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Jayanta Manoharmayum

Publications and source records attributed to Jayanta Manoharmayum.

7 recordsLinked to original sources
arXiv2024

The Artin--Rees lemma and size of spaces over nonassociative complete filtered rings

This paper studies nonassociative filtered rings using associated gradations. We show that a complete filtered ring $R$ with affine associated graded ring generated in degree $1$ is a local ring, and prove that the Artin--Rees lemma holds for $R$. Assuming finiteness of the residue field, we derive asymptotics for abelian groups with an operation of $R$, and identify classes of torsion elements for spaces over a central extension of $R$.

math.RA↗
arXiv2013

The inverse deformation problem

We show the inverse deformation problem has an affirmative answer: given a complete local noetherian ring $A$ with finite residue field $\pmb{k}$, we show that there is a topologically finitely generated profinite group $Γ$ and an absolutely irreducible continuous representation $\barρ:Γ\to GL_n(\pmb{k})$ such that $A$ is the universal deformation ring for $Γ,\barρ$.

math.RA↗
arXiv2013

A structure theorem for subgroups of $GL_n$ over complete local Noetherian rings with large residual image

Given a complete local Noetherian ring $(A,\m_A)$ with finite residue field and a subfield $\pmb{k}$ of $A/\m_A$, we show that every closed subgroup $G$ of $GL_n(A)$ such that $G\mod{\m_A}\supseteq SL_n(\pmb{k})$ contains a conjugate of $SL_n(W(\pmb{k})_A)$ under some small restrictions on $\pmb{k}$. Here $W(\pmb{k})_A$ is the closed subring of $A$ generated by the Teichmüller lifts of elements of the subfield $\pmb{k}$.

math.RA↗
arXiv2013

Higher congruence companion forms

For a rational prime $p \geq 3$ we consider $p$-ordinary, Hilbert modular newforms $f$ of weight $k\geq 2$ with associated $p$-adic Galois representations $ρ_f$ and $\mod{p^n}$ reductions $ρ_{f,n}$. Under suitable hypotheses on the size of the image, we use deformation theory and modularity lifting to show that if the restrictions of $ρ_{f,n} $ to decomposition groups above $p$ split then $f$ has a companion form $g$ modulo $p^n$ (in the sense that $ρ_{f,n}\sim ρ_{g,n}\otimesχ^{k-1}$).

math.NT↗
arXiv2013

Lifting $N$-dimensional Galois representations to characteristic zero

Let $F$ be a number field, let $N\geq 3$ be an integer, and let $k$ be a finite field of characteristic $\ell$. We show that if $\rb:G_F\longrightarrow GL_N(k)$ is a continuous representation with image of $\rb$ containing $SL_N(k)$ then, under moderate conditions at primes dividing $\ell\infty$, there is a continuous representation $ρ:G_F\longrightarrow GL_N(W(k))$ unramified outside finitely many primes with $\rb\simρ\mod{\ell}$.

math.NT↗
arXiv2007

On the modularity of supersingular elliptic curves over certain totally real number fields

We study generalisations to totally real fields of methods originating with Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some applications of our results, for example proving the modularity of all semistable elliptic curves over $\mathbb{Q}(\sqrt{2})$.

math.NT↗