Elliptic curves and continued fractions

TL;DR

This paper explores the continued fraction expansion of square roots of quartic polynomials in function fields, revealing connections to elliptic sequences and Somos relations, and providing new insights into their properties.

Contribution

It introduces new results on elliptic sequences derived from continued fractions of quartic polynomial square roots, with detailed analysis and a comprehensive exposition.

Findings

Elliptic sequences satisfy Somos relations.

Continued fraction expansion relates to divisor addition at infinity.

New properties of elliptic sequences are established.

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.