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I would like to know if Grothendieck published something about this conjecture?

Is there some book (or expository article) about this conjecture?

Is there any connection between this conjecture and others important conjectures (Tate, Hodge, Standadrd, Beilinson,...)?

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You'll find a detailed history of the conjecture, including a discussion of Grothendieck's original contribution, in this paper of Bost and Charles.

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You might look at Joseph Ayoub's 'Periods and the conjectures of Grothendieck and Kontsevich-Zagier.' Newsletter of the European Mathematical Society, March 2014, Issue 91 (pdf)

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