The Arithmetic of the Fourier Coefficients of Automorphic Forms

Abstract

This thesis studies modular forms from a classical and adelic viewpoint. We use this interplay to obtain results about the arithmetic of the Fourier coefficients of modular forms and their generalisations. In Chapter 2, we compute lower bounds for the $p$-adic valuation of local Whittaker newforms with non-trivial central character. We obtain these bounds by using the local Fourier analysis of these local Whittaker newforms and the $p$-adic properties of $\eps$-factors for $\GL_1$. In Chapter 3, we study the fields generated by the Fourier coefficients of Hilbert newforms at arbitrary cusps. Precisely, given a cuspidal Hilbert newform $f$ and a matrix $\sigma$ in (a suitable conjugate of) the Hilbert modular group, we give a cyclotomic extension of the field generated by the Fourier coefficients at infinity which contains all the Fourier coefficients of $f||_k\sigma$. Chapters 2 and 3 are independent of each other and can be read in either order. In Chapter 4, we briefly discuss the relation between these two chapters and mention potential future work.