1 Introduction↩︎

Let \(X\) be a geometrically irreducible smooth projective curve over a field \(k\). In [1], Yamazaki et al.construct a pairing \[\langle-,-\rangle \colon \operatorname{Pic}(X)_\mathrm{tors} \times \operatorname{Pic}(X)_\mathrm{tors} \to k^\times \otimes \mathbb{Q}/\mathbb{Z}.\] The pairing is symmetric and biadditive, and also satisfies the following:

1. (Functoriality) Let \(\phi \colon X \to Y\) be a map of schemes, \([D] \in \operatorname{Pic}(X)_\mathrm{tors}\) and \([E] \in \operatorname{Pic}(Y)_\mathrm{tors}\). Then \[\langle [D], \phi^\ast [E]\rangle_X = \langle \phi_\ast [D], [E]\rangle_Y.\]

2. (Base change) If \(k'/k\) is an extension of fields, \(X'\) is the base change of \(X\) to \(k'\), and \(\phi^\ast\) denotes the map induced by \(k \hookrightarrow k'\), we have \(\phi^\ast\langle [D],[E]\rangle = \langle \phi^\ast [D], \phi^\ast [E]\rangle'\) (where \(\langle-,-\rangle'\) is the pairing on \(\operatorname{Pic}(X')_\mathrm{tors}\)).

The kernel of the pairing is the intrinsic subgroup, \[\operatorname{Pic}(X)_\mathrm{tors}^\mathrm{is} := \left\{[D] \in \operatorname{Pic}(X)_\mathrm{tors} : \langle [D],[E]\rangle = 0 \:\forall [E] \in \operatorname{Pic}(X)_\mathrm{tors} \right\}.\] When \(X\) is an elliptic curve \(E\), we identify \(E(k) \simeq \operatorname{Pic}^0(E)\), and so the pairing is on the torsion points of \(E\) itself.

Yamazaki et al.consider the case where \(E\) is an elliptic curve over \(\mathbb{Q}\). Using an explicit analytic method, they arrive at a classification which can be summarized as follows:

In Section 4, we detail a procedure which, given the Weierstrass coefficients of an elliptic curve \(E/\mathbb{Q}\) and the coordinates of a set of generators for \(E(\mathbb{Q})_\mathrm{tors}\), returns the order of \(E(\mathbb{Q})_\mathrm{tors}^\mathrm{is}\) and a generator for \(E(\mathbb{Q})_\mathrm{tors}^\mathrm{is}\). An implementation of this algorithm at [2] was used to compute the intrinsic subgroup orders for every elliptic curve over \(\mathbb{Q}\) cataloged in the LMFDB [3].

In Section 2, we work over a general field \(k\), and show that all elliptic curves admitting cyclic isogenies of a certain type must have intrinsic subgroup of a certain size. This section effectively generalizes the results of [1]. However, while they use analytic methods restricted to \(\mathbb{Q}\) and perform casework on each isomorphism class of \(E(\mathbb{Q})_\mathrm{tors}^\mathrm{is}\), our algebraic methods allow us to work over a general field and consider \(E(\mathbb{Q})_\mathrm{tors}^\mathrm{is} \cong \mathbb{Z}/N\mathbb{Z}\) for arbitrary \(N\).

Let \(M,N \in \mathbb{Z}\) with \(M \mid N\). Our results in Section 2 give a partial solution to the moduli problem of classifying elliptic curves over a general field \(k\) having torsion \(\mathbb{Z}/N\mathbb{Z}\) and intrinsic subgroup \(\mathbb{Z}/M\mathbb{Z}\). Using a simple algebraic method, we give a sufficient (but not necessary) condition for \(E\) to have intrinsic subgroup \(\mathbb{Z}/M\mathbb{Z}\) in terms of isogenies admitted by \(E\). As a consequence, we find that the non-cuspidal \(k\)-points of a particular modular curve, which we name \(X_1(N)_M^{+1}\) (following the conventions of [1]), correspond to pairs \((E,P)\) where \(E\) is an elliptic curve and \(P \in E(k)_\mathrm{tors}\) such that \(\langle P \rangle \cong \mathbb{Z}/N\mathbb{Z}\) and \((N/M)\langle P,P\rangle = 0\). In the case where \(E(k)_\mathrm{tors} = \langle P\rangle\), we find \(E(k)_\mathrm{tors}^\mathrm{is} \cong \mathbb{Z}/M\mathbb{Z}\). However, not every curve \(E\) with the desired torsion and intrinsic subgroup arises as a rational point of the modular curve \(X_1(N)_M^{+1}\). In Section 3, we generalize the methods of [1] to show that, at least in the case where \(k\) is a subfield of \(\mathbb{C}\), every curve \(E\) with \(E(k)_\mathrm{tors} \cong \mathbb{Z}/N\mathbb{Z}\) and \(E(k)_\mathrm{tors}^\mathrm{is} \cong \mathbb{Z}/M\mathbb{Z}\) arises as a rational point of some twist \(X_1(N)_M^\zeta\) of the curve \(X_1(N)_M^{+1}\) by a root of unity \(\zeta\).

2 Cyclic isogenies and the intrinsic subgroup↩︎

Purely using the functoriality of the pairing, we are able to give a sufficient condition in terms of an isogeny admitted by a curve to describe its intrinsic subgroup structure.

3 Galois images and the intrinsic subgroup↩︎

In this section, let \(k\) be a subfield of \(\mathbb{C}\). We can phrase the hypotheses of Theorem 2 as a condition on the adelic Galois image of \(E\). Suppose \(E\) is an elliptic curve with adelic Galois image lying inside the subgroup \[\Gamma :=\Gamma_{1,0}(N,MN) := \Gamma_1(N) \cap \Gamma_0(MN)= \left\{T: T \equiv \begin{pmatrix} 1 + N\ast & \ast \\ 0 & \ast \end{pmatrix} \bmod MN\right\} \leq \operatorname{GL}_2(\widehat \mathbb{Z}).\] The modular curve \(X_{1,0}(N,MN) := X_\Gamma\) parameterizes triples \((E,P,C)\) where \(E\) is such an elliptic curve, \(P\) is a point of \(E\), and \(C\) is a \(k\)-rational cyclic subgroup of order \(MN\) containing \(P\). Theorem 2 shows that all \((E,P,C)\) appearing as points of \(X_{1,0}(N,MN)\) have intrinsic subgroup containing \(\langle (N/M)P\rangle\). However, the converse is not true: the curve \(X_{1,0}(N,MN)\) does not parameterize all elliptic curves with the given intrinsic subgroup. In general, as we will see, we need to consider several twists of \(X_{1,0}(N,MN)\). We will show the following result.

The case \(k=\mathbb{Q}\) was considered in [1], where we see that if \(E\) is an elliptic curve with cyclic torsion of order \(N \in \{1,\dots,10,12\}\) and intrinsic subgroup of order \(M \mid N\), then \(E\) lies on either the modular curve \(X_1(N)_M^{+}\), which is the same as our \(X_{1,0}(N,MN)\), or the curve \(X_1(N)_M^{-}\), which is a twist of \(X_{1,0}(N,MN)\). They give an explicit planar model of \(X_1(N)_M^\varepsilon\) in the form \(f_N(t) = \varepsilon s^M\), where the \(f_N\) are polynomials which they explicitly compute for each \(N \in \{1,\dots,10,12\}\). Many aspects of their argument are applicable to a general field \(k \subset\mathbb{C}\). We follow their argument broadly. In the place of the curves \(X_1(N)_M^{\pm}\), we construct a family of curves \(\{X_1(N)_M^\zeta : \zeta \in \mu(k)\}\), by giving completely explicit models. It will be evident from these models that they are all twists of \(X_1(N)_M^{+1}\).

Let \(E\) be an elliptic curve with a nonzero rational torsion point \(P\) of order \(N\). Through an affine transformation, we can put \((E,P)\) into Tate normal form. That is, there exists an isomorphism \(\phi \colon E \overset{\sim}{\to} E^{(N)}_{a,b}\) for some \(a, b \in k\), with \[E^{(N)}_{a,b} \colon \begin{cases} y^2 = x^3 + ax^2 + bx, \quad b(a^2 - 4b) \neq 0 & \text{if N = 2;} \\ y^2 + axy + by = x^3, \quad b(a^3 - 27b) \neq 0 & \text{if N = 3;} \\ y^2 + (1 + a)xy + by = x^3 + bx^2, \quad b \neq 0 & \text{ if N \notin \{2,3\}.} \end{cases}\] and \(\phi(P) = (0,0)\). In the case \(N \notin \{2,3\}\), the parameters \(a,b \in k\) are uniquely determined by the ordered pair \((E,P)\), so \(a,b = a(E,P), b(E,P)\).

If \(N = 2\), we can ensure through an affine transformation \(x \mapsto \lambda x\) that either \(E \simeq E_{0,b}^{(2)} : y^2 = x^3 + bx\) or \(E \simeq E_{1,b}^{(2)} \colon y^2 = x^3 + x^2 + bx\), i.e., that \(\psi_2(a,b) := a^2 - a = 0\). Similarly, if \(N = 3\), we can ensure by sending \(y \mapsto \lambda y\) that either \(a = 0\) or \(a = 1\), so again \(\psi_3(a,b) := a^2 - a = 0\).

For each \(N > 3\), we can explicitly compute the coordinates \(N(0,0) = (x_N/z_N,y_N/z_N)\) where \(x_N,y_N,z_N \in k[a,b]\). Dividing out factors from \(z_N\), we get a polynomial \(\psi_N\) such that \((0,0)\) has order exactly \(N\) on the curve \(E_{a,b}^{(0)}\) if and only if \(\psi_N(a,b) = 0\). Furthermore every such \((a,b) \in Z(\psi_N)\) defines an elliptic curve \(E_{a,b}\) where \((0,0)\) has order \(N\), except when this curve would be singular, i.e., when \(\Delta(a,b) = 0\).

Then for all \(N \geq 2\), the quasi-affine plane curve \(Z(\psi_N) \setminus Z(\Delta)\) explicitly gives a plane model of the modular curve \(Y_1(N)\). Then \(\mathbb{C}(X_1(N)) \simeq \mathbb{C}(a,b)\).

Now, we can also choose a point \(\tau \in \mathcal{H}\) so that \((E,P) \cong (\mathbb{C}/(\mathbb{Z}+ \tau \mathbb{Z}), 1/N)\). Two points \(\tau, \tau'\) identify the same pair \((E,P)\) if and only if they lie in the same \(\Gamma_1(N)\)-orbit; this is the standard construction of \(X_1(N)\) as \((\Gamma_1(N))\backslash\mathcal{H}\). In [1], it is shown that we can write \(a=a(E,P)=a(\tau)\) and \(b=b(\tau)\) as functions of \(\tau\) in terms of the Jacobi theta function \(\vartheta_1(z) = \vartheta_1(\tau;z)\), \[a(\tau) = \frac{\vartheta_1(1/N)^4 \vartheta_1(4/N)}{\vartheta_1(2/N)^5}, \qquad b(\tau) = \frac{\vartheta_1(1/N)^5 \vartheta_1(3/N)^3}{\vartheta_1(2/N)^8}.\] We have \(\psi_N(a(\tau),b(\tau)) = 0\) for every elliptic curve \(E_\tau\), hence as functions in \(\mathbb{C}(X_1(N))\). So we can write \(\mathbb{C}(X_1(N)) = \mathbb{C}(a(\tau),b(\tau))\).

With \(E,P,\tau\) as above, it is shown in [1] that \[-\langle P,P\rangle = f_N(\tau) \otimes 1/N \in k^\times \otimes \mathbb{Q}/\mathbb{Z}\] where \[f_N(\tau) = \left(\frac{\vartheta_1(2/N)^{2}}{\vartheta_1(1/N)\vartheta_1(3/N)}\right)^N.\] We also get \(f_N \in \mathbb{C}(X_1(N))\), and \(f\) has \(\mathbb{Q}\)-rational Fourier coefficients; thus it can be written as a rational function \(f_N(\tau) = f_N(a,b) \in \mathbb{Q}(a,b)\).

Now let \(M \mid N\) and define \[s(\tau) = \left(\frac{\vartheta_1(2/N)^{2}}{\vartheta_1(1/N)\vartheta_1(3/N)}\right)^{N/M}.\] Now \(s(\tau)\) is not in general a \(\Gamma_1(N)\)-modular function, but in [1], it is checked that \(s\) is \(\Gamma_{1,0}(N,MN)\)-modular. So \(s \in \mathbb{C}(X_{1,0}(N,MN))\). Then \(\mathbb{C}(s,a,b) \subseteq \mathbb{C}(X_{1,0}(N,MN))\). Also \(s^M \in \mathbb{C}(X_1(N))\), and so \([\mathbb{C}(s,a,b):\mathbb{C}(a,b)] = M\). But the map \(X_{1,0}(N,MN) \to X_1(N)\) has degree \(M\). Thus \(\mathbb{C}(s,a,b) = \mathbb{C}(X_{1,0}(N,MN))\), and the two equations \(z_N(a,b) = 0\), \(f_N(a,b) = s^M\) define \(X_{1,0}(N,MN)\) over \(k\).

Now we can prove Theorem 3.

Notice that not every elliptic curve parameterized by the curve \(X_1(N)_M^\zeta\) has \(E(k)_\mathrm{tors}^\mathrm{is} \simeq \mathbb{Z}/M\mathbb{Z}\); namely, if \(E(k)_\mathrm{tors}\) is strictly larger than \(\langle P\rangle\), then \(E(k)_\mathrm{tors}^\mathrm{is}\) may be smaller than \(\langle (N/M)P\rangle\).

Write \(\Gamma := \Gamma_0(MN) \cap \Gamma_1(N)\). Another way to view this result is a condition on the Galois image of \(E\). We showed in Section 2 that when the Galois image of \(E\) is \(\Gamma\), then \(E(k)_\mathrm{tors}^\mathrm{is} \simeq \mathbb{Z}/M\mathbb{Z}\); we can now see that the converse is true up to conjugacy.

4 Computing Intrinsic Subgroups Over \(\mathbb{Q}\).↩︎

In this section, we outline an algorithm to give \(E(\mathbb{Q})_\mathrm{tors}^\mathrm{is}\), in the case where \(E(\mathbb{Q})_\mathrm{tors}\) is cyclic of order at least \(4\). Along with Theorem 1, which gives the classification of possibilities for \(E(\mathbb{Q})_\mathrm{tors}^\mathrm{is}\), we use the following result, which is involved in the proof of that classification:

The function \(f_N(t(a,b))\) is the same as the function \(f_N(a,b)\) which we defined in a more general context in Section 3. The existence of the single parameter \(t\) is due to the fact that all of the curves we are working with have genus \(0\).

The proposition yields a straightforward procedure to compute \(\langle P,P\rangle\) given \((E,P)\), and thereby to compute \(E(\mathbb{Q})_\mathrm{tors}^\mathrm{is}\).

To see that the output is correct, notice that \[Q \in E(\mathbb{Q})_\mathrm{tors}^\mathrm{is} \iff \langle P,Q \rangle = 0 \iff \exists \mu \in \mu(\mathbb{Q}) = \{\pm 1\}, s \in \mathbb{Q},\: f_N(t(a,b)) = \mu s^M.\] A similar, but more involved, procedure allows us to compute the intrinsic subgroup in the cases \(N = 2,3\), and when \(E(\mathbb{Q})_\mathrm{tors}\) is not cyclic. The full algorithm is implemented in [2].

5 Discussion↩︎

We found that the curve \(X := X_{1,0}(N,MN)\) gives a family of curves having (with a few exceptions) \(E(k)_\mathrm{tors} \simeq \mathbb{Z}/N\mathbb{Z}\) and \(E(k)_\mathrm{tors}^\mathrm{is} \simeq \mathbb{Z}/M\mathbb{Z}\). We also showed that, when \(k \subset \mathbb{C}\), every such curve lies on one of several twists of \(X\). It seems natural to expect that the same curves also solve the moduli problem over a general field; however, our methods are not sufficient to show this. That is, the following seems likely (at least if \(\operatorname{char}(k) \nmid N\)):

For simplicity, we restrict our attention in Section 2 to elliptic curves. However, it seems likely that our results can be phrased for a general curve \(X\), in terms of isogenies admitted by the Jacobian \(J=\operatorname{Jac}(X)\). In particular, Lemma 1 could be proven similarly. This is exactly what we showed for subfields of \(\mathbb{C}\) in Section 3.

Our algorithm in Section 4 uses very little that is specific to \(\mathbb{Q}\); most of the steps would work just as well using the rational functions \(f_N(a,b)\), which have the necessary properties over any field. The largest source of additional complexity is that if we vary the field, \(N\) is no longer uniformly bounded; we can no longer precompute the \(f_N\). The functions \(f_N\) can still be computed using their Fourier expansions, but the degree grows linearly with \(N\). The other step which does not generalize immediately is identity testing in \(k^\times \otimes \mathbb{Q}/\mathbb{Z}\): given \(z \in k^\times\) and \(n \in \mathbb{\mathbb{Z}}_{>0}\), we must determine whether \(z \otimes \frac{1}{n} = 1\) in \(k^\times \otimes \mathbb{Q}/\mathbb{Z}\), or equivalently whether \(z \in \mu(k) \cdot (k^\times)^n\). This can be done quickly for many classes of fields, including global fields and local fields; however, for other \(k\) it may not be easy.

Throughout, we focused on the case with \(E(k)_\mathrm{tors}\) cyclic. The analytic approach used in Section 3 and in [1] does not immediately generalize to the noncyclic case. However, it seems reasonable to expect that there is still a relationship between \(E(k)_\mathrm{tors}^\mathrm{is}\) and the Galois image of \(E\), even when \(E(k)_\mathrm{tors}\) is noncyclic. What is this relationship? Are there cases in which \(E(k)_\mathrm{tors}\) may not be cyclic, but \(E(k)_\mathrm{tors}^\mathrm{is}\) admits a description purely in terms of the isogenies of \(E\), analogous to Theorem 2?

Some other basic questions also remain unanswered. In general, given a field \(k\), which groups can appear as \(E(k)_\mathrm{tors}^\mathrm{is}\)? Is it true that, given any \(G = \mathbb{Z}/N_1\mathbb{Z}\oplus \mathbb{Z}/N_2\mathbb{Z}\) and \(H \leq G\), there exists a field \(k\) and an elliptic curve \(E/k\) with \(E(k)_\mathrm{tors} \simeq G\) and \(E(k)_\mathrm{tors}^\mathrm{is} \simeq H\)? When there exists such a curve, are there infinitely many such curves? For example, in the case \(G = \mathbb{Z}/N\mathbb{Z}\) and \(H = \mathbb{Z}/M\mathbb{Z}\), Faltings’ theorem implies that, whenever the genus of \(X_{1,0}(N,MN)\) is at least \(2\), there are finitely many such curves over any fixed number field.

Acknowldgements↩︎

The author thanks Prof. Andrew Sutherland for his continuous support in this exploration.

References↩︎