Extensions of CM elliptic curves and orbit counting on the projective line
 Julian Rosen^{1} and
 Ariel Shnidman^{2}Email authorView ORCID ID profile
Received: 19 August 2016
Accepted: 17 January 2017
Published: 1 May 2017
Abstract
There are several formulas for the number of orbits of the projective line under the action of subgroups of \(\mathrm{GL}_2\). We give an interpretation of two such formulas in terms of the geometry of elliptic curves, and prove a more general formula for a large class of congruence subgroups of Bianchi groups. Our formula involves the number of walks on a certain graph called an isogeny volcano. Underlying our results is a complete description of the group of extensions of a pair of CM elliptic curves, as well as the group of extensions of a pair of lattices in a quadratic field.
Keywords
Kleinian groups Elliptic curves Complex multiplication1 Introduction
We can connect these two formulas using the theory of elliptic curves. For this, we let E and \(E'\) be elliptic curves over \({\mathbb {C}}\), and we consider the number \(N(E,E')\) of elliptic curves on the abelian surface \(A = E \times E'\) up to the action of \(\mathrm{Aut}(A)\). This number is finite by [5], and in fact \(N(E,E') = 2\) unless there exists an isogeny \(\lambda :E \rightarrow E'\). This raises the question: how do we compute \(N(E,E')\) if E and \(E'\) are isogenous?
If \(\mathrm{End}(E) = {\mathbb {Z}}\), then \(\mathrm{Hom}(E,E') = {\mathbb {Z}}\lambda \), for a certain minimal isogeny \(\lambda \), whose degree we will denote by N. Then \(\mathrm{Aut}(A) \simeq \Gamma _0(N)\) and we have \(N(E,E') =\#\Gamma _0(N) \backslash {\mathbb {P}}^1({\mathbb {Q}})\) [9, Prop. 3.7]. So the number \(N(E,E')\) is given by (1.1). On the other hand, if E has complex multiplication (CM) by an imaginary quadratic field K, we may think of \(\mathrm{Aut}(A)\) as a subgroup of \(\mathrm{GL}_2(K)\). As before, we have \(N(E,E') =\# \mathrm{Aut}(A) \backslash {\mathbb {P}}^1(K)\); see Lemma 3.1. In the special case where \(E = E'\) and \(\mathrm{End}(E) = {\mathcal {O}}_{K}\), we have \(\mathrm{Aut}(A) \simeq \mathrm{GL}_2({\mathcal {O}}_{K})\) and Bianchi’s formula (1.2) gives \(N(E,E) = h\).
Our main result is a formula for \(N(E,E')\) for any two elliptic curves \(E,E'\) with CM by K. Equivalently, we compute \(\#\mathrm{Aut}(M) \backslash {\mathbb {P}}^1(K)\) for any lattice \(M \subset K^2\). To state the result, we follow Kani [4] and define the econductor of an elliptic curve E with CM by K to be the index \({[{\mathcal {O}}_K:\mathrm{End}(E)]}\). Thus, if E has econductor c, then \(\mathrm{End}(E)\) is isomorphic to \({\mathcal {O}}_c\), the unique subring of index c inside \({\mathcal {O}}_K\). Concretely, if \(E \simeq {\mathbb {C}}/{\mathfrak {a}}\), for some lattice \({\mathfrak {a}}\) in K, then c is the index of the ring of multipliers \(\{ \alpha \in K :\alpha {\mathfrak {a}}\subset {\mathfrak {a}}\}\) in \({\mathcal {O}}_K\).
Theorem 1.1

\(\displaystyle h_d = \#\mathrm{Pic}({\mathcal {O}}_d) = \# \mathrm{Pic}({\mathcal {O}}_K) \cdot d \prod _{p  d} \left( 1  \chi _{K}(p)/p\right) ,\)

\(\chi _{K}\) is the quadratic Dirichlet character associated to K,

\(\omega (n)\) is the number of distinct prime factors of n, and

\(r_{K}(a,b,N)\) is the number of cyclic subgroups of order N of a fixed elliptic curve of econductor a such that the quotient has econductor b.
The mysterious quantity in our formula is the function \(r_K(a,b,N)\). This is a 3variable multiplicative function which is made completely explicit in Corollary 4.3. For each prime p, the values of \(r_K\) on powers of p depend only on how p splits in K, so the number \(N(E,E')\) depends only on the two integers c and \(c'\) and the values of \(\chi _K\) on primes dividing \(cc'\). The explicit formulas for \(r_K(a,b,N)\) are derived by interpreting these numbers as counting certain walks on graphs called pisogeny volcanoes. The structure of the pisogeny volcano then allows one to compute \(r_K(a,b,N)\) simply by looking at the graph (see Theorem 4.2). We also provide a Sage script for computing the numbers \(N(E,E')\) [10].
Corollary 1.2
In favorable cases, the formula in Corollary 1.2 simplifies. For example, if \({\mathfrak {a}}= {\mathfrak {a}}' = {\mathcal {O}}_f\) for some \(f \ge 1\), then \(\Gamma ({\mathfrak {a}},{\mathfrak {a}}') = \mathrm{GL}_2({\mathcal {O}}_f)\) and the right hand side becomes a simple Dirichlet convolution:
Corollary 1.3
Proof
See Sect. 4. \(\square \)
These results have application to other counting problems in geometry. For example, \(\#\Gamma ({\mathfrak {a}},{\mathfrak {a}}')\backslash {\mathbb {P}}^1(K)\) is the number of cusps on the hyperbolic 3manifold \(\Gamma ({\mathfrak {a}},{\mathfrak {a}}')\backslash {\mathbb {H}}^3\). It is also the number of equivalence classes of contractions of the abelian surface \(A = E \times E'\), in the sense of the minimal model program. Our formula can be used to study the asymptotics of these quantities as A varies, and should be helpful in individual computations as well.
The proof of Theorem 1.1 involves a careful study of \(\mathrm{Ext}\)groups in the category of products of elliptic curves with complex multiplication by K, or equivalently, in the category of lattices in imaginary quadratic fields. These results, found in Sect. 2, are interesting in their own right and should find other applications.
2 Extensions of CM elliptic curves
Fix an imaginary quadratic number field \(K\subset {\mathbb {C}}\). A Klattice of rank n is a free abelian subgroup \(L\subset K^n\) of rank 2n. The quotient \({\mathbb {C}}^n/L\) is a complex torus, which is known to be algebraic.
2.1 Singular abelian surfaces
For context, we recall a basic fact about the abelian surfaces we are considering.
Proposition 2.1
 (1)
\(A \simeq {\mathbb {C}}^2/\Lambda \), with \(\Lambda \) a Klattice of rank 2.
 (2)
A is isomorphic to a product of two elliptic curves, both having CM by K.
 (3)
A is isogenous to a product of two elliptic curves, both having CM by K.
If these equivalent conditions hold, we say that A is singular and that A has CM by K.
2.2 Rank 1 Klattices
Recall that an order in K is a subring \(R\subset {\mathcal {O}}_{K}\) such that \(\mathrm {Frac}(R)=K\). Every order has the form \(R={\mathbb {Z}}+f{\mathcal {O}}_{K}\) for a unique positive integer f, called the conductor of R. For each lattice \(L \subset K^n\), the ring of multipliers R(L) is the order \(\{ \alpha \in K : \alpha L \subset L\}\). The conductor of L is defined to be the conductor of R(L).
If \({\mathfrak {a}}\) is a rank 1 Klattice, then \({\mathfrak {a}}\) is projective as an \(R({\mathfrak {a}})\)module. Two rank 1 Klattices \({\mathfrak {a}}\) and \({\mathfrak {a}}'\) are homothetic if \({\mathfrak {a}}= \gamma {\mathfrak {a}}'\) for some \(\gamma \in K^\times \). The set of homothety classes of lattices of conductor f forms a group under multiplication of lattices, which is denoted \(\mathrm{Pic}({\mathcal {O}}_f)\). The set of homothety classes of lattices in K is therefore in bijection with \(\coprod _{f \ge 1} \mathrm{Pic}({\mathcal {O}}_f)\). If \(E = {\mathbb {C}}/{\mathfrak {a}}\), then we define the econductor of E to be the conductor of \({\mathfrak {a}}\).
Proposition 2.2
Proof
Since A is singular with CM by K, we may write \(A = E \times E'\) with \(E = {\mathbb {C}}/{\mathfrak {a}}\) and \(E' = {\mathbb {C}}/{\mathfrak {a}}'\) for rank 1 Klattices \({\mathfrak {a}}\) and \({\mathfrak {a}}'\). Let c and \(c'\) be the econductors of these elliptic curves. Then \(E \times E' \simeq {\mathbb {C}}/{\mathcal {O}}_f \times {\mathbb {C}}/{\mathfrak {a}}{\mathfrak {a}}'\), where \(f = \mathrm{lcm}(c_1,c_2)\). Moreover, if two lattices \({\mathfrak {a}}, {\mathfrak {b}}\subset K\) have multiplication by \({\mathcal {O}}_f\), then \({\mathbb {C}}^2/({\mathcal {O}}_f \oplus {\mathfrak {a}}) \simeq {\mathbb {C}}^2/({\mathcal {O}}_f \oplus {\mathfrak {b}})\) if and only if \({\mathfrak {a}}\) and \({\mathfrak {b}}\) are homothetic (see [6] or [4]). \(\square \)
2.3 Extensions of rank 1 lattices
Let \(L_{1}, L_{2} \subset K\) be Klattices of conductors \(f_{1}\) and \(f_{2}\), and let \(E_i = {\mathbb {C}}/L_i\) (for \(i=1,2\)) be the corresponding elliptic curves. Suppose also that \(L_{1}={\mathbb {Z}}+{\mathbb {Z}}\tau _{1}\) and \(L_{2}={\mathbb {Z}}+{\mathbb {Z}}\tau _{2}\) for elements \(\tau _{1}\) and \(\tau _{2}\) of K; up to homothety, we may always choose such a basis.
The following result shows that the extensions of \(E_1\) by \(E_2\) are controlled by a third elliptic curve \(\tilde{E} := {\mathbb {C}}/L_1L_2\).
Proposition 2.3
Remark
There is a similar result in [8, Thm. 6.1], attributed to Lichtenbaum, but the result is not stated correctly there.
Proposition 2.3 shows that \(\mathrm{Ext}^1_{\mathrm{alg}}(E_{1},E_{2}) \simeq ({\mathbb {Q}}/{\mathbb {Z}})^2\) as a group. But it is not clear which (or how many) extension classes correspond to some fixed abelian surface S. The following theorem gives this extra information.
Theorem 2.4

For any integer c divisible by f, the lattices \(L_{1}\) and \(L_{2}\) can be considered as \({\mathcal {O}}_c\)modules, and we have the group \(\mathrm{Ext}^1_{{\mathcal {O}}_c}(L_{1},L_{2})\) of extensions of \({\mathcal {O}}_c\)modules;

The group \(\mathrm{Ext}^1(L_{1},L_{2})\) of extensions of Klattices;

The group \(\mathrm{Ext}^1_\mathrm{alg}({\mathbb {C}}/L_{1},{\mathbb {C}}/L_{2})\) of extensions of abelian varieties;

The group \(\mathrm{Ext}^1_{\mathrm{an}}({\mathbb {C}}/L_{1},{\mathbb {C}}/L_{2})\) of extensions of complex tori.
Proposition 2.5
Proof
Remark
Proposition 2.5 holds if K is a real quadratic field as well.
Corollary 2.6
The map \(\gamma _1\) of (2.1) takes \(\mathrm{Ext}^1_{{\mathcal {O}}_c}(L_{1},L_{2})\) isomorphically onto the (c / f)torsion in \(\mathrm{Ext}^1(L_{1},L_{2})\).
Proof
Note that as an abelian group, \(\mathrm{Hom}_{{\mathcal {O}}_c}(L_1, L_2)\) is free of rank 2. Hence, by Proposition 2.5, \(\mathrm{Ext}^1_{{\mathcal {O}}_c}(L_{1},L_{2})\) is an (c / f)torsion group of size \((c{/}f)^2\) and \(\gamma _1\) maps into the (c / f)torsion in \(\mathrm{Ext}^1(L_{1},L_{2})\). Proposition 2.3 implies that the (c / f)torsion in \(\mathrm{Ext}^1_\mathrm{alg}({\mathbb {C}}/L_{1}, {\mathbb {C}}/L_2)\) has cardinality \((c{/}f)^2\), hence the (c / f)torsion in \(\mathrm{Ext}^1(L_{1},L_{2})\) also has cardinality \((c{/}f)^2\) because \(\gamma _2\) is an isomorphism. As \(\gamma _{1}\) induces an injective map between two sets of the same cardinality, it must be an isomorphism onto the (c / f)torsion of \(\mathrm{Ext}^1(L_{1},L_{2})\). \(\square \)
Lemma 2.7
Proof
The conductor of L is the minimal c such that the class in \(\mathrm{Ext}^1(L_{1},L_{2})\) representing L is in the image of \(\mathrm{Ext}^1_{{\mathcal {O}}_c}(L_{1},L_{2})\). Corollary 2.6 implies that this value is nf. \(\square \)
Proof of Theorem 2.4
By Proposition 2.2, the abelian surface S is isomorphic to \({\mathbb {C}}/ {\mathcal {O}}_N \times {\mathbb {C}}/{\mathfrak {a}}\) for some integer N and some lattice \({\mathfrak {a}}\subset K\) with \({\mathcal {O}}_N\subset R({\mathfrak {a}})\). Since the conductor of \({\mathcal {O}}_N \oplus {\mathfrak {a}}\) is N, we must have \(N = nf\) by the previous lemma. Thus, it remains to show that \({\mathbb {C}}/{\mathfrak {a}}\simeq \tilde{E} / \langle P \rangle \). We do this by computing the exterior power of the lattice corresponding to S in two different ways.
3 Proof of Theorem 1.1 and Corollary 1.2
Let \(A = E \times E'\) be a product of two elliptic curves with CM by the same imaginary quadratic field K. Then we may think of \(\mathrm{Aut}(A)\) as a subgroup of \(\mathrm{GL}_2(K)\). Explicitly, choose an isogeny \(\lambda :E \rightarrow E'\), and identify \(\mathrm{Hom}(E,A) \otimes _{\mathbb {Z}}{\mathbb {Q}}\) with \(K^2\) via the basis (1, 0) and \((0, \lambda )\). Then \(\mathrm{Aut}(A)\) acts faithfully on this 2dimensional Kvector space by postcomposition. On the other hand, \(\mathrm{GL}_2(K)\) acts on \({\mathbb {P}}^1(K)\) by fractional linear transformation.
Lemma 3.1
The orbits of \({\mathbb {P}}^1(K)\) under the action of \(\mathrm{Aut}(A)\subset \mathrm{GL}_2(K)\) are in bijection with the \(\mathrm{Aut}(A)\)orbits of elliptic curves contained in A.
Proof
Every elliptic curve on \(A = E \times E'\) is isogenous to E, so is the image of some nonzero map \(E \rightarrow A\). Two maps \(a,b: E \rightarrow A\) have the same image if and only if there are nonzero \(x, y \in \mathrm{End}(E)\) with \(ax = by\). In other words, a and b have the same image if and only if they determine the same class in the Kprojectivization \({\mathbb {P}}_K(\mathrm{Hom}(E, A) \otimes _{\mathbb {Z}}{\mathbb {Q}}) = {\mathbb {P}}^1(K)\). So we have constructed an injective map from the set of elliptic curves on A to \({\mathbb {P}}^1(K)\), which is also surjective since every nonzero element of \(\mathrm{Hom}(E, A) \otimes _{\mathbb {Z}}{\mathbb {Q}}\) is a Kmultiple of an element of \(\mathrm{Hom}(E, A)\). This bijection is evidently compatible with the pushforward action of \(\mathrm{Aut}(A)\) on elliptic curves and the action of \(\mathrm{Aut}(A)\) on \({\mathbb {P}}^1(K)\) defined above, so the lemma follows. \(\square \)
Lemma 3.2
If \(E_1 \subset A\) is an elliptic curve, then the econductor of \(E_1\) divides f.
Proof
Proposition 3.3
Proof
The previous lemma shows that any elliptic curve \(E_{1} \subset A\) has econductor dividing f. Dualizing, we see that \(E_{2} = A/E_{1}\) is also an elliptic curve in \(\hat{A} \simeq A\), so has econductor dividing f as well. \(\square \)
Now fix integers \(f_{1}\) and \(f_{2}\) dividing f, and set \(k=\gcd (f_{1},f_{2})\), \(d=\mathrm{lcm}(f_{1},f_{2})\). Recall that an isogeny of elliptic curves is cyclic if its kernel is a cyclic group. We call a cyclic isogeny based if the kernel is equipped with a distinguished generator. By Theorem 2.4, classes in \(\mathrm{Ext}^1_{{\mathcal {O}}_f}(L_{2},L_{1})_A\) correspond to based cyclic isogenies \({\mathbb {C}}/L_{1}L_{2}\rightarrow {\mathbb {C}}/{\mathfrak {a}}\) of degree f / d. Dualizing, we find that these are equinumerous to based cyclic (f / d)isogenies \({\mathbb {C}}/{\mathfrak {a}}\rightarrow {\mathbb {C}}/L_{1}L_{2}\).
4 Computation of \(r_K(a,b,c)\)
To make Theorem 1.1 explicit, we need to compute the number \(r_K(a,b,c)\) of cyclic subgroups C of order c of a fixed CM elliptic curve E of econductor a such that E / C has econductor b. We will see that this does not depend on the choice of the elliptic curve E of econductor a. First we reduce to the case where a, b, and c are powers of a prime p:
Lemma 4.1
Proof
Next, we fix a prime p and show how to compute \(r_K(p^a,p^b,p^c)\). We will use pisogeny volcanoes [11], a tool typically used in the study of elliptic curves over finite fields. We define a graph \(G_p\), whose vertex set is the set of isomorphism classes of elliptic curves \(E/{\mathbb {C}}\) with CM by K (we omit the dependence on K from the notation). The edge set of \(G_p\) is the set of isomorphism classes of pisogenies between two such elliptic curves. For each isogeny \(\phi :E \rightarrow E'\) there is a dual isogeny \(\hat{\phi }:E' \rightarrow E\), so we identify these two edges and consider \(G_p\) as an undirected graph.

\(H_p\) is \((p+1)\)regular.

The surface \(H_{p,0}\) is a \((1 + \chi _K(p))\)regular graph on t vertices, where t is the order of the class \([{\mathfrak {p}}]\) in \(\mathrm{Pic}({\mathcal {O}}_K)\), for any prime ideal \({\mathfrak {p}}\) above p.

For \(k \ge 1\), each vertex in \(H_{p,k}\) has a unique edge leading to \(H_{p,k1}\), and this accounts for every edge not in the surface.
Definition
A walk is called nonbacktracking if \(e_{i} \ne e_{i+1}\) for all i.
Theorem 4.2
Let a, b, c be nonnegative integers. Then \(r_K(p^a, p^b,p^c)\) is the number of nonbacktracking walks of length c in \(H_p\) starting at a fixed vertex of level a and ending at a vertex of level b.
Proof
Suppose E is an elliptic curve with CM by K of econductor \(p^a\). Then isomorphism classes of pisogenies out of E are in bijection with subgroups of E of order p, since we work in characteristic 0, and since \(\mathrm{Disc}(K) < 4\). It follows that subgroups of E of order p are in bijection with length 1 walks on \(H_p\) starting at E. An isogeny of degree \(p^c\) is a composition of pisogenies, and thus corresponds to a walk along \(H_p\) of length c. Backtracking amounts to composing with the dual of the previous pisogeny, which would make the the composite isogeny divisible by p and hence not cyclic. The theorem follows. \(\square \)
Theorem 4.2 allows us to compute \(r_K(p^a,p^b,p^c)\) explicitly:
Corollary 4.3
Proof
The key point is that nonbacktracking walks on \(H_p\) are very constrained: once a nonbacktracking walk begins to descend down the volcano, it cannot reascend. Moreover, the only ‘horizontal’ edges on \(H_p\) are at the surface. It follows that a nonbacktracking walk on \(H_p\) consists of (at most) three stages in the following order: ascend up the volcano, move horizontally at the surface, and then descend down the volcano.
We give the proof of the corollary in the case \(a \le b\) and \(c > b + a + 1\), and leave the proofs of the other cases to the reader, as they are similar. We must choose a vertex in level a and count the number of nonbacktracking walks of length c to a vertex of level b. Note that \(b \ge 1\) in this case. Since \(c > a + b\), any nonbacktracking walk from level a to level b of length c on \(H_p\) must begin by ascending via the unique path to the surface. Then the walk must take \(c  b a \ge 2\) horizontal steps along the surface before descending down to level b. This is only possible, without backtracking, if p splits in K, and in that case there are exactly two ways to traverse across the ncycle at the surface. There are then \((p1)p^{b1}\) ways to descend to level b. Thus, in the split case there are a total of \(2(p1)p^{b1}\) nonbacktracking walks, whereas in the nonsplit cases there are \(0 = \chi _K(p) + \chi _K(p)\) such walks, as claimed. \(\square \)
As an example of how to compute these numbers “by eye”, we now give the proof of Corollary 1.3. This is the special case where \(A = {\mathbb {C}}^2/{\mathcal {O}}_f^2\), and hence \(\mathrm{Aut}(A) = \mathrm{GL}_2({\mathcal {O}}_f)\).
Proof of Corollary 1.3
In the general case, the formula for \(N(E,E')\) obtained by combining Theorem 1.1 and Corollary 4.3 does not seem to simplify much further.
Acknowledgements
AS thanks Andrew Snowden for a helpful conversation and Andrew Sutherland for help with the figures. AS was partially supported by NSF Grant DMS0943832.
Our definition of \(\Gamma _0(N)\) is not the traditional one, as we allow elements of determinant −1.
There are analogous formulas for the two imaginary quadratic fields with more roots of unity, but we omit these cases for simplicity.
Declarations
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