Change of Selmer group under isogeny, Iwasawa invariants of Λ-adic representation, and coefficient field of newforms

This thesis consists of three parts, the first is about change of Selmer groups under isogeny. In 1987, Schneider [23] proved a formula relating the mu-invariants of isogenous elliptic curves over Q . In 1989, Perrion-Riou [20] generalized this result to p-adic representations. In 2008, Ochiai [19] generalized the result to Zp [[T]]-adic Galois representations. Our work generalizes the result further to Galois representations over any complete Noetherian local ring with finite residue field. We also show that under the so-called p-critical condition, the Selmer groups do not change under the natural R-isogeny induced by the inclusion R → R˜, where R˜ is the integral closure of R.;The second part concerns density result for the mu and lambda invariants attached to Λ-adic representations. It is a classical result that the number of primes ℓ for which tau(ℓ) vanishes has Dirichlet density 0, where tau(ℓ) is the Ramanujan tau function. We study an analogous question for Λ-adic representation with full image. In particular, we show that the set of primes ℓ for which the trace of the Frobenius at ℓ has positive mu-invariant has Dirichlet density 0. We also discuss analogous Dirichlet densities of lambda-invariants.;The third part deals with about coefficients of newforms. Let f be a non-CM newform of weight k ≥ 2. Let L be a subfield of the coefficient field of f. We completely settle the question of the density of the set of primes p such that the p-th coefficient of f generates the field L. This density is determined by the inner twists of f. As a particular case, we obtain that in the absence of non-trivial inner twists, the density is 1 for L equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation.