Congruences related to Hilbert modular forms of integer and half-integer weights.

Let K be a totally real quadratic field of narrow class number 1. In this thesis, we investigate congruences between Fourier coefficients of classical modular forms and then generalise these congruences to Hilbert modular forms of parallel weight over K. Given an odd prime p, we first prove mod p congruences between Fourier coefficients of integer weight ordinary Hilbert eigenforms that are at the same level but whose weights differ by an odd multiple of (p-1). These eigenforms belong to Hida’s p-adic family of eigenforms. We are then able to lift these congruences to congruences between Fourier coefficients of half-integer weight ordinary Hilbert eigenforms that are at the same level but whose weights differ by an odd multiple of (p −1)/2 . As an example, we briefly work with a real quadratic field and see the significance of Fourier coefficients of half-integer weight Hilbert modular forms and the related congruences.