THE ARCHIMEDEAN COMPONENT

OF THE CANONICAL HEIGHT

ON ELLIPTIC CURVES

Armand Brumer and Oisín McGuinness

Fordham University

June 15, 1990

§1 Introduction

Suppose that E is an elliptic curve defined over Q given by the equation

(1)  y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

where we assume that a_{i} ∈ Z. The canonical height

ĥ : E(Q) → R

is a non-degenerate quadratic form on E(Q) which may be defined globally by the limit formula

ĥ(P) = lim_{n→∞} 4^{−n}h(x(2^{n}P))

where for x = m/n ∈ Q, h(x) = log max(|m|, |n|), and for P ∈ E(Q), x(P) denotes the x-coordinate of P in the model (1) above. It is clear from this formula that ĥ(P) = 0 if and only if P is a torsion point. Recall (see [SI2]) that the most efficient way to compute the height function is to use the decomposition of the canonical height as a sum of local heights

(2)  ĥ(P) = ∑_{v} λ̂_{v}(P)

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 11G40, Secondary 11D25, 11G05, 11-04, 14K15.

Key words and phrases. Elliptic curve, Canonical Height.

where the sum is over all places v of Q, and λ̂_{v} : E(Q_{v}) − {O} → R is the local height function at v. For example, suppose E has type I_{1} reduction at all places of bad reduction. Then if d is the denominator of x = x(P), for P ∈ E(Q),

(3)  ĥ(P) = λ̂_{∞}(P) + \frac{1}{2} log |d|,

so that computing the height of P reduces to computing the archimedean component. This formula is easily seen from the explicit formulas for the local height functions in [SI2]. The assumption on E is satisfied if the discriminant Δ of E is square-free, as is true of the curves studied in [BrMcG]. It should be remarked that the local height functions are unique up to addition of a constant, and we follow the conventions of [SI2], as recalled below in section 2 for λ̂_{∞}.

In the work reported on briefly in [BrMcG], many interesting problems to do with the values of the height function arose. One question that became of interest to us is the following: where does the archimedean height function λ̂_{∞} achieve its minimum on E(R)? If one can predict this from the equation of E, then perhaps one will attain some understanding of the much more difficult problem of locating integral points on elliptic curves. (Compare [HS].) In this paper we shed some partial light on this problem, determining the exact location of the minimum for the elliptic curves over R with two components (Δ > 0), and reporting some experimental evidence on the location of the minimum in the other case.

The paper is organised as follows: In section 2 we recall several definitions of λ̂_{∞}, and compute some special values. In section 3 we determine the location of the minimum in the case of two components, while in section 4 we try to say something useful about the other case.

We would like to thank Ken Kramer for useful conversations. The work reported here was not supported by the NSF, except in that an NSF SCREMS grant paid for the Macintosh II computers on which the numerical computations were carried out.

§2 Definitions

We use the usual notation for the invariants attached to elliptic curves, see the formulaire in [Mod4] or [SI1]. The paper [SI2] is our reference for the basic properties of height functions and our notation generally is as used there. The field of definition of the elliptic curves considered in this paper is taken to be R unless otherwise stated.

According to Néron and Tate, the local height function λ̂_{∞} : E(R) − {O} → R is uniquely determined by the three properties:

(1) λ̂_{∞}(2P) = 4λ̂_{∞}(P) − log |2y + a_{1}x + a_{3}|,  P = (x, y), 2P ≠ O;

(2)  lim_{P →O} (λ̂_{∞}(P) − \frac{1}{2} log |x(P)|) exists;

and lastly

(3)  λ̂_{∞} is bounded on any open subset of E(R) − {O}.

The duplication formula

(4)  x(2P) = \frac{x^{4} − b_{4}x^{2} − 2b_{6}x − b_{8}}{(2y + a_{1}x + a_{3})^{2}}

can be used to rewrite (1) as

λ̂_{∞}(2P) − \frac{1}{2} log |x(2P)| = 4λ̂_{∞}(P) − \frac{1}{2} log |x^{4} − b_{4}x^{2} − 2b_{6}x − b_{8}|.

Define

(5)  μ(P) = λ̂_{∞}(P) − \frac{1}{2} log |x(P)|

(this is 8 times the μ(P) in [SI2]), so that

(6)  μ(2P) = 4μ(P) − \frac{1}{2} log |1 − \frac{b_{4}}{x^{2}} − \frac{2b_{6}}{x^{3}} − \frac{b_{8}}{x^{4}} | .

From this it is clear that μ(P) → 0 as P → O in E(R).

These defining properties can be used to calculate some special values of λ̂_{∞}(P). Suppose first that P_{2} is a two-division point of E, with e = x(P_{2}) in either R or in C. Then in (6) let P → P_{2}, and deduce, using the definition of μ, that

(7)  λ̂_{∞}(P_{2}) = \frac{1}{8} log |e^{4} − b_{4}e^{2} − 2b_{6}e − b_{8}| .

If b_{2} = 0 this simplifies to

(8)  λ̂_{∞}(P_{2}) = \frac{1}{4} log |3e^{2} + b_{4}/2| ,

as in this case we have the identity e^{4} − b_{4}e^{2} − 2b_{6}e − b_{8} = e^{4} − b_{4}e^{2} − 2b_{6}e − b_{8} + 2e(4e^{3} + 2b_{4}e + b_{6}) = (3e^{2} + b_{4}/2)^{2}. Note that b_{2} = 0 can always be achieved by a suitable translation.

If now P_{3} is a 3-division point, then 2P_{3} = −P_{3}, and we have x(2P_{3}) = x(−P_{3}) = x(P_{3}), so that

μ(P_{3}) = \frac{1}{6} log |1 − \frac{b_{4}}{x^{2}} − \frac{2b_{6}}{x^{3}} − \frac{b_{8}}{x^{4}} | ,

and then

(9)  λ̂_{∞}(P_{3}) = \frac{1}{6} log |\frac{x^{4} − b_{4}x^{2} − 2b_{6}x − b_{8}}{x} | .

If the model is such that x = 0 gives a 3-division point, so that b_{8} = 0 from the explicit formula for multiplication by 3, (see [SI2, equation (29)], this formula is modified: λ̂_{∞}(P_{3}) = (1/6) log |2b_{6}|.

One can similarly derive explicit formulas for the local height of any torsion point. Although our focus is on the archimedean component of the height, the exact analogues of the properties above are true for the other places of Q, with the archimedean absolute value | | replaced by the v-adic absolute value. The resulting formulas for λ̂_{v}(P_{2}) and λ̂_{v}(P_{3}) are the same formally as those we have derived. The product formula then shows that the canonical height of these torsion points is 0, as we remarked in the introduction.

The computation of λ̂_{∞}(P) for non-torsion points is best done using the method of [SI2]. A special case is that given by Tate in his letter to Serre of October 1, 1979. Suppose that there is an ϵ > 0 so that |x(P)| > ϵ for all P ∈ E(R) − {O}. Then we have

(10)  λ̂_{∞}(P) = \frac{1}{2} log |x(P)| + \frac{1}{8} ∑_{n=0}^{∞} 4^{−n} log |z(2^{n}P)|,

where t = t(P) = 1/x, z = z(P) = 1−b_{4}t^{2}−2b_{6}t^{3}−b_{8}t^{4}, w = 4t+b_{2}t+2b_{4}t^{3}+ b_{6}t^{4}, so that x(2P) = z/w and t(2P) = w/z. This series converges extremely rapidly; we refer to [SI2, Theorem 4.2] for explicit estimates. (Note that if P is torsion, then the sequence {2^{n}P } repeats, and the infinite series collapses into a finite sum of explicitly summable geometric series. The formulas (7) and (9) for λ̂_{∞}(P_{2}) and λ̂_{∞}(P_{3}) are also obtainable in this way.) It is easy to check that the series on the right side of (10) satisfies the axioms for λ̂_{∞} given at the beginning of this section, so one could regard (10) as an existence proof for the local height. If the condition that |x(P)| be bounded away from 0 is not satisfied, then the series needs to be modified. We refer to [SI2, section 2] for the details. This is the case if the base field is C, for example. Over R one can always perform a suitable translation to obtain the condition on |x|.

We have implemented this series in an MPW tool, ShowHeightArch, that given an elliptic curve and a list of x-coordinates of points on the curve, calculates the corresponding values of λ̂_{∞}. This was used in the experimental calculations described later.

There are two other viewpoints on the local height function that are of interest, the second of which we will use later. Firstly, following [GR], the height function can be described using the idea of Green’s Functions on Riemann Surfaces. (This is also the context in which the local height function appears in a corner of Superstring theory; some references are given below.) Suppose that X is a curve over C. Regard X(C) as a Riemann surface. Then for each x ∈ X(C) there exists a Green’s function g_{(x)} : X(C)−{x} → R characterised by proposition 6.1 in [GR]. One of its properties is that g_{(x)} − log |z| is real analytic near x where z is a uniformizing parameter at x. Then one defines G(x, y) = g_{(x)}(y) = g_{(y)}(x) for x ≠ y, and the height pairing is defined on relatively prime divisors of degree 0 on X by ⟨∑ m_{x}(x), ∑ m_{y}(y)⟩ = ∑ m_{x}m_{y}G(x, y). The height pairing for an elliptic curve may then be constructed explicitly by constructing a Green’s function, see section 8 of [GR]. We omit the details of passing from the height pairing to the height function.

Without using this general theory, the local height function on E(R) can be described as follows. By the theory of elliptic functions (see [LAN] for example) there is a lattice L = Zω_{1} + Zω_{2} with ω_{1} > 0 and τ = ω_{2}/ω_{1} in the upper half plane H, so that over C there is a complex analytic isomorphism

ϕ : C/L → E(C).

Furthermore ϕ(0) = O, and ϕ(z) = (x, y) where the connection with the Weierstraß ℘-function is: x + b_{2}/12 = ℘(z; L), 2y + a_{1}x + a_{3} = ℘′(z; L). There are functions ζ(z; L) and σ(z; L) such that −ζ′ = ℘, and ζ = σ′/σ. The Weierstraß sigma function is entire with simple zeros at the lattice points:

σ(z; L) = z ∏_{0≠ω∈L} (1 − \frac{z}{ω} ) e^{\frac{z}{ω} + \frac{1}{2}(\frac{z}{ω})^{2}}

so then

ζ(z; L) = \frac{1}{z} + ∑_{0≠ω∈L} (\frac{1}{z − ω} + \frac{1}{ω} + \frac{z}{ω^{2}} )

These functions are not L-periodic but almost are: there are constants η_{1} ∈ R and η_{2} satisfying the Legendre relation η_{1}ω_{2} − η_{2}ω_{1} = 2πi, and so that

ζ(z + ω_{1}; L) = ζ(z; L) + η_{1}, ζ(z + ω_{2}; L) = ζ(z; L) + η_{2}.

Define the real-linear function η(z) so that η(ω_{1}) = η_{1} and η(ω_{2}) = η_{2}. Finally we define the Klein form k(z; L) = e^{−zη(z)/2}σ(z)Δ^{1/12}, and define for z ∈ C − L:

(11)  λ(z; L) = − log |k(z; L)|.

Near z = 0 we have λ(z; L) = − log |z|+O(1), and since log |x| = −2 log |z|+ O(1) from x = ℘(z; L) + b_{2}/12, we have λ(z; L) − \frac{1}{2} log |x| bounded near 0. This is axiom 2 for the local height function. It turns out that λ and λ̂_{∞} differ by a constant:

(12)  λ(z; L) = λ̂_{∞}(ϕ(z)) − \frac{1}{12} log Δ.

In the next section we will use this expression for the height function.

§3 Two components

Suppose that E has two components over R. Then Δ > 0, and all of the 2-division points are defined over R. In terms of the complex analytic parametrisation, we have ω_{2} ∈ iR, the lattice L has a rectangular fundamental domain. The Legendre relation then shows η_{2} ∈ iR. The 2-division points are at x = e_{1} = ϕ(ω_{1}/2), e_{2} = ϕ(ω_{2}/2) and e_{3} = ϕ((ω_{1} + ω_{2})/2). In the usual picture of E(R), we have e_{1} > e_{2} > e_{3}, and the line segment [0, ω_{1}] in C is mapped by ϕ to the connected component of the identity in E(R), and the line segment [ω_{2}/2, ω_{1} + ω_{2}/2] maps to the “oval”. In this situation, numerical experiments showed unequivocally that the minimum of λ̂_{∞} on E(R) was achieved at e_{1}. And the actual minimum value was calculated in section 1.

We prove this by showing that the minimum of λ(z; L) on the line segments (0, ω_{1}/2] and on [ω_{2}/2, (ω_{1} + ω_{2})/2] occur at the 2-division points. (Note that we need only look at half of the original segments.) First a convenient normalization is to take ω_{1} = 1, ω_{2} = τ. This is possible since one easily verifies that for γ > 0 one has the homogeneity relation

λ(γz; γL) = λ(z; L).

(This was the reason for including the Δ factor in the definition of k(z; L)). Now we set, as is usual in modular function theory, q = e^{2πiτ}. The wellknown q-expansions of the elliptic functions defined above give: (1)

λ(z; 1, τ) = − log |e^{−zη(z)/2}e^{η_{1}z^{2}/2} \frac{(u^{1/2} − u^{−1/2})}{2πi} ∏_{n=1}^{∞} \frac{(1 − q^{n}u)(1 − q^{n}u^{−1})}{(1 − q^{n})^{2}} |

where we have put u = e^{2πiz}. (Compare equation (15) in [HS], and also formulas on pages 16, 73, 224 and 234 of [GSW].) Since τ = it with t > 0, we have q = e^{−2πt} and so 0 < q < 1. Along the line segment 0 < z = s ≤ 1/2, the function reduces to (throwing away the contribution of Δ, namely the factors (1 − q^{n}) that don’t vary):

(2)  − log |\frac{1}{π} sin(πs) ∏_{n=1}^{∞} (1 − 2 cos(2πs)q^{n} + q^{2n})|

Note that η(s) = η_{1}s here, so the η(z) terms cancelled. We want to maximize the quantity inside the logarithm. The maximum of sin(πs) certainly occurs at s = 1/2. And since 0 < q < 1, the maximum of each term in the infinite product occurs when s = 1/2 also. We have proved our claim for the first line segment. The argument even gives, if we are careful, a formula for the minimum height. A comparison of this formula with that in (1.7) gives a formula well known from θ-function theory. (See the appendix to chapter 8 of [GSW], for an expression of λ in terms of θ-functions.)

It remains to take care of the segment z = s + it/2, for 0 ≤ s ≤ 1/2. Here η(z) = sη_{1} + η_{2}/2. Then −zη(z)/2 + η_{1}z^{2}/2 has real part −η_{1}t^{2}/8 which does not depend on s. Here u = e^{2πis−πt}, and removing constant factors as before we want to minimize: (3)

− log |\frac{1}{π} (e^{πis−πt/2} − e^{−πis+πt/2}) ∏_{n=1}^{∞} (1 − q^{n+1/2}e^{2πis}) (1 − q^{n−1/2}e^{−2πis})|

Removing a factor of −e^{−πis}q^{−1/4} from the term in front of the infinite product, and shifting indices by 1 in the second term in the infinite product we see this becomes:

(4)  − log |q^{−1/4} \frac{1}{π} ∏_{n=0}^{∞} (1 − q^{n+1/2}e^{2πis}) (1 − q^{n+1/2}e^{−2πis})|

or

(5)  − log |q^{−1/4} \frac{1}{π} ∏_{n=0}^{∞} (1 − 2 cos(2πs)q^{n+1/2} + q^{2n+1})|

which clearly has its minimum at s = 1/2, as we needed to check. Now we know the minimum of λ̂_{∞} occurs at one of the three 2-division points. From equation (1.8) we see that the smallest of the three values of λ̂_{∞} is at x = e_{1}, as was claimed. (Oops, I think I need more details here!)

§4 One component

By contrast with the last section, here we will see that the formula for the archimedean height that comes from the complex analytic viewpoint of elliptic curves does not appear to be useful for finding the location of the minimum height in the case where E(R) has one component.

References

[Mod4] B. Birch and W. Kuyk, editors, Modular Functions of One Variable IV, volume 476 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1975.

[BrMcG] A. Brumer and O. McGuinness, The behavior of the Mordell-Weil group of elliptic curves, Bull. Amer. Math. Soc. ?? (1990), ???.

[BGZ] J. Buhler, B. H. Gross, and D. B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3, Math. Comp. 44 (1985), 473–481.

[GR]B.H. Gross, Local Heights on Curves, Arithmetic Geometry, edited by G. Cornell and

J. Silverman, Springer-Verlag, New York, 1986, pp. 327–339.

[GSW] M. Green, J. Schwarz, and E. Witten, Superstring Theory: 2 Loop amplitudes, Anomalies and Phenomenology, vol. 2, Cambridge University Press, Cambridge and New York, 1987.

[HS]M. Hindry and J. Silverman, The canonical height and integral points on elliptic

curves, Invent. Math. 93 (1988), 419–450.

[LAN] S. Lang, Elliptic Functions, Springer Verlag, New York, 1987. [SI1]J. H. Silverman, The arithmetic of elliptic curves, volume 106, Graduate Texts in

Mathematics, Springer Verlag, New York, 1986. [SI2]J. H. Silverman, Computing heights on elliptic curves, Math. Comp. 51 (1988), 339–

358. [TA]J. Tate, The arithmetic of elliptic curves., Invent. Math. 23 (1974), 179–206.

Mathematics Department, Fordham University, Bronx NY 10458, USA

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