Coleman’s L-invariant and Families of Modular Forms

Let p be a prime > 2 and N be a positive integer with p 6 |N . Let f be a classical newform over Γ0(Np) of even weight k0 + 2 ≥ 2 and assume f is split multiplicative at p, thus ap(f) = p0 where ap(f) is the eigenvalue of the U -operator at p acting on f . Under these hypotheses, Coleman [3] defined an L-invariant L(f) which he conjectured to be equal to the higher weight Mazur-TateTeitelbaum L-invariant [16]. In this paper we will prove Coleman’s conjecture. More precisely, let X := Z/(p− 1)Z×Zp with Z embedded in X diagonally and let Lp(f,−) : X −→ Cp be the p-adic L-function attached to f as in [16]. We will prove the following theorem.