A conditional construction of artin representations for real analytic siegel cusp forms of weight (2, 1)

Let F be a vector-valued real analytic Siegel cusp eigenform of weight (2, 1) with the eigenvalues −5/12 and 0 for the two generators of the center of the algebra consisting of all Sp₄ (R)-invariant differential operators on the Siegel upper half plane of degree 2. Under natural assumptions in analogy of holomorphic Siegel cusp forms, we construct a unique symplectically odd Artin representation ρ_F: G_Q − GSp₄ (C) associated to F. For this, we develop the arithmetic theory of vector-valued real analytic Siegel modular forms. Several examples which satisfy these assumptions are given by using various transfers and automorphic induction.