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Authors: Jennifer Balakrishnan, Maarten Derickx, Hershy Kisilevsky, William A. Stein, Andrew Sutherland, Jan Vonk

References

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 LL-value of an elliptic curve LL-function twisted by a character of odd prime order (called kk). The analytic value of Sha (divided by the corresponding value over Q\QQ) is just the product of these values over all the Galois conjugates, and so os the norm from Q(χk)\QQ(\chi_k) to Q\QQ of this value. Since the nE(χ)n_E(\chi) all lie in the real subfield, this shows that this norm is KK ×\times a square in Q\QQ for all curves for which the sign wE=1w_E = -1, and a square if wE=+1w_E = +1. You might want to verify this numerical in examples. This would say that for any EE of sign wE=1w_E = -1, you should find a non-square She (extra kk-factor) for any character of order k.k. Best, Hershy