Critical slope p-adic L-functions

Let g be an eigenform of weight k+2 on Γ0(p)∩Γ1(N) with p N . If g is non-critical (i.e. of slope less than k + 1), using the methods of [1, 20], 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.