Constructing semisimple p -adic Galois representations with prescribed properties

Khare, Chandrashekhar; Larsen, Michael; Ramakrishna, Ravi

doi:10.1353/ajm.2005.0026

Cited by 27 publications

(57 citation statements)

References 19 publications

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“…The proof of Fact 2.3 below can be found in [R1, §3]. See also [KLR1,§2]. 1) for i = 0, 1, 2 and have dimensions 1, 2 and 1, respectively.…”

Section: Recollectionsmentioning

confidence: 99%

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Constructing Galois Representations with Very Large Image

Ramakrishna¹

2008

Can. j. math.

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“…The proof of Fact 2.3 below can be found in [R1, §3]. See also [KLR1,§2]. 1) for i = 0, 1, 2 and have dimensions 1, 2 and 1, respectively.…”

Section: Recollectionsmentioning

confidence: 99%

“…Proof That Q comes from a Chebotarev condition follows from [R1,Lemma 10] which gives that the splitting conditions of Proposition 2.2 are independent of those imposed by ζ and the ζ i . See also [KLR1,Lemma 6].…”

Section: The Setupmentioning

confidence: 99%

Constructing Galois Representations with Very Large Image

Ramakrishna¹

2008

Can. j. math.

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“…Part of the length of the argument below is because of this. That we do not know whether we have to allow ramification at one or two nice primes to remove obstructions to deformation problems here and in [8] is the same phenomenon. 5.5.1.…”

Section: 5mentioning

confidence: 96%

“…The proof of Theorem 20 is more involved than the proof of Theorem 11. Essential use is made of the techniques of [8].…”

Section: Introductionmentioning

confidence: 99%

Lifting Torsion Galois Representations

Khare¹,

Ramakrishna

2015

Forum math. Sigma

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Let p 5 be a prime, and let O be the ring of integers of a finite extension K of Q p with uniformizer π. Let ρ n : G Q → GL 2 (O/(π n )) have modular mod-π reductionρ, be ordinary at p, and satisfy some mild technical conditions. We show that ρ n can be lifted to an O-valued characteristic-zero geometric representation which arises from a newform. This is new in the case when K is a ramified extension of Q p . We also show that a prescribed ramified complete discrete valuation ring O is the weight-2 deformation ring forρ 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 semistableρ of weight 2 can have arbitrarily large ramification index at p.

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“…We remark that Conjecture 1.1 is certainly false for many infinitely ramified Galois representations, as by [6] it is possible to control the Frobenius polynomials (and thus the a p (ρ)) at a set of primes of density one.…”

Section: Conjecture 11 Let ρ Be a Motivic Galois Representation As Amentioning

confidence: 99%

Power Residues of Fourier Coefficients of Modular Forms

Weston¹

2005

Can. j. math.

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Abstract. Let ρ : G Q → GL n (Q ℓ ) be a motivic ℓ-adic Galois representation. For fixed m > 1 we initiate an investigation of the density of the set of primes p such that the trace of the image of an arithmetic Frobenius at p under ρ is an m-th power residue modulo p. Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals 1/m whenever the image of ρ is open. We further conjecture that for such ρ the set of these primes p is independent of any set defined by Cebatorev-style Galois-theoretic conditions (in an appropriate sense). We then compute these densities for certain m in the complementary case of modular forms of CM-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem (which does apply in the CM case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at p in abelian extensions of imaginary quadratic fields unramified away from p.

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Constructing semisimple p -adic Galois representations with prescribed properties

Cited by 27 publications

References 19 publications

Constructing Galois Representations with Very Large Image

Constructing Galois Representations with Very Large Image

Lifting Torsion Galois Representations

Power Residues of Fourier Coefficients of Modular Forms

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