Elliptic Curves and Continued Fractions

We detail the continued fraction expansion of the square root of the general monic quartic polynomial. We note that each line of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. The paper includes a detailed ’reminder exposition’ on continued fractions of quadratic irrationals in function fields. A delightful ‘essay’ [16] by Don Zagier explains why the sequence (Bh)h∈Z , defined by B−2 = 1, B−1 = 1, B0 = 1, B1 = 1, B2 = 1 and the recursion (1) Bh−2Bh+3 = Bh+2Bh−1 +Bh+1Bh , 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 an elliptic curve and it is it fairly plain how to relate the surprising integer sequences and the elliptic curves from which they arise. A brief reminder exposition on continued fractions in quadratic function fields appears as §4, starting at page 77 below. 1. 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 ] . Set (2) C : Y 2 = D(X) := (X + f) + 4v(X − w), and for brevity write A = X + f and R = v(X − w). For h = 0, 1, 2, . . . we denote the complete quotients of Y0 by (3) Yh = (Y +A+ 2eh)/vh(X − wh) , Typeset April 16, 2008 [0:24] . 2000 Mathematics Subject Classification. Primary: 11A55, 11G05; Secondary: 14H05, 14H52.