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 $\mathbb{Q}$ that possess an isogeny of degree $3$.
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 $\mathbb{Q}$ with nontrivial torsion are comparatively rare. Recently, Harron–Snowden Reference 3 have refined this result by counting elliptic curves over $\mathbb{Q}$ with prescribed torsion, as follows. Every elliptic curve $E$ over $\mathbb{Q}$ is defined uniquely up to isomorphism by an equation of the form
$$\begin{equation} E \colon y^2=f(x)=x^3+Ax+B \cssId{texmlid1}{\tag{1.1}} \end{equation}$$
with $A,B \in \mathbb{Z}$ such that $4A^3+27B^2 \neq 0$ and there is no prime $\ell$ such that $\ell ^4 \mid A$ and $\ell ^6 \mid B$. We define the height of such $E$ by
for $d(G) \in \mathbb{Q}$ explicitly given, and $f(X) \asymp g(X)$ means that there exist $a_1,a_2 \in \mathbb{R}_{>0}$ such that $a_1 g(X) \leq f(X) \leq a_2 g(X)$ for $X$ large. In the case $G \simeq \mathbb{Z}/2\mathbb{Z}$, i.e., the case of 2-torsion, they show the count is $cX^{1/2}+O(X^{1/3})$ for an explicit constant $c\approx 3.1969$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 $\mathbb{Q}$. 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 $3$.
For $X \in \mathbb{R}_{\geq 1}$, let $N_3(X)$ count the number of elliptic curves $E$ over $\mathbb{Q}$ in the form Equation 1.1 with $\operatorname {ht}(E) \leq X$ that possess a $3$-isogeny defined over $\mathbb{Q}$. Our main result is as follows.
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 $c_2 \approx 0.16$. If we count without these conditions, for the coefficient of the $X^{1/3}$ term we find the explicit constant $c_6 =1.1204\dots$, given in Equation 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 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 $\mathcal{E}$ denote the set of elliptic curves $E$ over $\mathbb{Q}$ in the form Equation 1.1 (minimal, with nonzero discriminant). For $X \in \mathbb{R}_{\geq 1}$, let
be the set of elliptic curves $E$ over $\mathbb{Q}$ with height at most $X$. We are interested in asymptotics for the functions
$$\begin{equation} \begin{aligned} N_3(X) \coloneq & \#\{E \in \mathcal{E}_{\leq X} : \text{$E$ has a $3$-isogeny defined over $\mathbb{Q}$}\}, \\ N_3'(X) \coloneq & \#\{(E, \pm \phi ) : E \in \mathcal{E}_{\leq X} \text{, $\phi \colon E \to E'$ is a $3$-isogeny defined over $\mathbb{Q}$}\}. \end{aligned} \cssId{texmlid4}{\tag{2.2}} \end{equation}$$
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 $y$-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 $E=E_{A,B} \in \mathcal{E}$, with $A,B \in \mathbb{Z}$. The $3$-division polynomial of $E$Reference 7, Exercise 3.7 is equal to
the roots of $\psi (x)$ are the $x$-coordinates of nontrivial $3$-torsion points on $E$.
Although the polynomial $\psi (x)$ is irreducible in $\mathbb{Z}[A,B][x]$, the special case where $A=0$ gives $\psi _{0,B}(x)=3x(x^3+4B)$, and so $a=0$ is automatically a root. We count these easily.
With these lemmas in hand, we define our explicit counting function. For $X>0$, let $N(X)$ denote the number of ordered triples $(A,B,a) \in \mathbb{Z}^3$ satisfying:
(N1)
$A\ne 0$ and $\psi _{A,B}(a)=0$;
(N2)
$\mathopen{\lvert \hspace{0.1ex}}4A^3\mathclose{\rvert }\leq X$ and $\mathopen{\lvert \hspace{0.1ex}}27B^2\mathclose{\rvert }\leq X$;
(N3)
$4A^3+27B^2\ne 0$; and
(N4)
there is no prime $\ell$ with $\ell ^4\mid A$ and $\ell ^6\mid B$.
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 $N_3'(X)$.
In light of the above, our main result will follow from an asymptotic for the easier function $N(X)$ defined in Equation 2.8, and so we proceed to study this function.
3. 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 Equation 2.8, the definition of $N(X)$.
Before proving Theorem 3.1, we begin with a few observations and lemmas. If $A,B,a \in \mathbb{Z}$ with $A \neq 0$, and $\psi _{A,B}(a)=0$, then $a \neq 0$ and
In this section, we prove an asymptotic for $N(X)$. We recall some notation introduced in the proof of Theorem 3.1. Let $(A,B,a) \in \mathbb{Z}^3$ satisfy (N1), so $a \neq 0$ and $B$ is determined by $A,a$ as in Lemma 3.3. Write
$$\begin{equation} \begin{aligned} a &=uv^2,\\ A &=uvw, \end{aligned} \cssId{texmlid13}{\tag{4.1}} \end{equation}$$
with $u \in \mathbb{Z}$ squarefree, $v \in \mathbb{Z}_{>0}$, and $w \in \mathbb{Z}_{\ne 0}$. Then
the transition points $\beta _1,\beta _2$ are algebraic numbers. Then $h(\beta )=g(\beta )$ on the intervals $(-\infty ,\beta _4)$,$(\beta _3,\beta _2)$, and $(\beta _1,\infty )$ and $h(\beta )=f(\beta )$ on the complementary intervals $(\beta _4,\beta _3)$ and $(\beta _2,\beta _1)$.
The relevance of these quantities (as well as their weighting) is made plain by the following lemma.
We then have the following first version of our main result.
5. 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 Equation 4.13, namely, the triples $(u,v,w) \in \mathbb{Z}^3$ such that $u$ is squarefree, $v>0$, and $\mathopen{\lvert \hspace{0.1ex}}\alpha \mathclose{\rvert } \leq h(\beta )$ where $\alpha ,\beta$ are defined by Equation 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.
Before proving our main theorem, we prove one lemma, generalizing Lemma 5.1. For $i\mid 6$ with $i>0$, let
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 $\lfloor \frac{11}{8}X^{1/6} \rfloor$ (valid as in the proof of Lemma 3.4) and keep only squarefree $u$. Then we loop over $v$ from $0$ up to $\lfloor \sqrt {\tfrac{11}{8}X^{1/6}/u} \rfloor$. This gives us the value of $a=uv^2$. Then plugging into $h$ gives
Then we loop over $w$ from $-\beta _{\text{max}}uv^3$ to $\beta _{\text{max}}uv^3$, ignoring $w=0$, and we take $A=uvw$. We then check that $\mathopen{\lvert \hspace{0.1ex}}4A^3\mathclose{\rvert } \leq X$, and letting
we check that $\mathopen{\lvert \hspace{0.1ex}}27B^2\mathclose{\rvert } \leq X$ 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 $m \leq 25$. We compute an approximate value for the constant $c_2 \approx 0.16$ as indicated in the fourth column.
Acknowledgments
The authors thank John Cullinan for useful conversations and Ed Schaefer for helpful corrections.
$$\begin{equation} E \colon y^2=f(x)=x^3+Ax+B \cssId{texmlid1}{\tag{1.1}} \end{equation}$$
Theorem 1.3.
Equation (2.2)
$$\begin{equation} \begin{aligned} N_3(X) \coloneq & \#\{E \in \mathcal{E}_{\leq X} : \text{$E$ has a $3$-isogeny defined over $\mathbb{Q}$}\}, \\ N_3'(X) \coloneq & \#\{(E, \pm \phi ) : E \in \mathcal{E}_{\leq X} \text{, $\phi \colon E \to E'$ is a $3$-isogeny defined over $\mathbb{Q}$}\}. \end{aligned} \cssId{texmlid4}{\tag{2.2}} \end{equation}$$
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