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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 Swinnerton-Dyer 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.

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Reference can be found here: en.wikipedia.org/wiki/Special_values_of_L-functions –  Marc Palm Apr 18 '13 at 16:00
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and here: mathoverflow.net/questions/96542/… –  Marc Palm Apr 18 '13 at 16:01
    
@ Marc Palm: Does there is a set of special L-functions where this conjecture apply just like the case of the Generalized Riemann hypothesis. –  Shpigle Apr 18 '13 at 16:18
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Yes this class of functions are called motivic L-functions. Following Scholl you can attach a motive to the new form so those L-functions are motivic. Examples include Artin L-function and Hasse-Weil L-functions. –  Arijit Apr 18 '13 at 16:40
    
@ Arijit: Thank you very much for clarification. –  Shpigle Apr 18 '13 at 16:46

The Bloch-Kato 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.uni-muenster.de/u/peter.schneider/publ/pap/beilin.ps

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I believe this is more special then the equivariant Tamagawa conjecture? –  Marc Palm Apr 18 '13 at 16:10
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Yes you are absolutely right. The equivariant version is of course general than TMC. –  Arijit Apr 18 '13 at 16:17
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More references math.caltech.edu/papers/baltimore-final.pdf –  Arijit Apr 18 '13 at 16:18

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

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@ Timo Keller: Received many thanks. –  Shpigle Oct 7 at 17:47

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