CORRIGENDUM: ELLIPTIC CURVES AND QUADRATIC RECURRENCE SEQUENCES

Extract

The statement of the numerical values $\kappa$ and $z_0$ on page 167 of [1, Section 3] contained an error. The values that were actually used were (to nine decimal places): $$\begin{eqnarray*} \kappa &=& 2 \omega_1 -\int_1^\infty (4t^3-4t+1)^{- {1}/{2}}\,dt \\[4pt] &=&1.859\,185\,431, \\[6pt] z_0&=&2\omega_3+\int_{-1}^\infty (4t^3-4t+1)^{- {1}/{2}}\,dt \\[4pt] &=&0.204\,680\,500+1.225\,694\,691i, \end{eqnarray*}$$ these being shifted, by the periods $2\omega_1$ and $2\omega_3$ respectively, compared with the values given in [1] (with $\omega_1=1.496\,729\,323$ and $\omega_3=1.225\,694\,691i$). With $\tau_0=\tau_1 =1$ and $\sigma (z)$ denoting the sigma function $\sigma (z;g_2,g_3)$ with invariants $g_2=4$, $g_3=-1$ associated with the elliptic curve given by equation (3.2), these values of $\kappa$ and $z_0$ yield $$\begin{eqnarray*} \sigma (\kappa )&=&1.555\,836\,426, \\[4pt] A=\frac{\tau_0}{\si (z_0 )} &=& 0.112\,724\,016-0.824\,911\,686i, \\[4pt] B=\frac{\si (\ka )\si (z_0 )\,\tau_1} {\si (z_0+\ka )\,\tau_0}&=&0.215\,971\,963+0.616\,028\,193i. \end{eqnarray*}$$ and the latter three values all agree with those stated in the paper (apart from rounding down the last digit in the imaginary part of $A$).