Here means that the quotient, the left-hand side divided by the right-hand side, is an algebraic number; pK denotes Shimura's period symbol, which we will explain in section 2. In this case, pK(id, id) is a period of an elliptic curve defined over Q with complex ultiplication by K. Absolute CM-periods concern a generalization of this formula. In the main conjecture, on the right-hand side of (1.2), we have Shimura's period symbol for a general CM-field and on the left-hand side, we have the product of special values of the multiple gamma function. In this exposition, we will explain a series of conjectures systematically.
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