We know that the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and Generalized Riemann hypothesis. My question is about the existence of a similar generalisation of the Birch and SwinnertonDyer conjecture.
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The equivariant Tamagawa number conjecture generalizes the BSD conjecture. I am not sure if this is the most general conjecture available. 


The BlochKato and Beilinson's conjectures. Here is an extremely pleasant write up on that http://www.claymath.org/programs/summer_school/2009/BellaicheNotes.pdf http://www.math.jussieu.fr/~nekovar/pu/mot.pdf http://wwwmath.unimuenster.de/u/peter.schneider/publ/pap/beilin.ps 

A proof of ETNC => BSD can be found here: Guido Kings http://www.uniregensburg.de/Fakultaeten/nat_Fak_I/preprints/Preprints2009/262009.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 

