The author connects continued fractions with elliptic curves. This was first done by W. W. Adams and M. J. Razar [Proc. Lond. Math. Soc., III. Ser. 41, 481–498 (1980; Zbl 0403.14002)]. The arguments of these authors are made explicit in the present paper. Later on D. Zagier [Problems posed at the St. Andrews Colloquium, 1996, Solutions, 5th day] has found out that the integrality of a certain sequence \((B_h)_{h\in\mathbb{Z}}\) has to do with the multiples of a certain point on an elliptic curve.
Anyway, the author of this paper considers the continued fraction expansion of the square root \(Y\) of a quartic polynomial \(D\in\mathbb{F}[X]\) over a field \(\mathbb{F}\) of characteristic \(n\neq 2,3\), where \(D\) is not a square (of course) and is supposed to be monic and have zero trace. Hence, we obtain the quartic elliptic curve \(C: Y^2= D(X)\) with two points at infinity \(S\) and \(O\), where \(O\) is taken as the zero of the group law. Moreover, \(D\) may be assumed to have the form \[ D(X)= (X^2+ f)^2+ 4v(X- w). \] The continued fraction expansion of \(Y\) leads to a sequence \[ Y_h= {Y+ A+ 2e_h\over v_h(X- w_h)}\qquad (h\in \mathbb{Z}\geq 0) \] with \(A= X^2+ f\) and a fortiori to so-called “elliptic” sequences \((A_h)\), \((W_h)\) and so on, which satisfy certain identities (see D. Zagier [loc. cit.]). The results are related to two dissertations, especially to C. Swart [Elliptic curves and related sequences, Ph.D. Thesis, Royal Holloway and Bedford New College, University of London (2003)] but also to [R. Shipsey, Elliptic divisibility sequences, Ph.D. Thesis, Goldsmiths College, University of London (2000)].
The paper is somewhat difficult to read, in particular for readers not so acquainted with the details of the theory of continued fractions.
Since the publication of M. Ward [Memoir on elliptic divisibility sequences, Am. J. Math. 70, 31–74 (1948; Zbl 0035.03702)] is referred to and the characteristics 2 and 3 are excluded, we mention also the multiplication formulae in [J. W. S. Cassels, A note on the division values of \(gs(u)\), Proc. Camb. Philos. Soc. 45, 167–172 (1949; Zbl 0032.26103)] for characteristic \(\neq 2,3\), [H. G. Zimmer, Computational aspects of the theory of elliptic curves, Number Theory and Applications, R. A. Mollin (ed.), 279–324 (1989; Zbl 0702.14029)] for characteristic \(\neq 2\) and [N. Koblitz, Constructing elliptic curve cryptosystems in characteristic 2, Lect. Notes Comput. Sci. 537, 156–167 (1991; Zbl 0788.94012)].
The continued fraction expansion of quadratic irrationalities in function fields was introduced by Artin. The method of Artin can be turned into efficient algorithms [B. Weis and H. G. Zimmer, Artins Theorie der quadratischen Kongruenzfunktionenkörper und ihre Anwendung auf die Berechnung der Einheiten- und Klassengruppen, Mitt. Math. Ges. Hamb. 12, No. 2, 261–286 (1991; Zbl 0757.11046)]. We recall this citation also to supplement §6 of the paper under consideration because it contains an exposition of the theory of continued fractions in quadratic function fields.