Tunnel's result on the congruent number problem hinges on the fact that there are modular forms with fourier coefficients related to the values $L(E_n,1)$.
Is there an interesting function that has coefficients related to $L'(E_n,1)$ instead? (for a reasonable definition of "interesting" and "related")
This is interesting since it can potentially strengthen the solution to the congruent number problem - it might give an effective algorithm to decide if rank$(E_n)>1$ (which is a step before actually constructing the points, as asked in "constructing non-torsion points ...").
(of course BSD is assumed, along with any other interesting and related conjectures)