Periodic Continued Fractions in Elliptic Function Fields - Macquarie ...

Periodic Continued Fractions in Elliptic Function Fields 111
with κ12(t) =4t(2t − 1) 2 (3t 2 − 3t +1) 3 /(t − 1) 11 ; and
(t−1) 4
2t(2t−1)(2t2−2t+1)(3t2
6t
−3t+1) X − 4 −8t 3 +2t 2 +2t−1
2(t−1) 3
2 X + 6t4−10t 3 +8t 2 −4t+1
2(t−1) 3
(t−1) 3
2t(2t−1)(3t2
2t
−3t+1) X − 4 +2t 3 −6t 2 +4t−1
2(t−1) 3
,
− 2(3t2−3t+1) (t−1)(2t2
2t
−2t+1) X + 4 −4t 3 +6t 2 −4t+1
2(t−1) 3
,
Y12(X, t) =[X 2 + u12(t) ,
, −
−
(t−1) 6
2t(2t−1)(3t2−3t+1) 2
(2t−1)(2t
X − 2 −2t+1)
2(t−1) 3
,
(2t−1)(2t
X − 2 −2t+1)
,
2(2t−1)(3t 2 −3t+1)
(t−1) 5
2(t−1) 3
(t−1) 10
2t(2t−1) 2 (2t2−2t+1)(3t2−3t+1) 2
2t X + 4 −4t 3 +6t 2 −4t+1
2(t−1) 3
− 2(2t−1)(3t2−3t+1) 2
(t−1) 8
2t X − 4 +2t 3 −6t 2 +4t−1
2(t−1) 3
,
2(t−1) 11
2t(2t−1) 2 (3t2−3t+1) 3
6t X + 4 −10t 3 +8t 2 −4t+1
2(t−1) 3
,
2(2t−1)(3t 2 −3t+1) 2
(t−1) 7 (2t 2 −2t+1)
6t X − 4 −8t 3 +2t 2 +2t−1
2(t−1) 3
(t−1) 11
2t(2t−1) 2 (3t 2 −3t+1) 3
,
,
2
X + u12(t) ,...].
Much as before, when m =12weconsider t(t − 1)(3t 2 − 3t +1)=w 2 with
w ∈ Q, which expands to w 2 =3t 4 −6t 3 +4t 2 −t. This quartic has a rational point
(1, 0). The transformation t ↦→ −1/(t − 1) and w ↦→ w/(t − 1) 2 transforms the
equation to w 2 = t 3 +7t 2 +17t+14. Its global minimal model is y 2 = x 3 +x 2 +x,
which is curve 48A4(A) of Cremona’s tables [4]. That curve has rank 0 and its
only rational point is the torsion point (0, 0) of order 2. Hence there is no t such
that κ12(t) isasquare, except if t =0,1,which give singular curves. ⊓⊔
5 Pseudo-elliptic Integrals
Listing the fundamental unit in each of the function fields Q Ym(X, t) is mere
teratology4 (teratology: the science or study of monstrosities ... ), so we provide
only examples. Note that to compute a unit x(X)+y(X)Y one either computes
the relevant convergent xm−2(X)/ym−2(X) ofthe cited expansions or, more
elegantly, one recalls that the unit is the product of the complete quotients
(Y + Ph)/Qh for h =1,... m− 1.
The following is a list of example pseudo-elliptic integrals, see [11],
fm(z) dz
Dm(z) = logxm(z)+ym(z) Dm(z) .
In each case the reader might verify that indeed x ′ = fy and x 2 −Dy 2 is constant.
4 The truly interested reader will learn more from computing them for herself than
from studying a list — in any case, the length and complexity of such a list would
have forced me to exceed my page limit.
,