Hyperelliptic Curves, Continued Fractions and Somos Sequences

Elliptic Divisibility Sequences
Now consider the singular case, M = S : thus the expansion of Z . It
will be convenient to write e in place of d , and — in honour of Morgan
Ward — (Wh) in place of (Ah). A brief computation reveals a0(X) = A,
e1 = 0, Q1(X) := u(X − w), e2 = −A(w), sufficing — using the
recursion for the sequence (dh) — to set W1 = 1, W2 = u , leading to
W3 = −u 2 A(w), W4 = −u 4` u + 2wA(w) ´ , . . . and one then notices
that (Wh) supplies the coefficients in
Ah−2Ah+2 = W 2 2 Ah−1Ah+1 − W1W3A 2 h .
I realised much later that this was obvious and did not need any work.
The point is that the recursion leading to the Ah is independent of the
translation M . So, the coefficients above must be precisely those that
fit the singular case Ah = Wh . I twigged to this self-determining
principle after Kate Stange first showed me her work on elliptic nets;
and then realised I had already used it in work with Chris Swart.
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