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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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up vote 3 down vote accepted

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: – Marc Palm Apr 18 '13 at 16:00
and here:… – 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. – China-Hong Kong Apr 18 '13 at 16:18
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. – China-Hong Kong Apr 18 '13 at 16:46

The Bloch-Kato and Beilinson's conjectures. Here is an extremely pleasant write up on that

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

A proof of ETNC => BSD can be found here: Guido Kings

For a generalisation of the BSD conjecture in positive characteristic to higher dimensional bases (over finite fields) see and

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@ Timo Keller: Received many thanks. – China-Hong Kong Oct 7 '14 at 17:47

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