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Bulletin of the London Mathematical Society 2005 37(2):161-171; doi:10.1112/S0024609304004163
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© London Mathematical Society

Elliptic Curves and Quadratic Recurrence Sequences

A. N. W. Hone

Institute of Mathematics, Statistics & Actuarial Science, University of Kent Canterbury CT2 7NF, United Kingdom anwh{at}kent.ac.uk

Received 6 March 2003. Revision received 15 June 2004.

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).


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