A proof of ETNC => BSD can be found here: Guido Kings http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/preprints/Preprints2009/26-2009.pdf
For a generalisation of the BSD conjecture in positive characteristic to higher dimensional bases (over finite fields) see http://arxiv.org/abs/1410.5293 and http://arxiv.org/abs/1410.5294