Structure of Hecke algebras in two scenarios: mod p modular forms and eigenvarieties

This thesis has two parts. In the first part, generalizing the recent results of Bellaiche and Khare for the level 1 case, we study the structure of the local components of the shallow Hecke algebras (i.e. Hecke algebras without Up and Uℓ for all primes ℓ dividing the level N) acting on the space of modular forms modulo p for Gamma0( N) and Gamma1(N). We relate them to pseudo-deformation rings and prove that in many cases, the local components are regular complete local algebras of dimension 2. In the second part, we show that the p-adic eigenvariety constructed by Andreatta-Iovita-Pilloni parameterizing cuspidal Hilbert modular eigenforms defined over a totally real field F is smooth at certain classical parallel weight one points which are regular at every place of F above p and also determine whether the map to the weight space at those points is etale or not. We prove these results assuming the Leopoldt conjecture for certain quadratic extensions of F in some cases, assuming the p-adic Schanuel conjecture in some cases and unconditionally in some cases, using the deformation theory of Galois representations. As a consequence, we also determine whether the 1-dimensional parallel weight eigenvariety, constructed by Kisin-Lai, is smooth or not at those points.