Congruences between cusp forms

Citation

Khare, Chandrashekhar B. (1995) Congruences between cusp forms. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/wgmk-nk78. https://resolver.caltech.edu/CaltechETD:etd-10122007-073759

Abstract

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In this thesis we study the ring of modular deformations of an absolutely irreducible mod p representation which is modular by studying the congruences between new- forms of weight 2 and varying p power levels. This fills in a missing case in the literature of the study of congruences between modular forms of varying levels. The results of [...]1.3, [...]1.4 and [...]1.5 give a thorough analysis of congruences in the (p, p) case. The results we prove along the way in Chapter 1 shed light on the multiplicities with which certain 2 dimensional representations arise in the Jacobians of modular curves. In [...]1.7 we apply the study of congruences in the (p, p) case to prove lower bounds on the ring of modular deformations. This lower bound has been proven earlier in Gouvea.

In Chapter 2 we study local components of Hecke algebras which arise by studying Hecke action on the space of mod p modular forms of fixed level and all weights. We relate the computation of dimensions of ring of modular deformations to certain properties of Hecke exact sequences. These exact sequences arise from the phenomenon that mod p there are inclusions between modular forms (identified with their q-expansions) of different weights.

In Chapter 3, which is joint work with D. Prasad, we raise a natural question about the nature of Fourier coefficients of cuspidal eigenforms which may be viewed as asking for a version of the Chinese Remainder Theorem for automorphic representations and answer it in some simple cases.

Thesis Files