Rigid Analytic Modular Symbols

Let p be a prime > 3 and consider the Tate algebra A := Q p z defined to be the Banach algebra of formal power series over Q p that converge on the closed unit disk B[0, 1] ⊆ C p with the supremum norm ||f || := sup z∈B[0,1] |f (z)| for f ∈ A. Equivalently, we have (0.1) A = f (z) = ∞ k=0 a k z k a k ∈ Q p and lim k→∞ a k = 0 and the norm is given by (0.2) ||f || = sup k |a k |, for f = ∞ k=0 a k z k. The pair (A, || ||) defines a Banach algebra over Q p. We let D denote the Banach space of Q p-valued continuous linear functionals on A. We will often regard the elements of A as analytic functions on Z p and the elements of D as distributions on Z p. In this spirit we will often use measure-theoretic conventions and write f (z)dµ := µ(f) for the value of a linear functional µ ∈ D on a power series f ∈ A. Let κ : Z × p −→ Z × p be a locally analytic character and consider the semigroup Σ 0 (p) := a b c d ∈ Mat(2 × 2, Z p) a ∈ Z × p , c ∈ pZ p , ad − bc = 0. We define the weight κ action of Σ 0 (p) on A by the formulas (0.3) (γ κ f)(z) := κ(a + cz) · f b + dz a + cz , for f ∈ A and γ = a b c d ∈ Σ 0 (p). A simple calculation shows that this defines a continuous action of Σ 0 (p) on the Tate algebra A. Hence by duality, we also obtain a continuous action of Σ 0 (p) on D, µ → µ| κ γ (µ ∈ D, γ ∈ Σ 0 (p)), which we call the dual weight κ action.