Motives and Siegel modular forms


For a motive M over Q, we define the fundamental periods of M using invariant theory. Our definition generalizes Deligne's periods. We show that if a motive M is constructed from motives M1,M2,...,Mn by a standard algebraic operation, then the fundamental periods of M can be expressed as monomials of the fundamental periods of M1,M2,...,Mn. Applying this theory, we discuss two (hypothetical) motives attached to a Siegel modular form. We show that a Siegel modular form of degree m has at most m + 1 period invariants.