Critical slope p-adic L-functions

Let g be an eigenform of weight k+2 on Γ₀(p) ∩Γ₁(N) with p | 2224; N. If g is non-critical (that is, of slope less than k+1), using the methods of Amice-Vélu and Višik, one can attach ['Distributions p-adiques associées aux séries de Hecke', Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974), Astérisque 24-25 (Soc. Math. France, Paris, 1975) 119-131 (French)] and Višik [Mat. Sb. (N.S.) 99 (1976) 248-260], then one can attach a p-adic L-function to g which is uniquely determined by its interpolation property together with a bound on its growth. However, in the critical slope case, the corresponding growth bound is too large to uniquely determine the p-adic L-function with its standard interpolation property.In this paper, using the theory of overconvergent modular symbols, we give a natural definition of p-adic L-functions in this critical slope case. If, moreover, the modular form is not in the image of theta, then the p-adic L-function satisfies the standard interpolation property.