Counting elliptic curves with an isogeny of degree three

By Maggie Pizzo, Carl Pomerance, and 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,

since and the polynomial on is positive and takes the maximum value . Thus, all conditions are satisfied.

We now count the choices for with the above conditions: we have

The inner sum on is , so that , which completes the proof of the lower bound.

4. An asymptotic

In this section, we prove an asymptotic for . We recall some notation introduced in the proof of Theorem 3.1. Let satisfy (N1), so and is determined by as in Lemma 3.3. Write

with squarefree, , and . Then

We rewrite condition (N4) and the conditions in Lemma 3.3 in terms of the quantities as follows.

Lemma 4.3.

Conditions (B1)(B3) and (N4) hold if and only if all of the following conditions hold:

(W1)

;

(W2)

not both and occur;

(W3)

not all of , , and occur;

(W4)

not all of , , and occur;

(W5)

;

(W6)

not both and occur; and

(W7)

for each prime , not both and occur.

Proof.

This lemma can be proven by a tedious case-by-case analysis. Alternatively, the conditions (B1)(B3) are determined by congruence conditions modulo and , so we may also just loop over the possibilities by computer.

Lemma 4.4.

The proportion among (with squarefree) satisfying the conditions (W1)(W7) is .

Proof.

For the conditions (W1)(W6), we just count residue classes (as in Lemma 4.3): we find proportions for conditions (W1)(W4) and for (W5)(W6). For condition (W7), the proportion of cases where and is , so the correction factor is

Thus, the total proportion is

Let , and suppose is counted by . Define by

(The quantity arose in the proof of Lemma 3.4.) Moreover, define the functions

The function is plotted in Figure 1.

The transition points for the piecewise function occur at

the transition points are algebraic numbers. Then on the intervals , , and and on the complementary intervals and .

We compute numerically that

The relevance of these quantities (as well as their weighting) is made plain by the following lemma.

Lemma 4.9.

The triple satisfies (N2) if and only if

Proof.

Since , 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 Equation 4.10 and Equation 4.11.

We then have the following first version of our main result.

Theorem 4.12.

We have

where

and is defined in Equation 4.8.

Proof.

Via Equation 4.1Equation 4.2, counts with squarefree, positive, , such that conditions (N2)(N3) hold as well as the local conditions (W1)(W7) (which implies (N4)). We may ignore condition (N3) as negligible: for each choice of there are choices of where (N3) fails, subtracting at most from the count.

We first show how to count triples satisfying (N2), not necessarily the local conditions, and define

We suppress the reminder that is taken to be squarefree. The number of triples with is negligible, so we ignore this condition.

Let . For counted by , we organize by the value of . Taking in an interval that does not contain a transition point in its interior, the integers are constrained by

(with minimal on , taking left or right endpoint) by Lemma 4.9. Given , we have , giving approximately possible values of . Repeating this argument with Riemann sum estimates, we obtain

as . (For a more refined approach with an error term, see Equation 5.7 below.)

We now evaluate this integral. Recall that

Inputting this into Equation 4.14 and letting , 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 ,

Further, the proportion of triples with and for some tends to 0 as . So, imposing (W7) introduces the factor as in Lemma 4.4. We conclude that

as , as claimed.

5. Secondary term

In this section, we work on a secondary term for (giving a tertiary term for ).

We start by explaining how this works for the function defined in Equation 4.13, namely, the triples such that is squarefree, , and where are defined by Equation 4.5. We discuss the modifications to this approach for below.

We begin by working out an analog of Euler’s constant for the squarefree harmonic series.

Lemma 5.1.

For real numbers we have

where

and is Euler’s constant.

Proof.

The integer variables in this proof are positive. We have

The -terms add up to . Since

and

the result follows.

Theorem 5.3.

There exists such that

where is defined in Equation 4.8. More precisely, we have

where is defined in Equation 5.2 and is Euler’s constant.

Proof.

We return to the derivation of the integral expression Equation 4.14 and consider the contribution of a single term . With , the contribution of to the integral is

Note that is continuous. Let and . Then is strictly increasing and is strictly decreasing. Letting be the inverses of , respectively, we have for any that

Plugging Equation 5.6 into the integral Equation 5.5, we obtain .

For a choice of , we count the number of nonzero integers with : this is equal to

So, the error when considering the integral in Equation 4.14 is , i.e.,

We next consider the evaluation of the integrand

(with the continued understanding that is squarefree). Let , so that if , then either or . Let be the contribution to the integrand when , let be the contribution when , and let be the contribution when both and . Then

Using that , for a given value of with ,

Summing this over squarefree numbers with and using Lemma 5.1, we get

Next we consider . For a given value of , we have

using that the number of squarefree numbers up to a bound is and partial summation. Summing for we get

Finally, for we have

Since , combining Equation 5.9, Equation 5.11, and Equation 5.12 we obtain

The expression Equation 5.13 is then to be integrated over all to obtain as in Equation 5.7. In this integral we may suppose that , since , and we may suppose that . Thus, integrating the first error term gives , and integrating the second gives . We conclude that

We compute numerically that

and so the coefficient of the secondary term of is .

Before proving our main theorem, we prove one lemma, generalizing Lemma 5.1. For with , let

Lemma 5.17.

We have

where

and is Euler’s constant. Moreover, for .

Proof.

The proof follows the same lines as Lemma 5.1.

We now prove our main result.

Proof of Theorem 1.3.

The asymptotic for was proven in Theorem 4.12, and a secondary term with power-saving error term for was proven in Theorem 5.3. To finish, we claim that the local conditions (W1)(W7) that move us from to can be applied in the course of the argument for Theorem 5.3 to obtain an (effectively computable) constant.

Let satisfy: , squarefree and coprime to 6, , and . Let denote the number of triples counted by with , , and . Then with running over triples consistent with conditions (W1)(W6), a signed sum of the counts gives . For example, take the case of coprime to 6, which satisfies (W1)(W6). The contribution of these triples to is

We have similar expressions for other portions of the -domain of triples.

We now estimate and control the contribution to from large . For the latter, since , we have , so we may suppose that is so bounded. Getting a good estimate for follows in exactly the same way as with . In particular, we have the analog of Equation 5.7:

where it is understood that is squarefree and . The sum here is estimated in the same way by first considering the contribution when , where , then the contribution when , and finally the contribution when both and . To accomplish this, we use the following estimates:

We also need the sum of , accomplished in Lemma 5.17.

Putting these ingredients together, we get that

where uniformly, and summing these contributions gives the result.

6. Computations

We conclude with some computations that give numerical verification of our asymptotic expression.

We computed the functions and as follows. First, we restrict to (still squarefree), since this gives exactly half the count. Second, we loop over up to (valid as in the proof of Lemma 3.4) and keep only squarefree . Then we loop over from up to . This gives us the value of . Then plugging into gives

Then we loop over from to , ignoring , and we take . We then check that , and letting

we check that and, if so, add to the count for . For , 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 with . We compute an approximate value for the constant as indicated in the fourth column.

Acknowledgments

The authors thank John Cullinan for useful conversations and Ed Schaefer for helpful corrections.

Figures

Figure 1.

Graph of the function , defined in Equation 4.6

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{overpic}[width=0.8\textwidth]{h_alternating} \put(23,70){$h(\beta)$} \put(100,1.25){$\beta$} \put(14.5,-1.5){$\beta_4$} \put(15.5,1.75){$\shortmid$} \put(18.5,-1.5){$\beta_3$} \put(20,1.75){$\shortmid$} \put(23.5,-1.5){$\beta_2$} \put(24.5,1.75){$\shortmid$} \put(81.75,-1.5){$\beta_1$} \put(82.75,1.75){$\shortmid$} \end{overpic}
Table 6.2.

Data before and after applying local conditions.

Mathematical Fragments

Equation (1.1)
Theorem 1.3.

There exist such that for ,

Moreover,

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

Equation (2.2)
Lemma 2.4.

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

Lemma 2.5.

If is a root of , then .

Lemma 2.7.

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

Equation (2.8)
Proposition 2.9.

We have

Theorem 3.1.

We have .

Equation (3.2)
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 .

Lemma 3.4.

Let satisfy conditions (N1)(N2). Then

Equation (3.6)
Equation (3.7)
Equation (4.1)
Equation (4.2)
Lemma 4.3.

Conditions (B1)(B3) and (N4) hold if and only if all of the following conditions hold:

(W1)

;

(W2)

not both and occur;

(W3)

not all of , , and occur;

(W4)

not all of , , and occur;

(W5)

;

(W6)

not both and occur; and

(W7)

for each prime , not both and occur.

Lemma 4.4.

The proportion among (with squarefree) satisfying the conditions (W1)(W7) is .

Equation (4.5)
Equation (4.6)
Equation (4.8)
Lemma 4.9.

The triple satisfies (N2) if and only if

Equation (4.10)
Equation (4.11)
Theorem 4.12.

We have

where

and is defined in Equation 4.8.

Equation (4.13)
Equation (4.14)
Lemma 5.1.

For real numbers we have

where

and is Euler’s constant.

Theorem 5.3.

There exists such that

where is defined in Equation 4.8. More precisely, we have

where is defined in Equation 5.2 and is Euler’s constant.

Equation (5.5)
Equation (5.6)
Equation (5.7)
Equation (5.9)
Equation (5.11)
Equation (5.12)
Equation (5.13)
Lemma 5.17.

We have

where

and is Euler’s constant. Moreover, for .

References

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Article Information

MSC 2010
Primary: 11G05 (Elliptic curves over global fields)
Secondary: 14H52 (Elliptic curves)
Keywords
  • Elliptic curves
Author Information
Maggie Pizzo
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
magdalene.r.pizzo.19@dartmouth.edu
Carl Pomerance
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
carl.pomerance@dartmouth.edu
MathSciNet
John Voight
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
jvoight@gmail.com
ORCID
MathSciNet
Additional Notes

The first author was supported by the Jack Byrne Scholars program at Dartmouth College.

The third author was supported by a Simons Collaboration grant (550029).

Communicated by
Rachel Pries
Journal Information
Proceedings of the American Mathematical Society, Series B, Volume 7, Issue 3, ISSN 2330-1511, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , and published on .
Copyright Information
Copyright 2020 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
Article References
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  • DOI 10.1090/bproc/45
  • MathSciNet Review: 4071798
  • Show rawAMSref \bib{4071798}{article}{ author={Pizzo, Maggie}, author={Pomerance, Carl}, author={Voight, John}, title={Counting elliptic curves with an isogeny of degree three}, journal={Proc. Amer. Math. Soc. Ser. B}, volume={7}, number={3}, date={2020}, pages={28-42}, issn={2330-1511}, review={4071798}, doi={10.1090/bproc/45}, }

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