Journal of Lie Theory Volume 14 (2004) 555–561 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

A Splitting Criterion for Two-Dimensional Semi-Tori

Jörg Winkelmann^{∗}

Communicated by F. Knop

Abstract. We investigate conditions under which a two-dimensional complex semi-torus splits into a direct product of ℂ^{∗} and a one-dimensional compact complex torus.

1. Introduction

A semi-torus is a complex Lie group arising as a quotient of the additive group of a complex vector space V by a discrete subgroup Γ with the property that Γ generates V as complex vector space. Semi-tori without non-constant holomorphic functions are also known as Cousin groups. They have been studied by many mathematicians since the work of Cousin about a century ago [1] and continue to be the focus of ongoing research.

In this article we investigate a special aspect of two-dimensional semi-tori. Let Γ be a discrete subgroup of ℤ-rank three in (ℂ^{2}, +). Then T = ℂ^{2}/Γ is a semi-torus. Let Γ_{ℝ} denote the ℝ-span of Γ and H = Γ_{ℝ} ∩ iΓ_{ℝ}. Then for every γ ∈ Γ \ H the quotient S_{γ} = ℂγ/(ℂγ ∩ Γ) is a closed complex Lie subgroup of T which is isomorphic to ℂ^{∗}. The quotient E_{γ} = T/S_{γ} is an elliptic curve. Usually, the isomorphism class and even the isogeny class of this elliptic curve E_{γ} will depend on γ. For example, if T is the quotient of ℂ^{2} by the lattice

Γ = ⟨(\begin{matrix}1 \\ 0 \end{matrix}) , (\begin{matrix}0 \\ 1 \end{matrix}) , (\begin{matrix}τ \\ σ \end{matrix})⟩_{ℤ} ,

with τ, σ ∈ H^{+} = {z ∈ ℂ : ℑ(z) > 0}, then there is a quotient elliptic curve isomorphic to ℂ/ ⟨1, σ⟩ as well as one isomorphic to ℂ/ ⟨1, τ⟩. On the other hand, if T ≃ ℂ^{∗} ×E for some elliptic curve E, then the restriction of the projection map T → E_{γ} to {1} × E yields an isogeny between E and E_{γ}. Thus in this case all these quotient elliptic curves E_{γ} must be isogenous. In this paper we investigate to what extent this property characterizes those T which split into a direct product of ℂ^{∗} and an elliptic curve. It turns out that besides such a splitting also a certain arithmetic property may cause all the quotient elliptic curves to be isogenous.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Theorem 1.1. Let Γ be a discrete subgroup of ℤ-rank three in (ℂ^{2}, +) and T = ℂ^{2}/Γ. Assume that all elliptic curves that are images of surjective homomorphisms from T have the same isogeny class.

∗ The author wants to thank the Korea Institute for Advanced Study in Seoul. The research for this article was done during the stay of the author at this institute.

Then one of the following conditions hold:

1. T is isomorphic to a direct product of ℂ^{∗} and an elliptic curve, or

2. There is a number field k of degree 3 over ℚ such that Γ ⊂ k^{2} after some linear change of coordinates on ℂ^{2}.

Conversely, if one of these two conditions is fulfilled, then there exists an elliptic curve E_{0} such that E is isogenous to E_{0} for every surjective complex Lie group homomorphism from T to an elliptic curve E.

It should be remarked that T = ℂ^{2}/Γ must be a direct product of ℂ^{∗} and an elliptic curve if Γ ⊂ k^{2} for some quadratic number field k (see lemma 2.1).

On the other hand, if k is a cubic number field, then there do exist examples of such Γ ⊂ k^{2} such that ℂ^{2}/Γ is not isomorphic to a direct product of ℂ^{∗} and an elliptic curve (see example 2.5 below).

One might inquire what happens if one asks for isomorphisms instead of mere isogenies between all quotient elliptic curves. As it turns out this simply is too much to ask for: There are always non-isomorphic quotient elliptic curves, as we will see in the last section.

2. Proofs

Before we prove the theorem, we need to deduce some auxiliary results.

Lemma 2.1. Let k be a quadratic number field, O_{k} its ring of algebraic integers and Γ ⊂ k^{2} a subgroup of ℤ-rank three which is discrete in ℂ^{2}.

Then there exists an elliptic curve E (isogenous to ℂ/O_{k}) such that ℂ^{2}/Γ ≃ ℂ^{∗} × E.

Proof. Since Γ is discrete, k cannot be totally real. Hence k = ℚ[\sqrt{−n}] for some n ∈ ℕ. Let τ = \sqrt{−n} ∈ k. Let Γ_{ℚ} = Γ ⊗_{ℤ} ℚ. Consider H = Γ_{ℚ} ∩ τ(Γ_{ℚ}). Since H is the intersection of two ℚ-hyperplanes in k^{2}, it is clear that dim_{ℚ}(H) = 2. Moreover H is stable under scalar multiplication with elements of k. Hence H ⊗_{ℚ} ℝ is a complex line which we will call H_{ℂ}. The intersection H ∩ Γ is a lattice of ℤ-rank 2 in H_{ℂ}. Since H is k-vector space, and since O_{k} is a lattice of ℤ-rank 2, it follows that H_{ℂ}/(H ∩ Γ) is isogenous to ℂ/O_{k}. On the other hand, rank_{ℤ}(H ∩ Γ) = 2 implies that rank_{ℤ} π(Γ) = 1 where π denotes the natural projection ℂ^{2} → ℂ^{2}/H_{ℂ}. Therefore π(Γ) ≃ ℤ and Γ contains an element γ_{0} such that π(γ_{0}) generates π(Γ). It follows that

ℂ^{2}/Γ ≃ (ℂγ_{0}/ℤγ_{0}) × (H_{ℂ}/(H ∩ Γ)).

This implies the statement.

Lemma 2.2. Let k be a cubic number field. Then GL_{2}(ℚ) has exactly two orbits in ℙ_{1}(k), namely ℙ_{1}(ℚ) and its complement.

Proof. We start by the claim: P GL_{2}(ℚ) acts freely on Ω = ℙ_{1}(k) \ ℙ_{1}(ℚ). Indeed, let x ∈ k \ ℚ. Then ( \begin{matrix}a & b \\ c & d \end{matrix}) is in the isotropy at [x : 1] iff

x(cx + d) = ax + b ⟺ cx^{2} + (d − a)x − b = 0.

Now x ∈ k \ ℚ and deg(k/ℚ) = 3 imply that 1, x, x^{2} are all ℚ-linearly independent. Thus the above equation implies that c = b = 0 and a = d. This yields the claim.

Next we consider the Borel group

B = \{(\begin{matrix}a & b \\ 0 & d \end{matrix}) : a, d ∈ ℚ^{∗}, b ∈ ℚ\} .

Such an element in B maps [x : 1] to [ax + b : d] = [\frac{a}{d}x + \frac{b}{d} : 1]. Fix a generator (“primitive element”) τ for the field extension k/ℚ. Then every element of k be written uniquely as a + bτ + cτ^{2} with a, b, c ∈ ℚ. We define a map ϕ : Ω → ℙ_{1}(ℚ) as follows:

[a + bτ + cτ^{2} : 1] → [b : c].

The fibers of ϕ are precisely the B -orbits in Ω.

We continue with a second claim: There exists a cubic polynomial Q ∈ ℚ[X] such that for all λ ∈ ℚ with Q(λ) ≠ 0 there exists an element A ∈ GL_{2}(ℚ) such that [λτ + τ^{2} : 1] = A([τ : 1]).

Let τ^{3} = p_{2}τ^{2} + p_{1}τ + p_{0} (p_{i} ∈ ℚ)). Choose c = 1, d = −(λ + p_{2}), a = p_{1} − λ(λ + p_{2}) and b = p_{0}.

Then

det (\begin{matrix}a & b \\ c & d \end{matrix}) = −p_{1}(λ + p_{2}) + λ(λ + p_{2})^{2} − p_{0}

and

(λτ + τ^{2})(cτ + d) = (aτ + b).

This proves the second claim with A = ( \begin{matrix}a & b \\ c & d \end{matrix}) and Q(λ) = −p_{1}(λ + p_{2}) + λ(λ + p_{2})^{2} − p_{0}.

Now we can prove the lemma. Let G = P GL_{2}(ℚ). By the last claim, the G-orbit through [τ : 1] contains all of Ω with the possible exception of at most three B -orbits. But G acts freely on Ω and G/B ≃ ℙ_{1}(ℚ) is infinite. Hence a union of finitely many B -orbits cannot be a G-orbit. It follows that G acts transitively on Ω.

Corollary 2.3. Let k be a cubic number field, and Γ_{1}, Γ_{2} lattices in ℂ which are contained in k. Then the two elliptic curves ℂ/Γ_{1} and ℂ/Γ_{2} are isogenous.

Proof. This follows because two lattice Γ_{1}, Γ_{2} in ℂ have isogenous quotient elliptic curves if and only if there is an element A ∈ P GL_{2}(ℚ) such that the associated fractional linear transformation carries Γ_{1} to Γ_{2}.

Corollary 2.4. Let k be a cubic number field, τ a primitive element for the field extension k/ℚ and Γ a lattice in ℂ^{2} which is contained in k^{2}. Let E be an elliptic curve for which there exists a surjective holomorphic Lie group homomorphism π : ℂ^{2}/Γ → E.

Then E is isogenous to ℂ/ ⟨1, τ⟩_{ℤ}.

Proof. Such a surjective homomorphism π is induced by a linear map π̃ : ℂ^{2} → ℂ. Since Γ has ℤ-rank 3, there is a non-zero element of Γ in the kernel of π̃. Therefore there are μ ∈ ℂ^{∗} and a_{i} ∈ k such that π̃(x_{1}, x_{2}) = μ(a_{1}x_{1} + a_{2}x_{2}). It follows that

\frac{1}{μ}π̃(Γ) ⊂ k.

Thus E is isogenous to ℂ/ ⟨1, τ⟩_{ℤ} by the preceding claim.

Here we would like to mention that in the situation of the above corollary in general ℂ^{2}/Γ is not a direct product:

Example 2.5. Let τ = ω\sqrt[3]{2}, where ω is a primitive third root of unity, say ω = −\frac{1}{2} +\sqrt{\frac{3}{4}}i. Let k = ℚ[τ]. Now τ^{3} = 2. Thus k is a cubic number field. Then

Γ = ⟨(\begin{matrix}1 \\ 0 \end{matrix}) , (\begin{matrix}0 \\ 1 \end{matrix}) , (\begin{matrix}τ \\ τ^{2} \end{matrix})⟩_{ℤ}

is discrete ℤ-module of rank three in ℂ^{2} with Γ ⊂ k^{2}. Let L = Γ ⊗ ℝ ∩ iΓ ⊗ ℝ. Then L is a complex line in ℂ^{2}, and ℂ^{2}/Γ is a direct product of ℂ^{∗} and an elliptic curve precisely if Γ ∩ L is cocompact (i.e. of ℤ-rank two) in L. However,

L = ⟨(\begin{matrix}ℑ(τ) \\ ℑ(τ^{2}) \end{matrix})⟩_{ℂ} = ⟨(\begin{matrix}^{3} & \sqrt{2} \\ −\sqrt[3]{4} \end{matrix})⟩_{ℂ} = ⟨(\begin{matrix}1 \\ −\sqrt[3]{2} \end{matrix})⟩_{ℂ}

and −\sqrt[3]{2} ∉ k, because k ∩ ℝ = ℚ. This implies L ∩ Γ ⊂ L ∩ k^{2} = \{0\}. Therefore ℂ^{2}/Γ cannot be isogenous to a direct product of ℂ^{∗} and an elliptic curve.

We are now able to prove the theorem:

Proof. For R ∈ {ℚ, ℝ, ℂ} the R-module generated by Γ will be denoted by Γ_{R}. Let H = Γ_{ℝ}∩ iΓ_{ℝ}. Then for every γ ∈ Γ \ H we obtain a surjective Lie group homomorphism onto an elliptic curve as follows: Let L_{γ} denote the quotient map from ℂ^{2} to Q = ℂ^{2}/ℂ · γ. By construction the image L_{γ}(Γ) has rank at most two (because ℤ · γ ⊂ ker L_{γ}). On the other hand γ ∉ H implies L_{γ}(H) = Q and therefore L_{γ}(Γ_{ℝ}) = Q. It follows that L_{γ}(Γ) is a lattice in Q and Q/L_{γ}(Γ) is an elliptic curve.

By choosing a basis for the complex vectorspace ℂ^{2} inside Γ \ H we may assume that

Γ = ⟨(\begin{matrix}1 \\ 0 \end{matrix}) (\begin{matrix}0 \\ 1 \end{matrix}) (\begin{matrix}α \\ β \end{matrix})⟩_{ℤ}

with α, β ∈ ℂ \ ℝ.

Consider

γ = γ_{m,n,p} = n (\begin{matrix}1 \\ 0 \end{matrix}) + m (\begin{matrix}0 \\ 1 \end{matrix}) + p (\begin{matrix}α \\ β \end{matrix})

for (m, n, p) ∈ ℚ^{3}. Let I denote the ℚ vector subspace of ℚ^{3} of all (m, n, p) for which γ_{m,n,p} ∈ H. Evidently dim_{ℚ}(I) ∈ {0, 1, 2}.

Let us first deal with the case dim_{ℚ}(I) = 2. Then H ∩ Γ has ℤ-rank two and H/(H ∩ Γ) is compact. Consider the linear projection π : ℂ^{2} → ℂ^{2}/H. Then π(Γ) is a subgroup of (ℂ, +) of ℤ-rank one. Hence π(Γ) ≃ ℤ. It follows that T ≃ ℂ^{∗} × H/(H ∩ Γ).

Thus we may assume that dim_{ℚ}(I) ∈ {0, 1}.

We claim that this implies ⟨1, α⟩_{ℚ} ≠ ⟨1, β⟩_{ℚ}. Indeed, if α = xβ + y for some x, y ∈ ℚ, then

(\begin{matrix}xβ \\ β \end{matrix}) = (\begin{matrix}xβ + y \\ β \end{matrix}) − (\begin{matrix}y \\ 0 \end{matrix}) ∈ Γ_{ℚ}.

Consequently H = {(xt, t) : t ∈ ℂ} and rank_{ℤ}(H ∩ Γ) = dim_{ℚ}(I) = 2. For (m, n, p) ∈ ℚ^{3} \ {(0, 0, 0)} we define E_{m,n,p} as the ℚ-vector space

E_{m,n,p} = ⟨m + pβ, n + pα, mα − nβ⟩_{ℚ} .

One verifies easily that for all (m, n, p) ∈ ℚ^{3} \ {(0, 0, 0)} the matrix

(\begin{matrix}m & 0 & p \\ n & p & 0 \\ 0 & m & −n \end{matrix})

has rank 2 which implies that E_{m,n,p} is a ℚ-plane in V = ⟨1, α, β⟩_{ℚ}.

Thus (m, n, p) ↦ E_{m,n,p} defines a map from ℚ^{3} \ {(0, 0, 0)} to the Grassmann variety G_{ℚ} of ℚ-planes in V .

If (m, n, p) ∈ ℚ^{3} \ I we have a projection L_{m,n,p} : ℂ^{2} → ℂ^{2}/ℂγ_{m,n,p} which in coordinates can be described as

(z_{1}, z_{2}) ↦ (m + pβ)z_{1} − (n + pα)z_{2}.

In this case E_{m,n,p} can be identified with the image of Γ_{ℚ} under the projection L_{m,n,p}.

Let E be an arbitrary ℚ-plane in V . There are two possibilities: Either 1 ∈ E or 1 ∉ E. In the first case E = ⟨1, mα − nβ⟩_{ℚ} for some (m, n) ∈ ℤ^{2} \ {(0, 0)}. Then E = ψ(m, n, 0). Let us now discuss the second case. Since E is a ℚhyperplane in V , we have that both E ∩⟨1, α⟩_{ℚ} and E ∩⟨1, β⟩_{ℚ} have ℚ-dimension one. Therefore E is the direct sum of two ℚ-lines which arise as intersections of E with ⟨1, α⟩_{ℚ} resp. ⟨1, β⟩_{ℚ}. Thus

E = ⟨m + pα, n + pβ⟩_{ℚ}

for some (m, n, p) ∈ ℚ^{3} \ {(0, 0, 0)}. It follows that E = ψ(m, n, p).

In this way we have shown that the map ψ : ℚ^{3} \ {0} → G_{ℚ} is surjective. Observe that I is at most one-dimensional and that ψ(m, n, p) depends

only on [m : n : p]. Therefore we see: With at most one exception every element of G_{ℚ} is in the image of ℚ^{3} \ I under ψ. Now let ℤ′ denote the set of all k ∈ ℤ for which ⟨1, β + kα⟩_{ℚ} is contained in the image ψ(ℚ^{3} \ I). Then ℤ \ ℤ′ contains at most one element.

Now ℂ/ ⟨1, α⟩_{ℤ} is isogenous to ℂ/ ⟨1, β + kα⟩_{ℤ} for all k ∈ ℤ′.

Thus for every k ∈ ℤ′ there is a matrix

A_{k} = (\begin{matrix}a_{k} & b_{k} \\ c_{k} & d_{k} \end{matrix}) ∈ GL_{2}(ℚ)

such that A_{k}(α) = \frac{a_{k}α+b_{k}}{c_{k}α+d_{k}} = β + kα. Then A_{k}(α) − A_{n}(α) = (k − n)α for all k, n ∈ ℤ′. It follows that

(a_{k}α + b_{k})(c_{n}α + d_{n}) − (a_{n}α + b_{n})(c_{k}α + d_{k}) = (k − n)α(c_{k}α + d_{k})(c_{n}α + d_{n})

Thus for all (k, n) ∈ ℤ′ × ℤ′ there is a ℚ-polynomial P_{k,n} of degree at most two such that

α^{3}(k − n)c_{k}c_{n} = P_{k,n}(α).

Therefore either deg(α) ≤ 3 or (k − n)c_{k}c_{n} = 0 for all k, n ∈ ℤ′.

However, A_{k}(α) = β + kα combined with ⟨1, α⟩_{ℚ} ≠ ⟨1, β⟩_{ℚ} implies that c_{k} ≠ 0.

Therefore deg(α) ≤ 3.

Furthermore A_{k}(α) = β + kα implies β ∈ ℚ(α).

For this reason we may deduce that Γ ⊂ k^{2} for some number field k of degree at most three.

Finally we recall that T splits into a direct product of ℂ^{∗} and an elliptic curve if deg(k/ℚ) = 2 (lemma 2.1).

3. Isogeny vs. isomorphism

Lemma 3.1. Let z, w be complex numbers with ℑ(z), ℑ(w) > 1.

Then the elliptic curves ℂ/ ⟨1, z⟩_{ℤ} and ℂ/ ⟨1, w⟩_{ℤ} are biholomorphic if and only if z − w ∈ ℤ.

Proof. This follows easily from the well-known fact that

F = \{z ∈ ℂ : |z| > 1, ℑ(z) > 0 and |ℜ(z)| < \frac{1}{2}\}

is a fundamental domain for the P SL_{2}(ℤ)-action on the upper halfplane.

Proposition 3.2. Let Γ be a discrete subgroup of ℂ^{2} of ℤ-rank three. Then there exists non-isomorphic quotient elliptic curves.

Proof. Without loss of generality we may assume that

Γ = ⟨(\begin{matrix}1 \\ 0 \end{matrix}) , (\begin{matrix}0 \\ 1 \end{matrix}) , (\begin{matrix}α \\ τ \end{matrix})⟩_{ℤ}

with ℑ(τ) > 0 and α ∈ ℂ. For every m ∈ ℤ the group Γ contains (m, 1). The quotient by the complex line through this element is given by

(z_{1}, z_{2}) ↦ mz_{2} − z_{1}

The image of Γ in ℂ under this projection is

Λ_{m} = ⟨−1, m, mτ − α⟩_{ℤ} = ⟨1, mτ − α⟩_{ℤ} .

By the preceding lemma the quotients ℂ/Λ_{m} and ℂ/Λ_{n} are not isomorphic for integers m, n with

m > n > \frac{1 + ℑ(α)}{ℑ(τ)}

because n > \frac{1+ℑ(α)}{ℑ(τ)} is equivalent to ℑ(nτ −α) > 1 and mτ −nτ is never contained in ℤ.

Winkelmann 561

Example 3.3. Let Γ = ℤ × ℤ[i] ⊂ ℂ × ℂ. Then (z_{1}, z_{2}) ↦ z_{1} − mz_{2} maps ℂ^{2}/Γ onto the elliptic curve ℂ/ ⟨1, mi⟩_{ℤ}.

References

[1] Cousin, P., Sur les fonctions triplement périodiques de deux variables, Acta Math 33 (1905), 105–232.

Jörg Winkelmann Institut Elie Cartan (Mathématiques) Université Henri Poincaré Nancy 1 B.P. 239 F-54506 Vandœuvre-les-Nancy Cedex France jwinkel@member.ams.org http://www.math.unibas.ch/˜ winkel/

Received June 13, 2003 and in final form January 28, 2004