Elliptic Curves and Continued Fractions

Alfred J. van der Poorten^{1} Centre for Number Theory Research 1 Bimbil Place Killara, Sydney NSW 2071 Australia

                      alf@math.mq.edu.au

Abstract

We detail the continued fraction expansion of the square root of the general monic quartic polynomial, noting that each line of the expansion corresponds to addition of the divisor at infinity. We analyse the data yielded by the general expansion. In that way we obtain “elliptic sequences” satisfying Somos relations. I mention several new results on such sequences. The paper includes a detailed “reminder exposition” on continued fractions of quadratic irrationals in function fields.

1 Introduction

A delightful essay [18] by Don Zagier explains why the sequence (B_{h})_{h∈ℤ}, defined by B_{−2} = 1, B_{−1} = 1, B_{0} = 1, B_{1} = 1, B_{2} = 1 and the recursion

B_{h−2}B_{h+3} = B_{h+2}B_{h−1} + B_{h+1}B_{h} ,  (1)

consists only of integers. Zagier comments that “the proof comes from the theory of elliptic curves, and can be expressed either in terms of the denominators of the co-ordinates of the multiples of a particular point on a particular elliptic curve, or in terms of special values of certain Jacobi theta functions.”

In the present note I study the continued fraction expansion of the square root of a quartic polynomial, inter alia obtaining sequences generated by recursions such as (1). Here, however, it is clear that I have also constructed the co-ordinates of the shifted multiples of a point on a cubic curve and it is it fairly plain how to relate the surprising integer sequences and the curves from which they arise.

A brief reminder exposition on continued fractions in quadratic function fields appears as §6, starting at page 14 below.

It turns out that several of my results related to sequences à la (1) also appear in the recent thesis [16] of Christine Swart; for further comment see §5. Michael Somos, see [5], had inter alia asked for the inner meaning of the behaviour of the sequences (B_{h}), above, and of (C_{h}) defined by C_{h−2}C_{h+2} = C_{h−1}C_{h+1} + C_{h}^{2} and C_{−2} = 1, C_{−1} = 1, C_{0} = 1, C_{1} = 1: of the sequences 5-Somos \underline{A006720} and 4-Somos \underline{A006721}. More generally, of course, one may both vary the initial values and coefficients and generalise the “gap” to 2m or 2m + 1 by studying Somos 2m sequences, respectively Somos 2m + 1 sequences, namely sequences satisfying the respective recursions

D_{h−m}D_{h+m} = ∑_{i=1}^{m} κ_{i}D_{h−m+i}D_{h+m−i} or D_{h−m}D_{h+m+1} = ∑_{i=1}^{m} κ_{i}D_{h−m+i}D_{h+m−i+1} .

I show in passing, a footnote on page 6, that a Somos 4 is always a Somos 6, while Theorem 2 points out it is always a Somos 5. After seeing [16], I added a somewhat painful proof, Theorem 4 on page 14, that it is also always a Somos 8. For example, 4-Somos satisfies all of

C_{h−3}C_{h+3} = C_{h−1}C_{h+1} + 5C_{h}^{2} ,
C_{h−2}C_{h+3} = −C_{h−1}C_{h+2} + 5C_{h}C_{h+1} ,
C_{h−4}C_{h+4} = 25C_{h−1}C_{h+1} − 4C_{h}^{2} .

In the light of such results one can be confident, see §3.3, page 6, that in general if (A_{h}) is a Somos 4, equivalently an elliptic sequence satisfying A_{h−2}A_{h+2} = W_{2}^{2}A_{h−1}A_{h+1} − W_{1}W_{3}A_{h}^{2}, then for all m both

W_{1}W_{2}A_{h−m}A_{h+m+1} = W_{m}W_{m+1}A_{h−1}A_{h+2} − W_{m−1}W_{m+2}A_{h}A_{h+1}

and

W_{1}^{2}A_{h−m}A_{h+m} = W_{m}^{2}A_{h−1}A_{h+1} − W_{m−1}W_{m+1}A_{h}^{2} .

2 Continued Fraction Expansion of the Square Root of a Quartic

We suppose the base field 𝔽 is not of characteristic 2 because that case requires nontrivial changes throughout the exposition and not of characteristic 3 because that requires some trivial changes to parts of the exposition. We study the continued fraction expansion of a quartic polynomial D ∈ 𝔽[X]; where D is not a square and for convenience suppose D to be monic and with zero trace. Set

C : Y^{2} = D(X) := (X^{2} + f)^{2} + 4v(X − w),  (2)

and for brevity write A = X^{2} + f and R = v(X − w). Set \overline{Y} for the conjugate −Y of Y . For h = 0, 1, 2, … we denote the complete quotients of Y_{0} by

Y_{h} = (Y + A + 2e_{h})/v_{h}(X − w_{h}) ,  (3)

noting that^{2} the Y_{h} all are reduced, namely deg Y_{h} > 0 but deg \overline{Y}_{h} < 0. The upshot is that the h-th line of the continued fraction expansion of Y_{0} satisfies

\frac{Y + A + 2e_{h}}{v_{h}(X − w_{h})} = \frac{2(X + w_{h})}{v_{h}} − \frac{\overline{Y} + A + 2e_{h+1}}{v_{h}(X − w_{h})} .

Thus evident recursion formulas, see (32) at page 14, yield

f + e_{h} + e_{h+1} = −w_{h}^{2}  (4)

and −v_{h}v_{h+1}(X − w_{h})(X − w_{h+1}) = (Y + A + 2e_{h+1})(\overline{Y} + A + 2e_{h+1}). Hence

v_{h}v_{h+1}(X − w_{h})(X − w_{h+1}) = −4(X^{2} + f + e_{h+1})e_{h+1} + 4v(X − w).  (5)

Equating coefficients in (5), and then dividing by −4e_{h+1} = v_{h}v_{h+1}, we get

−4e_{h+1} = v_{h}v_{h+1} ;  X^{2}:
v/e_{h+1} = w_{h} + w_{h+1} ;  X^{1}:
f + e_{h+1} + vw/e_{h+1} = w_{h}w_{h+1} .  X^{0}:

The five displayed equations immediately above readily lead by several routes to

e_{h}e_{h+1} = v(w − w_{h}) .  (6)

For example, apply the remainder theorem to the right hand side of (5) after noting it is divisible by X − w_{h}, and recall (4).

Theorem 1. Denote the two points at infinity on the quartic curve (2) by S and O, with O the zero of its group law. The points M_{h+1} := (w_{h}, e_{h} − e_{h+1}) all lie on C. Set M_{1} = M, and M_{h+1} =: M + S_{h}. Then S_{h} = hS.

Proof. The points M_{h+1} lie on the curve C : Y^{2} = D(X) because

(e_{h} − e_{h+1})^{2} − (w_{h}^{2} + f)^{2} = (e_{h} − (e_{h+1} + w_{h}^{2} + f))((e_{h} + w_{h}^{2} + f) − e_{h+1})

= −4e_{h}e_{h+1} = 4v(w_{h} − w) .

The birational transformations

X = (V − v)/U,  Y = 2U − (X^{2} + f) ;  (7)

conversely,

2U = Y + X^{2} + f ,  2V = XY + X^{3} + fX + 2v ,  (8)

2Conditions apply. For instance we must choose the constants e_{0} and w_{0} so that X − w_{0} divides the norm (Y + A + 2e_{0})(\overline{Y} + A + 2e_{0}) of its numerator. See page 14 for more details.

Note also that I presume that the v_{h}, equivalently the e_{h}, all are nonzero. See comments in §3.3, page 6 on the “singular case”.

move the point S to (0, 0), leave O at infinity, and change the quartic model to a Weierstrass model

W : V^{2} − vV = U^{3} − fU^{2} + vwU .  (9)

Specifically, one sees that U(M_{h+1}) = −e_{h+1}, and V (M_{h+1}) = v − w_{h}e_{h+1}. We also note that U(−M_{h+1}) = −e_{h+1}, V (−M_{h+1}) = w_{h}e_{h+1}.

To check S + (M + S_{h−1}) = M + S_{h} on W it suffices for us to show that the three points (0, 0), (−e_{h}, v − w_{h−1}e_{h}), and (−e_{h+1}, w_{h}e_{h+1}) lie on a straight line. But that is (v − w_{h−1}e_{h})/e_{h} = w_{h}. So w_{h−1} + w_{h} = v/e_{h} proves the claim.

One might view the preceding discussion culminating in Theorem 1 as making explicit the argument of Adams and Razar [1].

3 Elliptic sequences

Theorem 2. Let (A_{h}) be the sequence defined by the “initial” values A_{0}, A_{1} and the recursive definition

A_{h−1}A_{h+1} = e_{h}A_{h}^{2} .  (10)

Then, given A_{0}, A_{1}, A_{2}, A_{3}, A_{4} satisfying (10), the recursive definition

A_{h−2}A_{h+2} = v^{2}A_{h−1}A_{h+1} + v^{2}(f + w^{2})A_{h}^{2}  (11)

defines the same sequence as does (10). Just so, also

A_{h−2}A_{h+3} = −v^{2}(f + w^{2})A_{h−1}A_{h+2} + v^{3}(v + 2w(f + w^{2}))A_{h}A_{h+1}  (12)

defines that sequence.

Proof. By (6) we obtain

e_{h−1}e_{h}^{2}e_{h+1} = v^{2}(w − w_{h−1})(w − w_{h}) = v^{2}(w_{h−1}w_{h} − w(w_{h−1} + w_{h}) + w^{2}) = v^{2} ((f + e_{h} + vw/e_{h}) − w · (v/e_{h}) + w^{2}) .

Thus

e_{h−1}e_{h}^{2}e_{h+1} = v^{2} (e_{h} + (f + w^{2})) .  (13)

However, A_{h−1}A_{h+1} = e_{h}A_{h}^{2} entails

A_{h−2}A_{h}A_{h−1}A_{h+1}A_{h}A_{h+2} = e_{h−1}e_{h}e_{h+1}A_{h−1}^{2}A_{h}^{2}A_{h+1}^{2} ,

and so A_{h−2}A_{h+2} = e_{h−1}e_{h}e_{h+1}A_{h−1}A_{h+1}, which is

A_{h−2}A_{h+2} = e_{h−1}e_{h}^{2}e_{h+1}A_{h}^{2} .  (14)

On multiplying (13) by A_{h}^{2} we obtain (11).

Similarly (10) yields A_{h−1}A_{h+1}A_{h}A_{h+2} = e_{h}e_{h+1}A_{h}^{2}A_{h+1}^{2}, and so

A_{h−1}A_{h+2} = e_{h}e_{h+1}A_{h}A_{h+1} .  (15)

It follows readily that

A_{h−2}A_{h+3} = e_{h−1}e_{h}^{2}e_{h+1}^{2}e_{h+2}A_{h}A_{h+1} .  (16)

Moreover, (13) implies that

e_{h−1}e_{h}^{3}e_{h+1}^{3}e_{h+2} = v^{4} (e_{h}e_{h+1} + (f + w^{2})(e_{h} + e_{h+1}) + (f + w^{2})^{2}) .

However, by (4) we know that v^{2} (e_{h} + e_{h+1} + f + w^{2}) = v^{2}(w^{2} − w_{h}^{2}). Here v(w − w_{h}) = e_{h}e_{h+1} and v(w + w_{h}) = −v(w − w_{h}) + 2vw = −e_{h}e_{h+1} + 2vw. So

e_{h−1}e_{h}^{2}e_{h+1}^{2}e_{h+2} = v^{2} (−(f + w^{2})e_{h}e_{h+1} + v^{2} + 2vw(f + w^{2})) ,  (17)

which immediately allows us to see that also (12) yields the sequence (A_{h}).

The extraordinary feature of the identities (13) and (17) is their independence of the translation M: thus of the initial data v_{0}, w_{0}, and e_{0}.

3.1 Two-sided infinite sequences

It is plain that the various definitions of the elliptic sequence (A_{h}) encourage one to think of it as bidirectional infinite. Indeed, albeit that one does feel a need to start a continued fraction expansion — so one conventionally begins it at Y_{0}, one is not stopped from thinking of the tableau listing the lines of the expansion as being two-sided infinite; note the remark at the end of § 6.1, page 14. In summary: we may and should view the various sequences (e_{h}), …, defined above, as two-sided infinite sequences.

3.2 Vanishing

If say v_{k} = 0, then line k of the continued fraction expansion of Y_{0} makes no sense both because the denominator Q_{k}(X) := v_{k}(X −w_{k}) of the complete quotient Y_{k} seems to vanish identically and because the alleged partial quotient a_{k} := 2(X + w_{k})/v_{k} blows up.

The second difficulty is real. The vanishing of v_{k} entails a partial quotient blowing up to higher degree. We deal with vanishing by refusing to look at it. We move the point of impact of the issue by dismissing most of the data we have obtained, including the continued fraction tableau, and keep only a part of the sequence (e_{h}). That makes the first difficulty moot.^{3}

Remark 3. There is no loss of generality in taking k = 0. Then, up to an otherwise irrelevant normalisation, Y_{0} = Y + A. If more than one of the v_{h} vanish then it is a simple exercise to confirm that the continued fraction expansion of Y + A necessarily is purely periodic, see the discussion at page 17. If Y_{0} does not have a periodic continued fraction expansion then there is some h_{0}, namely h_{0} = 0, so that, for all h > h_{0}, line h of the expansion of Y_{0} does make sense.

Except of course when dealing explicitly with periodicity, we suppose in the sequel that if v_{k} = 0 then k = 0; we refer to this case as the singular case.

3In any case, the first apparent difficulty is just an artifact of our notation. If, from the start, we had written Q_{h} = v_{h}X + y_{h}, as we might well have done at the cost of nasty fractions in our formulas, we would not have entertained the thought that v_{k} = 0 entails y_{k} = 0. Plainly, we must allow v_{k} = 0 yet v_{k}w_{k} ≠ 0.

Note, exercise, that v_{k}(X − w_{k}) never does vanish identically.

3.3 The singular case

We remark that in the singular case the sequence (e_{h})_{h≥1} defines antisymmetric doublesided sequences (W_{h}), that is with W_{−h} = −W_{h}, by W_{h−1}W_{h+1} = e_{h}W_{h}^{2} and so that, for all integers h, m, and n,

W_{h−m}W_{h+m}W_{n}^{2} + W_{n−h}W_{n+h}W_{m}^{2} + W_{m−n}W_{m+n}W_{h}^{2} = 0 .  (18′)

Actually, one may find it preferable to forego an insistence on antisymmetry in favour of rewriting (18′) less elegantly as

W_{h−m}W_{h+m}W_{n}^{2} = W_{h−n}W_{h+n}W_{m}^{2} − W_{m−n}W_{m+n}W_{h}^{2} ,  (18)

just for h ≥ m ≥ n. In any case, (18) seems more dramatic than it is. An easy exercise confirms that, if W_{1} = 1, (18) is equivalent to just

W_{h−m}W_{h+m} = W_{m}^{2}W_{h−1}W_{h+1} − W_{m−1}W_{m+1}W_{h}^{2}  (19)

for all integers h ≥ m. Indeed, (19) is just a special case of (18). However, given (19), obvious substitutions in (18) quickly show one may return from (19) to the apparently more general (18).

But there is a drama here. As already remarked in a near identical situation, the recurrence relation W_{h−2}W_{h+2} = W_{2}^{2}W_{h−1}W_{h+1} − W_{1}W_{3}W_{h}^{2}, and four nonzero initial values, already suffices to produce (W_{h}). Thus (19) for all m is apparently entailed by its special case m = 2.

I can show this directly^{4}, by way of new relations on the e_{h}, for m = 3. But the case m = 4 already did not seem worth the effort. Whatever, my approach gave me no hint as to how to concoct an inductive argument leading to general m. Plan B, to look it up, fared little better. In her thesis [12], Rachel Shipsey shyly refers the reader back to Morgan Ward’s opus [17]; but Ward does not comment on the matter at all, having defined his sequences by (19). Well, perhaps Ward does comment. The issue is whether (19) is coherent: do different m yield the one sequence? Ward notes that if σ is the Weierstraß σ-function then a sequence (σ(hu)/σ(u)^{h^{2}} ) satisfies (19) for all m. Whatever, a much more direct argument would be much more satisfying.^{5}

4Plainly e_{h−2}e_{h−1}^{2}e_{h}^{3}e_{h+1}^{2}e_{h+2} · e_{h} = v^{4}(e_{h−1} + (f + w^{2}))(e_{h+1} + (f + w^{2}))e_{h}^{2} . Now notice that (e_{h−1}e_{h} + e_{h}e_{h+1})e_{h} = v(w − w_{h−1} + w − w_{h})e_{h} = 2vwe_{h} − v^{2} and recall that e_{h−1}e_{h}^{2}e_{h+1} = v^{2}(e_{h} + (f + w^{2})). The upshot is a miraculous cancellation yielding

e_{h−2}e_{h−1}^{2}e_{h}^{3}e_{h+1}^{2}e_{h+2} · e_{h} = v^{4}((f + w^{2})^{2}e_{h}^{2} + v(v + 2w(f + w^{2}))e_{h})

and allowing us to divide by the auxiliary e_{h}. Thus the bottom line is

A_{h−3}A_{h+3} = v^{4}((f + w^{2})^{2}A_{h−1}A_{h+1} + v(v + 2w(f + w^{2}))A_{h}^{2}), which is A_{h−3}A_{h+3} = W_{3}^{2}A_{h−1}A_{h+1} − W_{2}W_{4}A_{h}^{2} .

5For additional remarks, and a dissatisfying proof for the case m = 4, see §5.2 on page 12.

Note Added in Proof: Christine Swart and the author have recently succeeded in applying the ideas of her thesis and of this paper to obtain a succinct inductive proof (thus, a much more satisfying direct argument)

of the conjectures hinted at in §3.3, and stated at the end of §1.

Theorem 2 shows that certainly W_{h−2}W_{h+2} = W_{2}^{2}W_{h−1}W_{h+1} − W_{1}W_{3}W_{h}^{2} for h = 1, 2, …, in which case (19) apparently follows by arguments in [17] and anti-symmetry; (18) is then just an easy exercise.

Up to multiplying the expansion by a constant, the singular case is initiated by v_{1} = 4v, w_{1} = w, e_{1} = 0, e_{2} = −(f + w^{2}). For temporary convenience set x = v/(f + w^{2}). From the original continued fraction expansion of Y + A or, better, the recursion formulas of page 3, we fairly readily obtain v_{2} = 1/x, w_{2} = w − x, e_{3} = −x(x + 2w), e_{4} = v(x^{2}(x + 2w) − v)/x^{2}(x + 2w)^{2}.

We are now free to choose, say W_{1} = 1, W_{2} = v, leading to W_{3} = −v^{2}(f + w^{2}), W_{4} = −v^{4}(v + 2w(f + w^{2})), W_{5} = −v^{6}(v(v + 2w(f + w^{2})) − (f + w^{2})^{3}), . . ..

That allows us to notice that (12) apparently is

W_{1}W_{2}A_{h−2}A_{h+3} = W_{2}W_{3}A_{h−1}A_{h+2} − W_{1}W_{4}A_{h}A_{h+1}

and that (11) of course is

A_{h−2}A_{h+2} = W_{2}^{2}A_{h−1}A_{h+1} − W_{1}W_{3}A_{h}^{2} .

Given that also A_{h−3}A_{h+3} = W_{3}^{2}A_{h−1}A_{h+1} − W_{2}W_{4}A_{h}^{2}, it is of course tempting to guess that more is true. Certainly, more is true in the special case (A_{h}) = (W_{h}), that’s the point of the discussion above. Moreover, the same “more” is true, see for example [16, Theorem 8.1.2, p. 191], for sequence translates: thus (A_{h}) = (W_{h+k}).

3.4 Elliptic divisibility sequences

Recall that in the singular case and for h = 1, 2, … the −e_{h} are in fact the U co-ordinates of the multiples hS of the point S = (0, 0) on the curve V^{2} − vV = U^{3} − fU^{2} + vwU.

Suppose we are working in the ring Z = ℤ[f, v, vw] of “integers”. If gcd(v, vw) = 1, so the exact denominator of the “rational” w is v, then our choices W_{1} = 1, W_{2} = v lead the definition W_{h−1}W_{h+1} = e_{h}W_{h}^{2} to be such that W_{h}^{2} is always the exact denominator of the “rational” e_{h}. It is this that is shown in detail by Rachel Shipsey [12]. In particular it follows that (W_{h}) is an elliptic divisibility sequence as described by Ward [17]. A convenient recent introductory reference is Chapter 10 of the book [3].

Set hS = (U_{h}/W_{h}^{2}, V_{h}/W_{h}^{3}), thus defining sequences (U_{h}), (V_{h}), and (W_{h}) of integers, with W_{h} chosen minimally. Shipsey notices, provided that indeed gcd(v, vw) = 1, that gcd(U_{h}, V_{h}) = W_{h−1} and W_{h−1}W_{h+1} = −U_{h}. Here, I have used this last fact to define the sequence (W_{h}).

Starting, in effect, from the definition (18), Ward [17] shows that with W_{0} = 0, W_{1} = 1, and W_{2}|W_{4}, the sequence (W_{h}) is a divisibility sequence; that is, if a|b then W_{a}|W_{b}. A little more is true. If also gcd(W_{3}, W_{4}) = 1 then in fact gcd(W_{a}, W_{b}) = W_{gcd(a,b)}. On the other hand, a prime dividing both W_{3} and W_{4} divides W_{h} for all h ≥ 3.

3.5 Quasi-periodicity of the continued fraction expansion

Suppose now that the sequence (W_{h}) has a zero additional to its zero W_{0} = 0. From the continued fraction expansion and, say, [8], we find that v = 0 (but w′ = vw ≠ 0 if our curve is to be elliptic) is the case of the continued fraction having quasi-period r = 1 and the divisor at infinity on the curve having torsion m = 2. Just so, f + w^{2} = 0, thus W_{3} = 0, signals r = 2 and m = 3, and x + 2w = 0, or W_{4} = 0, is r = 3 and m = 4. And so on; for more see [8]. In summary, m > 0 is minimal with W_{m} = 0 if and only if the continued fraction expansion of, say, Z = \frac{1}{2}(Y + A) has a minimal quasi-period of length r = m − 1.

Note 1. It has been suggested in my hearing that “Mathematics is the study of degeneracy”. Given that slogan, it is equivalent to W_{m} = 0 that the sequence (e_{h}) be periodic with period m. However, recall that W_{0} = 0 and W_{±1} = ±1 entails e_{0} must be infinite. Just so then, W_{m} = 0 entails e_{m} infinite and e_{m+1} = 0. It also follows, see §6, that the word e_{1} , e_{2} , ⋯, e_{m−1} is a palindrome. Note that, in effect, we define (W_{h}) by way of antisymmetry, its initial values W_{1} = 1, W_{2}, W_{1}W_{3} = e_{2}W_{2}^{2}, W_{2}W_{4} = e_{3}W_{3}^{2}, and the recurrence relation W_{h−2}W_{h+2} = W_{2}^{2}W_{h−1}W_{h+1} − W_{1}W_{3}W_{h}^{2} — plainly that relation allows us to “jump over” zeros of the sequence. Note that, in contrast, Christine Swart [16] declares her elliptic sequences as undefined beyond a 0.

As for the singular continued fraction expansion, our notation has us set Z_{1} := −Z/v(X− w). In consequence, we are committed to the line Z = A − \overline{Z} in effect functioning as the pair of lines m − 1 and m; just so then it must also be the pair of lines \overline{1} and 0. These matters are no great issue here; but they will matter in generalising the present work to hyperelliptic curves of higher genus.

The periodicity of (e_{h}) is necessary, but not at all sufficient for the periodicity of (W_{h}). Indeed, Ward [17] shows and one fairly readily confirms that precisely the periods 1, 2, 3, 4, 5, 6, 8, or 10 are possible for an integral elliptic divisibility sequence defined by (19).

4 Examples

4.1

Consider the curve C : Y^{2} = (X^{2} − 29)^{2} − 4 · 48(X + 5); here a corresponding cubic model is E : V^{2} + 48V = U^{3} + 29U^{2} + 240U. Set A = X^{2} − 29. The first several preceding and succeeding steps in the continued fraction expansion of Y_{0} = (Y + A + 16)/8(X + 3) are^{6}

\frac{Y + A + 18}{16(X + 2)/3} = \frac{X − 2}{8/3} − \frac{\overline{Y} + A + 32}{16(X + 2)/3}  line \overline{3} :
\frac{Y + A + 32}{12(X + 1)} = \frac{X − 1}{6} − \frac{\overline{Y} + A + 24}{12(X + 1)}  line \overline{2} :
\frac{Y + A + 24}{4(X + 3)} = \frac{X − 3}{2} − \frac{\overline{Y} + A + 16}{4(X + 3)}  line \overline{1} :
\frac{Y + A + 16}{8(X + 3)} = \frac{X − 3}{4} − \frac{\overline{Y} + A + 24}{8(X + 3)}  line 0 :
\frac{Y + A + 24}{6(X + 1)} = \frac{X − 1}{3} − \frac{\overline{Y} + A + 32}{6(X + 1)}  line 1 :
\frac{Y + A + 32}{32(X + 2)/3} = \frac{X − 2}{16/3} − \frac{\overline{Y} + A + 18}{32(X + 2)/3}  line 2 :
\frac{Y + A + 18}{9(3X + 10)/8} = ⋯

where elegance has suggested we write “line \overline{h}” as short for “line −h”.

The feature motivating this example is the six integral points (−2, ±7), (−1, ±4), and (−3, ±4) on C. With M_{C} = (−3, 4) and S_{C} the “other” point at infinity these are in fact the six points M_{C} + hS_{C} for h = −3, −2, −1, 0, 1, and 2.

Correspondingly, on E we have the integral points M + 2S = (−16, −32) and M − 2S = (−16, −16), M − S = (−12, −36) and M + S = (−12, −12); here M = M_{E} = (−8, −24); S = S_{E} = (0, 0). Of course E is not minimal; nor, for that matter was C. In fact the replacements X ← 2X + 1, Y ← 4Y yield

Y^{2} = (X^{2} + X − 7)^{2} − 4 · 6(X + 3) ,  (20)

correctly suggesting we need a more general treatment than that presented in the discussion above. It turns out to be enough for present purposes to replace e_{h} ← 4e_{h} obtaining

…, e_{−3} = \frac{9}{4}, e_{−2} = 4, e_{−1} = 3, e_{0} = 2, e_{1} = 3, e_{2} = 4, e_{3} = \frac{9}{4}, ….

Then A_{0} = 1, A_{1} = 1 and

A_{h−1}A_{h+1} = e_{h}A_{h}^{2}

yields the sequence …, A_{−4} = 2^{5}3^{5}, A_{−3} = 2^{5}3^{2}, A_{−2} = 2^{3}3, A_{−1} = 2, A_{0} = 1, A_{1} = 1, A_{2} = 3, A_{3} = 2^{2}3^{2}, A_{4} = 2^{2}3^{5}, . . .. Notice that we’re hit for six^{7} by increasingly high powers of primes dividing 6 appearing as factors of the A_{h}. However, we know that (12) derives from (17). With the original e_{h}s divided by 4 that yields

A_{h−2}A_{h+3} = 6^{2}A_{h−1}A_{h+2} + 6^{3}A_{h}A_{h+1} .

6Here my choice of v_{0} = 8 is arbitrary but not at random.

7My remark is guided by knowing that V^{2} + UV + 6V = U^{3} + 7U^{2} + 12U is a minimal model for E, and noticing that gcd(6, 12) = 6. Notice too that 6^{2} divides the discriminant of this model.

Remarkably, one may remove the effect of the 6 by renormalising to a sequence (B_{h}) of integers satisfying

B_{h−2}B_{h+3} = B_{h−1}B_{h+2} + B_{h}B_{h+1} .

Specifically, …, B_{−4} = 3, B_{−3} = 2, B_{−2} = 1, B_{−1} = 1, B_{0} = 1, B_{1} = 1, B_{2} = 1, B_{3} = 2, B_{4} = 3, B_{5} = 5, B_{6} = 11, B_{7} = 37, B_{8} = 83, …, and the sequence is symmetric about B = 0. Interestingly, the choice of each B_{h} as a divisor of A_{h} is forced, in the present case by the data A_{0} = A_{1} = 1 and the decision that the coefficient of B_{h}B_{h+1} be 1. Of course it is straightforward to verify that A_{h−2}A_{h+3} is always divisible by 6^{3} and A_{h−1}A_{h+2} always by 6. For a different treatment see §4.4 below.

4.2

Take v = ±1 and f + w^{2} = 1. Thus, by (13), e_{h−1}e_{h}^{2}e_{h+1} = e_{h} + 1 and so e_{0} = 1, e_{1} = 1 yields the sequence …, 2, 1, 1, 2, 3/4, 14/9, …, of values of e_{h}. As explained above, with C_{0} = 1 and C_{1} = 1, the definition C_{h−1}C_{h+1} = e_{h}C_{h}^{2} yields the symmetric sequence …, 2, 1, 1, 1, 1, 2, 3, 7, 23, 59, …, of values of C_{h} satisfying the recursion

C_{h−2}C_{h+2} = C_{h−1}C_{h+1} + C_{h}^{2} .

Set Y^{2} = A^{2} + 4v(X − w), where A = X^{2} + f. Then the recursion for (C_{h}) entails v^{2} = 1 and f + w^{2} = 1. Plainly, one can get four consecutive values 1 in a sequence (C_{h}) only by having two consecutive values 1 in the corresponding sequence (e_{h}). Thus (4) yields f + 2 = −w_{0}^{2} and then f + w^{2} = 1 entails w^{2} − w_{0}^{2} = 3. Hence w^{2} = 4 and f = −3. The identity (6) implies vw is positive.

So v = ±1, w = ±2, f = −3. Up to X ← −X, the sequence (C_{h}) is given by the curve C : Y^{2} = (X^{2} − 3)^{2} + 4(X − 2) and its points M_{C} + hS_{C}, M_{C} = (1, 0), S_{C} the “other point” at infinity; equivalently by

E : V^{2} − V = U^{3} + 3U^{2} + 2U with M = (−1, 1), S = (0, 0).

Indeed, M + S = (−1, 0), M + 2S = (−2, 1), M + 3S = (−3/4, 3/8), . . .. Note that it is impossible to have three consecutive values 1 in the sequence (e_{h}) if also v = ±1, except for trivial periodic cases, so the hoo-ha of the example at §4.1 above is in a sense unavoidable.

4.3 Remarks

The two examples get a rather woolly treatment in [15] and its preceding discussion; see [5] for context. Before seeing [16] I had also remarked that “the observation that a twist V^{2} −vV = dU^{3} −fU^{2} +vwU becomes V^{2} −dvV = U^{3} −fU^{2} +dvwU by U ← dU, V ← dV allows one to presume v = ±1. A suitable choice of e_{0}, e_{1} and A_{0}, A_{1} should now allow one to duplicate the result claimed in [15] in somewhat less brutal form.” Namely, one wishes to obtain elliptic curves yielding a sequence (A_{h}) with nominated A_{−1}, A_{0}, A_{1}, A_{2} and such that A_{h−2}A_{h+2} = κA_{h−1}A_{h+1} + λA_{h}^{2}; only the cases κ not a square are at issue. In fact, this issue is dealt with by Christine Swart at [16, p. 153ff ] in more straightforward fashion than I had in mind. In very brief, if A_{h−1}A_{h+1} = e_{h}A_{h}^{2}, then

B_{h} = κ^{\frac{1}{2} h(h+1)}A_{h} entails B_{h−1}B_{h+1} = κe_{h}B_{h}^{2} ,

and so B_{h−2}B_{h+2} = (1/κ)^{2}B_{h−1}B_{h+1} + (λ/κ^{4})B_{h}^{2}. Yet more simply, one may remark that there always is an elliptic curve defined over the base field with \sqrt{κ} adjoined; that is, over a quadratic twist — exactly as I had mooted.

4.4 Reprise

It seems appropriate to return to the example of §4.1 so as to discover the elliptic curve giving rise to (B_{h}) = (…, 1, 1, 1, 1, 1, …), given that B_{h−2}B_{h+3} = B_{h−1}B_{h+2} + B_{h}B_{h+1}. Recall we expect the squares of the integers B_{h} to be the precise denominators of the points M + hS on the minimal Weierstraß model W of the curve; here M is some point on that model and S = (0, 0).

Suppose e_{−2}, e_{−1}, e_{0}, e_{1}, e_{2} supply the five integer co-ordinates yielding B_{−2}, B_{−1}, B_{0}, B_{1}, B_{2}. Because no more than two of these e_{i} can be 1 we must have

e_{0}B_{−1}B_{1} = e_{0}B_{0}^{2} , e_{1}B_{0}B_{2} = e_{1}B_{1}^{2} , \frac{1}{2}e_{2}B_{1}B_{3} = e_{2}B_{2}^{2} ,

since of course the recursion for (B_{h}) entails B_{3} = 2. Suppose in general that

c_{h}B_{h−1}B_{h+1} = e_{h}B_{h}^{2} .

Then the identity (17),

e_{h−1}e_{h}^{2}e_{h+1}^{2}e_{h+2} = v^{2} (−(f + w^{2})e_{h}e_{h+1} + v^{2} + 2vw(f + w^{2})) ,

and B_{h−2}B_{h+3} = B_{h−1}B_{h+2} + B_{h}B_{h+1} entail

c_{h−1}c_{h}^{2}c_{h+1}^{2}c_{h+2} = −v^{2}(f + w^{2})c_{h}c_{h+1} = v^{3}(v + 2w(f + w^{2})).

Thus c_{h}c_{h+1} = kv, say, is independent of h and we have

k^{2} = −(f + w^{2}) and k^{3} + 2wk^{2} − v = 0 .

Note that if f + w^{2}, or 2w and v, are integers, also k must be an integer. Also,

e_{0}e_{1} = vk and e_{1}e_{2} = 2vk .  (21)

Remark 1. However, e_{h−1}e_{h}^{2}e_{h+1} = v^{2}e_{h} + v^{2}(f + w^{2}) implies

k^{2}B_{h−2}B_{h+2} = c_{h}B_{h−1}B_{h+1} + (f + w^{2})B_{h}^{2} ,

without the coefficients necessarily being independent^{8} of h. In particular, k^{2} = −(f + w^{2}) entails c_{0} = e_{0} = 2k^{2} and c_{1} = e_{1} = 3k^{2}.

8Both Christine Swart and Andy Hone have pointed out to me that c_{h}c_{h+1} constant, and thus both c_{2h} and c_{2h+1} constant, of course boils down to a Somos 5 corresponding to a pair of interlinked Somos 4.

On the other hand, the identity (6) now reports that k = w − w_{0} and 2k = w − w_{1}. By (5) we then have

w_{0} + w_{1} = v/e_{1} = 2w − 3k

whilst by (4) we see that f + e_{0} + e_{1} = −(w − k)^{2}, f + e_{1} + e_{2} = −(w − 2k)^{2}, so, recalling that e_{2} = 2e_{0}, also e_{0} = (2w − 3k)k. Hence

(2w − 3k)k = 2k^{2} and so 2w = 5k .  (22)

In summary, we then can quickly conclude that also

v = 6k^{3} , 4f = −29k^{2} , and 2w_{0} = 3k .  (23)

The normalisation k = −2 retrieves the continued fraction expansion in §4.1 on page 8. As shown in §7 on page 17 the corresponding minimal Weierstraß model is V^{2} + UV + 6V = U^{3} + 7U^{2} + 12U; and M = (−2, −2) is a point of order two.

5 Somos Sequences

5.1 Christine Swart’s thesis [16]

Much of the work reported by me here is sui generis with original intent to make explicit the ideas of Adams and Razar [1] and to rediscover the rational elliptic torsion surfaces (thus, the “pencils” of rational elliptic curves with, say (0, 0), a torsion point of given order m) by a new method, see [8] and its references. Eventually, I learned of Michael Somos’s sequences, see [5], and realised how they arise from my data. I had sort of heard of Christine Swart’s work from Nelson Stephens in 2003 but, fortunately as it turned out^{9}, did not have access to her thesis [16] until very recently when this paper was already essentially complete; see [9].

Christine Swart’s discussion of the interrelationship between elliptic sequences and elliptic curves is more detailed and complete than mine. Among many other things, she is careful to recognise that the formulas neither know nor care whether the given elliptic curve is in fact elliptic: thus, for example, also my quartics may have multiple zeros. If so, extra comment — mostly, quite straightforward — is required at a number of points above; but is neglected by me. Further, Christine Swart reports inter alia that Nelson Stephens (personal communication to her) had noticed, by completely different methods, identities equivalent to (13) and (11), see page 4; these are her Theorems 3.5.1 and 7.1.2 [16, p. 29 and p. 153].

5.2 Much more satisfying?

I can in fact show that an elliptic sequence (A_{h}) is also given by A_{h−4}A_{h+4} = W_{4}^{2}A_{h−1}A_{h+1}− W_{3}W_{5}A_{h}^{2}. However, I consider my argument as typical of the sort of thing that gives mathematics its bad name: and regret to have to admit that this sort of nonsense seemingly does generalise to proving that a Somos 4 also always is a three-term Somos (4 + n).

9It was fun to puzzle out not just the answers to questions, but also to attempt to guess what the questions ought to be.

Specifically, we know that

A_{h−2}A_{h+2} = W_{2}^{2}A_{h−1}A_{h+1} − W_{1}W_{3}A_{h}^{2} , by definition;  (24)
A_{h−3}A_{h+3} = W_{3}^{2}A_{h−1}A_{h+1} − W_{2}W_{4}A_{h}^{2} , see footnote page 6,  (25)

and intend to show that

A_{h−4}A_{h+4} = W_{4}^{2}A_{h−1}A_{h+1} − W_{3}W_{5}A_{h}^{2} .  (26)

Notice that, in particular, W_{1}W_{5} = W_{2}^{2}W_{2}W_{4} − W_{1}W_{3}W_{3}^{2}; so because W_{1} = 1, W_{3}^{4} = W_{2}^{3}W_{3}W_{4} − W_{3}W_{5}. Now observe that

A_{h−4}A_{h+2}A_{h−2}A_{h+4} = (W_{3}^{2}A_{h−2}A_{h} − W_{2}W_{4}A_{h−1}^{2})(W_{3}^{2}A_{h}A_{h+2} − W_{2}W_{4}A_{h+1}^{2}) .

In the product on the right, the first term is

(W_{2}^{3}W_{3}W_{4}A_{h}^{2} − W_{3}W_{5}A_{h}^{2})A_{h−2}A_{h+2}

and half of it contributes half of (26). Similarly, half the final term of the product, thus of

W_{4}^{2}A_{h−1}A_{h+1} · W_{2}^{2}A_{h−1}A_{h+1} = W_{4}^{2}A_{h−1}A_{h+1}(A_{h−2}A_{h+2} + W_{3}A_{h}^{2}) ,

provides the other half of (26). Thus it’s ugly but true that we have proved that (26) holds if and only if W_{3}W_{4} = 0 or

W_{2}^{3}A_{h−2}A_{h}^{2}A_{h+2} − W_{2}W_{3}(A_{h−2}A_{h}A_{h+1}^{2} + A_{h−1}^{2}A_{h}A_{h+2}) + W_{4}A_{h−1}A_{h}^{2}A_{h+1} = 0 .

I now compound this brutality by fiercely replacing the two occurrences of A_{h−2} by the evident relation

A_{h−2} = (W_{2}^{2}A_{h−1}A_{h+1} − W_{1}W_{3}A_{h}^{2})/A_{h+2} .

That necessitates our then multiplying by A_{h+2}. Fortunately, we can compensate for this cruelty by dividing by A_{h}. We are left with needing to show that

W_{2}^{5}A_{h−1}A_{h}A_{h+1}A_{h+2} − W_{3}^{2}W_{3}A_{h}^{3}A_{h+2} − W_{2}^{3}W_{3}A_{h−1}A_{h+1}^{3} − W_{2}W_{3}A_{h−1}^{2}A_{h+2}^{2} + W_{2}W_{3}^{2}A_{h}^{2}A_{h+1}^{2} + W_{4}A_{h−1}A_{h}A_{h+1}A_{h+2} = 0 . (27)

It’s now natural to despair, and to start looking for a Plan B. However, one might notice, on page 7, that W_{4} = −v^{4}(v + 2w(f + w^{2})); and W_{2} = v. Moreover W_{3} = −v^{2}(f + w^{2}). Thus, conveniently,

W_{2}^{5} + W_{4} = −2v^{4}w(f + w^{2}) = W_{2}^{2}W_{3} .

Hence, just as our result is trivial if W_{3}W_{4} = 0, so also it is trivial if W_{2} = 0. All this is a sign that we may not as yet have made an error. We may divide (27) by W_{2}W_{3}. Better yet, let’s also divide by A_{h}^{2}A_{h+1}^{2} by using the definitions

A_{h−1}A_{h+1} = e_{h}A_{h}^{2} , whence also A_{h−1}A_{h+2} = e_{h}e_{h+1}A_{h}A_{h+1} .

Then all that remains is a confirmation that

2vwe_{h}e_{h+1} − v^{2}(e_{h} + e_{h+1}) − e_{h}^{2}e_{h+1}^{2} − v^{2}(f + w^{2}) = 0 .  (28)

However (13), page 4, is e_{h}e_{h+1} = v(w − w_{h}), while e_{h} + e_{h+1} = −f − w_{h}^{2} is (4), page 3. Astonishingly, the claim (28) follows immediately.

Theorem 4. If (A_{h}) is a Somos 4 then it is a Somos 8 of the shape

A_{h−4}A_{h+4} = κA_{h−1}A_{h+1} + λA_{h}^{2} .

Proof. Given the argument above, it suffices to note that any Somos 4 is equivalent to an elliptic sequence.

6 Rappels

6.1 Continued fraction expansion of a quadratic irrational

Let Y = Y (X) be a quadratic irrational integral element of the field 𝔽((X^{−1})) of Laurent series

∑_{h=−d}^{∞} f_{−h}X^{−h}, some d ∈ ℤ  (29)

defined over some given base field 𝔽; that is, there are polynomials T and D defined over 𝔽 so that

Y^{2} = T (X)Y + D(X) .  (30)

Plainly, by translating Y by a polynomial if necessary, we may suppose that deg D ≥ 2 deg T + 2, with deg D = 2g + 2, say, and deg T ≤ g; then deg Y = g + 1. Recall here that a Laurent series (29) with f_{d} ≠ 0 has degree d.

Set Y_{0} = (Y + P_{0})/Q_{0} where P_{0} and Q_{0} are polynomials so that Q_{0} divides the norm (Y + P_{0})(\overline{Y} + P_{0}); notice here that an 𝔽[X]-module ⟨Q, Y + P ⟩ is an ideal in 𝔽[X, Y] if and only if Q|(Y + P)(\overline{Y} + P).

Further, suppose that deg Y_{0} > 0 and deg \overline{Y}_{0} < 0; that is, Y_{0} is reduced. Then the continued fraction expansion of Y_{0} is given by a sequence of lines, of which the h-th is

Y_{h} := (Y + P_{h})/Q_{h} = a_{h} − (\overline{Y} + P_{h+1})/Q_{h}; in brief Y_{h} = a_{h} − \overline{B}_{h} .  (31)

Here the polynomial a_{h} is a partial quotient, and the next complete quotient Y_{h+1} is the reciprocal of the preceding remainder −(\overline{Y} +P_{h+1})/Q_{h} . Plainly the sequences of polynomials (P_{h}) and (Q_{h}) are given by the recursion formulas

P_{h} + P_{h+1} + (Y + \overline{Y} ) = a_{h}Q_{h} and Y \overline{Y} + (Y + \overline{Y})P_{h+1} + P_{h+1}^{2} = −Q_{h}Q_{h+1} .  (32)

It is easy to see by induction on h that Q_{h} divides the norm (Y + P_{h})(\overline{Y} + P_{h}). We observe also that we have a conjugate expansion with h-th line

B_{h} := (Y + P_{h+1})/Q_{h} = a_{h} − (\overline{Y} + P_{h})/Q_{h} , that is, B_{h} = a_{h} − \overline{Y}_{h} .  (33)

Note that the next line of this expansion is the conjugate of the previous line of its conjugate expansion : conjugation reverses a continued fraction tableau. Because the conjugate of line 0 is the last line of its tableau we can extend the expansion forming the conjugate tableau, leading to lines h = 1, 2, …

(Y + P_{−h+1})/Q_{−h} = a_{−h} − (Y + P_{−h})/Q_{−h}; that is, B_{−h} = a_{−h} − \overline{Y}_{−h} .

Plainly the original continued fraction tableau also is two-sided infinite and our thinking of it as “starting” at Y_{0} is just convention.

6.2 Continued fractions

One writes Y_{0} = [ a_{0} , a_{1} , a_{2} , …], where formally

[ a_{0} , a_{1} , a_{2} , …, a_{h}] = a_{0} + 1/ [ a_{1} , a_{2} , …, a_{h−1} ] and [ ] = ∞ .  (34)

It follows, again by induction on h, that the definition

(\begin{matrix}a_{0} & 1 \\ 0 & 1 \end{matrix}) (\begin{matrix}a_{1} & 1 \\ 0 & 1 \end{matrix}) ⋯ (\begin{matrix}a_{h} & 1 \\ 0 & 1 \end{matrix}) =: (\begin{matrix}x_{h} & x_{h−1} \\ y_{h} & y_{h−1} \end{matrix})

entails [ a_{0} , a_{1} , a_{2} , …, a_{h}] = x_{h}/y_{h}. This provides a correspondence between the convergents x_{h}/y_{h} and certain products of 2 × 2 matrices (more precisely, between the sequences (x_{h}), (y_{h}) of continuants and those matrices). It is a useful exercise to notice that Y_{0} = [ a_{0} , a_{1} , …, a_{h} , Y_{h+1} ] implies that

Y_{h+1} = −(y_{h−1}Y − x_{h−1})/(y_{h}Y − x_{h})

and that this immediately gives

Y_{1}Y_{2} ⋯ Y_{h+1} = (−1)^{h}(x_{h} − y_{h}Y)^{−1}.  (35)

The quantity − deg(x_{h} − y_{h}Y) = deg y_{h+1} is a weighted sum giving a measure of the “distance” traversed by the continued fraction expansion to its (h + 1)-st complete quotient. Taking norms yields

(x_{h} − y_{h}Y)(x_{h} − y_{h}\overline{Y}) = (−1)^{h+1}Q_{h+1} .  (36)

6.3 Conjugation, symmetry, and periodicity

Each partial quotient a_{h} is the polynomial part of its corresponding complete quotient Y_{h}. Note, however, that the assertions above are independent of that conventional selection rule.

One readily shows that Y_{0} being reduced, to wit deg Y_{0} > 0 and deg \overline{Y}_{0} < 0, implies that each complete quotient Y_{h} is reduced. Indeed, it also follows that deg B_{h} > 0, while plainly deg \overline{B}_{h} < 0 since −\overline{B}_{h} is a remainder; so the B_{h} too are reduced. In particular a_{h}, the polynomial part of Y_{h}, is also the polynomial part of B_{h}.

Plainly, at least the first two leading terms of each polynomial P_{h} must coincide with the leading terms of Y − T. It also follows that the polynomials P_{h} and Q_{h} satisfy the bounds

deg P_{h} = g + 1 and deg Q_{h} ≤ g .  (37)

Thus, if the base field 𝔽 is finite the box principle entails the continued fraction expansion of Y_{0} is periodic. If 𝔽 is infinite, periodicity is just happenstance.

Suppose, however, that the function field 𝔽(X, Y) is exceptional in that Y_{0}, say, does have a periodic continued fraction expansion. If the continued fraction expansion of Y_{0} is periodic then, by conjugation, also the expansion of B_{0} is periodic. But conjugation reverses the order of the lines comprising a continued fraction tableau. Hence the conjugate of any preperiod is a “postperiod”, an absurd notion. It follows that, if periodic, the two conjugate expansions are purely periodic.

Denote by A the polynomial part of Y , and recall that Y + \overline{Y} = T. It happens that line 0 of the continued fraction expansion of Y + A − T is

Y + A − T = 2A − T − (\overline{Y} + A − T)  (38)

and is symmetric. In general, if the expansion of Y_{0} has a symmetry, and if the continued fraction expansion is periodic, its period must have a second symmetry^{1}. So if Y is exceptional in having a periodic continued fraction expansion then its period is of length 2s and has an additional symmetry of the first kind P_{s} = P_{s+1}, or its period is of length 2s + 1 and also has a symmetry of the second kind, Q_{s} = Q_{s+1}. Conversely, this is the point, if the expansion of Y has a second symmetry then it must be periodic as just described.

6.4 Units

It is easy to apply the Dirichlet box principle to prove that an order ℚ[ω] of a quadratic number field ℚ(ω) contains nontrivial units. Indeed, by that principle there are infinitely many pairs of integers (p, q) so that |qω−p| < 1/q, whence |p^{2}−(ω+\overline{ω})pq+ω\overline{ω}q^{2}| < (ω−\overline{ω})+ 1. It follows, again by the box principle, that there is an integer l with 0 < |l| < (ω − \overline{ω}) + 1 so that the equation p^{2} − (ω + \overline{ω})pq + ω\overline{ω}q^{2} = l has infinitely many pairs (p, q) and (p′, q′) of solutions with p ≡ p′ and q ≡ q′ (mod l). For each such distinct pair, xl = pp′ − ω\overline{ω}qq′, yl = pq′ − p′q + (ω + \overline{ω})qq′, yields (x − ωy)(x − \overline{ω}y) = 1.

In the function field case, we cannot apply the the box principle for a second time if the base field 𝔽 is infinite. So the existence of a nontrivial unit x(X)−y(y)Y (X) is exceptional. This should not be a surprise. By the definition of the notion “unit”, such a unit u(X) say, has a divisor supported only at infinity. Moreover, u is a function of the order 𝔽[X, Y], and is say of degree m, so the existence of u implies that the class containing the divisor at infinity is a torsion divisor on the Jacobian of the curve (30). The existence of such a torsion divisor is of course exceptional.

Suppose now that the function field 𝔽(X, Y) does contain a nontrivial unit u, say of norm −κ and degree m. Then deg(yY − x) = −m < − deg y, so x/y is a convergent of Y and so some Q is ±κ, say Q_{r} = κ with r odd. That is, line r of the continued fraction expansion of Y + A − T is

Y_{r} := (Y + A − T)/κ = 2A/κ − (\overline{Y} + A − T)/κ;  line r:

here we have used the fact that (Y + P_{r})/κ is reduced to deduce that necessarily P_{r} = P_{r+1} = A − T.

By conjugation of the (r + 1)-line tableau commencing with (38) we see that

Y_{2r} := Y + A − T = 2A − T − (\overline{Y} + A − T) ,  line 2r:

so that in any case if Y + A − T has a quasi-periodic continued fraction expansion then it is periodic of period twice the quasi-period. This result of Tom Berry [2] applies to arbitrary quadratic irrationals with polynomial trace. Other elements (Y + P)/Q of 𝔽(X, Y), with Q dividing the norm (Y + P)(\overline{Y} + P), may be honest-to-goodness quasi-periodic, that is, not also periodic.

1The case of period length 1 is an exception unless we count its one line as having two symmetries; alternatively unless we deem it to have period r = 2.

Further, if κ ≠ −1 then r must be odd. To see that, notice the identity

B[ Ca_{0} , Ba_{1} , Ca_{2} , Ba_{3} , …] = C[ Ba_{0} , Ca_{1} , Ba_{2} , Ca_{3} , …],  (39)

reminding one how to multiply a continued fraction expansion by some quantity; this cute formulation of the multiplication rule is due to Wolfgang Schmidt [11]. The “twisted symmetry” occasioned by division by κ, equivalent to the existence of a non-trivial quasi-period, is noted by Christian Friesen [4].

In summary, if the continued fraction expansion of Y is quasi-periodic it is periodic, and the expansion has the symmetries of the more familiar number field case, as well as twisted symmetries occasioned by a nontrivial κ.

One shows readily that if x/y = [ A , a_{1} , …, a_{r−1} ] and x − Y y is a unit of the domain 𝔽[X, Y] then, with a_{r−1} = κa_{1}, a_{r−2} = a_{2}/κ, a_{r−3} = κa_{3}, …,

[ \overline{2A − T , a_{1} , …, a_{r−1} , (2A − T)/κ , a_{r−1} , …, a_{1}} ]

is the quadratic irrational Laurent series Y + A − T. Alternatively, given the expansion of Y + A − T, and noting that therefore deg Q_{r} = 0, the fact that the said expansion of x/y yields a unit follows directly from (36).

7 Comments

7.1

According to Gauss (Disquisitiones Arithmeticæ, Art. 76) … veritates ex notionibus potius quam ex hauriri debebant^{2}. Nonetheless, one should not underrate the importance of notation; good notation can decrease the viscosity of the flow to truth. From the foregoing it seems clear that, given Y^{2} = A^{2} + 4v(X − w), one should study the continued fraction expansion of Z = \frac{1}{2}(Y + A), as is done in [1]. Moreover, it is a mistake to be frustrated by minimal models V^{2} + UV − vV = U^{3} − fU + vwU.

Specifically, we understand that V^{2} − 8vV = U^{3} − (4f − 1)U^{2} + 8v(2w − 1)U yields Y^{2} = (X^{2} + 4f − 1)^{2} + 4 · 8v(X − (2w − 1)) by way of 2U = X^{2} + Y + (4f − 1) and (V − 8v) = XU. Now X ← 2X + 1, Y ← 4Y means that, instead, we obtain Y^{2} = (X^{2} + X + f)^{2} + 4v(X − (w − 1)). This derives from V^{2} + UV − vV = U^{3} − fU + vwU by taking 2U = X^{2} + X + Y + f and V − v = XU.

7.2

The discussion above may have some interest for its own sake, but my primary purpose is to test ideas for generalisation to higher genus g. An important difficulty when g > 1 is that partial quotients may be of degree greater than one without that entailing periodicity, whence my eccentric aside on page 8. Happily, the generalisation to translating by a point (w_{0}, e_{0}−e_{1}) on the quartic model effected above also is a simplification in that one surely may always choose a translating divisor so as to avoid meeting singular steps in the continued fraction expansion. In that context one finds that the sequence (…, 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 8, 17, 50, …) satisfying the recursion T_{h−3}T_{h+3} = T_{h−2}T_{h+2} + T_{h}^{2} arises from adding multiples of the class of the divisor at infinity on the Jacobian of the curve Y^{2} = (X^{3} − 4X + 1)^{2} + 4(X − 2) of genus 2 to the class of the divisor defined by the pair of points (φ, 0) and (\overline{φ}, 0); here φ is the golden ratio.

2[mathematical] truths flow from notions rather than from notations.

References

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[2] T. G. Berry, On periodicity of continued fractions in hyperelliptic function fields, Arch. Math. 55 (1990), 259–266.

[3] Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Mathematical Surveys and Monographs 104, American Mathematical Society 2003, 318pp.

[4] Christian Friesen, Continued fraction characterization and generic ideals, in [6], 465– 474.

[5] David Gale, The strange and surprising saga of the Somos sequences, Mathematical Intelligencer 13.1 (1991), 40–42; and ‘Somos sequence update’, ibid. 13.4 (1991), 49– 50. For more see Jim Propp, ‘The Somos Sequence Site’, http://www.math.wisc.edu/

~propp/somos.html.

[6] David Goss, David R. Hayes and Michael I. Rosen eds., The Arithmetic of Function Fields, Proceedings of the workshop held at The Ohio State University, Columbus, Ohio, June 17–26, 1991. Ohio State University Mathematical Research Institute Publications,

2. Walter de Gruyter & Co., Berlin, 1992. viii+482 pp.

[7] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186.

[8] Alfred J. van der Poorten, Periodic continued fractions and elliptic curves, in High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Alf van der Poorten and Andreas Stein eds., Fields Institute Communications 42, American Mathematical Society, 2004, 353–365.

[9] , Elliptic curves and continued fractions, v0.9 of this manuscript: http://www.

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[10] Alfred J. van der Poorten and Xuan Chuong Tran, Quasi-elliptic integrals and periodic continued fractions, Monatshefte Math., 131 (2000), 155-169.

[11] Wolfgang M, Schmidt, On continued fractions and diophantine approximation in power series fields, Acta Arith. 95 (2000), 139–166.

[12] Rachel Shipsey, Elliptic divisibility sequences, Phd Thesis, Goldsmiths College, Univer-

    sity of London, 2000 (see http://homepages.gold.ac.uk/rachel/).

[13] Rachel Shipsey, talk at “The Mathematics of Public Key Cryptography”, Toronto 1999;

    see http://homepages.gold.ac.uk/rachel/toronto/sld006.htm.

[14] N. J. A. Sloane, The on-line encyclopedia of integer sequences, http://www.research.

    att.com/~njas/sequences/.

[15] David Speyer, E-mail concluding

    http://www.math.wisc.edu/~propp/somos/elliptic.

[16] Christine Swart, Elliptic curves and related sequences, PhD Thesis, Royal Holloway and Bedford New College, University of London, 2003; 226pp.

[17] Morgan Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31–74.

[18] Don Zagier, Problems posed at the St Andrews Colloquium, 1996, Solutions, 5th day;

    see http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Problems.html.

2000 Mathematics Subject Classification: Primary 11A55, 11G05; Secondary 14H05, 14H52. Keywords: continued fraction expansion, function field of characteristic zero, elliptic curve, Somos sequence.

(Concerned with sequences \underline{A006720} and \underline{A006721}.)

Received September 20, 2004; revised version received November 18 2004. Published in Journal of Integer Sequences, May 16 2005.

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