Deformations of Galois Representations and the Theorems of Sato-Tate, Lang-Trotter and others

We construct infinitely ramified Galois representations $\rho$ such that the $a_l(\rho )$'s have distributions in contrast to the statements of Sato-Tate, Lang-Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalising a result of Ramakrishna.