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Forum of Mathematics, Sigma
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Forum of Mathematics, Sigma
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LIFTING TORSION GALOIS REPRESENTATIONS

Lifting torsion Galois representations
Authors: CHANDRASHEKHAR KHARE; RAVI RAMAKRISHNA;

LIFTING TORSION GALOIS REPRESENTATIONS

Abstract

Let$p\geqslant 5$be a prime, and let${\mathcal{O}}$be the ring of integers of a finite extension$K$of$\mathbb{Q}_{p}$with uniformizer${\it\pi}$. Let${\it\rho}_{n}:G_{\mathbb{Q}}\rightarrow \mathit{GL}_{2}\left({\mathcal{O}}/({\it\pi}^{n})\right)$have modular mod-${\it\pi}$reduction$\bar{{\it\rho}}$, be ordinary at$p$, and satisfy some mild technical conditions. We show that${\it\rho}_{n}$can be lifted to an${\mathcal{O}}$-valued characteristic-zero geometric representation which arises from a newform. This is new in the case when$K$is a ramified extension of$\mathbb{Q}_{p}$. We also show that a prescribed ramified complete discrete valuation ring${\mathcal{O}}$is the weight-$2$deformation ring for$\bar{{\it\rho}}$for a suitable choice of auxiliary level. This implies that the field of Fourier coefficients of newforms of weight 2, square-free level, and trivial nebentype that give rise to semistable$\bar{{\it\rho}}$of weight 2 can have arbitrarily large ramification index at$p$.

Related Organizations
Keywords

11F80, Mathematics - Number Theory, Galois representations, 11R, Representation-theoretic methods; automorphic representations over local and global fields, QA1-939, FOS: Mathematics, 11F, Number Theory (math.NT), Mathematics

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
4
Average
Top 10%
Average
Green
Published in a Diamond OA journal