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

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

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