Abstract: For every prime p, a construction by Eichler-Shimura and Deligne attaches to a modular eigenform of weight k>1 a p-adic representation rho of the absolute Galois group of the rationals. The restriction of rho to a decomposition group at p is determined by k, by the Hecke eigenvalue a_p, and if the form is new at p, by its so-called L-invariant. When the weight is allowed to vary p-adically and a_p is nonzero, the theory of the Coleman-Mazur eigencurve describes all congruences between eigenforms and their associated representations modulo powers of p. On the other hand, congruences between eigenforms of the same weight are a mysterious phenomenon. Via explicit computations with Peter Gräf, we detect congruences between p-newforms modulo very high powers of p, whose exponents appear to be controlled by the valuation of the L-invariant of the forms involved. I will present the setting of our computations, our conjectures and some theoretical evidence for them.
