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.
The equivariant Tamagawa number conjecture generalizes the BSD conjecture. I am not sure if this is the most general conjecture available.

1$\begingroup$ Reference can be found here: en.wikipedia.org/wiki/Special_values_of_Lfunctions $\endgroup$ – Marc Palm Apr 18 '13 at 16:00

1$\begingroup$ and here: mathoverflow.net/questions/96542/… $\endgroup$ – Marc Palm Apr 18 '13 at 16:01

$\begingroup$ @ Marc Palm: Does there is a set of special Lfunctions where this conjecture apply just like the case of the Generalized Riemann hypothesis. $\endgroup$ – ChinaHong Kong Apr 18 '13 at 16:18

1$\begingroup$ Yes this class of functions are called motivic Lfunctions. Following Scholl you can attach a motive to the new form so those Lfunctions are motivic. Examples include Artin Lfunction and HasseWeil Lfunctions. $\endgroup$ – Arijit Apr 18 '13 at 16:40

$\begingroup$ @ Arijit: Thank you very much for clarification. $\endgroup$ – ChinaHong Kong Apr 18 '13 at 16:46
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

1$\begingroup$ I believe this is more special then the equivariant Tamagawa conjecture? $\endgroup$ – Marc Palm Apr 18 '13 at 16:10

1$\begingroup$ Yes you are absolutely right. The equivariant version is of course general than TMC. $\endgroup$ – Arijit Apr 18 '13 at 16:17

1$\begingroup$ More references math.caltech.edu/papers/baltimorefinal.pdf $\endgroup$ – Arijit Apr 18 '13 at 16:18
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

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