On Nov 13, 1:26 am, John Cremona <john.crem...@gmail.com> wrote: > No, you do not miss anything. As someone has already reminded us (and > I say clearly in the talk for which those are the slides), there is no > known way of proving that the 3rd (or higher) derivative of an > elliptic curve L-function is 0. Any computations of higher derivative > are always based on an assumption that some lower derivatives which > have been computed to some approximation and which look as if they are > zero, really are zero.
OK, just for posterity (the MathOverflow question points to this discussion), let's make clear here that to provide an *upper bound* on the order of vanishing (say, r), one needs to prove that *one* of L^(0)(1), L^(1)(1), ..., L^(r)(1) is nonzero, not any particular one. With proper error bounds that's doable in principle: as John points out, the error bounds used to prove that L^(r)(1) is non-zero assume that L^(0)(1)=...=L^(r-1)(1)=0, but if that assumption isn't valid then r is still a valid upper bound on the order of vanishing of L at 1, just not a sharp one. Every time that sage underreports the order of vanishing rather than not finish, run out of memory, or throw an error "too much work for me", I think we can consider it a bug. Of course *proving* that sage is underreporting the order of vanishing is hard :-). -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To post to this group, send email to sage-devel@googlegroups.com. To unsubscribe from this group, send email to sage-devel+unsubscr...@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel?hl=en.
