The Eisenstein ideal of weight kk and ranks of Hecke algebras

Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight $k$ and level $\Gamma_0(\ell)$ at the maximal Eisenstein ideal containing $p$. We give a necessary and sufficient condition for the $\mathbb{Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for $k=2$ using our methods. In addition, we prove some $R=\mathbb{T}$ theorems under certain hypothesis.Comment: 33 pages, Comments are welcom

On the Hilbert eigenvariety at exotic and CM classical weight 1 points

Let $F$ be a totally real number field and let $f$ be a classical cuspidal $p$-regular Hilbert modular eigenform over $F$ of parallel weight $1$. Let $x$ be the point on the $p$-adic Hilbert eigenvariety $\mathcal E$ corresponding to an ordinary $p$-stabilization of $f$. We show that if the $p$-adic Schanuel Conjecture is true, then $\mathcal E$ is smooth at $x$ if $f$ has CM. If we additionally assume that $F/\mathbb Q$ is Galois, we show that the weight map is \'etale at $x$ if $f$ has either CM or exotic projective image (which is the case for almost all cuspidal Hilbert modular eigenforms of parallel weight $1$). We prove these results by showing that the completed local ring of the eigenvariety at $x$ is isomorphic to a universal nearly ordinary Galois deformation ring.Comment: The material in the introduction and the final sections was reorganized. The sections on background material were substantially shortene

Dihedral Universal Deformations

This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic number theory, (3) modularity. As to (1), we prove that the universal deformation is dihedral if all infinitesimal deformations are dihedral. Concerning (2) in the setting of Galois representations of number fields, we give sufficient conditions to ensure that the universal deformation relatively unramified outside a finite set of primes is dihedral, and discuss in how far these conditions are necessary. As side-results, we obtain cases of the unramified Fontaine-Mazur conjecture, and in many cases positively answer a question of Greenberg and Coleman on the splitting behaviour at p of p-adic Galois representations attached to newforms. As to (3), we prove a modularity theorem of the form `R=T' for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral.Comment: 43 pages; minor corrections and improvements following referee's comment

Unramifiedness of weight one Hilbert Hecke algebras

We prove that the Galois pseudo-representation valued in the mod $p^n$ cuspidal Hecke algebra for GL(2) over a totally real number field $F$, of parallel weight $1$ and level prime to $p$, is unramified at any place above $p$. The same is true for the non-cuspidal Hecke algebra at places above $p$ whose ramification index is not divisible by $p-1$. A novel geometric ingredient, which is also of an independent interest, is the construction and study, in the case when $p$ ramifies in $F$, of generalised $\Theta$-operators using Reduzzi--Xiao's generalised Hasse invariants, including especially an injectivity criterion in terms of minimal weights.Comment: 29 pages; v2: main result made unconditional in the cuspidal case; introduction of partial Frobenius operator

Effect of increasing the ramification on pseudo-deformation rings

Given a continuous, odd, semi-simple $2$-dimensional representation of $G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$ and a prime $\ell$ not dividing $Np$, we study the relation between the universal deformation rings of the corresponding pseudo-representation for the groups $G_{\mathbb{Q},N\ell p}$ and $G_{\mathbb{Q},Np}$. As a related problem, we investigate when the universal pseudo-representation arises from an actual representation over the universal deformation ring. Under some hypotheses, we prove analogues of theorems of Boston and B\"{o}ckle for the reduced pseudo-deformation rings. We improve these results when the pseudo-representation is unobstructed and $p$ does not divide $\ell^2-1$. When the pseudo-representation is unobstructed and $p$ divides $\ell+1$, we prove that the universal deformation rings in characteristic $0$ and $p$ of the pseudo-representation for $G_{\mathbb{Q},N\ell p}$ are not local complete intersection rings. As an application of our main results, we prove a big $R=\mathbb{T}$ theorem.Comment: 41 Pages, v2: following referee's comments added some remarks and a subsection in the introduction, reorganized Section 2, modified some proofs and made some minor corrections, comments are welcom

On the μ\mu equals zero conjecture for the fine Selmer group in Iwasawa theory

We study the Iwasawa theory of the fine Selmer group associated to certain Galois representations. The vanishing of the $\mu$-invariant is shown to follow in some cases from a natural property satisfied by Galois deformation rings. We outline conditions under which the $\mu=0$ conjecture is shown to hold for various Galois representations of interest.Comment: Version 3: Final version, accepted for publication in Pure and applied math quaterl