Tunnel like theorem: is there an interesting function with fourier coefficients related to $L'(E_n,1)$ instead of $L(E_n,1)$?

Question

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)

Answer 1

Yes. Bruinier and Ono have shown in their paper "Heegner divisors, L-functions and harmonic weak Maass forms" that the vanishing or nonvanishing of central derivatives of twisted L-functions like this is related to the algebraicity properties of coefficients of a certain harmonic weak Maass form. You should also look at the classic paper of Gross-Kohnen-Zagier, and the recent paper of Darmon-Tornaria "Stark-Heegner points and the Shimura correspondence", for results of a similar flavor.