arXiv subjects

Stefan Nikoloski

Publications and source records attributed to Stefan Nikoloski.

2 recordsLinked to original sources

arXiv2025

Lifting residual Galois representations with the same semi-simplification

If a $p$-adic Galois representation $ρ_{f,ν}:Γ_{\mathbb Q} \to \GL_2(E_{f,ν})$ attached to some eigenform $f$ is residually reducible it will have 2 non-isomorphic reductions, which have the same semi-simplification. In this paper, we answer a version of the inverse question, first brought up by Toby Gee and Alice Pozzi. Starting with two modulo $p$ non-semi-simple representations $\overline{ρ_1},\overline{ρ_2}:Γ_{\mathbb Q} \to \GL_2(k)$, which have the same semi-simplification we show that under some mild conditions that they are reductions of representations attached to newforms of the same weight $r \ge 2$, the same level $N \ge 1$, and the same Neben character $\varepsilon$.

math.NT↗

arXiv2024

Forcing ramification in the relative deformation method

In this paper we generalize the forcing ramification argument of Khare-Ramakrishna to the setting of the lifting result by Fakhruddin-Khare-Patrikis. In particular, we show that in the relative deformation method the finite set of primes can be chosen so that the geometric characteristic 0 lift will be ramified at each of those primes. Moreover, we show that the restrictions of this lift to the local absolute Galois groups will correspond to formally smooth points in the generic fibers of the local universal lifting rings at each prime. Eventually, we prove that the local formal smoothness at each of the primes implies the vanishing of the geometric Bloch-Kato Selmer group associated to the adjoint representation of that characteristic 0 lift.

math.NT↗