THE SPECTRAL HALO

In this thesis, we construct a function defined over the space of integral overconvergent modular forms of Andreatta, Iovita and Pilloni, taking values in the module of continuous functions from p adic integers to a suitably constructed perfection of an overconvergence region of the formal weight space. The existence of this function is related to the possibility of proving Coleman's Halo conjecture about the distribution of eigenvalues of the Hecke operator at p acting on spaces of overconvergent modular forms. In the thesis, we construct the function, and we also compute explicitly the action of the operator over those continuous functions which are in the image of the map. Moreover, in chapter 3, we also show that the space of such continuous functions admits a Mahler basis.