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

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Periodic Continued Fractions in Elliptic Function Fields 109
Y4(X, t) =[X 2 + 1
1
1
4 (4t − 1) , 2(X − 2 )/4t ,2(X − 2 ) , 2X2 + 1
4 (4t − 1) /4t,
2(X − 1
1
2 ) , 2(X − 2 )/4t ,2X2 + 1
4 (4t − 1) ].
Thus κ4(t) =4t. This entails that Y4(X, 1
4s2 )/s has the periodic continued fraction
expansion of period length r =3:
[ X 2 + 1
4 (s2 − 1) /s , 2(X − 1
1
2 )/s , 2s(X − 2 ) , 2X2 + 1
4 (s2 − 1) /s ].
For m =6,and t ∈ Q \{0, −1}, the surface C6(t) isgiven by
u6(t) = 1
4 (3t2 +6t− 1), v6(t) =4t(t +1), w6(t) =− 1
2 (t − 1). (12)
and its continued fraction is detailed by
[ X 2 + 1
4 (3t2 +6t− 1) , X + 1
2 (t − 1) /2t(t +1), 2 X − 1
2 (t +1) ,
1 X − 2 (t +1) /2t ,2 X + 1
2 (t − 1) /(t +1),
2 X 2 + 1
4 (3t2 +6t− 1) /4t ,...].
Thus κ6(t) =4t. Itfollows that Y6(X, s2 )/2s has the periodic continued fraction
expansion of period length r =5:
[ X 2 + 1
4 (3s4 +6s 2 − 1) /2s , X + 1
2 (s2 − 1) /s(s2 +1),
1 X − 2 (s2 +1) /s , X − 1
2 (s2 +1) /s ,
1 X + 2 (s2 − 1) /s(s2 +1), 2 X2 + 1
4 (3t2 +6t− 1) /2s ].
Finally, because this provides the last of the cases with odd period length,
the elliptic surface C8(t) :Y 2
8 (X, t) =D8(X, t) isdefined by
u8(t) =(4t 4 +4t 3 − 16t 2 +8t − 1)/4t 2 ,
v8(t) =4(t − 1)(2t − 1), w8(t) =−(2t 2 − 4t +1)/2t, (13)
and, if t ∈ Q \{0, 1
2 , 1}, then Y8(X, t) has the continued fraction expansion
[ X 2 + u8(t) , 1
2
X +(2t 2 − 4t +1)/2t /(t − 1)(2t − 1) ,
2 X − (2t 2 − 4t +1)/2t , 1
2tX − (2t − 1)/2t /(t − 1)(2t − 1) ,
2(2t − 1) X − (2t − 1)/2t /t 2 , 1
2t3X − (2t 2 − 4t +1)/2t /(t − 1)(2t − 1) 2 ,
2(2t − 1) X +(2t 2 − 4t +1)/2t /t 3 , 1
2t3X 2 + u8(t) /(t − 1)(2t − 1) 2 ,...].
Thus κ8(t) =4(t−1)(2t − 1) 2 /t3
2 2 .Itfollows that Y8 X, 1/(1 − s ) /2s(1 + s )
has a continued fraction expansion with period r =7for s ∈ Q \{0, ±1}. For
example
1
20
Y8( 1
6
1 1
X, − 3 )= 720 X4 − 898X2 + 1920X + 245761
=[(X 2 − 449)/720 , 3(X − 23)/4 , (X + 17)/60 , −(X − 15)/4 ,
−(X − 15)/4 , (X + 17)/60 , 3(X − 23)/4 , 2(X 2 − 449)/720 ].