- Pollack-Weston: Paper defining theta elements for arbitrary modular forms on pages 7-8 (Sections 2.1 and 2.3)
- [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 $L(E,\chi,1)$ to the problem of constructing Sha of order p times a perfect square

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