Iwasawa invariants of galois deformations

On anticyclotomic μ-invariants of modular forms

“…Versions of this theorem appear in [15,13,33]. Our theorem differs from these results in that it allows for more general local conditions at p (not just an ordinary condition) and also allows for the possibility of primes splitting infinitely in the Z p -extension K ∞ /K.…”

A deformation problem for Galois representations over imaginary quadratic fields

“…Since it clearly has an unramified submodule, it would be split and thus the upper shoulder * would correspond to a non-Z p -torsion extension of F (Ψ ) unramified away from p, which does not exist by Proposition 5.21. On the other hand, ρ is nearly ordinary in the sense of Tilouine (see, for example, Definition 3.1 of [43]) with respect to the upper-triangular Borels at p andp. Since one has…”

On the structure of Selmer groups of p-ordinary modular forms over Z-extensions

“…The essential content of the theorem is that, under the various hypotheses, the λ-invariants of the respective modular forms depend only on the residual representations along with data coming from local cohomology at primes dividing the product of the tame levels. This result is formally and thematically similar to [EPW06, Theorem 4.3.3, 4.3.4], [Wes05, Theorem 3.1, 3.2], and [PW11, Theorem 7.1], which apply in the cases F = Q, F ∞ a cyclotomic Z pextension, and F ∞ the anticyclotomic Z p -extension of an imaginary quadratic F , respectively (the results in these references are stated in a somewhat different form from ours, using the framework of Hida families or Galois deformations, and while [EPW06] and [PW11] consider representations associated to modular forms as we do, [Wes05] considers higher-dimensional "nearly ordinary" Galois representations). The theorem is a direct generalization of (and was motivated by) [Gre10,p.…”

“…In order for the technique of proof, which is originally due to Greenberg as far as we can tell, to succeed in our setting, we require the hypothesis alluded to above that H 0 (F v , A * ) vanishes for each v ∈ Σ which splits completely in F ∞ (and this hypothesis accordingly is made in all the subsequent results which rely on Theorem 6.1). This hypothesis does not arise in [Wes05] because the Z p -extensions considered there are cyclotomic, so all finite primes are finitely decomposed. However, another version of this result is given as Proposition A.2 of [PW11], and in the generality of loc.…”