Elliptic curves and quadratic recurrence sequences. (English) Zbl 1166.11333

Hone, A. N. W.

Bull. Lond. Math. Soc. 37, No. 2, 161-171 (2005).

Summary: 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.

MSC:

11G05	Elliptic curves over global fields
11B83	Special sequences and polynomials
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K20	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions

Software:

OEIS