ELLIPTIC CURVES AND QUADRATIC RECURRENCE SEQUENCES

Abstract

The explicit solution of a general three-term bilinear recurrence relation of fourth order is constructed here in terms of the Weierstrass sigma function. The construction of the elliptic curve associated to the Somos 4 sequence is presented as an example. An interpretation via the theory of integrable systems is provided, leading to a conjecture relating certain higher-order recurrences with hyperelliptic curves of higher genus. 2000 Mathematics Subject Classification 11B37 (primary), 33E05, 37J35 (secondary).