References
[BIRS 2016](BIRS 2016 - Heuristics for the growth of Mordell-Weil ranks in big extensions of number fields.pdf)
[Dokchitser's L-functions algorithm](Dokchitser - Computing special values of motivic L-functions.pdf)
[Proof of Mazur-Rubin-Stein conjecture](kim-sun-Modular symbols and modular_L_values_beta.pdf)
[Barry's talk at Bhargava conference in Toronto](Toronto 2016 - The statistical behavior of modular symbols and arithmetic conjectures.pdf)
Paper applying nonvanishing of to the problem of constructing Sha of order p times a perfect square
Kisilevsky
Hi William, It was good to see you at the workshop. I am attaching two papers:
Also in the paper labelled DFKcyclicfields, the result of Theorem 2.1 gives a value for the algebraic value of the central -value of an elliptic curve -function twisted by a character of odd prime order (called ). The analytic value of Sha (divided by the corresponding value over ) is just the product of these values over all the Galois conjugates, and so os the norm from to of this value. Since the all lie in the real subfield, this shows that this norm is a square in for all curves for which the sign , and a square if . You might want to verify this numerical in examples. This would say that for any of sign , you should find a non-square She (extra -factor) for any character of order Best, Hershy