What can homotopy theory say about BSD?
Maybe use Serre style argument to get finiteness of Sha??
In work of Mitchell and maybe Dwyer, they study the homotopy type of the alg Kth spectrum of rings of integers, and expresses the answer in terms of Iwasawa theory. What happens if we replace the ROI with an elliptic curve here? The review of Snaith mentions implications of the L-Q conjecture, maybe BK has some interesting consequences here?
nLab page on Ideas on elliptic curves