THE HEIGHT PAIRING ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION

S. SIKSEK

Abstract. In this paper we show how to use the canonical height on elliptic curves with complex multiplication to construct a complex inner product on the curve. This can be used to check for the dependence of points on the curve over the order of complex multiplication.

The canonical height is an extremely powerful and versatile tool for the study of elliptic curves. The usefulness of the canonical height rests on two of its properties:

(1) That its difference from the logarithmic height is bounded. The logarithmic height is a function with measures the ‘size’ of the points on the curve (see [5], [6]).

(2) That the canonical height can be used to define a quadratic form which, roughly speaking, turns the elliptic curve in to a real inner product space.

We can make the second property above precise as follows: suppose that L is a number field and E is an elliptic curve defined over L. Consider the real vector space ℝ ⊗_{ℤ} E(L); it is well known that E(L)/E_{tors}(L) embeds injectively into ℝ ⊗_{ℤ} E(L) and that the canonical height ĥ : E(L) → ℝ can be extended uniquely to a positive definite quadratic form on ℝ ⊗_{ℤ} E(L), which we lazily denote by ĥ again (see [3, page 232]). Thus ĥ satisfies the following three properties:

(1) ĥ(Q) ≥ 0 for all Q ∈ ℝ ⊗_{ℤ} E(L); moreover ĥ(Q) = 0 if and only if Q = 0.

(2) ĥ(αQ) = α^{2}ĥ(Q) for all Q ∈ ℝ ⊗_{ℤ} E(L) and all real α.

(3) ĥ satisfies the parallelogram law which states that for all Q_{1}, Q_{2} in ℝ ⊗_{ℤ} E(L) then

ĥ(Q_{1} + Q_{2}) + ĥ(Q_{1} − Q_{2}) = 2ĥ(Q_{1}) + 2ĥ(Q_{2})

One can now define a real inner product on ℝ ⊗_{ℤ} E(L) as follows:

⟨Q_{1}, Q_{2}⟩ = \frac{1}{2}(ĥ(Q_{1} + Q_{2}) − ĥ(Q_{1}) − ĥ(Q_{2}))

.

Suppose now that E is an elliptic curve defined over a number field L, E having multiplication by O, an order in a complex quadratic field K. Further, suppose that E(L) is closed under the action of O, thus αP ∈ E(L) for all α ∈ O and P ∈ E(L). It follows that E(L) is an O-module and it is natural to consider the complex vector space ℂ ⊗_{O} E(L) which we denote by E(L).

1991 Mathematics Subject Classification. 11G05; Secondary 11G15.

Key words and phrases. Canonical Height, Elliptic Curves, Complex Multiplication.

This research was conducted while the author was a research assistant at the University of Kent, and funded by a grant from the EPSRC (UK).

In this paper we show how to use the canonical height on such an elliptic curve E to define a positive definite hermitian pairing on E(L), which in turn endows E(L) with the structure of a complex inner product (or unitary) space.

Theorem 1. E(L)/E_{tors}(L) embeds injectively into E(L). The canonical height extends uniquely to E(L); the map Q ↦ √(ĥ(Q)) makes E(L) into a complex normed vector space.

Proof. It is easy to see that E(L) and ℝ ⊗_{ℤ} E(L) are naturally isomorphic as vector spaces over ℝ, and we will identify them. The Theorem now almost follows from the fact that the canonical height extends to a positive definite quadratic form on the real vector space ℝ ⊗_{ℤ} E(L). We need only show that if α ∈ ℂ and Q ∈ E(L) then

ĥ(αQ) = N(α)ĥ(Q),  (1)

where N(α) = αᾱ is the norm of α as a complex number.

First we show that the function α ↦ ĥ(αQ) is continuous on ℂ for any fixed Q ∈ E(L). Fix an element ω ∈ O such that ω is non-real. Write ∥Q∥ = √ĥ(Q). Then ∥.∥ is a real norm. Suppose α, β are in ℂ. By the triangle inequality we have

−∥(α − β)Q∥ ≤ ∥βQ∥ − ∥αQ∥ ≤ ∥(α − β)Q∥

and if we write α − β = ϵ_{1} + ωϵ_{2} where ϵ_{1}, ϵ_{2} are real, then

|∥βQ∥ − ∥αQ∥| ≤ ∥(α − β)Q∥ ≤ |ϵ_{1}|∥Q∥ + |ϵ_{2}|∥ωQ∥.

If β tends to α then ϵ_{1}, ϵ_{2} will tend to 0 and so ∥βQ∥ tends to ∥αQ∥. We deduce that the function α ↦ ĥ(αQ) is continuous on ℂ.

Since K is dense in ℂ it will be sufficient to prove (1) just for α ∈ K, and so by “homogeneity” just for α ∈ O. This is simply Lemma 1 below. □

The following lemma completes the proof of the above theorem.

Lemma 1. Suppose α ∈ O and Q ∈ E(L), then

ĥ(αQ) = N(α)ĥ(Q)  (2)

where N(α) denotes the norm of α.

Proof. The proof divides into two steps:

Step 1: We prove the result (2) for α ∈ O and Q ∈ E(L). We would like to thank Professor J. Silverman for his help in proving this first step. We first recall that the map [α] : E → E given by Q ↦ αQ has degree N(α) (see [4, page 103]).

Now suppose f : E → E is a non-trivial endomorphism. We use the fact that the canonical height is defined with respect to the divisor (0). Write f^{∗}(0) = T_{1} + … + T_{m}, the T_{j} being cleary torsion points and m is the degree of f. Then

\begin{aligned}ĥ(f(Q)) & = & ĥ_{(0)}(f(Q)) \\ & = & ĥ_{(f^{∗}(0))}(Q) \\ & = & ĥ_{(T_{1})+...+(T_{m})}(Q) \\ & = & ∑_{j=1}^{m} ĥ(Q − T_{j}) \\ & = & mĥ(Q) \\ & = & deg(f)ĥ(Q). \end{aligned}  (3)

This completes the proof of the first step.

Step 2: Suppose now that α ∈ O and Q ∈ E(L); we want to prove (2). Suppose that Q_{1}, …, Q_{n} is a ℤ-basis for the free part of E(L). Then we may write Q = ∑ u_{j}Q_{j} where the u_{j} are real numbers. Thus

ĥ(αQ) = ĥ(∑ u_{j}αQ_{j}) = uH(αQ_{1}, …, αQ_{n})u^{t}  (4)

where u = (u_{1}, …, u_{n}) and H denotes the height pairing matrix (see [3, page 233]). The entries of the height pairing matrix appearing in (4) are of the form

\begin{aligned}⟨αQ_{i}, αQ_{j}⟩ & = & \frac{1}{2}(ĥ(αQ_{i} + αQ_{j}) − ĥ(αQ_{i}) − ĥ(αQ_{j})) \\ & = & \frac{1}{2}N(α)(ĥ(Q_{i} + Q_{j}) − ĥ(Q_{i}) − ĥ(Q_{j})) \\ & = & N(α)⟨Q_{i}, Q_{j}⟩ \end{aligned}  (5)

where here we have used the first step. Hence

ĥ(αQ) = N(α)uH(Q_{1}, … Q_{n})u^{t} = N(α)ĥ(Q)

This completes the proof. □

We are now ready to construct our new height pairing: define, for any Q_{1}, Q_{2} ∈ E(L)

⟨Q_{1}, Q_{2}⟩_{ℂ} = \frac{1}{4} ∑_{r=0}^{3} \frac{1}{i^{r}} ĥ(Q_{1} + i^{r}Q_{2})

Theorem 2. The above pairing makes E(L) into a complex inner product spaces.

Proof. This is the standard construction of a complex inner product from a norm which satisfies the parallelogram law (see for example [2, page 237]). □

Theorem 3. Suppose Q_{1}, …, Q_{r} ∈ E(L). Define the height pairing matrix of Q_{1}, …, Q_{r} by H_{ℂ} = (⟨Q_{j}, Q_{k}⟩_{ℂ})_{1≤j,k≤r}. Then H_{ℂ} is a hermitian matrix. Q_{1}, …, Q_{r} are independent over O if and only if det(H_{ℂ}) is non-zero.

Proof. If some non-trivial O-linear combination of the points is zero then it is straightforward to show that the rows of H_{ℂ} are dependent, and the result follows.

Conversely, suppose that the determinant of H_{ℂ} is zero. Thus the rows are dependent over ℂ and so there exists complex numbers α_{1}, …, α_{r}, not all zero, such that ∑ α_{j}⟨Q_{j}, Q_{k}⟩_{ℂ} = 0 for all k. It is easy to show that ⟨∑ α_{j}Q_{j}, ∑ α_{j}Q_{j}⟩ = 0. Hence ∑ α_{j}Q_{j} = 0 in E(ℂ). Let ω be non-real element of O; then we may write α_{j} = a_{j} + b_{j}ω where the a_{j}, b_{j} are real. Hence ∑ a_{j}Q_{j} + ∑ b_{j}ωQ_{j} = 0. Since Q_{j}, ωQ_{j} are all elements of E(L) it follows from the well-known properties of the canonical height that some non-trivial ℤ-linear combination of Q_{j}, ωQ_{j} is zero. Hence Q_{1}, …, Q_{r} are dependent over O. □

References

[1] J.W.S. Cassels, Lectures on Elliptic Curves, LMS Students Text 24, Cambridge University Press, 1991.

[2] H. G. Heuser, Functional Analysis, John Wiley, 1982.

[3] J. H. Silverman, The Arithmetic of Elliptic Curves, GTM 106, Springer-Verlag, 1986.

[4] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, Springer-Verlag, 1994.

[5] J. H. Silverman, The Difference between the Weil Height and the Canonical Height on Elliptic Curves, Math. Comp. 55 (1990), 723–743.

[6] S. Siksek, Infinite Descent on Elliptic Curves, Rocky Mountain Journal of Mathematics 25 (1995), 1501–1538.

4 S. SIKSEK

Department of Mathematics, King Khalid University, Abha, PO Box 157, Saudi Arabia