in alternative geometric language as follows. Let be the modular curve parametrizing (generalized) elliptic curves equipped with an isogeny of degree . Then counts rational points of bounded height on with respect to the height arising from the pullback of the natural height on the -line . From this vantage point, the main term corresponds to a single elliptic point of order on ! The modular curves 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 lower-order 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.

2. Setup

In this section, we set up the problem in a manner suitable for direct investigation. We continue the notation from the introduction.

Let denote the set of elliptic curves over in the form Equation 1.1 (minimal, with nonzero discriminant). For , let

be the set of elliptic curves over with height at most . We are interested in asymptotics for the functions

In defining , we note that we may always post-compose (or pre-compose) a -isogeny by the automorphism , giving a different isogeny (negating the -coordinate) 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 , with . The -division polynomial of Reference 7, Exercise 3.7 is equal to

the roots of are the -coordinates of nontrivial -torsion points on .

Lemma 2.4.

The elliptic curve has a -isogeny defined over if and only if has a root .

Proof.

For , let be a -isogeny defined over . Then is stable under the absolute Galois group , so . Thus, for all , and hence is a root of by definition. For , if with , then letting we obtain , a Galois stable subgroup of order , and accordingly the map is a -isogeny defined over .

Lemma 2.5.

If is a root of , then .

Proof.

By the rational root test, , and so

whence and .

Although the polynomial is irreducible in , the special case where gives , and so is automatically a root. We count these easily.

Lemma 2.7.

Let and be defined as in Equation 2.2 but restricted to with . Then

Proof.

In light of the above, we have

A standard sieve gives this count as ; see Pappalardi Reference 6. If such an elliptic curve had another unsigned -isogeny over (i.e., a -isogeny other than ), it would correspond to a root of , in which case is a cube; the count of such is .

With these lemmas in hand, we define our explicit counting function. For , let denote the number of ordered triples satisfying:

(N1)

and ;

(N2)

and ;

(N3)

; and

(N4)

there is no prime with and .

That is to say, we define

We have excluded from the count for from the function ; we have handled this in Lemma 2.7. To conclude this section, we summarize and compare and .

Proposition 2.9.

We have

Proof.

For the first equality, the difference counts elliptic curves with more than one unsigned -isogeny. Let be an elliptic curve with -isogenies such that and let for . Then , and so the image of acting on is a subgroup of the group of diagonal matrices in . This property is preserved by any twist of , so such elliptic curves are characterized by the form of their -invariant, explicitly Reference 8, Table 1, 3D-3a

for . Computing an elliptic surface for this -invariant, we conclude that every such is of the form for some , where

Then by Harron–Snowden Reference 3, Proposition 4.1 (with so and ), the number of such elliptic curves is bounded above (and below) by a constant times , 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 defined in Equation 2.8, and so we proceed to study this function.

3. Order of magnitude

In this section, we introduce new variables 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 Equation 2.8, the definition of .

Theorem 3.1.

We have .

Before proving Theorem 3.1, we begin with a few observations and lemmas. If with , and , then and

Lemma 3.3.

Let with . Then if and only if all of the following conditions hold:

(B1)

and ;

(B2)

have the same parity; and

(B3)

if are both even, then .

Proof.

The verification is straightforward.

Lemma 3.4.

Let satisfy conditions (N1)(N2). Then

Proof.

Let . Since , we have

The inequality for and Equation 3.2 imply that

so that

The inequality Equation 3.6 fails for large—in fact, we have —which proves the first part of Equation 3.5. To get the second part, note that the first part and condition (N2) imply that . And since Equation 3.2 implies that

we have .

Proof of Theorem 3.1.

We first prove the upper bound. Every nonzero can be written uniquely as , where is squarefree and . Replacing , we see that if and only if . Therefore with arbitrary. The inequalities in Equation 3.5 imply that there exist such that

Thus,

For , we have

For the lower bound, we let range over positive, odd, squarefree numbers with and let and as in the previous paragraph. These ensure that conditions (B1)–(B3) hold, so by Lemma 3.3 we have . Conditions (N1) and (N4) are also satisfied, and condition (N3) is negligible. To ensure (N2), we choose

Then so . Moreover,