Henri . Cohen on Sat, 20 Oct 2012 22:50:43 +0200

 Re: New ellinit interface



A question: computing eta1,et2 is currently done via E_2 (and Legendre

relation). Even for tau in the standard fundamental domain, computing E_2(tau)
in terms of \sum_{n>0} n q^n / (1-q^n) is expensive, in O~(prec^2),
assuming quasi-linear multiplication. Can one do better ?


Yes, I believe so, using the Weierstrass sigma function, which is also
a theta function: with
q=exp(2\pi i\tau) and u=exp(2\pi i z/\om_2) we have

sigma(z;L)=(\om_2/2\pi i)exp(\eta_2 z^2/(2\om_2))eta^{-3}(\tau)*
\sum_{n\in\Z}(kronecker(-4,n)q^{n^2/8}u^{n/2}

so by choosing any reasonable value of z (om_2/2 perhaps)
it should be faster.